Citation | Kilin A. A., Karavaev Y. L., Experimental observation of stabilization of tippe top spinning on a vibrating plane, Theoretical and Applied Mechanics, 2025, https://doi.org/10.2298/TAM241204001K |
---|---|
DOI: | 10.2298/TAM241204001K |
Full text: | pdf (840.04 Kb) |
Kilin A. A., Ivanova T. B., Karavaev Y. L., Yefremov K. S.
In this paper, we address the problem of the controlled motion of a roller racer on a plane. We assume that the angle between the platforms is a given periodic function of time (control function), and the no-slip conditions (nonholonomic constraint) and viscous friction forces act at the points of contact of the wheels with the plane. In this case, all trajectories of the reduced system tend asymptotically to a periodic solution. In this paper, we show that for a selected periodic control function there exists a motion of the system that is bounded (along a circle) and unbounded (along a straight line). Unbounded motion corresponds to the resonant case which takes place at zero average value of the control function. The theoretical dependence of the trajectory and the velocity of the roller racer on its parameters and the parameters of the selected control function is investigated. These dependences are confirmed experimentally.
Citation | Kilin A. A., Ivanova T. B., Karavaev Y. L., Yefremov K. S., Theoretical and experimental investigations of the controlled motion of a roller racer, Theoretical and Applied Mechanics, 2024, https://doi.org/10.2298/TAM241203010K |
---|---|
DOI: | 10.2298/TAM241203010K |
Full text: | pdf (1.37 Mb) |
Kilin A. A., Pivovarova E. N., Ivanova T. B.
This paper addresses the problem of a homogeneous ball rolling on the inner surface of a circular cylinder in a field of gravity parallel to its axis. It is assumed that the ball rolls without slipping on the surface of the cylinder, and that the cylinder executes plane-parallel motions in a circle perpendicular to its symmetry axis. The integrability of the problem by quadratures is proved. It is shown that in this problem the trajectories of the ball are quasiperiodic in the general case, and that an unbounded elevation of the ball is impossible. However, in contrast to a fixed (or rotating) cylinder, there exist resonances at which the ball moves on average downward with constant acceleration.
Keywords: | homogeneous ball, nonholonomic constraint, surface of revolution, moving cylinder, unbounded drift, nonautonomous system, quadrature, integrability |
---|---|
Citation: | Kilin A. A., Pivovarova E. N., Ivanova T. B., Rolling of a Homogeneous Ball on a Moving Cylinder, Regular and Chaotic Dynamics, 2024, https://doi.org/10.1134/S1560354724590027 |
DOI: | 10.1134/S1560354724590027 |
Full text: | pdf (710.83 Kb) |
Kilin A. A., Gavrilova A. M., Artemova E. M.
This paper is concerned with the plane-parallel motion of an elliptic foil with an attached vortex of constant strength in an ideal fluid. Special attention is given to the case in which the vortex lies on the continuation of one of the semiaxes of the ellipse. It is shown that in this case there exist no attracting solutions and the system is integrable by the Euler – Jacobi theorem. A complete qualitative analysis of the equations of motion is carried out for cases where the vortex lies on the continuation of the large or the small semiaxis of the ellipse. Possible types of trajectories of an elliptic foil with an attached vortex are established: quasi-periodic, unbounded (going to infinity) and periodic trajectories.
Keywords: | ideal fluid, elliptic foil, point vortex, integrable system, bifurcation analysis |
---|---|
Citation: | Kilin A. A., Gavrilova A. M., Artemova E. M., Dynamics of an Elliptic Foil with an Attached Vortex in an Ideal Fluid: The Integrable Case, Regular and Chaotic Dynamics, 2024, https://doi.org/10.1134/S1560354724590015 |
DOI: | 10.1134/S1560354724590015 |
Full text: | pdf (1.52 Mb) |
Artemova E. M., Lagunov D. A., Vetchanin E. V.
This paper is concerned with the motion of an elliptic foil in the field of a fixed point singularity. A complex potential of the fluid flow is constructed, and the forces and the torque which act on the foil from the fluid are obtained. It is shown that the equations of motion of the elliptic foil in the field of a fixed point vortex source can be represented as Lagrange – Euler equations. It is also shown that the system has an additional first integral due to the conservation of the angular momentum. An effective potential of the system under consideration is constructed. For the cases where the singularity is a vortex or a source, unstable relative equilibrium points corresponding to the circular motion of the foil around the singularity are found.
Keywords: | ideal fluid, elliptic foil, point vortex, point source, Lagrangian form, Hamiltonian form |
---|---|
Citation: | Artemova E. M., Lagunov D. A., Vetchanin E. V., The Motion of an Elliptic Foil in the Field of a Fixed Vortex Source, Rus. J. Nonlin. Dyn., 2024, https://doi.org/10.20537/nd241203 |
DOI: | 10.20537/nd241203 |
Full text: | pdf (656.91 Kb) |
This paper investigates the trajectories of light beams in a Kerr metric, which describes the gravitational field in the neighborhood of a rotating black hole. After reduction by cyclic coordinates, this problem reduces to analysis of a Hamiltonian system with two degrees of freedom. A bifurcation diagram is constructed and a classification is made of the types of trajectories of the system according to the values of first integrals. Relations describing the boundary of the shadow of the black hole are obtained for a stationary observer who rotates with an arbitrary angular velocity about the axis of rotation of the black hole.
Keywords: | Kerr metric, trajectories of light beams, shadow of a black hole, bifurcation diagram, gravitational lensing |
---|---|
Citation: | Bizyaev I. A., Trajectories of Light Beams in a Kerr Metric: the Influence of the Rotation of an Observer on the Shadow of a Black Hole, Rus. J. Nonlin. Dyn., 2025, https://doi.org/10.20537/nd250101 |
DOI: | 10.20537/nd250101 |
Full text: | pdf (5.33 Mb) |
In this paper we consider the dynamics of a roller bicycle on a horizontal plane. For this bicycle we derive a nonlinear system of equations of motion in a form that allows us to take into account the symmetry of the system in a natural form. We analyze in detail the stability of straight-line motion depending on the parameters of the bicycle. We find numerical evidence that, in addition to stable straight-line motion, the roller bicycle can exhibit other, more complex, trajectories for which the bicycle does not fall.
Keywords: | roller bicycle, nonholonomic system, stability, quasi-velocities, Poincaré map |
---|---|
Citation: | Bizyaev I. A., Mamaev I. S., Nonlinear Dynamics of a Roller Bicycle, Regular and Chaotic Dynamics, 2024, vol. 29, no. 5, |
DOI: | 10.1134/S1560354724530017 |
Full text: | pdf (2.93 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Klekovkin A. V., Karavaev Y. L., Nazarov A. V.
This paper is concerned with the experimental development of the stabilizing regulator for a spherical pendulum-type robot moving on an oscillating base. Using a mathematical model of the motion of the spherical robot with an internal pendulum mechanism, a regulator stabilizing the lower position of the pendulum is developed. The developed regulator has been tested in practice by means of a real prototype of the spherical robot. The results of real experiments are presented to assess the stabilization of the lower position of the pendulum of the spherical robot during its motion along a straight line on a plane executing longitudinal oscillations, and during the stabilization of the lower position of the pendulum, when the spherical shell remains fixed relative to the plane.
Keywords: | spherical robot, stabilization, rolling motion, vibrations |
---|---|
Citation: | Klekovkin A. V., Karavaev Y. L., Nazarov A. V., Stabilization of a Spherical Robot with an Internal Pendulum During Motion on an Oscillating Base, Russian Journal of Nonlinear Dynamics, 2024, vol. 20, no. 5, |
DOI: | 10.20537/nd241213 |
Full text: | pdf (2.17 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bizyaev I. A., Vetchanin E. V.
This paper addresses the problem of the Chaplygin sleigh moving on an inclined plane under the action of periodic controls. Periodic controls are implemented by moving point masses. It is shown that, under periodic oscillations of one point mass in the direction perpendicular to that of the knife edge, for a nonzero initial velocity there exists a motion with acceleration or a uniform motion (on average per period) in the direction opposite to that of the largest descent. It is shown that adding to the system two point masses which move periodically along some circle enables a period-averaged uniform motion of the system from rest.
Keywords: | Chaplygin sleigh, motion on an inclined plane, speedup, nonholonomic mechanics |
---|---|
Citation: | Bizyaev I. A., Vetchanin E. V., Climb of the Chaplygin Sleigh on an Inclined Plane under Periodic Controls: Speedup and Uniform Motion, Russian Journal of Nonlinear Dynamics, 2024, vol. 20, no. 4, |
DOI: | 10.20537/nd241202 |
Full text: | pdf (669.5 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
This paper addresses the problem of the motion of two point vortices of arbitrary strengths in an ideal incompressible fluid on a finite flat cylinder. A procedure of reduction to the level set of an additional first integral is presented. It is shown that, depending on the parameter values, three types of bifurcation diagrams are possible in the system. A complete bifurcation analysis of the system is carried out for each of them. Conditions for the orbital stability of generalizations of von Kármán streets for the problem under study are obtained.
Keywords: | point vortices, ideal fluid, flat cylinder, bifurcation diagram, phase portrait, von Kármán vortex street, stability, boundary, flow in a strip |
---|---|
Citation: | Kilin A. A., Artemova E. M., Bifurcation Analysis of the Problem of Two Vortices on a Finite Flat Cylinder, Russian Journal of Nonlinear Dynamics, 2024, vol. 20, no. 1, |
DOI: | 10.20537/nd231209 |
Full text: | pdf (1.11 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Vetchanin E. V., Valieva A. R.
The problem of describing the motion of a rigid body in a fluid is addressed by considering a symmetric Joukowsky foil. Within the framework of the model of an ideal fluid, the force and torque acting on an unsteady moving foil are calculated. The analytical results are compared with those obtained based on the numerical solution of the Navier – Stokes equations. It is shown that analytical expressions for the force and torque can be consistent with the results of numerical simulations using scaling and a delayed arguments.
Keywords: | motion of a body in a fluid, Joukowsky foil, complex potential, Kutta – Chaplygin condition, the Navier – Stokes equations |
---|---|
Citation: | Vetchanin E. V., Valieva A. R., Analysis of the Force and Torque Arising During the Oscillatory Motion of a Joukowsky Foil in a Fluid, Russian Journal of Nonlinear Dynamics, 2024, vol. 20, no. 1, |
DOI: | 10.20537/nd231210 |
Full text: | pdf (7.95 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Klekovkin A. V., Karavaev Y. L., Kilin A. A., Nazarov A. V.
This paper describes the design of an aquatic robot moving on the surface of a fluid and driven by two internal moving masses. The body of the aquatic robot in cross section has the shape of a symmetrical airfoil with a sharp edge. In this prototype, two internal masses move in circles and are rotated by a single DC motor and a gear mechanism that transmits torque from the motor to each mass. Angular velocities of moving masses are used as a control action, and the developed kinematic scheme for transmitting rotation from the motor to the moving masses allows the rotation of two masses with equal angular velocities in magnitude, but with a different direction of rotation. It is also possible to install additional tail fins of various shapes and sizes on the body of this robot. Also in the work for this object, the equations of motion are presented, written in the form of Kirchhoff equations for the motion of a solid body in an ideal fluid, which are supplemented by terms of viscous resistance. A mathematical description of the additional forces acting on the flexible tail fin is presented. Experimental studies on the influence of various tail fins on the speed of motion in the fluid were carried out with the developed prototype of the robot. In this work, tail fins of the same shape and size were installed on the robot, while having different stiffness. The experiments were carried out in a pool with water, over which a camera was installed, on which video recordings of all the experiments were obtained. Next processing of the video recordings made it possible to obtain the object’s movements coordinates, as well as its linear and angular velocities. The paper shows the difference in the velocities developed by the robot when moving without a tail fin, as well as with tail fins having different stiffness. The comparison of the velocities developed by the robot, obtained in experimental studies, with the results of mathematical modeling of the system is given.
Keywords: | mobile robot, aquatic robot, motion simulation, experimental investigations |
---|---|
Citation: | Klekovkin A. V., Karavaev Y. L., Kilin A. A., Nazarov A. V., The influence of tail fins on the speed of an aquatic robot driven by internal moving masses, Computer Research and Modeling, 2024, vol. 16, no. 4, |
DOI: | 10.20537/2076-7633-2024-16-4-869-882 |
Full text: | pdf (4.54 Mb) |
Impact-factor RSCI (2022): | 0.257 (Q4) |
---|---|
ISSN (print): | 2076-7633 |
ISSN (online): | 2077-6853 |
Site: | http://crm.ics.org.ru/ |
Finite-dimensional equations constructed earlier to describe the motion of an aquatic drop-shaped robot due to given rotor oscillations are studied. To study the equations of motion, we use the Poincaré map method, estimates of the Lyapunov exponents, and the parameter continuation method to explore the evolution of asymptotically stable solutions. It is shown that, in addition to the so-called main periodic solution of the equations of motion for which the robot moves in a circle in a natural way, an additional asymptotically stable periodic solution can arise under the influence of highly asymmetric impulsive control. This solution corresponds to the robot’s sideways motion near the circle. It is shown that this additional periodic solution can lose stability according to the Neimark–Sacker scenario, and an attracting torus appears in its vicinity. Thus, a quasiperiodic mode of motion can exist in the phase space of the system. It is shown that quasiperiodic solutions of the equations of motion also correspond to the quasiperiodic motion of the robot in a bounded region along a trajectory of a rather complex shape. Also, strange attractors were found that correspond to the drifting motion of the robot. These modes of motion were found for the first time in the dynamics of the drop-shaped robot.
Keywords: | aquatic robot, finite-dimensional model, invariant torus, strange attractor |
---|---|
Citation: | Vetchanin E. V., Mamaev I. S., Numerical Analysis of a Drop-Shaped Aquatic Robot , Mathematics, 2024, vol. 12, no. 2, 312, |
DOI: | 10.3390/math12020312 |
Full text: | pdf (3.41 Mb) |
Impact-factor WoS (2022): | 2.400 (Q1) |
---|---|
Impact-factor RSCI (2022): | 0.446 (Q2) |
ISSN (online): | 2227-7390 |
Site: | https://www.mdpi.com/journal/mathematics |
Artemova E. M., Vetchanin E. V.
A finite-dimensional model is developed, which describes the motion of a balanced circular foil with proper circulation in the field of a fixed vortex source. The motion of the foil has been studied in two special cases: that of a fixed vortex and that of a fixed source. It is shown that in the absence of proper circulation, the fixed vortex and the fixed source have the same impact on the motion of the foil. However, adding non- zero proper circulation leads to qualitative differences in the foil’s dynamics. For a fixed vortex, there exist three types of motions: the fall on a vortex in finite time, periodic and quasiperiodic motion around the vortex. The investigation of this case reduces to analysis of a Hamiltonian system with one degree of freedom. Typical phase portraits and graphs of the effective potential of the system are plotted vs the distance between the geometric center of the foil and the vortex. For a fixed source, two types of motions are possible: the fall on the source in finite time and unbounded escape from the source. For small intensities of the source, the asymptotics of escape to infinity is constructed.
Citation: | Artemova E. M., Vetchanin E. V., The motion of a circular foil in the field of a fixed point singularity: Integrability and asymptotic behavior, Physics of Fluids, 2024, vol. 36, 027139, |
---|---|
DOI: | 10.1063/5.0185865 |
Full text: | pdf (1.45 Mb) |
Impact-factor WoS (2022): | 4.600 (Q1) |
---|---|
Impact-factor RSCI (2022): | 1.083 (Q1) |
ISSN (print): | 1070-6631 |
ISSN (online): | 1089-7666 |
Site: | https://aip.scitation.org/journal/phf |
A mathematical model featuring the motion of a multilink wheeled vehicle is developed using a nonholonomic model. A detailed analysis of the inertial motion is made. Fixed points of the reduced system are identified, their stability is analyzed, and invariant manifolds are found. For the case of three platforms (links), a phase portrait for motion on an invariant manifold is shown and trajectories of the attachment points of the wheel pairs of the three-link vehicle are presented. In addition, an analysis is made of motion in the case where the leading platform has a rotor whose angular velocity is a periodic function of time. The existence of trajectories for which one of the velocity components increases without bound is established, and the asymptotics for it is found.
Keywords: | Wheeled vehicles, nonholonomic mechanics, stability, unbounded speedup |
---|---|
Citation: | Artemova E. M., Bizyaev I. A., Dynamics of a multilink wheeled vehicle: Partial solutions and unbounded speedup, International Journal of Non-Linear Mechanics, 2024, vol. 165, 104774, |
DOI: | 10.1016/j.ijnonlinmec.2024.104774 |
Full text: | pdf (1.17 Mb) |
Impact-factor WoS (2022): | 3.200 (Q2) |
---|---|
ISSN (print): | 0020-7462 |
ISSN (online): | 1878-5638 |
Site: | http://www.journals.elsevier.com/international-journal-of-non-linear-mechanics/ |
This paper is concerned with the problem of an ellipsoid of revolution rolling on a horizontal plane under the assumption that there is no slipping at the point of contact and no spinning about the vertical. A reduction of the equations of motion to a fixed level set of first integrals is performed. Permanent rotations corresponding to the rolling of an ellipsoid in a circle or in a straight line are found. A linear stability analysis of permanent rotations is carried out. A complete classification of possible trajectories of the reduced system is performed using a bifurcation analysis. A classification of the trajectories of the center of mass of the ellipsoid depending on parameter values and initial conditions is performed.
Keywords: | rubber ellipsoid, body of revolution, rolling motion, nonholonomic constraint, bifurcations, stability |
---|---|
Citation: | Kilin A. A., Pivovarova E. N., Bifurcation analysis of the problem of a “rubber” ellipsoid of revolution rolling on a plane, Nonlinear Dynamics, 2024, vol. 112, |
DOI: | 10.1007/s11071-024-09863-7 |
Full text: | pdf (1.04 Mb) |
Impact-factor WoS (2022): | 5.600 (Q1) |
---|---|
ISSN (print): | 0924-090X |
ISSN (online): | 1573-269X |
Site: | http://link.springer.com/journal/11071 |
This paper investigates the trajectories of neutral particles in the Schwarzschild-Melvin spacetime. After reduction by cyclic coordinates this problem reduces to investigating a two-degree-of-freedom Hamiltonian system that has no additional integral. A classification of regions of possible motion of a particle is performed according to the values of the momentum and energy integrals. Bifurcations of periodic solutions of the reduced system are analyzed using a Poincaré map.
Citation: | Bizyaev I. A., Classification of the trajectories of uncharged particles in the Schwarzschild-Melvin metric, Physical Review D, 2024, vol. 110, 104031, |
---|---|
DOI: | 10.1103/PhysRevD.110.104031 |
Full text: | pdf (4.31 Mb) |
Impact-factor WoS (2022): | 5.0 (Q1) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 2470-0010 |
ISSN (online): | 2470-0029 |
Site: | https://journals.aps.org/prd/ |
Ardentov A. A., Artemova E. M.
A symmetric mathematical model of a wheeled robot with a trailer is considered for various types of coupling between the robot and the trailer. It is shown that for fixed coupling parameters and fixed initial position of the robot with trailer there are two symmetric abnormal extremals. In motion along these trajectories the robot and the trailer traverse normal extremal trajectories for the sub-Riemannian problem on the group of motions of the plane; the coupling point always draws an inflectional elastica or a straight line.
Keywords: | robot with trailer, kinematic model, Pontryagin maximum principle, abnormal trajectories, sub-Riemannian geometry |
---|---|
Citation: | Ardentov A. A., Artemova E. M., Abnormal extremals in the sub-Riemannian problem for a general model of a robot with a trailer, Sbornik: Mathematics, 2023, vol. 214, no. 10, |
DOI: | https://doi.org/10.4213/sm9829 |
Full text: | pdf (524.4 Kb) |
Impact-factor WoS (2022): | 0.800 (Q3) |
---|---|
Impact-factor RSCI (2014): | 1.024 |
ISSN (print): | 0368-8666 |
ISSN (online): | 2305-2783 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=sm&option_lang=rus |
In this work, a model that describes the motion of point vortices in an ideal incompressible fluid on a finite flat cylinder is obtained. The case of two vortices is considered in detail. It is shown that the equations of motion of vortices can be represented in Hamiltonian form and have an additional first integral. A procedure of reduction to a fixed level of the first integral is proposed. For the reduced system, phase portraits are constructed, fixed points and singularities of the system are indicated.
Keywords: | point vortices, ideal fluid, fixed points, singularities, phase portrait |
---|---|
Citation: | Artemova E. M., Dynamics of two vortices on a finite flat cylinder, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2023, vol. 33, no. 4, |
DOI: | 10.35634/vm230407 |
Full text: | pdf (824.78 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
Kilin A. A., Ivanova T. B., Pivovarova E. N.
This paper treats the problem of a spherical robot with an axisymmetric pendulum drive rolling without slipping on a vibrating plane. The main purpose of the paper is to investigate the stabilization of the upper vertical rotations of the pendulum using feedback (additional control action). For the chosen type of feedback, regions of asymptotic stability of the upper vertical rotations of the pendulum are constructed and possible bifurcations are analyzed. Special attention is also given to the question of the stability of periodic solutions arising as the vertical rotations lose stability.
Keywords: | spherical robot, vibration, feedback, stabilization, damped Mathieu equation |
---|---|
Citation: | Kilin A. A., Ivanova T. B., Pivovarova E. N., Stabilization of Steady Rotations of a Spherical Robot on a Vibrating Base Using Feedback, Regular and Chaotic Dynamics, 2023, vol. 28, no. 6, |
DOI: | 10.1134/S1560354723060060 |
Full text: | pdf (1.41 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The problem of the rolling of a disk on a plane is considered under the assumption that there is no slipping in the direction parallel to the horizontal diameter of the disk and that the center of mass does not move in the horizontal direction. This problem is reduced to investigating a system of three first-order differential equations. It is shown that the reduced system is reversible relative to involution of codimension one and admits a two-parameter family of fixed points. The linear stability of these fixed points is analyzed. Using numerical simulation, the nonintegrability of the problem is shown. It is proved that the reduced system admits, even in the nonintegrable case, a two-parameter family of periodic solutions. A number of dynamical effects due to the existence of involution of codimension one and to the degeneracy of the fixed points of the reduced system are found.
Keywords: | nonholonomic constraint, unbalanced disk, omnidisk, permanent rotations, periodic solutions, stability, integrability, chaos, invariant manifolds, manifolds of fall |
---|---|
Citation: | Kilin A. A., Pivovarova E. N., Dynamics of an Unbalanced Disk with a Single Nonholonomic Constraint, Regular and Chaotic Dynamics, 2023, vol. 28, no. 1, |
DOI: | 10.1134/S1560354723010069 |
Full text: | pdf (2.23 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
In this paper we investigate a nonholonomic system with parametric excitation, a Roller Racer with variable gyrostatic momentum. We examine in detail the problem of the existence of regimes with unbounded growth of energy (nonconservative Fermi acceleration). We find a criterion for the existence of trajectories for which one of the velocity components increases withound bound and has asymptotics t1/3. In addition, we show that the problem under consideration reduces to analysis of a three-dimensional Poincaré map. This map exhibits both regular attractors (a fixed point, a limit cycle and a torus) and strange attractors.
Keywords: | nonholonomic mechanics, Roller Racer, Andronov –Hopf bifurcation, stability, central manifold, unbounded speedup, Poincaré map, limit cycle, strange attractor |
---|---|
Citation: | Bizyaev I. A., Mamaev I. S., Roller Racer with Varying Gyrostatic Momentum: Acceleration Criterion and Strange Attractors, Regular and Chaotic Dynamics, 2023, vol. 28, no. 1, |
DOI: | 10.1134/S1560354723010070 |
Full text: | pdf (6.63 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Kilin A. A., Karavaev Y. L., Shestakov V. A.
The article is devoted to the motion analysis of a highly maneuverable mobile robot with four omniwheels, taking into account the conditions for the appearance of wheel detachment from the surface, and the occurrence of wheel slipping. Within the motion analysis the task of determination support reactions for a mobile robot is considered. To solve this task, the design of a mobile robot is presented in the form of the frame with rods. To disclosure the static indeterminacy of the considered system the forces method is used. Dependences of support reactions from the position of the center mass are obtained. The feature of the considered system is that the obtained dependencies of the support reactions are nonlinear. Based on the obtained dependences of the support reactions, the influence of the position of the center of mass of a mobile robot with four wheels on the occurrence of detachment and slipping of wheels of a mobile robot was considered. Investigation was carried out within the framework of the dry friction model, according to which module of the friction force proportionally depends on the support reaction acting on the wheel from the side of the motion surface. Simulation was carried out, as a result of which the conditions for the position of the center of mass of a mobile robot were determined, in which wheels of a mobile robot do not detach from the motion surface, and there is no wheel slipping.
Keywords: | highly maneuverable mobile robot, friction force, normal reaction forces, static indeterminacy, force method, slipping, simulation, center of mass |
---|---|
Citation: | Kilin A. A., Karavaev Y. L., Shestakov V. A., Motion of a Four-Wheeled Omnidirectional Mobile Robot without Slipping and Detachment from the Surface, Mekhatronika, Avtomatizatsiya, Upravlenie, 2023, vol. 24, no. 8, |
DOI: | 10.17587/mau.24.403-411 |
Full text: | pdf (956.53 Kb) |
Impact-factor RSCI (2022): | 0.243 (Q3) |
---|---|
ISSN (print): | 1684-6427 |
ISSN (online): | 2619-1253 |
Site: | https://mech.novtex.ru/jour/index |
This paper investigates the problem of a sphere with axisymmetric mass distribution rolling on a horizontal plane. It is assumed that the sphere can slip in the direction of the projection of the symmetry axis onto the supporting plane. Equations of motion are obtained and their first integrals are found. It is shown that in the general case the system considered is nonintegrable and does not admit an invariant measure with smooth density. Some particular cases of the existence of an additional integral of motion are found and analyzed. In addition, the limiting case in which the system is integrable by the Euler – Jacobi theorem is established.
Keywords: | nonholonomic constraint, first integral, nonintegrability, Poincaré map |
---|---|
Citation: | Kilin A. A., Ivanova T. B., The Problem of the Rolling Motion of a Dynamically Symmetric Spherical Top with One Nonholonomic Constraint, Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 4, |
DOI: | 10.20537/nd231201 |
Full text: | pdf (591.72 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Klekovkin A. V., Karavaev Y. L., Mamaev I. S.
This paper presents the design of an aquatic robot actuated by one internal rotor. The robot body has a cylindrical form with a base in the form of a symmetric airfoil with a sharp edge. For this object, equations of motion are presented in the form of Kirchhoff equations for rigid body motion in an ideal fluid, which are supplemented with viscous resistance terms. A prototype of the aquatic robot with an internal rotor is developed. Using this prototype, experimental investigations of motion in a fluid are carried out.
Keywords: | mobile robot, aquatic robot, motion simulation |
---|---|
Citation: | Klekovkin A. V., Karavaev Y. L., Mamaev I. S., The Control of an Aquatic Robot by a Periodic Rotation of the Internal Flywheel, Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 2, |
DOI: | 10.20537/nd230301 |
Full text: | pdf (10 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
This paper addresses the problem of a sphere with axisymmetric mass distribution rolling on a horizontal plane. It is assumed that there is no slipping of the sphere as it rolls in the direction of the projection of the symmetry axis onto the supporting plane. It is also assumed that, in the direction perpendicular to the above-mentioned one, the sphere can slip relative to the plane. Examples of realization of the above-mentioned nonholonomic constraint are given. Equations of motion are obtained and their first integrals are found. It is shown that the system under consideration admits a redundant set of first integrals, which makes it possible to perform reduction to a system with one degree of freedom.
Keywords: | nonholonomic constraint, first integral, integrability, reduction |
---|---|
Citation: | Kilin A. A., Ivanova T. B., The Integrable Problem of the Rolling Motion of a Dynamically Symmetric Spherical Top with One Nonholonomic Constraint, Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 1, |
DOI: | 10.20537/nd221205 |
Full text: | pdf (642.75 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
In this paper, we study the plane-parallel motion of a circular foil interacting with two vortex pairs in an infinite volume of an ideal fluid. We assumed that the circulation of the velocity of the fluid around the foil was zero. We showed that the equations of motion possess an invariant submanifold such that the foil performed translational motion and the vortices were symmetric relative to the foil’s direction of motion. A qualitative analysis of the motion on this invariant submanifold was made. New relative equilibria were found, a bifurcation diagram was constructed, and a stability analysis is given. In addition, trajectories generalizing Helmholtz leapfrogging were found where the vortices passed alternately through each other, while remaining at a finite distance from the foil.
Keywords: | point vortices, ideal fluid, bifurcation diagram, stability, relative equilibria, Poincaré map, Helmholtz leapfrogging |
---|---|
Citation: | Bizyaev I. A., Mamaev I. S., Dynamics of a Circular Foil and Two Pairs of Point Vortices: New Relative Equilibria and a Generalization of Helmholtz Leapfrogging, Symmetry, 2023, vol. 15, no. 3, 698, |
DOI: | 10.3390/sym15030698 |
Full text: | pdf (1.96 Mb) |
Impact-factor WoS (2022): | 2.700 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.483 (Q2) |
ISSN (online): | 2073-8994 |
Site: | https://www.mdpi.com/journal/symmetry |
In this paper, we address the problem of an ellipsoid with axisymmetric mass distribution rolling on a horizontal absolutely rough plane under the assumption that the supporting plane performs periodic vertical oscillations. In the general case, the problem reduces to a system with one and a half degrees of freedom. In this paper, instead of considering exact equations, we use a vibrational potential that describes approximately the dynamics of a rigid body on a vibrating plane. Since the vibrational potential is invariant under rotation about the vertical, the resulting problem with the additional potential is integrable. For this problem, we analyze the influence of vibrations on the linear stability of vertical rotations of the ellipsoid.
Keywords: | axisymmetric ellipsoid; vibrating plane; nonholonomic constraint; permanent rotations; vertical rotations; stability; vibrational potential; integrability |
---|---|
Citation: | Kilin A. A., Pivovarova E. N., Stability of Vertical Rotations of an Axisymmetric Ellipsoid on a Vibrating Plane, Mathematics, 2023, vol. 11, no. 18, 3948, |
DOI: | 10.3390/math11183948 |
Full text: | pdf (1.15 Mb) |
Impact-factor WoS (2022): | 2.400 (Q1) |
---|---|
Impact-factor RSCI (2022): | 0.446 (Q2) |
ISSN (online): | 2227-7390 |
Site: | https://www.mdpi.com/journal/mathematics |
We investigate the dynamics of particles in a Kerr metric which describes the gravitational field in a neighborhood of a rotating black hole. After elimination of cyclic coordinates this problem reduces to investigating a Hamiltonian system with 2 degrees of freedom. This system possesses an additional Carter integral quadratic in momenta and hence is integrable by the Liouville-Arnold theorem. A bifurcation diagram is constructed and a classification of the types of trajectories of the system is carried out according to the values of first integrals. In particular, it is shown that there are seven different regions of values of first integrals which differ in the topological type of the integral submanifold. Pro-and-retrograde trajectories of particles are found.
Citation: | Bizyaev I. A., Mamaev I. S., Bifurcation diagram and a qualitative analysis of particle motion in a Kerr metric, Physical Review D, 2022, vol. 105, 063003, |
---|---|
DOI: | 10.1103/PhysRevD.105.063003 |
Full text: | pdf (2.68 Mb) |
Impact-factor WoS (2022): | 5.0 (Q1) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 2470-0010 |
ISSN (online): | 2470-0029 |
Site: | https://journals.aps.org/prd/ |
A model governing the motion of an aquatic robot with a shell in the form of a symmetrical airfoil NACA0040 is considered. The motion is controlled by periodic oscillations of the rotor. It is numerically shown that for physically admissible values of the control parameters in the phase space of the system, there exists only one limit cycle. The limit cycle that occurs under symmetric control corresponds to the motion of the robot near a straight line. In the case of asymmetric controls, the robot moves near a circle. An algorithm for controlling the course of the robot motion is proposed. This algorithm uses determined limit cycles and transient processes between them.
Keywords: | motion in a fluid, aquatic robot, control algorithm, limit cycles |
---|---|
Citation: | Vetchanin E. V., Mamaev I. S., Numerical analysis of the periodic controls of an aquatic robot, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2022, vol. 32, no. 4, |
DOI: | 10.35634/vm220410 |
Full text: | pdf (1.1 Mb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
The dynamics of a system governing the controlled motion of an unbalanced circular foil in the presence of point vortices is considered. The foil motion is controlled by periodically changing the position of the center of mass, the gyrostatic momentum, and the moment of inertia of the system. A derivation of the equations of motion based on Sedov's approach is proposed, the equations of motion are presented in the Hamiltonian form. A periodic perturbation of the known integrable case is considered.
Keywords: | motion in an ideal fluid, point vortices, period perturbation, vortex-body interaction |
---|---|
Citation: | Vetchanin E. V., Mamaev I. S., Periodic perturbation of motion of an unbalanced circular foil in the presence of point vortices in an ideal fluid, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2022, vol. 32, no. 4, |
DOI: | 10.35634/vm220409 |
Full text: | pdf (459.19 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
Artemova E. M., Kilin A. A., Korobeinikova Y. V.
This paper addresses the problem of a roller-racer rolling on an oscillating plane. Equations of motion of the roller-racer in the form of a system of four nonautonomous differential equations are obtained. Two families of particular solutions are found which correspond to rectilinear motions of the roller-racer along and perpendicular to the plane's oscillations. Numerical estimates are given for the multipliers of solutions corresponding to the motion of the robot along the oscillations. Also, a special case is presented in which it is possible to obtain analytic expressions of the multipliers. In this case, it is shown that the motion along oscillations of a “folded” roller-racer is linearly orbitally stable as it moves with its joint ahead, and that all other motions are unstable. It is shown that, in a linear approximation, the family corresponding to the motion of the robot is perpendicular to the plane's oscillations, that is, it is unstable.
Keywords: | roller-racer, nonholonomic constraints, vibrating plane, monodromy matrix, orbital stability |
---|---|
Citation: | Artemova E. M., Kilin A. A., Korobeinikova Y. V., Investigation of the orbital stability of rectilinear motions of roller-racer on a vibrating plane, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2022, vol. 32, no. 4, |
DOI: | 10.35634/vm220408 |
Full text: | pdf (513.14 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
This paper is concerned with the study of permanent rotations of a rigid body rolling without slipping on a horizontal plane (i. e., the velocity of the point of contact of the ellipsoid with the plane is zero). By permanent rotations we will mean motions of a rigid body on a horizontal plane such that the angular velocity of the body remains constant and the point of contact does not change its position. A more detailed analysis is made of permanent rotations of an omnirotational ellipsoid whose characteristic feature is the possibility of permanent rotations about any point of its surface.
Keywords: | nonholonomic mechanics, poincare map, stability, permanent rotations |
---|---|
Citation: | Bizyaev I. A., Mamaev I. S., Permanent Rotations in Nonholonomic Mechanics. Omnirotational Ellipsoid, Regular and Chaotic Dynamics, 2022, vol. 27, no. 6, |
DOI: | 10.1134/S1560354722060016 |
Full text: | pdf (2.99 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
In this paper, we consider the dynamics of two interacting point vortex rings in a Bose – Einstein condensate. The existence of an invariant manifold corresponding to vortex rings is proved. Equations of motion on this invariant manifold are obtained for an arbitrary number of rings from an arbitrary number of vortices. A detailed analysis is made of the case of two vortex rings each of which consists of two point vortices where all vortices have same topological charge. For this case, partial solutions are found and a complete bifurcation analysis is carried out. It is shown that, depending on the parameters of the Bose –Einstein condensate, there are three different types of bifurcation diagrams. For each type, typical phase portraits are presented.
Keywords: | Bose – Einstein condensate, point vortices, vortex rings, bifurcation analysis |
---|---|
Citation: | Artemova E. M., Kilin A. A., Dynamics of Two Vortex Rings in a Bose – Einstein Condensate, Regular and Chaotic Dynamics, 2022, vol. 27, no. 6, |
DOI: | 10.1134/S1560354722060089 |
Full text: | pdf (1.49 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Artemova E. M., Vetchanin E. V.
Describing the phenomena of the surrounding world is an interesting task that has long attracted the attention of scientists. However, even in seemingly simple phenomena, complex dynamics can be revealed. In particular, leaves on the surface of various bodies of water exhibit complex behavior. This paper addresses an idealized description of the mentioned phenomenon. Namely, the problem of the plane-parallel motion of an unbalanced circular disk moving in a stream of simple structure created by a point source (sink) is considered. Note that using point sources, it is possible to approximately simulate the work of skimmers used for cleaning swimming pools. Equations of coupled motion of the unbalanced circular disk and the point source are derived. It is shown that in the case of a fixed-position source of constant intensity the equations of motion of the disk are Hamiltonian. In addition, in the case of a balanced circular disk the equations of motion are integrable. A bifurcation analysis of the integrable case is carried out. Using a scattering map, it is shown that the equations of motion of the unbalanced disk are nonintegrable. The nonintegrability found here can explain the complex motion of leaves in surface streams of bodies of water.
Keywords: | ideal fluid, motion in the presence of a source, nonintegrability, scattering map, chaotic scattering |
---|---|
Citation: | Artemova E. M., Vetchanin E. V., The Motion of an Unbalanced Circular Disk in the Field of a Point Source, Regular and Chaotic Dynamics, 2022, vol. 27, no. 1, |
DOI: | 10.1134/S1560354722010051 |
Full text: | pdf (1019.43 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper discusses conditions for the existence of polynomial (in velocities)
first integrals of the equations of motion of mechanical systems in a nonpotential force field
(circulatory systems). These integrals are assumed to be single-valued smooth functions on
the phase space of the system (on the space of the tangent bundle of a smooth configuration
manifold). It is shown that, if the genus of the closed configuration manifold of such a system
with two degrees of freedom is greater than unity, then the equations of motion admit no
nonconstant single-valued polynomial integrals. Examples are given of circulatory systems with
configuration space in the form of a sphere and a torus which have nontrivial polynomial laws
of conservation. Some unsolved problems involved in these phenomena are discussed.
Keywords: | circulatory system, polynomial integral, genus of surface |
---|---|
Citation: | Kozlov V. V., On the Integrability of Circulatory Systems, Regular and Chaotic Dynamics, 2022, vol. 27, no. 1, |
DOI: | 10.1134/S1560354722010038 |
Full text: | pdf (378.07 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper describes the existing designs of spherical robots and reviews studies devoted to investigating their dynamics and to developing algorithms for controlling them. An analysis is also made of the key features and the historical aspects of the development of their designs, in particular, taking into account various areas of application.
Keywords: | spherical robot, rolling, design, modeling |
---|---|
Citation: | Karavaev Y. L., Spherical Robots: An Up-to-Date Overview of Designs and Features, Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 4, |
DOI: | 10.20537/nd221207 |
Full text: | pdf (51.9 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
This paper is concerned with the controlled motion of a three-link wheeled snake robot propelled by changing the angles between the central and lateral links. The limits on the applicability of the nonholonomic model for the problem of interest are revealed. It is shown that the system under consideration is completely controllable according to the Rashevsky – Chow theorem. Possible types of motion of the system under periodic snake-like controls are presented using Fourier expansions. The relation of the form of the trajectory in the space of controls to the type of motion involved is found. It is shown that, if the trajectory in the space of controls is centrally symmetric, the robot moves with nonzero constant average velocity in some direction.
Keywords: | nonholonomic mechanics, wheeled vehicle, snake robot, controllability, periodic control |
---|---|
Citation: | Artemova E. M., Kilin A. A., A Nonholonomic Model and Complete Controllability of a Three-Link Wheeled Snake Robot, Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 4, |
DOI: | 10.20537/nd221204 |
Full text: | pdf (582.78 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Vetchanin E. V., Portnov E. A.
In this paper we present a method for constructing inhomogeneous velocity fields of an incompressible fluid using expansions in terms of eigenfunctions of the Laplace operator whose weight coefficients are determined from the problem of minimizing the integral of the squared divergence. A number of examples of constructing the velocity fields of plane-parallel and axisymmetric flows are considered. It is shown that the problem of minimizing the integral value of divergence is incorrect and requires regularization. In particular, we apply Tikhonov’s regularization method. The method proposed in this paper can be used to generate different initial conditions in investigating the nonuniqueness of the solution to the Navier – Stokes equations.
Keywords: | inhomogeneous velocity field, expansion in terms of eigenfunctions, ill-conditioned system of linear algebraic equations |
---|---|
Citation: | Vetchanin E. V., Portnov E. A., Construction of Inhomogeneous Velocity Fields Using Expansions in Terms of Eigenfunctions of the Laplace Operator, Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 3, |
DOI: | 10.20537/nd220308 |
Full text: | pdf (3.56 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Karavaev Y. L., Klekovkin A. V., Mamaev I. S., Tenenev V. A., Vetchanin E. V.
This paper is concerned with the motion of an aquatic robot whose body has the form of a sharp-edged foil. The robot is propelled by rotating the internal rotor without shell deformation. The motion of the robot is described by a finite-dimensional mathematical model derived from physical considerations. This model takes into account the effect of added masses and viscous friction. The parameters of the model are calculated from comparison of experimental data and numerical solution to the equations of rigid body motion and the Navier–Stokes equations. The proposed mathematical model is used to define controls implementing straight-line motion, motion in a circle, and motion along a complex trajectory. Experiments for estimation of the efficiency of the model have been conducted.
Keywords: | aquatic robot, propulsion in a fluid, periodic control action, motion planning, identification of parameters of the model |
---|---|
Citation: | Karavaev Y. L., Klekovkin A. V., Mamaev I. S., Tenenev V. A., Vetchanin E. V., A Simple Physical Model for Control of an Propellerless Aquatic Robot, Journal of Mechanisms and Robotics, 2022, vol. 14, no. 1, 011007, |
DOI: | 10.1115/1.4051240 |
Full text: | pdf (981.69 Kb) |
Impact-factor WoS (2022): | 2.600 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.756 (Q1) |
ISSN (print): | 1942-4302 |
ISSN (online): | 1942-4310 |
Site: | https://asmejmr.org/ |
Kilin A. A., Karavaev Y. L., Ivanova T. B.
The paper presents the model of rolling resistance and the application of this model for the control of a pendulum actuated spherical robot on a horizontal plane. Control actions are derived in the form of maneuvers (gaits) which ensure the transition between two steady motions of the system. The experiments confirming the applicability of the model of viscous rolling friction and a method for determining coefficients of rolling resistance from experimental data are presented.
Keywords: | Spherical robot, Control, Nonholonomic constraint, Rolling resistance |
---|---|
Citation: | Kilin A. A., Karavaev Y. L., Ivanova T. B., Rolling Resistance Model and Control of Spherical Robot, Climbing and Walking Robots Conference: Robotics for Sustainable Future, CLAWAR 2021, 2022, |
DOI: | 10.1007/978-3-030-86294-7_35 |
Full text: | pdf (693.88 Kb) |
Site: | https://link.springer.com/book/10.1007/978-3-030-86294-7 |
---|
Kilin A. A., Karavaev Y. L., Yefremov K. S.
In this paper we presents the results of experimental investigations and simulations of the motion of the Roller Racer. We assume that the angle φ(t) between the platforms is a prescribed function of time and that the no-slip condition (nonholonomic constraint) and viscous friction force act at the points of contact of the wheels. The results of theoretical and experimental investigations are compared for various values of the parameters of the control action and design parameters of mobile robot.
Keywords: | Roller Racer, Nonholonomic constraint, Viscous friction, Control, Periodic solution |
---|---|
Citation: | Kilin A. A., Karavaev Y. L., Yefremov K. S., Experimental Investigations of the Controlled Motion of the Roller Racer Robot, Climbing and Walking Robots Conference: Robotics for Sustainable Future, CLAWAR 2021, 2022, |
DOI: | 10.1007/978-3-030-86294-7_38 |
Full text: | pdf (710.39 Kb) |
Site: | https://link.springer.com/book/10.1007/978-3-030-86294-7 |
---|
Ivanova T. B., Karavaev Y. L., Kilin A. A.
We investigate the model of controlledmotion of a pendulum-actuated spherical robot on a horizontal plane, taking rolling resistance into account. We derive equations of motion and obtain partial steady-state solutions.We present algorithms for designing elementary maneuvers (gaits) which ensure transition between two steady motions of the system. These gaits correspond to the acceleration along a straight line and to the turn through a given angle.
Keywords: | spherical robot, control, nonholonomic constraint, rolling resistance, gait, steady-state solutions |
---|---|
Citation: | Ivanova T. B., Karavaev Y. L., Kilin A. A., Control of a pendulum-actuated spherical robot on a horizontal plane with rolling resistance, Archive of Applied Mechanics, 2022, vol. 92, |
DOI: | 10.1007/s00419-021-02045-6 |
Full text: | pdf (618.92 Kb) |
Impact-factor WoS (2022): | 2.800 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.594 (Q2) |
Site: | https://www.springer.com/journal/419/ |
This paper investigates the controlled motion of a spherical robot on a plane performing horizontal periodic oscillations. The spherical robot is modeled by a balanced dynamically asymmetric sphere (the Chaplygin sphere) with three noncoplanar gyrostats placed inside it. The motion of the sphere is controlled by the controlled rotation of the internal gyrostats. The paper addresses two control problems concerning the construction of controls which generate motion along a trajectory given either on a moving plane or in a fixed frame of reference. It is shown that, using a control torque constant in the fixed frame of reference, the general problem can be reduced to the problem of control on the zero level set of the angular momentum integral. It is proved that, on the zero level set of the angular momentum integral, the system under consideration is completely controllable according to the Rashevsky – Chow theorem. Control algorithms for the motion of the sphere along an arbitrary prescribed trajectory are constructed. Examples are given of controls for the sphere rolling in a straight line in an arbitrary direction and in a circle, and for the sphere turning so that the position of the center of mass, both relative to the moving plane and relative to the fixed frame of reference, remains unchanged.
Keywords: | Chaplygin sphere, motion control, vibrating plane, rolling motion, spherical robot, angular momentum integral |
---|---|
Citation: | Kilin A. A., Pivovarova E. N., Motion control of the spherical robot rolling on a vibrating plane, Applied Mathematical Modelling, 2022, vol. 109, |
DOI: | 10.1016/j.apm.2022.05.002 |
Full text: | pdf (1.26 Mb) |
Impact-factor WoS (2022): | 5.0 (Q1) |
---|---|
Impact-factor RSCI (2022): | 1.080 (Q1) |
ISSN (print): | 0307-904X |
Site: | https://www.journals.elsevier.com/applied-mathematical-modelling |
This paper investigates the dynamics of a point vortex and a balanced circular foil in an ideal fluid. An explicit reduction to quadratures is performed. A bifurcation diagram is constructed and a classification of the types of integral manifolds is carried out. The stability of critical solutions is studied in which the foil and the vortex move in a circle or in a straight line.
Keywords: | point vortex, ideal fluid, bifurcation diagram, stability, relative equilibria |
---|---|
Citation: | Bizyaev I. A., Mamaev I. S., Qualitative Analysis of the Dynamics of a Balanced Circular Foil and a Vortex, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, |
DOI: | 10.1134/S1560354721060058 |
Full text: | pdf (1.34 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
In this paper we investigate the motion of a Chaplygin sphere rolling without slipping on a plane performing horizontal periodic oscillations. We show that in the system under consideration the projections of the angular momentum onto the axes of the fixed coordinate system remain unchanged. The investigation of the reduced system on a fixed level set of first integrals reduces to analyzing a three-dimensional period advance map on SO(3). The analysis of this map suggests that in the general case the problem considered is nonintegrable. We find partial solutions to the system which are a generalization of permanent rotations and correspond to nonuniform rotations about a body- and space-fixed axis. We also find a particular integrable case which, after time is rescaled, reduces to the classical Chaplygin sphere rolling problem on the zero level set of the area integral.
Keywords: | Chaplygin sphere, rolling motion, nonholonomic constraint, nonautonomous dynamical system, periodic oscillations, permanent rotations, integrable case, period advance map |
---|---|
Citation: | Kilin A. A., Pivovarova E. N., A Particular Integrable Case in the Nonautonomous Problem of a Chaplygin Sphere Rolling on a Vibrating Plane, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, |
DOI: | 10.1134/S1560354721060149 |
Full text: | pdf (679.77 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper addresses the problem of conditions for the existence of conservation laws (first integrals) of circulatory systems which are quadratic in velocities (momenta), when the external forces are nonpotential. Under some conditions the equations of motion are reduced to Hamiltonian form with some symplectic structure and the role of the Hamiltonian is played by a quadratic integral. In some cases the equations are reduced to a conformally Hamiltonian rather than Hamiltonian form. The existence of a quadratic integral and its properties allow conclusions to be drawn on the stability of equilibrium positions of circulatory systems.
Keywords: | circulatory system, polynomial integrals, Hamiltonian system, property of being conformally Hamiltonian, indices of inertia, asymptotic trajectories, Ziegler’s pendulum |
---|---|
Citation: | Kozlov V. V., Integrals of Circulatory Systems Which are Quadratic in Momenta, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, |
DOI: | 10.1134/S1560354721060046 |
Full text: | pdf (374.52 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Mamaev I. S., Kilin A. A., Karavaev Y. L., Shestakov V. A.
In this paper we present a study of the dynamics of a mobile robot with omnidirectional wheels taking into account the reaction forces acting from the plane. The dynamical equations are obtained in the form of Newton – Euler equations. In the course of the study, we formulate structural restrictions on the position and orientation of the omnidirectional wheels and their rollers taking into account the possibility of implementing the omnidirectional motion. We obtain the dependence of reaction forces acting on the wheel from the supporting surface on the parameters defining the trajectory of motion: linear and angular velocities and accelerations, and the curvature of the trajectory of motion. A striking feature of the system considered is that the results obtained can be formulated in terms of elementary geometry.
Keywords: | omnidirectional mobile robot, reaction force, simulation, nonholonomic model |
---|---|
Citation: | Mamaev I. S., Kilin A. A., Karavaev Y. L., Shestakov V. A., Criteria of Motion Without Slipping for an Omnidirectional Mobile Robot, Russian Journal of Nonlinear Dynamics, 2021, vol. 17, no. 4, |
DOI: | 10.20537/nd210412 |
Full text: | pdf (1.24 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Shaura A. S., Tenenev V. A., Vetchanin E. V.
This paper addresses the problem of balancing an inverted pendulum on an omnidirectional platform in a three-dimensional setting. Equations of motion of the platform – pendulum system in quasi-velocities are constructed. To solve the problem of balancing the pendulum by controlling the motion of the platform, a hybrid genetic algorithm is used. The behavior of the system is investigated under different initial conditions taking into account a necessary stop of the platform or the need for continuation of the motion at the end point of the trajectory. It is shown that the solution of the problem in a two-dimensional setting is a particular case of three-dimensional balancing.
Keywords: | balancing of an inverted pendulum, omnidirectional platform, hybrid genetic algorithm, Poincaré equations in quasi-velocities |
---|---|
Citation: | Shaura A. S., Tenenev V. A., Vetchanin E. V., The Problem of Balancing an Inverted Spherical Pendulum on an Omniwheel Platform, Russian Journal of Nonlinear Dynamics, 2021, vol. 17, no. 4, |
DOI: | 10.20537/nd210411 |
Full text: | pdf (501.49 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
This paper is concerned with the analysis of the influence of the friction model on the existence of additional integrals of motion in a system describing the sliding of a spherical top on a plane. We consider a model in which the friction is described not only by the force applied at the point of contact, but also by an additional friction torque. It is shown that, depending on the chosen friction model, the system admits various first integrals. In particular, we give examples of friction models in which either the Jellett integral or the Lagrange integral or the area integral is preserved.
Keywords: | spherical top, friction model, friction, integrals of motion |
---|---|
Citation: | Kilin A. A., Pivovarova E. N., Conservation laws for a spherical top on a plane with friction, International Journal of Non-Linear Mechanics, 2021, vol. 129, 103666, |
DOI: | 10.1016/j.ijnonlinmec.2020.103666 |
Full text: | pdf (353.6 Kb) |
Impact-factor WoS (2022): | 3.200 (Q2) |
---|---|
ISSN (print): | 0020-7462 |
ISSN (online): | 1878-5638 |
Site: | http://www.journals.elsevier.com/international-journal-of-non-linear-mechanics/ |
In this paper, we analyze the effect which the choice of a frictionmodel has on tippe top inversion in the case where the resulting action of all dissipative forces is described not only by the force applied at the contact point, but also by the additional rolling resistance torque. We show that the possibility or impossibility of tippe top inversion depends on the existence of specific integrals of the motion of the system. In this paper, we consider an example of the law of rolling resistance by which the area integral is preserved in the system.We examine in detail the case where the action of all dissipative forces reduces to the horizontal rolling resistance torque. This model describes fast rotations of the top between two horizontal smooth planes. For this case, we find permanent rotations of the system and analyze their linear stability. The stability analysis suggests that no tippe top inversion is possible under fast rotations between two planes.
Keywords: | spherical tippe top, rolling resistance torque, integrals of motion, tippe top inversion, reduction, stability analysis, partial solutions, bifurcations |
---|---|
Citation: | Kilin A. A., Pivovarova E. N., The influence of the first integrals and the rolling resistance model on tippe top inversion, Nonlinear Dynamics, 2021, vol. 103, no. 1, |
DOI: | 10.1007/s11071-020-06161-w |
Full text: | pdf (494.9 Kb) |
Impact-factor WoS (2022): | 5.600 (Q1) |
---|---|
ISSN (print): | 0924-090X |
ISSN (online): | 1573-269X |
Site: | http://link.springer.com/journal/11071 |
Bizyaev I. A., Bolotin S. V., Mamaev I. S.
This paper investigates nonholonomic systems (the Chaplygin sleigh and the Suslov system) with periodically varying mass distribution. In these examples, the behavior of velocities is described by a system of the form
dvdτ=f2(τ)u2+f1(τ)u+f0(τ),dudτ=−uv+g(τ),
where the coefficients are periodic functions of time τ with the same period. A detailed analysis is made of the problem of the existence of modes of motion for which the system speeds up indefinitely (an analog of Fermi’s acceleration). It is proved that, depending on the choice of coefficients, variable v has the asymptotics t1/k,k=1,2,3. In addition, we show regions of the phase space for which the system, when the trajectories are started from them, is observed to speed up. The proof uses normal forms and averaging in a slightly unusual form since unusual form averaging is performed over a variable that is not fast.
This paper continues a series of studies [1–5] of the
dynamics of nonholonomic systems with vary-
ing mass distribution due to the prescribed pe-
riodic motion of some structural components (ro-
tor, point masses etc.). Depending on the choice
of the law of variation of mass distribution, such
systems generally exhibit a large variety of be-
havior, both regular and chaotic. In addition,
it turns out that nonholonomic systems are one
of the simplest mathematical models of mechan-
ical systems exhibiting a phenomenon known as
unbounded speedup, which is due to redistribu-
tion of internal masses. In this paper, for a cer-
tain class of nonholonomic systems (including the
Chaplygin sleigh and the Suslov system) we find a
criterion which must be satisfied by the periodic
variation of mass distribution for the existence of
speeding-up trajectories.
Citation: | Bizyaev I. A., Bolotin S. V., Mamaev I. S., Normal forms and averaging in an acceleration problem in nonholonomic mechanics, Chaos, 2021, vol. 31, 013132, |
---|---|
DOI: | 10.1063/5.0030889 |
Full text: | pdf (1.05 Mb) |
Impact-factor WoS (2022): | 2.900 (Q1) |
---|---|
ISSN (print): | 1054-1500 |
ISSN (online): | 1089-7682 |
Site: | http://scitation.aip.org/content/aip/journal/chaos |
This paper addresses the problem of the motion of an unbalanced circular foil and point vortices in an ideal incompressible fluid. Using Bernoulli’s theorem for unsteady potential flow, the force due to the pressure from the fluid on the foil is obtained for an arbitrary vortex motion. A detailed analysis is made of the case of free vortex motion in which a Hamiltonian reduction by symmetries is performed. For the resulting system, relative equilibria corresponding to the motion of an unbalanced foil and a vortex in a circle or in a straight line are found and their stability is investigated. New examples of stationary configurations of a vortex and a foil are given. Using a Poincare map, it is also shown that in the general case of an unbalanced circular foil the reduced system exhibits chaotic trajectories.
Citation: | Mamaev I. S., Bizyaev I. A., Dynamics of an unbalanced circular foil and point vortices in an ideal fluid, Physics of Fluids, 2021, vol. 33, 087119, |
---|---|
DOI: | 10.1063/5.0058536 |
Full text: | pdf (2.36 Mb) |
Impact-factor WoS (2022): | 4.600 (Q1) |
---|---|
Impact-factor RSCI (2022): | 1.083 (Q1) |
ISSN (print): | 1070-6631 |
ISSN (online): | 1089-7666 |
Site: | https://aip.scitation.org/journal/phf |
The problem of stability of rotating regular vortex N-gons (Thomson’s configurations) in a Bose – Einstein condensate in a harmonic trap is considered. A reduction procedure on the level set of the momentum integral is proposed. The dependence of the velocity of rotation ω of vortex polygon about the center of the trap is obtained as a function of the number of vortices N and the radius of the configuration, R. The analysis of the orbital linear and nonlinear stability of the motion of such configurations is carried out. For N⩽6, regions of orbital stability of configurations in the parameter space are constructed. It is shown that vortex N-gons for N>6 are unstable for any parameters of the system. In this paper, we study the stability of rotating regular vortex N-gons in a Bose – Einstein condensate in a harmonic trap. The analysis of the orbital linear and nonlinear stability of motion is carried out. The dependence of the stability of regular vortex N-gons on the number of vortices N and the parameters of the system is given.
Citation: | Artemova E. M., Kilin A. A., Nonlinear stability of regular vortex polygons in a Bose – Einstein condensate, Physics of Fluids, 2021, vol. 33, no. 12, 127105, |
---|---|
DOI: | 10.1063/5.0070763 |
Full text: | pdf (610.67 Kb) |
Impact-factor WoS (2022): | 4.600 (Q1) |
---|---|
Impact-factor RSCI (2022): | 1.083 (Q1) |
ISSN (print): | 1070-6631 |
ISSN (online): | 1089-7666 |
Site: | https://aip.scitation.org/journal/phf |
This paper investigates the rolling motion of a spherical top with an axisymmetric mass distribution on a smooth horizontal plane performing periodic vertical oscillations. For the system under consideration, equations of motion and conservation laws are obtained. It is shown that the system admits two equilibrium points corresponding to uniform rotations of the top about the vertical symmetry axis. The equilibrium point is stable when the center of mass is located below the geometric center, and is unstable when the center of mass is located above it. The equations of motion are reduced to a system with one and a half degrees of freedom. The reduced system is represented as a small perturbation of the problem of the motion of the Lagrange top. Using Melnikov’s method, it is shown that the stable and unstable branches of the separatrix intersect transversally with each other. This suggests that the problem is nonintegrable. Results of computer simulation of the top dynamics near the unstable equilibrium point are presented.
Keywords: | spherical top, vibrating plane, Lagrange case, separatrix splitting, Melnikov’s integral, nonintegrability, chaos, period advance map |
---|---|
Citation: | Kilin A. A., Pivovarova E. N., Separatrix splitting in the problem of a spherical top rolling on a vertically vibrating plane, 2021 International Conference ''Nonlinearity, Information and Robotics'' – IEEE, 2021, |
DOI: | 10.1109/NIR52917.2021.9666059 |
Full text: | pdf (135.06 Kb) |
Site: | https://ieeexplore.ieee.org/xpl/conhome/9666040/proceeding |
---|
Shestakov V. A., Mamaev I. S., Karavaev Y. L.
In this paper, the object of study is a highly maneuverable mobile robot with omni wheels. The paper presents the investigation of the dynamics of omni wheels as part of a mobile platform. The design of a mobile platform with omni wheels is analyzed, taking into account the dynamics of an individual wheel as a part of a mobile platform. The equations of dynamics are presented in the form of Lagrange equations of the second kind with undetermined multipliers. As a result, several conditions of design constraints were identified, under which the omnidirectional motion of the mobile platform is impossible.
Keywords: | omnidirectional mobile robot, reaction force, simulation, nonholonomic model |
---|---|
Citation: | Shestakov V. A., Mamaev I. S., Karavaev Y. L., Influence of the design features of omni-wheeled mobile robots on the possibility of motion without slipping, 2021 International Conference ''Nonlinearity, Information and Robotics'' – IEEE, 2021, |
DOI: | 10.1109/NIR52917.2021.9666137 |
Full text: | pdf (187.28 Kb) |
Site: | https://ieeexplore.ieee.org/xpl/conhome/9666040/proceeding |
---|
Klekovkin A. V., Karavaev Y. L., Mamaev I. S.
This paper describes the design of the underwater robot which is moved by rotating the internal rotor. The design of the robot is described. A finite-dimensional mathematical model describing the robot’s motion is presented. The study of the derived equations of motion is carried out, the influence of the parameters of the control action on the mode of the robot motion is considered.
Keywords: | mobile robot, aquatic robot, motion simulation |
---|---|
Citation: | Klekovkin A. V., Karavaev Y. L., Mamaev I. S., Design and control for the underwater robot with internal rotor, 2021 International Conference ''Nonlinearity, Information and Robotics'' – IEEE, 2021, |
DOI: | 10.1109/NIR52917.2021.9666116 |
Full text: | pdf (139.59 Kb) |
Site: | https://ieeexplore.ieee.org/xpl/conhome/9666040/proceeding |
---|
This paper considers the application of the harmonic balance method for calculating the boundaries of the instability region of the Liouville problem. Explicit equations for the boundaries of the first four instability regions are given.
Keywords: | harmonic balance method, Liouville’s problem, equations with periodic coefficients |
---|---|
Citation: | Vetchanin E. V., Calculation of instability regions of the Liouville problem based on the harmonic balance method, 2021 International Conference ''Nonlinearity, Information and Robotics'' – IEEE, 2021, |
DOI: | 10.1109/NIR52917.2021.9666094 |
Full text: | pdf (117.07 Kb) |
Site: | https://ieeexplore.ieee.org/xpl/conhome/9666040/proceeding |
---|
Ardentov A. A., Yefremov K. S.
The work investigates the experimental problem of reparking trailer for a wheeled robot with a trailer. The geometric model with kinematic constraints leads to the sub-Riemannian problem. We solve this problem via nilpotent approximation. The corresponding solution is close to optimal and locally minimizes the total kinetic energy of the driving wheels. A full-scale model is designed in a way to avoid phase constraints usually appearing in trailer systems. We perform 64 different experiments with reparking trailer and obtain the satisfactory accuracy: for one maneuver we repark the trailer with maximum angle error equal to 4 degrees.
Keywords: | mobile robot, trailer, motion planning, Vicon, sub-Riemannian problem, nilpotent approximation |
---|---|
Citation: | Ardentov A. A., Yefremov K. S., Automatic reparking of the robot trailer along suboptimal paths, 2021 International Conference ''Nonlinearity, Information and Robotics'' – IEEE, 2021, |
DOI: | 10.1109/NIR52917.2021.9666086 |
Full text: | pdf (391.66 Kb) |
Site: | https://ieeexplore.ieee.org/xpl/conhome/9666040/proceeding |
---|
Equations governing the motion of vortex N-gons in a Bose – Einstein condensate enclosed in a harmonic trap are obtained. It is shown that these equations have two first integrals: the integral of energy and the integral of the moment. The reduction on the fixed level set of the integral of moment is carried out. The equilibrium positions of the system considered are found. Phase portraits of the system of two vortex N-gons, each of which consists of two vortices, are constructed.
Keywords: | vortex N-gons, Bose – Einstein condensate, integrability |
---|---|
Citation: | Artemova E. M., Kilin A. A., Dynamics of interacting two vortex N-gons in BECs, 2021 International Conference ''Nonlinearity, Information and Robotics'' – IEEE, 2021, |
DOI: | 10.1109/NIR52917.2021.9666121 |
Full text: | pdf (274.08 Kb) |
Site: | https://ieeexplore.ieee.org/xpl/conhome/9666040/proceeding |
---|
We consider a nonholonomic system that describes the rolling without slipping of a spherical shell inside which a frame rotates with constant angular velocity (this system is one of the possible generalizations of the problem of the rolling of a Chaplygin sphere). After a suitable scale transformation of the radius of the shell or the mass of the system the equations of motion can be represented as a perturbation of the integrable Euler case in rigid body dynamics. Using this representation, we explicitly calculate a Melnikov integral, which contains an isolated zero under some restrictions on the system parameters. Thereby we prove the absence of an additional integral in this system and the existence of chaotic trajectories. We conclude by presenting numerical experiments that illustrate the system dynamics depending on the behavior of the Melnikov function.
Keywords: | Nonholonomic mechanics, Melnikov integral, Separatrix splitting, Poincaré map |
---|---|
Citation: | Bizyaev I. A., Mamaev I. S., Separatrix splitting and nonintegrability in the nonholonomic rolling of a generalized Chaplygin sphere, International Journal of Non-Linear Mechanics, 2020, vol. 126, 103550, |
DOI: | 10.1016/j.ijnonlinmec.2020.103550 |
Full text: | pdf (2.22 Mb) |
Impact-factor WoS (2022): | 3.200 (Q2) |
---|---|
ISSN (print): | 0020-7462 |
ISSN (online): | 1878-5638 |
Site: | http://www.journals.elsevier.com/international-journal-of-non-linear-mechanics/ |
We consider the problem of the stability of rotating regular vortex N-gons (Thomson configurations) in a Bose-Einstein condensate in a harmonic trap. The dependence of the rotation velocity ω of the Thomson configuration around the center of the trap is obtained as a function of the number of vortices N and the radius of the configuration R. The analysis of the stability of motion of such configurations in the linear approximation is carried out. For N⩽6, regions of orbital stability of configurations in the parameter space are constructed. It is shown that vortex N-gons for N>6 are unstable for any parameters of the system.
Keywords: | vortex dynamics, Thomson configurations, Bose–Einstein condensate, linear stability |
---|---|
Citation: | Kilin A. A., Artemova E. M., Stability of regular vortex polygons in Bose–Einstein condensate, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2020, vol. 56, |
DOI: | 10.35634/2226-3594-2020-56-02 |
Full text: | pdf (185 Kb) |
Impact-factor WoS (2022): | 0.400 |
---|---|
Impact-factor RSCI (2022): | 0.318 (Q3) |
ISSN (print): | 2226-3594 |
ISSN (online): | 2410-1737 |
Site: | http://journals.udsu.ru/mathematics |
We consider a propellerless robot that moves on the surface of a fluid by rotating of the internal rotor. The robot shell has a symmetric shape of NACA 0040 airfoil. The equations of motion are written in the form of classical Kirchhoff equations with terms describing the viscous friction. The control action based on the derived model is proposed. The influences of various model parameters on the robot’s trajectory have been studied.
Keywords: | mobile robot, propellerless robot, aquatic robot, motion simulation, Kirchhoff equations |
---|---|
Citation: | Klekovkin A. V., Simulation of the motion of a propellerless mobile robot controlled by rotation of the internal rotor, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2020, vol. 30, no. 4, |
DOI: | 10.35634/vm200408 |
Full text: | pdf (210.33 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
This paper investigates the rolling motion of a spherical top with an axisymmetric mass distribution on a smooth horizontal plane performing periodic vertical oscillations. For the system under consideration, equations of motion and conservation laws are obtained. It is shown that the system admits two equilibrium points corresponding to uniform rotations of the top about the vertical symmetry axis. The equilibrium point is stable when the center of mass is located below the geometric center, and is unstable when the center of mass is located above it. The equations of motion are reduced to a system with one and a half degrees of freedom. The reduced system is represented as a small perturbation of the problem of the Lagrange top motion. Using Melnikov’s method, it is shown that the stable and unstable branches of the separatrix intersect transversally with each other. This suggests that the problem is nonintegrable. Results of computer simulation of the top dynamics near the unstable equilibrium point are presented.
Keywords: | spherical top, vibrating plane, Lagrange case, separatrix splitting, Melnikov’s integral, nonintegrability, chaos, period advance map |
---|---|
Citation: | Kilin A. A., Pivovarova E. N., Nonintegrability of the problem of a spherical top rolling on a vibrating plane, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2020, vol. 30, no. 4, |
DOI: | 10.35634/vm200407 |
Full text: | pdf (299.53 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
In this paper we obtain equations of motion for a vortex pair and a circular foil with parametric excitation due to the periodic motion of a material point. Undoubtedly, such problems are, on the one hand, model problems and cannot be used for an exact quantitative description of real trajectories of the system. On the other hand, in many cases such 2D models provide a sufficiently accurate qualitative picture of the dynamics and, due to their simplicity, an estimate of the influence of different parameters. We describe relative equilibria that generalize F¨oppl solutions and collinear configurations when the material point does not move. We show that a stochastic layer forms in the neighborhood of relative equilibria in the case of periodic motion of the foil’s center of mass.
Keywords: | point vortices, ideal fluid, Poincar´e map, foil in a fluid, 2D hydrodynamics |
---|---|
Citation: | Bizyaev I. A., Mamaev I. S., Dynamics of a pair of point vortices and a foil with parametric excitation in an ideal fluid, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2020, vol. 30, no. 4, |
DOI: | 10.35634/vm200406 |
Full text: | pdf (553.19 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
Artemova E. M., Vetchanin E. V.
The motion of a circular cylinder in an ideal fluid in the field of a fixed source is considered. It is shown that, when the source has constant strength, the system possesses a momentum integral and an energy integral. Conditions are found under which the equations of motion reduced to the level set of the momentum integral admit an unstable fixed point. This fixed point corresponds to circular motion of the cylinder about the source. A feedback is constructed which ensures stabilization of the above-mentioned fixed point by changing the strength of the source.
Keywords: | control, ideal fluid, feedback, motion in the presence of a source |
---|---|
Citation: | Artemova E. M., Vetchanin E. V., Control of the motion of a circular cylinder in an ideal fluid using a source, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2020, vol. 30, no. 4, |
DOI: | 10.35634/vm200405 |
Full text: | pdf (356.62 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
This paper addresses the problem of a spherical robot having an axisymmetric pendulum drive and rolling without slipping on a vibrating plane. It is shown that this system admits partial solutions (steady rotations) for which the pendulum rotates about its vertical symmetry axis. Special attention is given to problems of stability and stabilization of these solutions. An analysis of the constraint reaction is performed, and parameter regions are identified in which a stabilization of the spherical robot is possible without it losing contact with the plane. It is shown that the partial solutions can be stabilized by varying the angular velocity of rotation of the pendulum about its symmetry axis, and that the rotation of the pendulum is a necessary condition for stabilization without the robot losing contact with the plane.
Keywords: | spherical robot, vibrations, stability, stabilization, partial solutions, constraint reaction, Lagrange top, Kapitsa pendulum |
---|---|
Citation: | Kilin A. A., Pivovarova E. N., Stability and Stabilization of Steady Rotations of a Spherical Robot on a Vibrating Base, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, |
DOI: | 10.1134/S1560354720060155 |
Full text: | pdf (1.62 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Artemova E. M., Karavaev Y. L., Mamaev I. S., Vetchanin E. V.
The motion of a spherical robot with periodically changing moments of inertia, internal rotors and a displaced center of mass is considered. It is shown that, under some restrictions on the displacement of the center of mass, the system of interest features chaotic dynamics due to separatrix splitting. A stability analysis is made of the upper equilibrium point of the ball and of the periodic solution arising in its neighborhood, in the case of periodic rotation of the rotors. It is shown that the lower equilibrium point can become unstable in the case of fixed rotors and periodically changing moments of inertia.
Keywords: | nonholonomic constraint, rubber rolling, unbalanced ball, rolling on a plane |
---|---|
Citation: | Artemova E. M., Karavaev Y. L., Mamaev I. S., Vetchanin E. V., Dynamics of a Spherical Robot with Variable Moments of Inertia and a Displaced Center of Mass, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, |
DOI: | 10.1134/S156035472006012X |
Full text: | pdf (950.79 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The properties of the Gibbs ensembles of Hamiltonian systems describing the motion along geodesics on a compact configuration manifold are discussed.We introduce weakly ergodic systems for which the time average of functions on the configuration space is constant almost everywhere. Usual ergodic systems are, of course, weakly ergodic, but the converse is not true. A range of questions concerning the equalization of the density and the temperature of a Gibbs ensemble as time increases indefinitely are considered. In addition, the weak ergodicity of a billiard in a rectangular parallelepiped with a partition wall is established.
Keywords: | Hamiltonian system, Liouville and Gibbs measures, Gibbs ensemble, weak ergodicity, mixing, billiard in a polytope |
---|---|
Citation: | Kozlov V. V., Nonequilibrium Statistical Mechanics of Weakly Ergodic Systems, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, |
DOI: | 10.1134/S1560354720060118 |
Full text: | pdf (431.27 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper examines the motion of a balanced spherical robot under the action of periodically changing moments of inertia and gyrostatic momentum. The system of equations of motion is constructed using the model of the rolling of a rubber body (without slipping and twisting) and is nonconservative. It is shown that in the absence of gyrostatic momentum the equations of motion admit three invariant submanifolds corresponding to plane-parallel motion of the sphere with rotation about the minor, middle and major axes of inertia. The abovementioned motions are quasi-periodic, and for the numerical estimate of their stability charts of the largest Lyapunov exponent and charts of stability are plotted versus the frequency and amplitude of the moments of inertia. It is shown that rotations about the minor and major axes of inertia can become unstable at sufficiently small amplitudes of the moments of inertia. In this case, the so-called “Arnol’d tongues” arise in the stability chart. Stabilization of the middle unstable axis of inertia turns out to be possible at sufficiently large amplitudes of the moments of inertia, when the middle axis of inertia becomes the minor axis for a part of a period. It is shown that the nonconservativeness of the system manifests itself in the occurrence of limit cycles, attracting tori and strange attractors in phase space. Numerical calculations show that strange attractors may arise through a cascade of period-doubling bifurcations or after a finite number of torus-doubling bifurcations.
Keywords: | nonholonomic constraints, rubber rolling, periodic control, stability analysis, perioddoubling bifurcation, torus-doubling bifurcation |
---|---|
Citation: | Mamaev I. S., Vetchanin E. V., Dynamics of Rubber Chaplygin Sphere under Periodic Control, Regular and Chaotic Dynamics, 2020, vol. 25, no. 2, |
DOI: | 10.1134/S1560354720020069 |
Full text: | pdf (4.03 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Ivanova T. B., Kilin A. A., Mamaev I. S.
This paper is concerned with the problem of a dynamically symmetric heavy ball rolling without slipping on a cone which rotates uniformly about its symmetry axis. The equations of motion of the system are obtained, partial periodic solutions are found, and their stability is analyzed.
Citation: | Borisov A. V., Ivanova T. B., Kilin A. A., Mamaev I. S., Circular orbits of a ball on a rotating conical turntable, Acta Mechanica, 2020, vol. 231, no. 3, |
---|---|
DOI: | 10.1007/s00707-019-02556-y |
Full text: | pdf (323.69 Kb) |
Impact-factor WoS (2022): | 2.700 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.598 (Q2) |
ISSN (print): | 0001-5970 |
ISSN (online): | 1619-6937 |
Site: | https://link.springer.com/journal/707 |
Borisov A. V., Vetchanin E. V., Mamaev I. S.
This paper considers the plane-parallel motion of an elliptic foil in a fluid with a nonzero constant circulation under the action of external periodic forces and torque. The existence of the first integral is shown for the case in which there is no external torque and an external force acts along one of the principal axes of the foil. It is shown that, in the general case, in the absence of friction, an extensive stochastic layer is observed for the period advance map. When dissipation is added to the system, strange attractors can arise from the stochastic layer.
Citation: | Borisov A. V., Vetchanin E. V., Mamaev I. S., Motion of a Smooth Foil in a Fluid under the Action of External Periodic Forces. II, Russian Journal of Mathematical Physics, 2020, vol. 27, no. 1, |
---|---|
DOI: | 10.1134/S106192082001001X |
Full text: | pdf (1.98 Mb) |
Impact-factor WoS (2022): | 1.400 (Q3) |
---|---|
ISSN (print): | 1061-9208 |
ISSN (online): | 1555-6638 |
Site: | http://link.springer.com/journal/11503 |
Hricko J., Havlík Š., Karavaev Y. L.
The paper is focused to design, simulation and modeling of the compact compliant structures widely used in construction of robotic devices. As the illustrative example it is proposed mechanism for reduction of motion, which enables to improve the accuracy of the positioning system. The physical model is fabricated by 3D printing technology. Its proposed performance characteristics are verified by measurement on the experimental test bed by using laser distance sensors and image sensing/processing technology.
Keywords: | compact compliant mechanisms, 3D printing, modeling and simulation, HIL simulations Received |
---|---|
Citation: | Hricko J., Havlík Š., Karavaev Y. L., Verifying the Performance Characteristics of the (micro) Robotic Devices, Russian Journal of Nonlinear Dynamics, 2020, vol. 16, no. 1, |
DOI: | 10.20537/nd200112 |
Full text: | pdf (2.54 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bozek P., Karavaev Y. L., Ardentov A. A., Yefremov K. S.
This article is concerned with developing an intelligent system for the control of a wheeled robot. An algorithm for training an artificial neural network for path planning is proposed. The trajectory ensures steering optimal motion from the current position of the mobile robot to a prescribed position taking its orientation into account. The proposed control system consists of two artificial neural networks. One of them serves to specify the position and the size of the obstacle, and the other forms a continuous trajectory to reach it, taking into account the information received, the coordinates, and the orientation at the point of destination. The neural network is trained on the basis of samples obtained by modeling the equations of motion of the wheeled robot which ensure its motion along trajectories in the form of Euler’s elastica.
Keywords: | Wheeled mobile robots, robust, adaptive and optimal control, collision avoidance and multi-vehicle systems, path planning and navigation, robot learning |
---|---|
Citation: | Bozek P., Karavaev Y. L., Ardentov A. A., Yefremov K. S., Neural network control of a wheeled mobile robot based on optimal trajectories, International Journal of Advanced Robotic Systems, 2020, |
DOI: | 10.1177/1729881420916077 |
Full text: | pdf (911.75 Kb) |
Impact-factor WoS (2022): | 2.300 (Q4) |
---|---|
Impact-factor RSCI (2022): | 0.550 (Q2) |
ISSN (print): | 1729-8814 |
ISSN (online): | 1729-8814 |
Site: | https://uk.sagepub.com/en-gb/asi/node/859241 |
This paper is concerned with the study of a nonholonomic system with parametric excitation, the Suslov problem with variable gyrostatic momentum. A detailed analysis is made of the problem of the existence of regimes with unbounded growth of energy (an analog of Fermi’s acceleration). The existence of trajectories is shown for which the angular velocity of the rigid body increases indefinitely and has the asymptotics t12
Keywords: | nonholonomic mechanics, Fermi’s acceleration, Suslov problem, unbounded speedup, strange attractor, rotor |
---|---|
Citation: | Bizyaev I. A., Mamaev I. S., Dynamics of the nonholonomic Suslov problem under periodic control: unbounded speedup and strange attractors, Journal of Physics A: Mathematical and Theoretical, 2020, vol. 53, 185701, |
DOI: | 10.1088/1751-8121/ab7e52 |
Full text: | pdf (1.24 Mb) |
Impact-factor WoS (2022): | 2.100 (Q1) |
---|---|
Impact-factor RSCI (2022): | 0.718 (Q2) |
ISSN (print): | 1751-8113 |
ISSN (online): | 1751-8121 |
Site: | http://iopscience.iop.org/1751-8121/ |
In this paper, we investigate the motion of a homogeneous heavy ball rolling without slipping on the surface of a rotating cylinder in two settings: a setting without dissipation and a setting with rolling friction torque which is proportional to the angular velocity of the ball. In both caseswe assume that there exists a non-holonomic constraint that corresponds to the condition that there be no slipping at the point of contact. In the first case, the system of five differential equations on the level set of first integrals is reduced to quadratures. To define possible types of motion, we carry out a bifurcation analysis of the reduced system. In the case with friction we show that all trajectories shift on average downward and the ball falls.
Keywords: | bifurcation analysis, cylinder, non-holonomic constraint, periodic solutions, qualitative analysis, rolling resistance, rotating surface |
---|---|
Citation: | Ivanova T. B., Non-holonomic rolling of a ball on the surface of a rotating cylinder, ZAMM - Zeitschrift fur Angewandte Mathematik und Mechanik, 2020, vol. 100, no. 12, e202000067, |
DOI: | 10.1002/zamm.202000067 |
Full text: | pdf (476.51 Kb) |
Impact-factor WoS (2022): | 2.300 (Q1) |
---|---|
Impact-factor RSCI (2022): | 0.410 (Q2) |
ISSN (print): | 0044-2267 |
ISSN (online): | 1521-4001 |
Site: | https://onlinelibrary.wiley.com/journal/15214001 |
This paper addresses a conservative system describing the motion of a smooth body in an ideal fluid under the action of an external periodic torque with nonzero mean and of an external periodic force. It is shown that, in the case where the body is circular in shape, the angular velocity of the body increases indefinitely (linearly in time), and the projection of the phase trajectory onto the plane of translational velocities is attracted to a circle. Asymptotic orbital stability (or asymptotic stability with respect to part of variables) exists in the system. It is shown numerically that, in the case of an elliptic body, the projection of the phase trajectory onto the plane of translational velocities is attracted to an annular region.
Citation: | Vetchanin E. V., Mamaev I. S., Asymptotic behavior in the dynamics of a smooth body in an ideal fluid, Acta Mechanica, 2020, vol. 231, |
---|---|
DOI: | 10.1007/s00707-020-02791-8 |
Full text: | pdf (519.06 Kb) |
Impact-factor WoS (2022): | 2.700 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.598 (Q2) |
ISSN (print): | 0001-5970 |
ISSN (online): | 1619-6937 |
Site: | https://link.springer.com/journal/707 |
Karavaev Y. L., Mamaev I. S., Kilin A. A., Pivovarova E. N.
This paper presents a review of papers devoted to the creation and investigation of spherical robots. Owing to their structural features, namely, geometric symmetry and resistance of actuating mechanisms and elements of the control system against aggressive environmental conditions, robotic systems of this type have a high potential of being used in problems of monitoring, reconnaissance, and transportation on Earth and other planets. A detailed description is given of the structures of prototypes of spherical robots which use different actuation principles: a spherical robot with an internal pendulum mechanism, a spherical robot with an internal omniwheel platform, a spherical robot with internal rotors, and a spherical robot of combined type. Experimental results are presented to give an estimate of the possibility and efficiency of controlled motion. The applied use of spherical robots depending on the type of the actuating mechanism is discussed.
Citation: | Karavaev Y. L., Mamaev I. S., Kilin A. A., Pivovarova E. N., Spherical rolling robots: Different designs and control algorithms, Robots in Human Life: Proceedings of the 23rd International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines, CLAWAR 2020, 2020, |
---|---|
DOI: | 10.13180/clawar.2020.24-26.08.47 |
Full text: | pdf (415.75 Kb) |
ISSN (print): | 978-1-9164490-4-6 |
---|---|
Site: | https://clawar.org/clawar2020/ |
Klekovkin A. V., Mamaev I. S., Vetchanin E. V., Tenenev V. A., Karavaev Y. L.
This paper is devoted to investigations of the motion of the propellerless aquatic robots. There are two models of aquatic robots under consideration that move due to rotation of internal rotors. Mathematical models to describe the motion of the robots are proposed. Experiments with different control actions for fabricated prototypes to verify mathematical models have been conducted.
Citation: | Klekovkin A. V., Mamaev I. S., Vetchanin E. V., Tenenev V. A., Karavaev Y. L., Propellerless aquatic robots, Robots in Human Life: Proceedings of the 23rd International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines, CLAWAR 2020, 2020, |
---|---|
DOI: | 10.13180/clawar.2020.24-26.08.54 |
Full text: | pdf (633.84 Kb) |
ISSN (print): | 978-1-9164490-4-6 |
---|---|
Site: | https://clawar.org/clawar2020/ |
Shestakov V. A., Mamaev I. S., Karavaev Y. L.
This paper presents the control algorithm of an omnidirectional mobile robot to implement motion along curvilinear trajectories. The trajectory is defined by the Bezier curves of the third order. An algorithm is proposed to generate control actions for the robot’s motion along a given curvilinear trajectory, taking into account the processes of acceleration and deceleration. The experimental results confirm the applicability of the proposed method.
Keywords: | omnidirectional mobile robot, Bezier curves, kinematics, modeling, omnidirectional motion, nonholonomic model |
---|---|
Citation: | Shestakov V. A., Mamaev I. S., Karavaev Y. L., Controlled motion of a highly maneuverable mobile robot along curvilinear trajectories, 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, |
DOI: | 10.1109/NIR50484.2020.9290159 |
Full text: | pdf (3.15 Mb) |
Site: | https://ieeexplore.ieee.org/xpl/conhome/9290150/proceeding |
---|
Karavaev Y. L., Kilin A. A., Klekovkin A. V., Pivovarova E. N.
This paper is devoted to investigations of a spherical robot rolling on an oscillating underlying surface. The model of spherical robot of combined type is considered. Based on analysis of equations of motion taking into account the oscillation of the underlying surface, a control algorithm for stabilization of the spherical robot is proposed. The influences of oscillations in horizontal and vertical directions are evaluated. The design of the spherical robot and its control system for fabrication of a prototype are described.
Keywords: | spherical robot, nonholonomic model, oscillations of the underlying surface, feedback |
---|---|
Citation: | Karavaev Y. L., Kilin A. A., Klekovkin A. V., Pivovarova E. N., Stabilization of a spherical robot rolling on an oscillating underlying surface, 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, |
DOI: | 10.1109/NIR50484.2020.9290225 |
Full text: | pdf (1.09 Mb) |
Site: | https://ieeexplore.ieee.org/xpl/conhome/9290150/proceeding |
---|
Yefremov K. S., Ivanova T. B., Kilin A. A., Karavaev Y. L.
In this paper we address the problem of the motion of the Roller Racer. We assume that the angle φ(t) between the platforms is a prescribed function of time and that the no-slip condition (nonholonomic constraint) and viscous friction force act at the points of contact of the wheels. In this case, all trajectories of the reduced system asymptotically tend to a periodic solution. In this paper it is shown analytically and experimentally that the chosen control defines periodic trajectories of the attachment point of the platforms on an average along a straight line. We determine the conditions for optimal control when the system moves along a straight line depending on the mass and geometric characteristics of the system and control parameters.
Keywords: | Roller Racer, nonholonomic constraint, viscous friction, control, periodic solution |
---|---|
Citation: | Yefremov K. S., Ivanova T. B., Kilin A. A., Karavaev Y. L., Theoretical and experimental investigations of the controlled motion of the Roller Racer, 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, |
DOI: | 10.1109/NIR50484.2020.9290220 |
Full text: | pdf (408.24 Kb) |
Site: | https://ieeexplore.ieee.org/xpl/conhome/9290150/proceeding |
---|
The motion of a spherical robot with periodically changing moments of inertia and gyrostatic momentum is considered. Equations of motion are derived within the framework of the model of "rubber" rolling (without slipping and twisting). The stability of partial solutions of the system is studied numerically. It is shown that the system is nonconservative, and, as a consequence, limit cycles and strange attractors exist in the phase space of the system.
Keywords: | stability of the motion, periodic controls, nonholonomic model, rubber rolling |
---|---|
Citation: | Mamaev I. S., Vetchanin E. V., Dynamics of a spherical robot with periodically changing moments of inertia, 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, |
DOI: | 10.1109/NIR50484.2020.9290229 |
Full text: | pdf (310.64 Kb) |
Site: | https://ieeexplore.ieee.org/xpl/conhome/9290150/proceeding |
---|
This paper addresses the problem of the motion of a circular foil and point vortices in an ideal fluid. A complete qualitative analysis of the motion of a balanced foil and one vortex is carried out. In particular, periodic solutions are found and their stability is investigated. Using a Poincaré map, it is shown that for an unbalanced foil a reduced system exhibits chaotic trajectories.
Keywords: | point vortices, ideal fluid, integrability, bifurcation diagram, Poincaré map |
---|---|
Citation: | Mamaev I. S., Bizyaev I. A., Dynamics of point vortices and a cylinder in an ideal fluid, 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, |
DOI: | 10.1109/NIR50484.2020.9290156 |
Full text: | pdf (241.98 Kb) |
Site: | https://ieeexplore.ieee.org/xpl/conhome/9290150/proceeding |
---|
The inertial motion of a three-link symmetric wheeled vehicle on a plane is considered. A special case where the system admits an additional first integral of motion and invariant measure with nonsmooth density is found.
Keywords: | nonholonomic mechanics, wheeled vehicle, inertial motion |
---|---|
Citation: | Artemova E. M., Kilin A. A., An integrable case in the dynamics of a three-link vehicle, 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, |
DOI: | 10.1109/NIR50484.2020.9290238 |
Full text: | pdf (1.17 Mb) |
Site: | https://ieeexplore.ieee.org/xpl/conhome/9290150/proceeding |
---|
The dynamics of a body with a fixed point is considered in the case where the moments of inertia of the system depend periodically on time. A stability of permanent rotations is estimated by a numerical approach. In a neighborhood of permanent rotations the linearization of equations of motion results in Hill's equation, and the stability is determined by the eigenvalues of a monodromy matrix. It is shown that stable rotations may be destabilized by periodically changing the moments of inertia due to parametric resonance.
Citation: | Vetchanin E. V., Stabilization of rotations of a rigid body with a fixed point by periodic perturbations, 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, |
---|---|
DOI: | 10.1109/NIR50484.2020.9290224 |
Full text: | pdf (1.06 Mb) |
Site: | https://ieeexplore.ieee.org/xpl/conhome/9290150/proceeding |
---|
Klekovkin A. V., Karavaev Y. L., Mamaev I. S., Vetchanin E. V., Tenenev V. A.
This paper is devoted to investigations of the motion of an aquatic propeller-less robot. The robot motion implemented by rotation of internal rotor. A simple finite-dimensional mathematical model to describe the motion of the robot is proposed. Experiments with control actions providing the motion along a straight line and a circle have been conducted.
Citation: | Klekovkin A. V., Karavaev Y. L., Mamaev I. S., Vetchanin E. V., Tenenev V. A., Experimental evaluation of simplified physical model for control of aquatic robot with internal rotor, 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, |
---|---|
DOI: | 10.1109/NIR50484.2020.9290211 |
Full text: | pdf (330.17 Kb) |
Site: | https://ieeexplore.ieee.org/xpl/conhome/9290150/proceeding |
---|
In this paper we consider the controlled motion of a symmetric wheeled three-link vehicle on a plane without slipping. Similar systems have been considered in [1] – [6] . We assume that the vehicle consists of three identical platforms (links). The platforms are connected to each other by joints, and a wheel pair is rigidly fastened to the center of mass of each platform. By a wheel pair we mean two wheels rotating independently on the same axis.
Citation: | Artemova E. M., Kilin A. A., Dynamics and control of a three-link wheeled vehicle, 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, |
---|---|
DOI: | 10.1109/NIR50484.2020.9290222 |
Full text: | pdf (226.65 Kb) |
Site: | https://ieeexplore.ieee.org/xpl/conhome/9290150/proceeding |
---|
Borisov A. V., Kilin A. A., Pivovarova E. N.
In this paper we consider the control of the motion of a dynamically asymmetric unbalanced ball (Chaplygin top) by means of two perpendicular rotors. We propose a mechanism for control by periodically changing the gyrostatic momentum of the system, which leads to an unbounded speedup. We then formulate a general hypothesis of the mechanism for speeding up spherical bodies on a plane by periodically changing the system parameters.
Keywords: | nonholonomic constraint, speedup, Chaplygin top, periodic oscillations |
---|---|
Citation: | Borisov A. V., Kilin A. A., Pivovarova E. N., Speedup of the Chaplygin Top by Means of Rotors, Doklady Physics, 2019, vol. 64, no. 3, |
DOI: | 10.1134/S1028335819030145 |
Full text: | pdf (486.65 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
This paper addresses the problem of the rolling of a spherical shell with a frame
rotating inside, on which rotors are fastened. It is assumed that the center of mass of the entire
system is at the geometric center of the shell.
For the rubber rolling model and the classical rolling model it is shown that, if the angular
velocities of rotation of the frame and the rotors are constant, then there exists a noninertial
coordinate system (attached to the frame) in which the equations of motion do not depend
explicitly on time. The resulting equations of motion preserve an analog of the angular
momentum vector and are similar in form to the equations for the Chaplygin ball. Thus, the
problem reduces to investigating a two-dimensional Poincaré map.
The case of the rubber rolling model is analyzed in detail. Numerical investigation of its
Poincaré map shows the existence of chaotic trajectories, including those associated with a
strange attractor. In addition, an analysis is made of the case of motion from rest, in which the
problem reduces to investigating the vector field on the sphere S2.
Keywords: | nonholonomic mechanics, Chaplygin ball, rolling without slipping and spinning, strange attractor, straight-line motion, stability, limit cycle, balanced beaver-ball |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., Different Models of Rolling for a Robot Ball on a Plane as a Generalization of the Chaplygin Ball Problem, Regular and Chaotic Dynamics, 2019, vol. 24, no. 5, |
DOI: | 10.1134/S1560354719050071 |
Full text: | pdf (1.93 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Ardentov A. A., Karavaev Y. L., Yefremov K. S.
This paper is concerned with the problem of optimal path planning for a mobile wheeled robot. Euler elasticas, which ensure minimization of control actions, are considered as optimal trajectories. An algorithm for constructing controls that realizes the motion along the trajectory in the form of an Euler elastica is presented. Problems and special features of the application of this algorithm in practice are discussed. In particular, analysis is made of speedup and deceleration along the elastica, and of the influence of the errors made in manufacturing the mobile robot on the precision with which the prescribed trajectory is followed. Special attention is also given to the problem of forming optimal trajectories of motion along Euler elasticas to a preset point at different angles of orientation. Results of experimental investigations are presented.
Keywords: | mobile wheeled robot, Euler’s elastica, optimal control, experimental investigations |
---|---|
Citation: | Ardentov A. A., Karavaev Y. L., Yefremov K. S., Euler Elasticas for Optimal Control of the Motion of Mobile Wheeled Robots: the Problem of Experimental Realization, Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, |
DOI: | 10.1134/S1560354719030055 |
Full text: | pdf (2.58 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
This paper is a small review devoted to the dynamics of a point on a paraboloid. Specifically, it is concerned with the motion both under the action of a gravitational field and without it. It is assumed that the paraboloid can rotate about a vertical axis with constant angular velocity. The paper includes both well-known results and a number of new results.
We consider the two most widespread friction (resistance) models: dry (Coulomb) friction and viscous friction. It is shown that the addition of external damping (air drag) can lead to stability of equilibrium at the saddle point and hence to preservation of the region of bounded motion
in a neighborhood of the saddle point. Analysis of three-dimensional Poincaré sections shows that limit cycles can arise in this case in the neighborhood of the saddle point.
Keywords: | parabolic pendulum, Paul trap, rotating paraboloid, internal damping, external damping, friction, resistance, linear stability, Hill's region, bifurcational diagram, Poincaré section, bounded trajectory, chaos, integrability, nonintegrability, separatrix splitting |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness, Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, |
DOI: | 10.1134/S1560354719030067 |
Full text: | pdf (1.48 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper presents a qualitative analysis of the dynamics in a fixed reference frame of a wheel with sharp edges that rolls on a horizontal plane without slipping at the point of contact and without spinning relative to the vertical. The wheel is a ball that is symmetrically truncated on both sides and has a displaced center of mass. The dynamics of such a system is described by the model of the ball’s motion where the wheel rolls with its spherical part in contact with the supporting plane and the model of the disk’s motion where the contact point lies on the sharp edge of the wheel. A classification is given of possible motions of the wheel depending on whether there are transitions from its spherical part to sharp edges. An analysis is made of the behavior of the point of contact of the wheel with the plane for different values of the system parameters, first integrals and initial conditions. Conditions for boundedness and unboundedness of the wheel’s motion are obtained. Conditions for the fall of the wheel on the plane of sections are presented.
Keywords: | integrable system, system with discontinuity, nonholonomic constraint, bifurcation diagram, body of revolution, sharp edge, wheel, rubber body model, permanent rotations, dynamics in a fixed reference frame, resonance, quadrature, unbounded motion |
---|---|
Citation: | Kilin A. A., Pivovarova E. N., Qualitative Analysis of the Nonholonomic Rolling of a Rubber Wheel with Sharp Edges, Regular and Chaotic Dynamics, 2019, vol. 24, no. 2, |
DOI: | 10.1134/S1560354719020072 |
Full text: | pdf (883.49 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper is concerned with the problem of the interaction of vortex lattices, which is equivalent to the problem of the motion of point vortices on a torus. It is shown that the dynamics of a system of two vortices does not depend qualitatively on their strengths. Steadystate configurations are found and their stability is investigated. For two vortex lattices it is also shown that, in absolute space, vortices move along closed trajectories except for the case of a vortex pair. The problems of the motion of three and four vortex lattices with nonzero total strength are considered. For three vortices, a reduction to the level set of first integrals is performed. The nonintegrability of this problem is numerically shown. It is demonstrated that the equations of motion of four vortices on a torus admit an invariant manifold which corresponds to centrally symmetric vortex configurations. Equations of motion of four vortices on this invariant manifold and on a fixed level set of first integrals are obtained and their nonintegrability is numerically proved.
Keywords: | vortices on a torus, vortex lattices, point vortices, nonintegrability, chaos, invariant manifold, Poincarґe map, topological analysis, numerical analysis, accuracy of calculations, reduction, reduced system |
---|---|
Citation: | Kilin A. A., Artemova E. M., Integrability and Chaos in Vortex Lattice Dynamics, Regular and Chaotic Dynamics, 2019, vol. 24, no. 1, |
DOI: | 10.1134/S1560354719010064 |
Full text: | pdf (1.61 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
It is well known that the maximal value of the central moment of inertia of a closed homogeneous thread of fixed length is achieved on a curve in the form of a circle. This isoperimetric property plays a key role in investigating the stability of stationary motions of a flexible thread. A discrete variant of the isoperimetric inequality, when the mass of the thread is concentrated in a finite number of material particles, is established. An analog of the isoperimetric inequality for an inhomogeneous thread is proved.
Keywords: | moment of inertia, Sundman and Wirtinger inequalities, articulated polygon |
---|---|
Citation: | Kozlov V. V., Isoperimetric Inequalities for Moments of Inertia and Stability of Stationary Motions of a Flexible Thread, Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 4, |
DOI: | 10.20537/nd190410 |
Full text: | pdf (252.39 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
This paper presents the results of the study of the dynamics of a real spherical robot of combined type in the case of control using small periodic oscillations. The spherical robot is set in motion by controlled change of the position of the center of mass and by generating variable gyrostatic momentum. We demonstrate how to use small periodic controls for stabilization of the spherical robot during motion. The results of numerical simulation are obtained for various initial conditions and control parameters that ensure a change in the position of the center of mass and a variation of gyrostatic momentum. The problem of the motion of a spherical robot of combined type on a surface that performs flat periodic oscillations is also considered. The results of numerical simulation are obtained for different initial conditions, control actions and parameters of oscillations.
Keywords: | spherical robot, nonholonomic constraint, small periodic control actions, stabilization |
---|---|
Citation: | Karavaev Y. L., Kilin A. A., The Dynamics of a Spherical Robot of Combined Type by Periodic Control Actions, Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 4, |
DOI: | 10.20537/nd190408 |
Full text: | pdf (528.42 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Karavaev Y. L., Shestakov V. A., Yefremov K. S.
This paper presents experimental investigations of the control algorithm of a highly maneuverable mobile manipulation robot. The kinematics of a mobile manipulation robot, the algorithm of trajectory planning of the mobile robot to the point of object gripping are considered. By realization of the algorithm, the following tasks are solved: solution of the inverse positional task for the mobile manipulation robot; motion planning of the mobile manipulator taking into account the minimization of energy and time consumption per movement. The result of the algorithm is a movement to the point of gripping of the manipulation object; grasping and loading of the object. Experimental investigations of the developed algorithms are given.
Keywords: | mobile manipulation robot, motion planning, trajectory discretization, Kinect |
---|---|
Citation: | Karavaev Y. L., Shestakov V. A., Yefremov K. S., Experimental Investigations of the Control Algorithm of a Mobile Manipulation Robot, Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 4, |
DOI: | 10.20537/nd190407 |
Full text: | pdf (1.23 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Vetchanin E. V., Mikishanina E. A.
The dynamics of a body with a fixed point, variable moments of inertia and internal rotors are considered. A stability analysis of permanent rotations and periodic solutions of the system is carried out. In some simplest cases the stability analysis is reduced to investigating the stability of the zero solution of Hill’s equation. It is shown that by periodically changing the moments of inertia it is possible to stabilize unstable permanent rotations of the system. In addition, stable dynamical regimes can lose stability due to a parametric resonance. It is shown that, as the oscillation frequency of the moments of inertia increases, the dynamics of the system becomes close to an integrable one.
Keywords: | Liouville equations, Euler –Poisson equations, Hill’s equation, Mathieu equation, parametric resonance, vibrostabilization, Euler – Poinsot case, Joukowski –Volterra case |
---|---|
Citation: | Vetchanin E. V., Mikishanina E. A., Vibrational Stability of Periodic Solutions of the Liouville Equations, Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 3, |
DOI: | 10.20537/nd190312 |
Full text: | pdf (775.13 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
This paper is concerned with the rolling of a homogeneous ball with slipping on a uniformly rotating horizontal plane. We take into account viscous friction forces arising when there is slipping at the contact point. It is shown that, as the coefficient of viscosity tends to infinity, the solution of the generalized problem on each fixed time interval tends to a solution of the corresponding nonholonomic problem.
Keywords: | rotating surface, turntable, nonholonomic constraint, rolling ball, sliding, viscous friction |
---|---|
Citation: | Ivanova T. B., The Rolling of a Homogeneous Ball with Slipping on a Horizontal Rotating Plane, Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 2, |
DOI: | 10.20537/nd190206 |
Full text: | pdf (443.65 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The motion of a circular cylinder in a fluid in the presence of circulation and external periodic force and torque is studied. It is shown that for a suitable choice of the frequency of external action for motion in an ideal fluid the translational velocity components of the body undergo oscillations with increasing amplitude due to resonance. During motion in a viscous fluid no resonance arises. Explicit integration of the equations of motion has shown that the unbounded propulsion of the body in a viscous fluid is impossible in the absence of external torque. In the general case, the solution of the equations is represented in the form of a multiple series.
Keywords: | rigid body dynamics, ideal fluid, viscous fluid, propulsion in a fluid, resonance |
---|---|
Citation: | Vetchanin E. V., The Motion of a Balanced Circular Cylinder in an Ideal Fluid Under the Action of External Periodic Force and Torque, Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 1, |
DOI: | 10.20537/nd190105 |
Full text: | pdf (408.7 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Kuznetsov S. P.
For a Chaplygin sleigh moving in the presence of weak friction, we present and investigate two mechanisms of arising acceleration due to oscillations of an internal mass. In certain parameter regions, the mechanism induced by small oscillations determines acceleration which is on average one-directional. The role of friction is that the velocity reached in the process of the acceleration is stabilized at a certain level. The second mechanism is due to the effect of the developing oscillatory parametric instability in the motion of the sleigh. It occurs when the internal oscillating particle is comparable in mass with the main platform and the oscillations are of a sufficiently large amplitude. In the nonholonomic model the magnitude of the parametric oscillations and the level of mean energy achieved by the system turn out to be bounded if the line of the oscillations of the moving particle is displaced from the center of mass; the observed sustained motion is in many cases associated with a chaotic attractor. Then, the motion of the sleigh appears to be similar to the process of two-dimensional random walk on the plane.
Keywords: | nonholonomic mechanics, Chaplygin sleigh, parametric oscillator, strange attractor, Lyapunov exponent, chaotic dynamics |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Kuznetsov S. P., The Chaplygin sleigh with friction moving due to periodic oscillations of an internal mass, Nonlinear Dynamics, 2019, vol. 95, no. 1, |
DOI: | 10.1007/s11071-018-4591-5 |
Full text: | pdf (2.97 Mb) |
Impact-factor WoS (2022): | 5.600 (Q1) |
---|---|
ISSN (print): | 0924-090X |
ISSN (online): | 1573-269X |
Site: | http://link.springer.com/journal/11071 |
Ivanova T. B., Kilin A. A., Pivovarova E. N.
In this paper, we develop a model of a controlled spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum, with a feedback that stabilizes given partial solutions for a free system at the final stage of motion. According to the proposed approach, feedback depends on phase variables (current position, velocities) and does not depend on the specific type of trajectory. We present integrals of motion and partial solutions, analyze their stability, and give examples of computer simulations of motion with feedback that demonstrate the efficiency of the proposed model.
Keywords: | Spherical robot, Nonholonomic constraint, Control, Feedback |
---|---|
Citation: | Ivanova T. B., Kilin A. A., Pivovarova E. N., Controlled Motion of a Spherical Robot with Feedback. II, Journal of Dynamical and Control Systems, 2019, vol. 25, no. 1, |
DOI: | 10.1007/s10883-017-9390-7 |
Full text: | pdf (803.07 Kb) |
Impact-factor WoS (2022): | 0.900 (Q3) |
---|---|
Impact-factor RSCI (2022): | 0.382 (Q3) |
ISSN (print): | 1079-2724 |
ISSN (online): | 1573-8698 |
Site: | http://link.springer.com/journal/10883 |
Borisov A. V., Kilin A. A., Karavaev Y. L., Klekovkin A. V.
The paper is concerned with the problem of stabilizing a spherical robot of combined type during its motion. The focus is on the application of feedback for stabilization of the robot which is an example of an underactuated system. The robot is set in motion by an inter- nal wheeled platform with a rotor placed inside the sphere. The results of experimental investigations for a prototype of the spherical robot are presented.
Keywords: | Spherical robot, Dynamical model, Stabilization, Nonholonomic constraint, Feedback |
---|---|
Citation: | Borisov A. V., Kilin A. A., Karavaev Y. L., Klekovkin A. V., Stabilization of the motion of a spherical robot using feedbacks, Applied Mathematical Modelling, 2019, vol. 69, |
DOI: | 10.1016/j.apm.2019.01.008 |
Full text: | pdf (1005.01 Kb) |
Impact-factor WoS (2022): | 5.0 (Q1) |
---|---|
Impact-factor RSCI (2022): | 1.080 (Q1) |
ISSN (print): | 0307-904X |
Site: | https://www.journals.elsevier.com/applied-mathematical-modelling |
Borisov A. V., Kilin A. A., Mamaev I. S.
In this paper, we address the free (uncontrolled) dynamics of a snakeboard consisting of two wheel pairs fastened to a platform. The snakeboard is one of the well-known sports vehicles on which the sportsman executes necessary body movements. From the theoretical point of view, this system is a direct generalization of the classical nonholonomic system of the Chaplygin sleigh. We carry out a topological and qualitative analysis of trajectories of this dynamical system. An important feature of the problem is that the common level set of first integrals is a compact two-dimensional surface of genus 5. We specify conditions under which the reaction forces infinitely increase during motion and the so-called phenomenon of nonholonomic jamming is observed. In this case, the nonholonomic model ceases to work and it is necessary to use more complex mechanical models incorporating sliding, elasticity, etc.
Keywords: | Nonholonomic mechanics, snakeboard, qualitative analysis, bifurcations, regularization (blowing up singularities), system on a torus, nonholonomic jamming, bifurcation analysis |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Invariant Submanifolds of Genus 5 and a Cantor Staircase in the Nonholonomic Model of a Snakeboard, International Journal of Bifurcation and Chaos, 2019, vol. 29, no. 3, 1930008, |
DOI: | 10.1142/S0218127419300088 |
Full text: | pdf (2.79 Mb) |
Impact-factor WoS (2022): | 2.200 (Q2) |
---|---|
ISSN (print): | 0218-1274 |
ISSN (online): | 1793-6551 |
Site: | http://www.worldscientific.com/worldscinet/ijbc |
Bizyaev I. A., Borisov A. V., Kozlov V. V., Mamaev I. S.
This paper is concerned with a nonholonomic system with parametric excitation—the Chaplygin sleigh with time-varying mass distribution. A detailed analysis is made of the problem of the existence of regimes with unbounded growth of energy (an analogue of Fermi’s acceleration) in the case where excitation is achieved by means of a rotor with variable angular momentum. The existence of trajectories for which the translational velocity of the sleigh increases indefinitely and has the asymptotics τ13 is proved. In addition, it is shown that, when viscous friction with a nondegenerate Rayleigh function is added, unbounded speed-up disappears and the trajectories of the reduced system asymptotically tend to a limit cycle.
Keywords: | nonholonomic mechanics, Fermi’s acceleration, Chaplygin sleigh, unbounded speed-up, limit cycle, rotor, viscous friction |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Kozlov V. V., Mamaev I. S., Fermi-like acceleration and power-law energy growth in nonholonomic systems, Nonlinearity, 2019, vol. 32, |
DOI: | 10.1088/1361-6544/ab1f2d |
Full text: | pdf (974.09 Kb) |
Impact-factor WoS (2022): | 1.700 (Q2) |
---|---|
ISSN (print): | 0951-7715 |
ISSN (online): | 1361-6544 |
Site: | http://iopscience.iop.org/0951-7715 |
Borisov A. V., Ivanova T. B., Kilin A. A., Mamaev I. S.
This paper investigates the rolling without slipping of a homogeneous heavy ball on the surface of a rotating cone in two settings: without dissipation in a nonholonomic setting and with rolling friction torque which is proportional to the angular velocity of the ball. In the nonholonomic setting, the resulting system of five differential equations on the level set of first integrals is reduced to quadratures. A bifurcation analysis of the above system is carried out to determine the possible types of motion. In the second case, it is shown that there are not only trajectories emanating from the lower point of the cone (its vertex), but also trajectories to the vertex of the cone (fall). An analysis of the dependence of the type of terminal motion of the center of mass of the ball on initial conditions is carried out.
Keywords: | Rotating surface, Cone, Nonholonomic constraint, Rolling friction, Jacobi integral, Bifurcation analysis |
---|---|
Citation: | Borisov A. V., Ivanova T. B., Kilin A. A., Mamaev I. S., Nonholonomic rolling of a ball on the surface of a rotating cone, Nonlinear Dynamics, 2019, vol. 97, no. 2, |
DOI: | 10.1007/s11071-019-05086-3 |
Full text: | pdf (1.04 Mb) |
Impact-factor WoS (2022): | 5.600 (Q1) |
---|---|
ISSN (print): | 0924-090X |
ISSN (online): | 1573-269X |
Site: | http://link.springer.com/journal/11071 |
Borisov A. V., Kilin A. A., Mamaev I. S.
This paper discusses two approaches for deriving the equations of motion for a ball that rolls without slipping on the surface of a rotating hyperbolic paraboloid. We analyze two possible methods for defining the surface on which the ball rolls, and show the relationship between the two methods. We describe how the stability of the ball’s rotation at the saddle point depends on the radius of the ball, in the case where the stability analysis is made in dimensionless parameters.
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Comment on “Confining rigid balls by mimicking quadrupole ion trapping” [Am. J. Phys. 85, 821 (2017)], American Journal of Physics, 2019, vol. 87, no. 11, |
---|---|
DOI: | 10.1119/10.0000006 |
Full text: | pdf (2.18 Mb) |
Impact-factor WoS (2022): | 0.900 (Q4) |
---|---|
Impact-factor RSCI (2022): | 0.524 (Q2) |
ISSN (print): | 0002-9505 |
ISSN (online): | 1943-2909 |
Site: | https://aapt.scitation.org/journal/ajp |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
This paper addresses the problem of the motion of a sleigh with a free rotor. It is shown that this system exhibits chaotic and regular motions. The case in which the system is balanced relative to the knife edge is of particular interest because it has an additional integral. In this case, the problem reduces to investigating a vector field on a torus and to classifying singular points on it.
Keywords: | Chaplygin sleigh, unbalanced rotor, nonholonomic mechanics, strange attractor, regular and chaotic trajectories, invariant measure, integrable systems, system of two bodies |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., Dynamics of a Chaplygin sleigh with an unbalanced rotor: regular and chaotic motions, Nonlinear Dynamics, 2019, vol. 98, |
DOI: | 10.1007/s11071-019-05325-7 |
Full text: | pdf (829 Kb) |
Impact-factor WoS (2022): | 5.600 (Q1) |
---|---|
ISSN (print): | 0924-090X |
ISSN (online): | 1573-269X |
Site: | http://link.springer.com/journal/11071 |
Borisov A. V., Vetchanin E. V., Mamaev I. S.
A plane-parallel motion of a circular foil is considered in a fluid with a nonzero constant circulation under the action of external periodic force and torque. Various integrable cases are treated. Conditions for the existence of resonances of two types are found. In the case of resonances of the first type, the phase trajectory of the system and the trajectory of the foil are unbounded. In the case of resonances of the second type, the foil trajectory is unbounded, while the phase trajectory of the system remains bounded during the motion.
Citation: | Borisov A. V., Vetchanin E. V., Mamaev I. S., Motion of a Smooth Foil in a Fluid under the Action of External Periodic Forces. I, Russian Journal of Mathematical Physics, 2019, vol. 26, no. 4, |
---|---|
DOI: | 10.1134/S1061920819040022 |
Full text: | pdf (963.73 Kb) |
Impact-factor WoS (2022): | 1.400 (Q3) |
---|---|
ISSN (print): | 1061-9208 |
ISSN (online): | 1555-6638 |
Site: | http://link.springer.com/journal/11503 |
Ivanova T. B., Kilin A. A., Pivovarova E. N.
In this work we consider the controlled motion of a pendulum spherical robot on an inclined plane. The algorithm for determining the control actions for the motion along an arbitrary trajectory and examples of numerical simulation of the controlled motion are given.
Citation: | Ivanova T. B., Kilin A. A., Pivovarova E. N., Control of the Rolling Motion of a Spherical Robot on an Inclined Plane, Doklady Physics, 2018, vol. 63, no. 10, |
---|---|
DOI: | 10.1134/S1028335818100099 |
Full text: | pdf (509.36 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
Ivanova T. B., Kilin A. A., Pivovarova E. N.
This paper is concerned with a model of the controlled motion of a spherical robot with an axisymmetric pendulum actuator on an inclined plane. First integrals of motion and partial solutions are presented and their stability is analyzed. It is shown that the steady solutions exist only at an inclination angle less than some critical value and only for constant control action.
Citation: | Ivanova T. B., Kilin A. A., Pivovarova E. N., Controlled Motion of a Spherical Robot of Pendulum Type on an Inclined Plane, Doklady Physics, 2018, vol. 63, no. 7, |
---|---|
DOI: | 10.1134/S1028335818070091 |
Full text: | pdf (568.83 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
Borisov A. V., García-Naranjo L. C., Mamaev I. S., Montaldi J.
We consider the two-body problem on surfaces of constant nonzero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each q>0 we show there are two relative equilibria where the masses are separated by a distance q. One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart. On the sphere of positive curvature, if the masses are different, there is a unique relative equilibrium (RE) for every angular separation except π/2. When the angle is acute, the RE is elliptic, and when it is obtuse the RE can be either elliptic or linearly unstable. We show using a KAM argument that the acute ones are almost always nonlinearly stable. If the masses are equal, there are two families of relative equilibria: one where the masses are at equal angles with the axis of rotation (‘isosceles RE’) and the other when the two masses subtend a right angle at the centre of the sphere. The isosceles RE are elliptic if the angle subtended by the particles is acute and is unstable if it is obtuse. At π/2, the two families meet and a pitchfork bifurcation takes place. Right-angled RE are elliptic away from the bifurcation point. In each of the two geometric settings, we use a global reduction to eliminate the group of symmetries and analyse the resulting reduced equations which live on a five-dimensional phase space and possess one Casimir function.
Keywords: | Reduction, Relative equilibria, Hamiltonian systems, Stability, Two-body problem, Energy–momentum bifurcation diagram |
---|---|
Citation: | Borisov A. V., García-Naranjo L. C., Mamaev I. S., Montaldi J., Reduction and relative equilibria for the two-body problem on spaces of constant curvature, Celestial Mechanics and Dynamical Astronomy, 2018, vol. 130, no. 6, |
DOI: | 10.1007/s10569-018-9835-7 |
Full text: | pdf (1.38 Mb) |
Impact-factor WoS (2022): | 1.600 (Q3) |
---|---|
ISSN (print): | 0923-2958 |
ISSN (online): | 1572-9478 |
Site: | http://link.springer.com/journal/10569 |
Borisov A. V., Mamaev I. S., Vetchanin E. V.
This paper addresses the problem of the self-propulsion of a smooth body in a fluid by periodic oscillations of the internal rotor and circulation. In the case of zero dissipation and constant circulation, it is shown using methods of KAM theory that the kinetic energy of the system is a bounded function of time. In the case of constant nonzero circulation, the trajectories of the center of mass of the system lie in a bounded region of the plane. The method of expansion by a small parameter is used to approximately construct a solution corresponding to directed motion of a circular foil in the presence of dissipation and variable circulation. Analysis of this approximate solution has shown that a speed-up is possible in the system in the presence of variable circulation and in the absence of resistance to translational motion. It is shown that, in the case of an elliptic foil, directed motion is also possible. To explore the dynamics of the system in the general case, bifurcation diagrams, a chart of dynamical regimes and a chart of the largest Lyapunov exponent are plotted. It is shown that the transition to chaos occurs through a cascade of period-doubling bifurcations.
Keywords: | self-propulsion in a fluid, smooth body, viscous fluid, periodic oscillation of circulation, control of a rotor |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Vetchanin E. V., Self-propulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, |
DOI: | 10.1134/S1560354718070043 |
Full text: | pdf (4.03 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper addresses the problem of controlled motion of the Zhukovskii foil in a viscous fluid due to a periodically oscillating rotor. Equations of motion including the added mass effect, viscous friction and lift force due to circulation are derived. It is shown that only limit cycles corresponding to the direct motion or motion near a circle appear in the system at the standard parameter values. The chart of dynamical regimes, the chart of the largest Lyapunov exponent and a one-parameter bifurcation diagram are calculated. It is shown that strange attractors appear in the system due to a cascade of period-doubling bifurcations.
Keywords: | self-propulsion, Zhukovskii foil, foil with a sharp edge, motion in a viscous fluid, controlled motion, period-doubling bifurcation |
---|---|
Citation: | Mamaev I. S., Vetchanin E. V., The Self-propulsion of a Foil with a Sharp Edge in a Viscous Fluid Under the Action of a Periodically Oscillating Rotor, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, |
DOI: | 10.1134/S1560354718070055 |
Full text: | pdf (1.64 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper is concerned with the dynamics of a wheel with sharp edges moving on a horizontal plane without slipping and rotation about the vertical (nonholonomic rubber model). The wheel is a body of revolution and has the form of a ball symmetrically truncated on both sides. This problem is described by a system of differential equations with a discontinuous right-hand side. It is shown that this system is integrable and reduces to quadratures. Partial solutions are found which correspond to fixed points of the reduced system. A bifurcation analysis and a classification of possible types of the wheel’s motion depending on the system parameters are presented.
Keywords: | integrable system, system with a discontinuous right-hand side, nonholonomic constraint, bifurcation diagram, body of revolution, sharp edge, wheel, rubber model |
---|---|
Citation: | Kilin A. A., Pivovarova E. N., Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, |
DOI: | 10.1134/S1560354718070067 |
Full text: | pdf (2.02 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
In this paper we consider the problem of the motion of the Roller Racer.We assume that the angle φ(t) between the platforms is a prescribed function of time. We prove that in this case the acceleration of the Roller Racer is unbounded. In this case, as the Roller Racer accelerates, the increase in the constraint reaction forces is also unbounded. Physically this means that, from a certain instant onward, the conditions of the rolling motion of the wheels without slipping are violated. Thus, we consider a model in which, in addition to the nonholonomic constraints, viscous friction force acts at the points of contact of the wheels. For this case we prove that there is no constant acceleration and all trajectories of the reduced system asymptotically tend to a periodic solution.
Keywords: | Roller Racer, speed-up, nonholonomic mechanics, Rayleigh dissipation function, viscous friction, integrability by quadratures, control, constraint reaction force |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., Exotic Dynamics of Nonholonomic Roller Racer with Periodic Control, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, |
DOI: | 10.1134/S1560354718070122 |
Full text: | pdf (1.99 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
This paper addresses the problem of an inhomogeneous disk rolling on a horizontal plane. This problem is considered within the framework of a nonholonomic model in which there is no slipping and no spinning at the point of contact (the projection of the angular velocity of the disk onto the normal to the plane is zero). The configuration space of the system of interest contains singular submanifolds which correspond to the fall of the disk and in which the equations of motion have a singularity. Using the theory of normal hyperbolic manifolds, it is proved that the measure of trajectories leading to the fall of the disk is zero.
Keywords: | nonholonomic mechanics, regularization, blowing-up, invariant measure, ergodic theorems, normal hyperbolic submanifold, Poincaré map, first integrals |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane, Regular and Chaotic Dynamics, 2018, vol. 23, no. 6, |
DOI: | 10.1134/S1560354718060035 |
Full text: | pdf (662.06 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Bizyaev I. A.
This paper is concerned with the problem of three vortices on a sphere S2 and the Lobachevsky plane L2. After reduction, the problem reduces in both cases to investigating a Hamiltonian system with a degenerate quadratic Poisson bracket, which makes it possible to study it using the methods of Poisson geometry. This paper presents a topological classification of types of symplectic leaves depending on the values of Casimir functions and system parameters.
Keywords: | Poisson geometry, point vortices, reduction, quadratic Poisson bracket, spaces of constant curvature, symplectic leaf, collinear configurations |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Bizyaev I. A., Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability, Regular and Chaotic Dynamics, 2018, vol. 23, no. 5, |
DOI: | 10.1134/S1560354718050106 |
Full text: | pdf (3.34 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Vetchanin E. V.
This paper addresses the problem of self-propulsion of a smooth profile in a medium with viscous dissipation and circulation by means of parametric excitation generated by oscillations of the moving internal mass. For the case of zero dissipation, using methods of KAM theory, it is shown that the kinetic energy of the system is a bounded function of time, and in the case of nonzero circulation the trajectories of the profile lie in a bounded region of the space. In the general case, using charts of dynamical regimes and charts of Lyapunov exponents, it is shown that the system can exhibit limit cycles (in particular, multistability), quasi-periodic regimes (attracting tori) and strange attractors. One-parameter bifurcation diagrams are constructed, and Neimark – Sacker bifurcations and period-doubling bifurcations are found. To analyze the efficiency of displacement of the profile depending on the circulation and parameters defining the motion of the internal mass, charts of values of displacement for a fixed number of periods are plotted. A hypothesis is formulated that, when nonzero circulation arises, the trajectories of the profile are compact. Using computer calculations, it is shown that in the case of anisotropic dissipation an unbounded growth of the kinetic energy of the system (Fermi-like acceleration) is possible.
Keywords: | self-propulsion in a fluid, motion with speed-up, parametric excitation, viscous dissipation, circulation, period-doubling bifurcation, Neimark – Sacker bifurcation, Poincaré map, chart of dynamical regimes, chart of Lyapunov exponents, strange att |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Vetchanin E. V., Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation, Regular and Chaotic Dynamics, 2018, vol. 23, no. 4, |
DOI: | 10.1134/S1560354718040081 |
Full text: | pdf (2.88 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
In this paper, equations of motion for the problem of a ball rolling without slipping on a rotating hyperbolic paraboloid are obtained. Integrals of motions and an invariant measure are found. A detailed linear stability analysis of the ball’s rotations at the saddle point of the hyperbolic paraboloid is made. A three-dimensional Poincar´e map generated by the phase flow of the problem is numerically investigated and the existence of a region of bounded trajectories in a neighborhood of the saddle point of the paraboloid is demonstrated. It is shown that a similar problem of a ball rolling on a rotating paraboloid, considered within the framework of the rubber model, can be reduced to a Hamiltonian system which includes the Brower problem as a particular case.
Keywords: | Paul trap, stability, nonholonomic system, three-dimensional map, gyroscopic stabilization, noninertial coordinate system, Poincaré map, nonholonomic constraint, rolling without slipping, region of linear stability |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., A Nonholonomic Model of the Paul Trap, Regular and Chaotic Dynamics, 2018, vol. 23, no. 3, |
DOI: | 10.1134/S1560354718030085 |
Full text: | pdf (6.87 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
A chain of quadratic first integrals of general linear Hamiltonian systems that have not been represented in canonical form is found. Their involutiveness is established and the problem of their functional independence is studied. The key role in the study of a Hamiltonian system is played by an integral cone which is obtained by setting known quadratic first integrals equal to zero. A singular invariant isotropic subspace is shown to pass through each point of the integral cone, and its dimension is found. The maximal dimension of such subspaces estimates from above the degree of instability of the Hamiltonian system. The stability of typical Hamiltonian systems is shown to be equivalent to the degeneracy of the cone to an equilibrium point. General results are applied to the investigation of linear mechanical systems with gyroscopic forces and finite-dimensional quantum systems.
Keywords: | Hamiltonian system, quadratic integrals, integral cones, degree of instability, quantum systems, Abelian integrals |
---|---|
Citation: | Kozlov V. V., Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, |
DOI: | 10.1134/S1560354718010033 |
Full text: | pdf (693.23 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Mamaev I. S., Tenenev V. A., Vetchanin E. V.
This paper addresses the problem of plane-parallel motion of the Zhukovskii foil in a viscous fluid. Various motion regimes of the foil are simulated on the basis of a joint numerical solution of the equations of body motion and the Navier – Stokes equations. According to the results of simulation of longitudinal, transverse and rotational motions, the average drag coefficients and added masses are calculated. The values of added masses agree with the results published previously and obtained within the framework of the model of an ideal fluid. It is shown that between the value of circulation determined from numerical experiments, and that determined according to the model of and ideal fluid, there is a correlation with the coefficient R=0.722. Approximations for the lift force and the moment of the lift force are constructed depending on the translational and angular velocity of motion of the foil. The equations of motion of the Zhukovskii foil in a viscous fluid are written taking into account the found approximations and the drag coefficients. The calculation results based on the proposed mathematical model are in qualitative agreement with the results of joint numerical solution of the equations of body motion and the Navier – Stokes equations.
Keywords: | Zhukovskii foil, Navier – Stokes equations, joint solution of equations, finitedimensional model, viscous fluid, circulation, sharp edge |
---|---|
Citation: | Mamaev I. S., Tenenev V. A., Vetchanin E. V., Dynamics of a Body with a Sharp Edge in a Viscous Fluid, Russian Journal of Nonlinear Dynamics, 2018, vol. 14, no. 4, |
DOI: | 10.20537/nd180404 |
Full text: | pdf (835.37 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Vetchanin E. V., Gladkov E. S.
This paper is concerned with the motion of heavy toroidal bodies in a fluid. For experimental purposes, models of solid tori with a width of 3 cm and external diameters of 10 cm, 12 cm and 15 cm have been fabricated by the method of casting chemically solidifying polyurethane (density 1100 kg/m3). Tracking of the models is performed using the underwater Motion Capture system. This system includes 4 cameras, computer and specialized software. A theoretical description of the motion is given using equations incorporating the influence of inertial forces, friction and circulating motion of a fluid through the hole. Values of the model parameters are selected by means of genetic algorithms to ensure an optimal agreement between experimental and theoretical data.
Keywords: | fall through a fluid, torus, body with a hole, multiply connected body, finitedimensional model, object tracking, genetic algorithms |
---|---|
Citation: | Vetchanin E. V., Gladkov E. S., Identification of parameters of the model of toroidal body motion using experimental data, Russian Journal of Nonlinear Dynamics, 2018, vol. 14, no. 1, |
DOI: | 10.20537/nd1801009 |
Full text: | pdf (1.49 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Kilin A. A., Klenov A. I., Tenenev V. A.
This article is devoted to the study of self-propulsion of bodies in a fluid by the action of internal mechanisms, without changing the external shape of th e body. The paper presents an overview of theoretical papers that justify the possibility of this displacement in ideal and viscous liquids. A special case of self-propulsion of a rigid body along the surface of a liquid is considered due to the motion of two internal masses along the circles. The paper presents a mathematical model of the motion of a solid body with moving internal masses in a three-dime nsional formulation. This model takes into account the three-dimensional vibrations of the body during motion, which arise under the action of external forces-gravity force, Archimedes force and forces acting on the body, from the side of a viscous fluid. The body is a homogeneous elliptical cylinder with a k eel located along the larger diagonal. Inside the cylinder there are two material point masses moving along the circles. The centers of the circles lie on the smallest diagonal of the ellipse at an equal distance from the center of mass. Equations of motion of the system (a body with two mater ial points, placed in a fluid) are represented as Kirchhoff equations with the addition of external for ces and moments acting on the body. The phenomenological model of viscous friction is quadratic in velocity used to describe the forces of resistance to motion in a fluid. The coefficients of resistance to movement were determ ined experimentally. The forces acting on the keel were determined by numerical modeling of the keel oscillations in a viscous liquid using the Navier – Stokes equations. In this paper, an experimental verification of the proposed mathematical model was carried out. Several series of experiments on self-propulsion of a body in a liquid by means of rotation of internal masses with different speeds of rotation are presented. The dependence of the average propagation velocity, the amplitude of the transverse oscillations as a function of the rotational speed of internal masses is investigated. The obtained experimental data are compared with the results obtai ned within the framework of the proposed mathematical model.
Keywords: | motion in a fluid, self-promotion, the equations of movement, above-water screwless robot, Navier – Stokes equations |
---|---|
Citation: | Kilin A. A., Klenov A. I., Tenenev V. A., Controlling the movement of the body using internal masses in a viscous liquid, Computer Research and Modeling, 2018, vol. 10, no. 4, |
DOI: | 10.20537/2076-7633-2018-10-4-445-460 |
Full text: | pdf (523.34 Kb) |
Impact-factor RSCI (2022): | 0.257 (Q4) |
---|---|
ISSN (print): | 2076-7633 |
ISSN (online): | 2077-6853 |
Site: | http://crm.ics.org.ru/ |
Ivanova T. B., Kilin A. A., Pivovarova E. N.
In this paper, we develop a model of a controlled spherical robot with an axisymmetric pendulum-type actuator with a feedback system suppressing the pendulum’s oscillations at the final stage of motion. According to the proposed approach, the feedback depends on phase variables (the current position and velocities) and does not depend on the type of trajectory. We present integrals of motion and partial solutions, analyze their stability, and give examples of computer simulation of motion using feedback to illustrate compensation of the pendulum’s oscillations.
Keywords: | Spherical robot, Nonholonomic constraint, Control, Feedback |
---|---|
Citation: | Ivanova T. B., Kilin A. A., Pivovarova E. N., Controlled Motion of a Spherical Robot with Feedback. I, Journal of Dynamical and Control Systems, 2018, vol. 24, no. 3, |
DOI: | 10.1007/s10883-017-9387-2 |
Full text: | pdf (606.17 Kb) |
Impact-factor WoS (2022): | 0.900 (Q3) |
---|---|
Impact-factor RSCI (2022): | 0.382 (Q3) |
ISSN (print): | 1079-2724 |
ISSN (online): | 1573-8698 |
Site: | http://link.springer.com/journal/10883 |
Borisov A. V., Ivanova T. B., Karavaev Y. L., Mamaev I. S.
In this work we investigate the motion of a homogeneous ball rolling without slipping on uniformly rotating horizontal and inclined planes under the action of a constant external force supplemented with the moment of rolling friction, which depends linearly on the angular velocity of the ball. We systematise well-known results and supplement them with the stability analysis of partial solutions of the system. We also perform an experimental investigation whose results support the adequacy of the rolling friction model used. Comparison of numerical and experimental results has shown a good qualitative agreement.
Keywords: | rolling, rotating surface, tilted turntable, non-holonomic constraint, rolling ball, rolling friction, qualitative analysis |
---|---|
Citation: | Borisov A. V., Ivanova T. B., Karavaev Y. L., Mamaev I. S., Theoretical and experimental investigations of the rolling of a ball on a rotating plane (turntable), European Journal of Physics, 2018, vol. 39, no. 6, 065001, |
DOI: | 10.1088/1361-6404/aad763 |
Full text: | pdf (510.14 Kb) |
Impact-factor WoS (2022): | 0.700 (Q4) |
---|---|
ISSN (print): | 0143-0807 |
ISSN (online): | 1361-6404 |
Site: | http://iopscience.iop.org/journal/0143-0807 |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
This paper addresses the problem of the Chaplygin ball rolling on a horizontal plane which rotates with constant angular velocity. In this case, the equations of motion admit area integrals, an integral of squared angular momentum and the Jacobi integral, which is a generalization of the energy integral, and possess an invariant measure. After reduction the problem reduces to investigating a three-dimensional Poincaré map that preserves phase volume (with density defined by the invariant measure). We show that in the general case the system’s dynamics is chaotic.
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., Dynamics of the Chaplygin ball on a rotating plane, Russian Journal of Mathematical Physics, 2018, vol. 25, no. 4, |
---|---|
DOI: | 10.1134/S1061920818040027 |
Full text: | pdf (872.8 Kb) |
Impact-factor WoS (2022): | 1.400 (Q3) |
---|---|
ISSN (print): | 1061-9208 |
ISSN (online): | 1555-6638 |
Site: | http://link.springer.com/journal/11503 |
In the paper, a study of rolling of a dynamically asymmetrical unbalanced ball (Chaplygin top) on a horizontal plane under the action of periodic gyrostatic moment is carried out. The problem is considered in the framework of the model of a rubber body, i.e., under the assumption that there is no slipping and spinning at the point of contact. It is shown that, for certain values of the parameters of the system and certain dependence of the gyrostatic moment on time, an acceleration of the system, i.e., an unbounded growth of the energy of the system, is observed. Investigations of the dependence of the presence of acceleration on the parameters of the system and on the initial conditions are carried out. On the basis of the investigations of the dynamics of the frozen system, a conjecture concerning the general mechanism of acceleration at the expense to periodic impacts in nonholonomic systems is expressed.
Citation: | Kilin A. A., Pivovarova E. N., Chaplygin Top with a Periodic Gyrostatic Moment, Russian Journal of Mathematical Physics, 2018, vol. 25, no. 4, |
---|---|
DOI: | 10.1134/S1061920818040088 |
Full text: | pdf (1.14 Mb) |
Impact-factor WoS (2022): | 1.400 (Q3) |
---|---|
ISSN (print): | 1061-9208 |
ISSN (online): | 1555-6638 |
Site: | http://link.springer.com/journal/11503 |
Borisov A. V., Kilin A. A., Mamaev I. S.
In this paper, we show that the trajectories of a dynamical system with nonholonomic constraints can satisfy Hamilton’s principle. As the simplest illustration, we consider the problem of a homogeneous ball rolling without slipping on a plane. However, Hamilton’s principle is formulated either for a reduced system or for a system defined in an extended phase space. It is shown that the dynamics of a nonholonomic homogeneous ball can be embedded in a higher-dimensional Hamiltonian phase flow. We give two examples of such an embedding: embedding in the phase flow of a free system and embedding in the phase flow of the corresponding vakonomic system.
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Hamilton’s Principle and the Rolling Motion of a Symmetric Ball, Doklady Physics, 2017, vol. 62, no. 6, |
---|---|
DOI: | 10.1134/S1028335817060052 |
Full text: | pdf (206.42 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
Borisov A. V., Kilin A. A., Karavaev Y. L.
This paper presents results of theoretical and experi- mental research explaining the retrograde final-stage rolling of a disk under certain relations between its mass and geometric parameters. Modifying the no-slip model of a rolling disk by including viscous rolling friction provides a qualitative explana- tion for the disk's retrograde motion. At the same time, the simple experiments described in the paper completely reject the aerodynamical drag torque as a key reason for the retro- grade motion of a disk considered, thus disproving some recent hypotheses.
Keywords: | retrograde turn, rolling disk, nonholonomic model, rolling friction |
---|---|
Citation: | Borisov A. V., Kilin A. A., Karavaev Y. L., Retrograde motion of a rolling disk, Physics-Uspekhi, 2017, vol. 60, no. 9, |
DOI: | 10.3367/UFNr.2017.01.038049 |
Full text: | pdf (364.25 Kb) |
Impact-factor WoS (2022): | 2.700 (Q2) |
---|---|
Impact-factor RSCI (2014): | 1.496 |
ISSN (print): | 0042-1294 |
ISSN (online): | 1996-6652 |
Site: | http://ufn.ru/ |
Borisov A. V., Vetchanin E. V., Kilin A. A.
The motion of a body shaped as a triaxial ellipsoid and controlled by the rotation of three internal rotors is studied. It is proved that the motion is controllable with the exception of a few particular cases. Partial solutions whose combinations enable an unbounded motion in any arbitrary direction are constructed.
Keywords: | ideal fluid, motion of a rigid body, Kirchhoff equations, control by rotors, gate |
---|---|
Citation: | Borisov A. V., Vetchanin E. V., Kilin A. A., Control of the Motion of a Triaxial Ellipsoid in a Fluid Using Rotors, Mathematical Notes, 2017, vol. 102, no. 4, |
DOI: | 10.1134/S0001434617090176 |
Full text: | pdf (616.14 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,502 |
ISSN (print): | 0025-567X |
ISSN (online): | 2305-2880 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=mzm&option_lang=eng |
Borisov A. V., Mamaev I. S., Bizyaev I. A.
This is a survey of the main forms of equations of dynamical systems with non-integrable constraints, divided into two large groups. The first group contains systems arising in vakonomic mechanics and optimal control theory, with the equations of motion obtained from the variational principle, and the second contains systems in classical non-holonomic mechanics, when the constraints are ideal and therefore the D’Alembert–Lagrange principle holds.
Keywords: | non-integrable constraints, vakonomic mechanics, optimal control theory, sub-Riemannian geometry, non-holonomic mechanics, invariant measure |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Bizyaev I. A., Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics, Russian Mathematical Surveys, 2017, vol. 72, no. 5, |
DOI: | 10.4213/rm9783 |
Full text: | pdf (1.09 Mb) |
Impact-factor WoS (2022): | 0.900 (Q2) |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0036-0279 |
ISSN (online): | 1468-4829 |
Site: | http://www.mathnet.ru/umn |
Nonholonomic mechanical systems arise in the context of many problems of practical significance. A famous model in nonholonomic mechanics is the Chaplygin sleigh. The Chaplygin sleigh is a rigid body with a sharp weightless wheel in contact with the (supporting) surface. The sharp edge of the wheel prevents the wheel from sliding in the direction perpendicular to its plane. This paper is concerned with a Chaplygin sleigh with time-varying mass distribution, which arises due to the motion of a point in the direction transverse to the plane of the knife edge. Equations of motion are obtained from which a closed system of equations with time-periodic coefficients decouples. This system governs the evolution of the translational and angular velocities of the sleigh. It is shown that if the projection of the center of mass of the whole system onto the axis along the knife edge is zero, the translational velocity of the sleigh increases. The trajectory of the point of contact is, as a rule, unbounded.
Keywords: | nonholonomic mechanics, Chaplygin sleigh, acceleration, first integrals |
---|---|
Citation: | Bizyaev I. A., A Chaplygin sleigh with a moving point mass, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2017, vol. 27, no. 4, |
DOI: | 10.20537/vm170408 |
Full text: | pdf (314.6 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
This paper addresses the dynamics of a disk rolling on an absolutely rough plane. It is proved that the equations of motion have an invariant measure with continuous density only in two cases: a dynamically symmetric disk and a disk with a special mass distribution. In the former case, the equations of motion possess two additional integrals and are integrable by quadratures by the Euler-Jacobi theorem. In the latter case, the absence of additional integrals is shown using a Poincaré map. In both cases, the volume of any domain in phase space (calculated with the help of the density) is preserved by the phase flow. Nonholonomic mechanics is populated with systems both with and without an invariant measure.
Keywords: | nonholonomic mechanics, Schwarzschild-Littlewood theorem, manifold of falls, chaotic dynamics |
---|---|
Citation: | Bizyaev I. A., Invariant measure in the problem of a disk rolling on a plane, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2017, vol. 27, no. 4, |
DOI: | 10.20537/vm170407 |
Full text: | pdf (312.38 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
This paper is concerned with a system of equations that describes the motion of two point vortices in a flow possessing constant uniform vorticity and perturbed by an acoustic wave. The system is shown to have both regular and chaotic regimes of motion. In addition, simple and chaotic attractors are found in the system. Attention is given to bifurcations of fixed points of a Poincar´e map which lead to the appearance of these regimes. It is shown that, in the case where the total vortex strength changes, the “reversible pitch-fork” bifurcation is a typical scenario of emergence of asymptotically stable fixed and periodic points. As a result of this bifurcation, a saddle point, a stable and an unstable point of the same period emerge from an elliptic point of some period. By constructing and analyzing charts of dynamical regimes and bifurcation diagrams we show that a cascade of period-doubling bifurcations is a typical scenario of transition to chaos in the system under consideration.
Keywords: | point vortices, shear flow, perturbation by an acoustic wave, bifurcations, reversible pitch-fork, period doubling |
---|---|
Citation: | Vetchanin E. V., Mamaev I. S., Dynamics of Two Point Vortices in an External Compressible Shear Flow, Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, |
DOI: | 10.1134/S1560354717080019 |
Full text: | pdf (3.4 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
This paper is concerned with the Chaplygin sleigh with time-varying mass distribution (parametric excitation). The focus is on the case where excitation is induced by a material point that executes periodic oscillations in a direction transverse to the plane of the knife edge of the sleigh. In this case, the problem reduces to investigating a reduced system of two first-order equations with periodic coefficients, which is similar to various nonlinear parametric oscillators. Depending on the parameters in the reduced system, one can observe different types of motion, including those accompanied by strange attractors leading to a chaotic (diffusion) trajectory of the sleigh on the plane. The problem of unbounded acceleration (an analog of Fermi acceleration) of the sleigh is examined in detail. It is shown that such an acceleration arises due to the position of the moving point relative to the line of action of the nonholonomic constraint and the center of mass of the platform. Various special cases of existence of tensor invariants are found.
Keywords: | nonholonomic mechanics, Fermi acceleration, Chaplygin sleigh, parametric oscillator, tensor invariants, involution, strange attractor, Lyapunov exponents, reversible systems, chaotic dynamics |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration, Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, |
DOI: | 10.1134/S1560354717080056 |
Full text: | pdf (1.91 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
In this paper we investigate the dynamics of a system that is a generalization of the Chaplygin sleigh to the case of an inhomogeneous nonholonomic constraint. We perform an explicit integration and a sufficiently complete qualitative analysis of the dynamics.
Keywords: | Chaplygin sleigh, inhomogeneous nonholonomic constraints, conservation laws, qualitative analysis, resonance |
---|---|
Citation: | Borisov A. V., Mamaev I. S., An Inhomogeneous Chaplygin Sleigh, Regular and Chaotic Dynamics, 2017, vol. 22, no. 4, |
DOI: | 10.1134/S1560354717040062 |
Full text: | pdf (676.55 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper is concerned with the dynamics of a top in the form of a truncated ball as it moves without slipping and spinning on a horizontal plane about a vertical. Such a system is described by differential equations with a discontinuous right-hand side. Equations describing the system dynamics are obtained and a reduction to quadratures is performed. A bifurcation analysis of the system is made and all possible types of the top’s motion depending on the system parameters and initial conditions are defined. The system dynamics in absolute space is examined. It is shown that, except for some special cases, the trajectories of motion are bounded.
Keywords: | integrable system, system with discontinuity, nonholonomic constraint, bifurcation diagram, absolute dynamics |
---|---|
Citation: | Kilin A. A., Pivovarova E. N., The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, |
DOI: | 10.1134/S156035471703008X |
Full text: | pdf (399.6 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper addresses the problem of the inertial motion of a roller racer, which reduces to investigating a dynamical system on a (two-dimensional) torus and to classifying singular points on it. It is shown that the motion of the roller racer in absolute space is asymptotic. A restriction on the system parameters in which this motion is bounded (compact) is presented.
Keywords: | roller racer, invariant measure, nonholonomic mechanics, scattering map |
---|---|
Citation: | Bizyaev I. A., The Inertial Motion of a Roller Racer, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, |
DOI: | 10.1134/S1560354717030042 |
Full text: | pdf (3.26 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
This paper is concerned with the Hess case in the Euler–Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder.
Keywords: | invariant submanifold, rotation number, Cantor ladder, limit cycles |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., The Hess–Appelrot Case and Quantization of the Rotation Number, Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, |
DOI: | 10.1134/S156035471702006X |
Full text: | pdf (991.1 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
In this paper we investigate the dynamics of a system that is a generalization of the Chaplygin sleigh to the case of an inhomogeneous nonholonomic constraint. We perform an explicit integration and a sufficiently complete qualitative analysis of the dynamics.
Keywords: | Chaplygin sleigh, inhomogeneous nonholonomic constraints, conservation laws, qualitative analysis, resonance |
---|---|
Citation: | Borisov A. V., Mamaev I. S., An inhomogeneous Chaplygin sleigh, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, |
DOI: | 10.20537/nd1704014 |
Full text: | pdf (518.79 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The dynamics of a spherical robot of combined type consisting of a spherical shell and a pendulum attached at the center of the shell is considered. At the end of the pendulum a rotor is installed. For this system we carry out a stability analysis for a partial solution which in absolute space corresponds to motion along a circle with constant velocity. Regions of stability of a partial solution are found depending on the orientation of the spherical robot during the motion, its velocity and the radius of the circle traced out by the point of contact.
Keywords: | spherical robot, nonholonomic constraint, partial solution, stability |
---|---|
Citation: | Pivovarova E. N., Stability analysis of steady motions of a spherical robot of combined type, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, |
DOI: | 10.20537/nd1704013 |
Full text: | pdf (1.75 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The dynamics of a spherical robot of combined type consisting of a spherical shell and a pendulum attached at the center of the shell is considered. At the end of the pendulum a rotor is installed. For this system we carry out a stability analysis for a partial solution which in absolute space corresponds to motion along a circle with constant velocity. Regions of stability of a partial solution are found depending on the orientation of the spherical robot during the motion, its velocity and the radius of the circle traced out by the point of contact.
Keywords: | spherical robot, nonholonomic constraint, partial solution, stability |
---|---|
Citation: | Pivovarova E. N., Stability analysis of steady motions of a spherical robot of combined type, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, |
DOI: | 10.20537/nd1704013 |
Full text: | pdf (1.76 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Karavaev Y. L., Kilin A. A., Klekovkin A. V.
In this paper the model of rolling of spherical bodies on a plane without slipping is presented taking into account viscous rolling friction. Results of experiments aimed at investigating the influence of friction on the dynamics of rolling motion are presented. The proposed dynamical friction model for spherical bodies is verified and the limits of its applicability are estimated. A method for determining friction coefficients from experimental data is formulated.
Keywords: | rolling friction, dynamical model, spherical body, nonholonomic model, experimental investigation |
---|---|
Citation: | Karavaev Y. L., Kilin A. A., Klekovkin A. V., The dynamical model of the rolling friction of spherical bodies on a plane without slipping, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, |
DOI: | 10.20537/nd1704012 |
Full text: | pdf (3.54 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
This paper presents a comparative analysis of computations of the motion of heavy three-bladed screws in a fluid along with experimental results. Simulation of the motion is performed using the theory of an ideal fluid and the phenomenological model of viscous friction. For experimental purposes, models of three-bladed screws with various configurations and sizes were manufactured by casting from chemically hardening polyurethane. Comparison of calculated and experimental results has shown that the mathematical models considered essentially do not reflect the processes observed in the experiments.
Keywords: | motion in a fluid, helical body, experimental investigation |
---|---|
Citation: | Vetchanin E. V., Klenov A. I., Experimental investigation of the fall of helical bodies in a fluid, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, |
DOI: | 10.20537/nd1704011 |
Full text: | pdf (853.58 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
This paper is concerned with the Hess case in the Euler –Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder.
Keywords: | invariant submanifold, rotation number, Cantor ladder, limit cycles |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., The Hess–Appelrot case and quantization of the rotation number, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 3, |
DOI: | 10.20537/nd1703010 |
Full text: | pdf (556.32 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Kazakov A. O., Pivovarova E. N.
This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of perioddoubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasi-periodic.
Keywords: | Chaplygin top, nonholonomic constraint, rubber model, strange attractor, bifurcation, trajectory of the point of contact |
---|---|
Citation: | Borisov A. V., Kazakov A. O., Pivovarova E. N., Regular and chaotic dynamics in the rubber model of a Chaplygin top, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 2, |
DOI: | 10.20537/nd1702009 |
Full text: | pdf (2.3 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S.
This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector (3, 6, 14), the other is defined by two generatrices and growth vector (2, 3, 5, 8). Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.
Keywords: | sub-Riemannian geometry, Carnot group, Poincaré map, first integrals |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S., Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 1, |
DOI: | 10.20537/nd1701009 |
Full text: | pdf (2.82 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Vetchanin E. V., Tenenev V. A., Kilin A. A.
In this paper we consider the controlled motion of a helical body with three blades in an ideal fluid, which is executed by rotating three internal rotors. We set the problem of selecting control actions, which ensure the motion of the body near the predetermined trajectory. To determine controls that guarantee motion near the given curve, we propose methods based on the application of hybrid genetic algorithms (genetic algorithms with real encoding and with additional learning of the leader of the population by a gradient method) and artificial neural networks. The correctness of the operation of the proposed numerical methods is estimated using previously obtained differential equations, which define the law of changing the control actions for the predetermined trajectory.
In the approach based on hybrid genetic algorithms, the initial problem of minimizing the integral functional reduces to minimizing the function of many variables. The given time interval is broken up into small elements, on each of which the control actions are approximated by Lagrangian polynomials of order 2 and 3. When appropriately adjusted, the hybrid genetic algorithms reproduce a solution close to exact. However, the cost of calculation of 1 second of the physical process is about 300 seconds of processor time.
To increase the speed of calculation of control actions, we propose an algorithm based on artificial neural networks. As the input signal the neural network takes the components of the required displacement vector. The node values of the Lagrangian polynomials which approximately describe the control actions return as output signals . The neural network is taught by the well-known back-propagation method. The learning sample is generated using the approach based on hybrid genetic algorithms. The calculation of 1 second of the physical process by means of the neural network requires about 0.004 seconds of processor time, that is, 6 orders faster than the hybrid genetic algorithm. The control calculated by means of the artificial neural network differs from exact control. However, in spite of this difference, it ensures that the predetermined trajectory is followed exactly.
Keywords: | motion control, genetic algorithms, neural networks, motion in a fluid, ideal fluid |
---|---|
Citation: | Vetchanin E. V., Tenenev V. A., Kilin A. A., Optimal control of the motion in an ideal fluid of a screw-shaped body with internal rotors, Computer Research and Modeling, 2017, vol. 9, no. 5, |
DOI: | 10.20537/2076-7633-2017-9-5-741-759 |
Full text: | pdf (567.52 Kb) |
Impact-factor RSCI (2022): | 0.257 (Q4) |
---|---|
ISSN (print): | 2076-7633 |
ISSN (online): | 2077-6853 |
Site: | http://crm.ics.org.ru/ |
In this paper we study the controlled motion of an arbitrary two-dimensional body in an ideal fluid with a moving internal mass and an internal rotor in the presence of constant circulation around the body. We show that by changing the position of the internal mass and by rotating the rotor, the body can be made to move to a given point, and discuss the influence of nonzero circulation on the motion control. We have found that in the presence of circulation around the body the system cannot be completely stabilized at an arbitrary point of space, but fairly simple controls can be constructed to ensure that the body moves near the given point.
Keywords: | ideal fuid, controllability, Kirchhoff equations, circulation around the body |
---|---|
Citation: | Vetchanin E. V., Kilin A. A., Control of Body Motion in an Ideal Fluid Using the Internal Mass and the Rotor in the Presence of Circulation Around the Body, Journal of Dynamical and Control Systems, 2017, vol. 23, |
DOI: | 10.1007/s10883-016-9345-4 |
Full text: | pdf (1.49 Mb) |
Impact-factor WoS (2022): | 0.900 (Q3) |
---|---|
Impact-factor RSCI (2022): | 0.382 (Q3) |
ISSN (print): | 1079-2724 |
ISSN (online): | 1573-8698 |
Site: | http://link.springer.com/journal/10883 |
The motion controlled by the rotation of three internal rotors of a body with helical symmetry in an ideal liquid is considered. The problem is to select controls that ensure the displacement of the body with minimum effort. The optimality of particular solutions found earlier is studied.
Citation: | Vetchanin E. V., Mamaev I. S., Optimal control of the motion of a helical body in a liquid using rotors, Russian Journal of Mathematical Physics, 2017, vol. 24, no. 3, |
---|---|
DOI: | 10.1134/S1061920817030128 |
Full text: | pdf (582.92 Kb) |
Impact-factor WoS (2022): | 1.400 (Q3) |
---|---|
ISSN (print): | 1061-9208 |
ISSN (online): | 1555-6638 |
Site: | http://link.springer.com/journal/11503 |
Bizyaev I. A., Borisov A. V., Kuznetsov S. P.
We consider the movement of Chaplygin sleigh on a plane that is a solid body with imposed nonholonomic constraint, which excludes the possibility of motions transversal to the constraint element (“knife-edge”), and complement the model with an attached mass, periodically oscillating relatively to the main platform of the sleigh. Numerical simulations indicate the occurrence of either unrestricted acceleration of the sleigh, or motions with bounded velocities and momenta, depending on parameters. We note the presence of phenomena characteristic to nonholonomic systems with complex dynamics; in particular, attractors occur responsible for chaotic motions. In addition, quasiperiodic regimes take place similar to those observed in conservative nonlinear dynamics.
Citation: | Bizyaev I. A., Borisov A. V., Kuznetsov S. P., Chaplygin sleigh with periodically oscillating internal mass, EPL, 2017, vol. 119, no. 6, 60008, |
---|---|
DOI: | 10.1209/0295-5075/119/60008 |
Full text: | pdf (854.88 Kb) |
Impact-factor WoS (2022): | 1.800 (Q3) |
---|---|
ISSN (print): | 0295-5075 |
ISSN (online): | 1286-4854 |
Site: | http://iopscience.iop.org/journal/0295-5075 |
Kilin A. A., Bozek P., Karavaev Y. L., Klekovkin A. V., Shestakov V. A.
In this article, a dynamical model for controlling an omniwheel mobile robot is presented. The proposed model is used to construct an algorithm for calculating control actions for trajectories characterizing the high maneuverability of the mobile robot. A description is given for a prototype of the highly maneuverable robot with four omniwheels, for which an algorithm for setting the coefficients of the PID controller is considered. Experiments on the motion of the robot were conducted at different angles, and the orientation of the platform was preserved. The experimental results are analyzed and statistically assessed.
Keywords: | omniwheel, mobile robot, dynamical model, PID controller, experimental investigations |
---|---|
Citation: | Kilin A. A., Bozek P., Karavaev Y. L., Klekovkin A. V., Shestakov V. A., Experimental investigations of a highly maneuverable mobile omniwheel robot, International Journal of Advanced Robotic Systems, 2017, vol. 14, no. 6, |
DOI: | 10.1177/1729881417744570 |
Full text: | pdf (654.85 Kb) |
Impact-factor WoS (2022): | 2.300 (Q4) |
---|---|
Impact-factor RSCI (2022): | 0.550 (Q2) |
ISSN (print): | 1729-8814 |
ISSN (online): | 1729-8814 |
Site: | https://uk.sagepub.com/en-gb/asi/node/859241 |
Ivanova T. B., Erdakova N. N., Karavaev Y. L.
The experimental stand and the results of investigation of the motion of a brake shoe are described. In the noncritical region, the friction coefficient is determined experimentally. It is shown that its value corresponds to the condition of uniqueness of the solution for construction of this brake shoe. The dynamics observed in the paradoxical-motion region is described.
Citation: | Ivanova T. B., Erdakova N. N., Karavaev Y. L., Experimental Investigation of the Dynamics of a Brake Shoe, Doklady Physics, 2016, vol. 61, no. 12, |
---|---|
DOI: | 10.7868/S0869565216340089 |
Full text: | pdf (413.7 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
In this paper, we study the free and controlled motion of an arbitrary two-dimensional body with a moving internal material point through an ideal fluid in presence of constant circulation around the body. We perform bifurcation analysis of free motion (with fixed internal mass). We show that by changing the position of the internal mass the body can be made to move to a specified point. There are a number of control problems associated with the nonzero drift of the body in the case of fixed internal mass.
Citation: | Vetchanin E. V., Kilin A. A., Free and controlled motion of a body with moving internal mass though a fluid in the presence of circulation around the body, Doklady Physics, 2016, vol. 466, no. 3, |
---|---|
DOI: | 10.7868/S0869565216030129 |
Full text: | pdf (300.44 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
We consider the controlled motion in an ideal incompressible fluid of a rigid body with moving internal masses and an internal rotor in the presence of circulation of the fluid velocity around the body. The controllability of motion (according to the Rashevskii–Chow theorem) is proved for various combinations of control elements. In the case of zero circulation, we construct explicit controls (gaits) that ensure rotation and rectilinear (on average) motion. In the case of nonzero circulation, we examine the problem of stabilizing the body (compensating the drift) at the end point of the trajectory. We show that the drift can be compensated for if the body is inside a circular domain whose size is defined by the geometry of the body and the value of circulation.
Citation: | Vetchanin E. V., Kilin A. A., Controlled Motion of a Rigid Body with Internal Mechanisms in an Ideal Incompressible Fluid, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, |
---|---|
DOI: | 10.1134/S037196851604018X |
Full text: | pdf (1.36 Mb) |
Impact-factor WoS (2022): | 0.500 (Q4) |
---|---|
Impact-factor RSCI (2022): | 0.276 (Q3) |
ISSN (print): | 0081-5438 |
ISSN (online): | 1531-8605 |
Site: | http://www.maik.ru/ru/journal/trstekl/ |
We present the results of theoretical and experimental investigations of the motion of a spherical robot on a plane. The motion is actuated by a platform with omniwheels placed inside the robot. The control of the spherical robot is based on a dynamic model in the nonholonomic statement expressed as equations of motion in quasivelocities with indeterminate coefficients. A number of experiments have been carried out that confirm the adequacy of the dynamic model proposed.
Citation: | Karavaev Y. L., Kilin A. A., Nonholonomic Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform: Theory and Experiments, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, |
---|---|
DOI: | 10.1134/S0081543816080095 |
Full text: | pdf (366.98 Kb) |
Impact-factor WoS (2022): | 0.500 (Q4) |
---|---|
Impact-factor RSCI (2022): | 0.276 (Q3) |
ISSN (print): | 0081-5438 |
ISSN (online): | 1531-8605 |
Site: | http://www.maik.ru/ru/journal/trstekl/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
This paper provides a detailed description of various reduction schemes in rigid body dynamics. The analysis of one of such nontrivial reductions makes it possible to put the cases already found in order and to obtain new generalizations of the Kovalevskaya case to e(3). Note that the indicated reduction allows one to obtain in a natural way some singular additive terms that were proposed earlier by D.N. Goryachev.
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., Generalizations of the Kovalevskaya Case and Quaternions, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, |
---|---|
DOI: | 10.1134/S0371968516040038 |
Full text: | pdf (190.82 Kb) |
Impact-factor WoS (2022): | 0.500 (Q4) |
---|---|
Impact-factor RSCI (2022): | 0.276 (Q3) |
ISSN (print): | 0081-5438 |
ISSN (online): | 1531-8605 |
Site: | http://www.maik.ru/ru/journal/trstekl/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
This paper is concerned with the nonholonomic Suslov problem and its generalization proposed by Chaplygin. The issue of the existence of an invariant measure with singular density (having singularities at some points of phase space) is discussed.
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., The Hess–Appelrot system and its nonholonomic analogs, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 294, |
---|---|
DOI: | 10.1134/S0371968516030171 |
Full text: | pdf (1.23 Mb) |
Impact-factor WoS (2022): | 0.500 (Q4) |
---|---|
Impact-factor RSCI (2022): | 0.276 (Q3) |
ISSN (print): | 0081-5438 |
ISSN (online): | 1531-8605 |
Site: | http://www.maik.ru/ru/journal/trstekl/ |
The dynamics of a Painlevé–Appell system consisting of two point masses joined by a weightless rigid rodis studied within two mechanical models, which describe different motion regimes. One of the massescan slide or can be supported at rest on a rough straight line. The boundaries of the region of definitionof each of the models are presented, and the transitions between them are analysed for various frictioncoefficients.
Citation: | Ivanova T. B., Mamaev I. S., Dynamics of a Painlevé-Appel system, Journal of Applied Mathematics and Mechanics, 2016, vol. 80, no. 1, |
---|---|
Full text: | pdf (1.78 Mb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Borisov A. V., Kuznetsov S. P., Mamaev I. S., Tenenev V. A.
From analysis of time series obtained on the numerical solution of a plane problem on the motion of a body with an elliptic cross section under the action of gravity force in an incompressible viscous fluid, a system of ordinary differential equations approximately describing the dynamics of the body is reconstructed. To this end, coefficients responsible for the added mass, the force caused by the circulation of the velocity field, and the resisting force are found by the least square adjustment. The agreement between the finitedimensional description and the simulation on the basis of the Navier–Stokes equations is illustrated by images of attractors in regular and chaotic modes. The coefficients found make it possible to estimate the actual contribution of different effects to the dynamics of the body.
Citation: | Borisov A. V., Kuznetsov S. P., Mamaev I. S., Tenenev V. A., Describing the Motion of a Body with an Elliptical Cross Section in a Viscous Uncompressible Fluid by Model Equations Reconstructed from Data Processing, Technical Physics Letters, 2016, vol. 42, no. 9, |
---|---|
DOI: | 10.1134/S1063785016090042 |
Full text: | pdf (508.12 Kb) |
Impact-factor WoS (2022): | 0.600 (Q4) |
---|---|
ISSN (print): | 1063-7850 |
ISSN (online): | 1090-6533 |
Site: | http://link.springer.com/journal/11455 |
Vetchanin E. V., Kilin A. A., Mamaev I. S.
This paper is concerned with the motion of a helical body in an ideal fluid, which is controlled by rotating three internal rotors. It is proved that the motion of the body is always controllable by means of three rotors with noncoplanar axes of rotation. A condition whose satisfaction prevents controllability by means of two rotors is found. Control actions that allow the implementation of unbounded motion in an arbitrary direction are constructed. Conditions under which the motion of the body along an arbitrary smooth curve can be implemented by rotating the rotors are presented. For the optimal control problem, equations of sub-Riemannian geodesics on SE(3) are obtained.
Keywords: | ideal fluid, motion of a helical body, Kirchhoff equations, control of rotors, gaits, optimal control |
---|---|
Citation: | Vetchanin E. V., Kilin A. A., Mamaev I. S., Control of the Motion of a Helical Body in a Fluid Using Rotors, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, |
DOI: | 10.1134/S1560354716070108 |
Full text: | pdf (1.23 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kazakov A. O., Pivovarova E. N.
This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of period-doubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasi-periodic.
Keywords: | Chaplygin top, nonholonomic constraint, rubber model, strange attractor, bifurcation, trajectory of the point of contact |
---|---|
Citation: | Borisov A. V., Kazakov A. O., Pivovarova E. N., Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, |
DOI: | 10.1134/S156035471607011X |
Full text: | pdf (2.21 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Karavaev Y. L., Kilin A. A., Klekovkin A. V.
In this paper we describe the results of experimental investigations of the motion of a screwless underwater robot controlled by rotating internal rotors. We present the results of comparison of the trajectories obtained with the results of numerical simulation using the model of an ideal fluid.
Keywords: | screwless underwater robot, experimental investigations, helical body |
---|---|
Citation: | Karavaev Y. L., Kilin A. A., Klekovkin A. V., Experimental Investigations of the Controlled Motion of a Screwless Underwater Robot, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, |
DOI: | 10.1134/S1560354716070133 |
Full text: | pdf (6.91 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper is devoted to an experimental investigation of the motion of a rigid body set in motion by rotating two unbalanced internal masses. The results of experiments confirming the possibility of motion by this method are presented. The dependence of the parameters of motion on the rotational velocity of internal masses is analyzed. The velocity field of the fluid around the moving body is examined.
Keywords: | self-propulsion, PIV, vortex formation, above-water screwless robot |
---|---|
Citation: | Klenov A. I., Kilin A. A., Influence of Vortex Structures on the Controlled Motion of an Above-water Screwless Robot, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, |
DOI: | 10.1134/S1560354716070145 |
Full text: | pdf (6.5 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper is concerned with the problem of the integrable behavior of geodesics on homogeneous factors of the Lobachevsky plane with respect to Fuchsian groups (orbifolds). Locally the geodesic equations admit three independent Noether integrals linear in velocities (energy is a quadratic form of these integrals). However, when passing along closed cycles the Noether integrals undergo a linear substitution. Thus, the problem of integrability reduces to the search for functions that are invariant under these substitutions. If a Fuchsian group is Abelian, then there is a first integral linear in the velocity (and independent of the energy integral). Conversely, if a Fuchsian group contains noncommuting hyperbolic or parabolic elements, then the geodesic flow does not admit additional integrals in the form of a rational function of Noether integrals. We stress that this result holds also for noncompact orbifolds, when there is no ergodicity of the geodesic flow (since nonrecurrent geodesics can form a set of positive measure).
Keywords: | Lobachevsky plane, Fuchsian group, orbifold, Noether integrals |
---|---|
Citation: | Kozlov V. V., On the Extendability of Noether’s Integrals for Orbifolds of Constant Negative Curvature, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, |
DOI: | 10.1134/S1560354716070054 |
Full text: | pdf (630.35 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S.
This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector (3,6,14), the other is defined by two generatrices and growth vector (2,3,5,8). Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.
Keywords: | sub-Riemannian geometry, Carnot group, Poincaré map, first integrals |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S., Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, |
DOI: | 10.1134/S1560354716060125 |
Full text: | pdf (3.48 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Bizyaev I. A.
In this paper, we consider in detail the 2-body problem in spaces of constant positive curvature S2 and S3. We perform a reduction (analogous to that in rigid body dynamics) after which the problem reduces to analysis of a two-degree-of-freedom system. In the general case, in canonical variables the Hamiltonian does not correspond to any natural mechanical system. In addition, in the general case, the absence of an analytic additional integral follows from the constructed Poincaré section. We also give a review of the historical development of celestial mechanics in spaces of constant curvature and formulate open problems.
Keywords: | celestial mechanics, space of constant curvature, reduction, rigid body dynamics, Poincaré section |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Bizyaev I. A., The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity, Regular and Chaotic Dynamics, 2016, vol. 21, no. 5, |
DOI: | 10.1134/S1560354716050075 |
Full text: | pdf (1.45 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Bizyaev I. A.
In this historical review we describe in detail the main stages of the development of nonholonomic mechanics starting from the work of Earnshaw and Ferrers to the monograph of Yu.I. Neimark and N.A. Fufaev. In the appendix to this review we discuss the d’Alembert–Lagrange principle in nonholonomic mechanics and permutation relations.
Keywords: | nonholonomic mechanics, nonholonomic constraint, d’Alembert–Lagrange principle, permutation relations |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Bizyaev I. A., Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period, Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, |
DOI: | 10.1134/S1560354716040055 |
Full text: | pdf (1.87 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
This paper is concerned with the dynamics of vortex sources in a deformation flow. The case of two vortex sources is shown to be integrable by quadratures. In addition, the relative equilibria (of the reduced system) are examined in detail and it is shown that in this case the trajectory of vortex sources is an ellipse.
Keywords: | integrability, vortex sources, reduction, deformation flow |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., The Dynamics of Vortex Sources in a Deformation Flow, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, |
DOI: | 10.1134/S1560354716030084 |
Full text: | pdf (1.37 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The onset of adiabatic chaos in rigid body dynamics is considered. A comparison of the analytically calculated diffusion coefficient describing probabilistic effects in the zone of chaos with a numerical experiment is made. An analysis of the splitting of asymptotic surfaces is performed and uncertainty curves are constructed in the Poincaré – Zhukovsky problem. The application of Hamiltonian methods to nonholonomic systems is discussed. New problem statements are given which are related to the destruction of an adiabatic invariant and to the acceleration of the system (Fermi’s acceleration).
Keywords: | adiabatic invariants, Liouville system, transition through resonance, adiabatic chaos |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics, Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, |
DOI: | 10.1134/S1560354716020064 |
Full text: | pdf (941.28 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found. In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.
Keywords: | Chaplygin sleigh, invariant measure, nonholonomic mechanics |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., Dynamics of the Chaplygin Sleigh on a Cylinder, Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, |
DOI: | 10.1134/S1560354716010081 |
Full text: | pdf (268.54 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found. In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.
Keywords: | Chaplygin sleigh, invariant measure, nonholonomic mechanics |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., Dynamics of the Chaplygin sleigh on a cylinder, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 4, |
DOI: | 10.20537/nd1604010 |
Full text: | pdf (331.42 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
This paper is concerned with the motion of an unbalanced heavy three-axial ellipsoid in an ideal fluid controlled by rotation of three internal rotors. It is proved that the motion of the body considered is controlled with respect to configuration variables except for some special cases. An explicit control that makes it possible to implement unbounded motion in an arbitrary direction has been calculated. Directions for which control actions are bounded functions of time have been determined.
Keywords: | ideal fluid, motion of a rigid body, Kirchhoff equations, control by rotors, gaits |
---|---|
Citation: | Vetchanin E. V., Kilin A. A., Control of the motion of an unbalanced heavy ellipsoid in an ideal fluid using rotors, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 4, |
DOI: | 10.20537/nd1604009 |
Full text: | pdf (304.78 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Bizyaev I. A.
In this historical review we describe in detail the main stages of the development of nonholonomic mechanics starting from the work of Earnshaw and Ferrers to the monograph of Yu.I. Neimark and N.A. Fufaev. In the appendix to this review we discuss the d’Alembert–Lagrange principle in nonholonomic mechanics and permutation relations.
Keywords: | nonholonomic mechanics, nonholonomic constraint, d’Alembert–Lagrange principle, permutation relations |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Bizyaev I. A., Historical and critical review of the development of nonholonomic mechanics: the classical period, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 3, |
DOI: | 10.20537/nd1603007 |
Full text: | pdf (1.9 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Kazakov A. O.
In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems.We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.
Keywords: | Suslov problem, nonholonomic constraint, reversal, strange attractor |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Kazakov A. O., Dynamics of the Suslov problem in a gravitational field: reversal and strange attractors, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 2, |
DOI: | 10.20537/nd1602008 |
Full text: | pdf (3.54 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
In this paper, we develop the results obtained by J.Hadamard and G.Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasi-velocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a one-wheeled vehicle and a wheeled vehicle with two rotating wheel pairs.
Keywords: | nonholonomic constraint, wheeled vehicle, reduction, equations of motion |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., On the Hadamard–Hamel problem and the dynamics of wheeled vehicles, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 1, |
DOI: | 10.20537/nd1601009 |
Full text: | pdf (445.79 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Tenenev V. A., Vetchanin E. V., Ilaletdinov L. F.
This paper is concerned with the process of the free fall of a three-bladed screw in a fluid. The investigation is performed within the framework of theories of an ideal fluid and a viscous fluid. For the case of an ideal fluid the stability of uniformly accelerated rotations (the Steklov solutions) is studied. A phenomenological model of viscous forces and torques is derived for investigation of the motion in a viscous fluid. A chart of Lyapunov exponents and bifucation diagrams are computed. It is shown that, depending on the system parameters, quasiperiodic and chaotic regimes of motion are possible. Transition to chaos occurs through cascade of period-doubling bifurcations.
Keywords: | ideal fluid, viscous fluid, motion of a rigid body, dynamical system, stability of motion, bifurcations, chart of Lyapunov exponents |
---|---|
Citation: | Tenenev V. A., Vetchanin E. V., Ilaletdinov L. F., Chaotic dynamics in the problem of the fall of a screw-shaped body in a fluid, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 1, |
DOI: | 10.20537/nd1601007 |
Full text: | pdf (4.44 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
In this paper, using the Hojman construction, we give examples of various Poisson brackets which differ from those which are usually analyzed in Hamiltonian mechanics. They possess a nonmaximal rank, and in the general case an invariant measure and Casimir functions can be globally absent for them.
Keywords: | Hamiltonization, Poisson bracket, Casimir functions, invariant measure, nonholonomic hinge, Suslov problem, Chaplygin sleigh |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., The Hojman Construction and Hamiltonization of Nonholonomic Systems, Symmetry, Integrability and Geometry: Methods and Applications, 2016, vol. 12, 012, |
DOI: | 10.3842/SIGMA.2016.012 |
Full text: | pdf (571.09 Kb) |
Impact-factor WoS (2022): | 0.900 (Q4) |
---|---|
ISSN (print): | 1815-0659 |
Site: | http://www.emis.de/journals/SIGMA/ |
Vetchanin E. V., Kazakov A. O.
In this paper, we consider a system governing the motion of two point vortices in a flow excited by an external acoustic forcing. It is known that the system of two vortices is integrable in the absence of acoustic forcing. However, the addition of the acoustic forcing makes the system much more complex, and the system becomes nonintegrable and loses the phase volume preservation property. The objective of our research is to study chaotic dynamics and typical bifurcations. Numerical analysis has shown that the reversible pitchfork bifurcation is typical. Also, we show that the existence of strange attractors is not characteristic for the system under consideration.
Keywords: | reversible pitchfork, point vortices, acoustic forcing, chaos |
---|---|
Citation: | Vetchanin E. V., Kazakov A. O., Bifurcations and chaos in the dynamics of two point vortices in an acoustic wave, International Journal of Bifurcation and Chaos, 2016, vol. 26, no. 4, 1650063, |
DOI: | 10.1142/S0218127416500632 |
Full text: | pdf (1.38 Mb) |
Impact-factor WoS (2022): | 2.200 (Q2) |
---|---|
ISSN (print): | 0218-1274 |
ISSN (online): | 1793-6551 |
Site: | http://www.worldscientific.com/worldscinet/ijbc |
In this paper, we focus on the study of two-dimensional plate dynamics on the Lobachevskii plane L2. First of all, we consider the free motion of such a plate, which is a pseudospherical analog of dynamics of the Euler top, and also present an analog of the Euler–Poisson equations enabling us to study the motion of the body in potential force fields having rotational symmetry. We present a series of integrable cases, having analogs in Euclidean space, for different fields. Moreover, in the paper, a partial qualitative analysis of the dynamics of free motion of a plate under arbitrary initial conditions is made and a number of computer illustrations are presented which show a substantial difference of the motion from the case of Euclidean space. The study undertaken in the present paper leads to interesting physical onsequences, which enable us to detect the influence of curvature on the body dynamics.
Citation: | Borisov A. V., Mamaev I. S., Rigid Body Dynamics in Non-Euclidean Spaces, Russian Journal of Mathematical Physics, 2016, vol. 23, no. 4, |
---|---|
DOI: | 10.1134/S1061920816040026 |
Full text: | pdf (704.17 Kb) |
Impact-factor WoS (2022): | 1.400 (Q3) |
---|---|
ISSN (print): | 1061-9208 |
ISSN (online): | 1555-6638 |
Site: | http://link.springer.com/journal/11503 |
Borisov A. V., Mamaev I. S., Erdakova N. N.
This paper is concerned with the problem of a rigid body (tripod) moving with three points in contact with a horizontal plane under the action of dry friction forces. It is shown that the regime of asymptotic motion (final dy- namics) of the tripod can be pure rotation, pure sliding, or sliding and rotation can cease simultaneously, which is determined by the position of the tripod’s supports relative to the radius of inertia. In addition, the dependence of the trajectory of the center of mass on the system parameters is investigated. A comparison is made with the well-known theoretical and experimental studies on the motion of bodies with a flat base
Keywords: | dry friction, linear pressure distribution, planar motion, Coulomb’s law, asymptotic motion, dynamics, tripod |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Erdakova N. N., Dynamics of a body sliding on a rough plane and supported at three points, Theoretical and Applied Mechanics, 2016, vol. 43, no. 2, |
DOI: | 10.2298/TAM161130013B |
Full text: | pdf (1.81 Mb) |
ISSN (print): | 1450-5584 |
---|---|
ISSN (online): | 2406-0925 |
Site: | http://www.mi.sanu.ac.rs/tam/ |
A new integrable problem of nonholonomic mechanics is considered and its mechanical realization is proposed. This problem is a generalization of the well-known problem of А. P. Veselov and L. E. Veselova concerning the rolling motion of the Chaplygin ball in a straight line. Particular cases are found in which integration can be reduced to explicit quadratures.
Citation: | Borisov A. V., Mamaev I. S., A New Integrable System of Nonholonomic Mechanics, Doklady Physics, 2015, vol. 60, no. 6, |
---|---|
DOI: | 10.1134/S1028335815060087 |
Full text: | pdf (255.48 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
We consider differential equations with quadratic right-hand sides which admit two quadratic first integrals, one of which is a positive definite quadratic form. We present general conditions under which a linear change of variables reduces this system to some "canonical" form. Under these conditions the system turns out to be nondivergent and is reduced to Hamiltonian form, however, the corresponding linear Lie–Poisson bracket does not always satisfy the Jacobi identity. In the three-dimensional case the equations are reduced to the classical equations of the Euler top, and in the four-dimensional space the system turns out to be superintegrable and coincides with the Euler–Poincaré equations on some Lie algebra. In the five-dimensional case we find a reducing multiplier after multiplication with which the Poisson bracket satisfies the Jacobi identity. In the general case, we prove that there is no reducing multiplier for n>5. As an example, we consider a system of Lotka–Volterra type with quadratic right-hand sides, which was studied already by Kovalevskaya from the viewpoint of the conditions for uniqueness of its solutions as functions of complex time.
Keywords: | first integrals, conformally Hamiltonian system, Poisson bracket, Kovalevskaya system, dynamical systems with quadratic right-hand sides |
---|---|
Citation: | Bizyaev I. A., Kozlov V. V., Homogeneous systems with quadratic integrals, Lie–Poisson quasi-brackets, and the Kovalevskaya method, Sbornik: Mathematics, 2015, vol. 206, no. 12, |
Full text: | pdf (481.65 Kb) |
Impact-factor WoS (2022): | 0.800 (Q3) |
---|---|
Impact-factor RSCI (2014): | 1.024 |
ISSN (print): | 0368-8666 |
ISSN (online): | 2305-2783 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=sm&option_lang=rus |
This is a reply to the comment by V.F. Zhuravlev (see Usp. Fiz. Nauk 185 1337 (2015) [Phys. Usp. 58 (12) (2015)]) on the inadequacy of the nonholonomic model when applied to the rolling of rigid bodies. The model of nonholonomic mechanics is discussed. Using recent results as examples, it is shown that the validity and potential of the nonholonomic model are not inferior to those of other dynamics and friction models.
Keywords: | nonholonomic model, dry friction, rattleback, rolling motion of a rigid body |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Notes on new friction models and nonholonomic mechanics, Physics-Uspekhi, 2015, vol. 58, no. 12, |
DOI: | 10.3367/UFNe.0185.201512g.1339 |
Full text: | pdf (262.98 Kb) |
Impact-factor WoS (2022): | 2.700 (Q2) |
---|---|
Impact-factor RSCI (2014): | 1.496 |
ISSN (print): | 0042-1294 |
ISSN (online): | 1996-6652 |
Site: | http://ufn.ru/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with a stratified density and a steady-state velocity field. As in the classical formulation of the problem, it is assumed that the figures, or their layers, uniformly rotate about an axis fixed in space. It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with a constant angular velocity, is at equilibrium. Expressions are obtained for the gravitational potential, change in the angular velocity and pressure, and the conclusion is drawn that the angular velocity on the outer surface is the same as that of the corresponding Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification. We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis.
Keywords: | Self-gravitating fluid, Confocal stratification, Homothetic stratification, Chaplygin problem, Axisymmetric equilibrium figures, Space of constant curvature |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., Figures of equilibrium of an inhomogeneous self-gravitating fluid, Celestial Mechanics and Dynamical Astronomy, 2015, vol. 122, no. 1, |
DOI: | 10.1007/s10569-015-9608-5 |
Full text: | pdf (651.34 Kb) |
Impact-factor WoS (2022): | 1.600 (Q3) |
---|---|
ISSN (print): | 0923-2958 |
ISSN (online): | 1572-9478 |
Site: | http://link.springer.com/journal/10569 |
Bolsinov A. V., Kilin A. A., Kazakov A. O.
The phenomenon of a topological monodromy in integrable Hamiltonian and nonholonomic systems is discussed. An efficient method for computing and visualizing the monodromy is developed. The comparative analysis of the topological monodromy is given for the rolling ellipsoid of revolution problem in two cases, namely, on a smooth and on a rough plane. The first of these systems is Hamiltonian, the second is nonholonomic. We show that, from the viewpoint of monodromy, there is no difference between the two systems, and thus disprove the conjecture by Cushman and Duistermaat stating that the topological monodromy gives a topological obstruction for Hamiltonization of the rolling ellipsoid of revolution on a rough plane.
Keywords: | Topological monodromy, Integrable systems, Nonholonomic systems, Poincaré map, Bifurcation analysis, Focus–focus singularities |
---|---|
Citation: | Bolsinov A. V., Kilin A. A., Kazakov A. O., Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: Pro or contra? , Journal of Geometry and Physics, 2015, vol. 87, |
DOI: | 10.1016/j.geomphys.2014.08.003 |
Full text: | pdf (754.64 Kb) |
Impact-factor WoS (2022): | 1.500 (Q1) |
---|---|
Impact-factor RSCI (2022): | 0.455 (Q2) |
ISSN (print): | 0393-0440 |
Site: | http://www.sciencedirect.com/science/journal/03930440 |
Borisov A. V., Mamaev I. S., Karavaev Y. L.
This paper is an experimental investigation of a round uniform disk rolling on a horizontal surface. Two methods for experimentally determining the loss of contact of the rolling disk from the horizontal surface before its stop are proposed. Results of experiments for disks having different masses and manufactured from different materials are presented. Causes of “microlosses of contact” detected in the processes of motion are discussed.
Keywords: | Euler’s disk, Loss of contact, Experiment |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Karavaev Y. L., On the loss of contact of the Euler disk, Nonlinear Dynamics, 2015, vol. 79, no. 4, |
DOI: | 10.1007/s11071-014-1811-5 |
Full text: | pdf (829.12 Kb) |
Impact-factor WoS (2022): | 5.600 (Q1) |
---|---|
ISSN (print): | 0924-090X |
ISSN (online): | 1573-269X |
Site: | http://link.springer.com/journal/11071 |
Citation: | Borisov A. V., Mamaev I. S., Equations of motion of non-holonomic systems, Russian Mathematical Surveys, 2015, vol. 70, no. 6, |
---|---|
DOI: | 10.4213/rm9691 |
Full text: | pdf (335.26 Kb) |
Impact-factor WoS (2022): | 0.900 (Q2) |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0036-0279 |
ISSN (online): | 1468-4829 |
Site: | http://www.mathnet.ru/umn |
Pivovarova E. N., Klekovkin A. V.
This paper presents an experimental investigation of the influence of rolling friction on the dynamics of a robot wheel. The robot is set in motion by changing the proper gyrostatic momentum using the controlled rotation of a rotor installed in the robot. The problem is considered under the assumption that the center of mass of the system does not coincide with its geometric center. In this paper we derive equations describing the dynamics of the system and give an example of the controlled motion of a wheel by specifying a constant angular acceleration of the rotor. A description of the design of the robot wheel is given and a method for experimentally determining the rolling friction coefficient is proposed. For the verification of the proposed mathematical model, experimental studies of the controlled motion of the robot wheel are carried out. We show that the theoretical results qualitatively agree with the experimental ones, but are quantitatively different.
Keywords: | robot-wheel, rolling friction, displacement of the center of mass |
---|---|
Citation: | Pivovarova E. N., Klekovkin A. V., Influence of rolling friction on the controlled motion of a robot wheel, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, |
DOI: | 10.20537/vm150414 |
Full text: | pdf (395.52 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
Klenov A. I., Vetchanin E. V., Kilin A. A.
This paper is concerned with the experimental determination of the added masses of bodies completely or partially immersed in a fluid. The paper presents an experimental setup, a technique of the experiment and an underlying mathematical model. The method of determining the added masses is based on the towing of the body with a given propelling force. It is known (from theory) that the concept of an added mass arises under the assumption concerning the potentiality of flow over the body. In this context, the authors have performed PIV visualization of flows generated by the towed body, and defined a part of the trajectory for which the flow can be considered as potential. For verification of the technique, a number of experiments have been performed to determine the added masses of a spheroid. The measurement results are in agreement with the known reference data. The added masses of a screwless freeboard robot have been defined using the developed technique.
Keywords: | added mass, movement on a free surface, hydrodynamic resistance, method of towing |
---|---|
Citation: | Klenov A. I., Vetchanin E. V., Kilin A. A., Experimental determination of the added masses by method of towing, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, |
DOI: | 10.20537/vm150413 |
Full text: | pdf (1.88 Mb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
The paper is devoted to the experimental verification of the Andersen–Pesavento–Wang model describing the falling of a heavy plate through a resisting medium. As a main research method the authors have used video filming of a falling plate with PIV measurement of the velocity of surrounding fluid flows. The trajectories of plates and streamlines were determined and oscillation frequencies were estimated using experimental results. A number of experiments for plates of various densities and sizes were performed. The trajectories of plates made of plastic are qualitatively similar to the trajectories predicted by the Andersen–Pesavento–Wang model. However, measured and computed frequencies of oscillations differ significantly. For a plate made of high carbon steel the results of experiments are quantitatively and qualitatively in disagreement with computational results.
Keywords: | PIV — Particle Image Velocimetry, Maxwell problem, model of Andersen–Pesavento–Wang |
---|---|
Citation: | Vetchanin E. V., Klenov A. I., Optical measurement of a fluid velocity field around a falling plate, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, |
DOI: | 10.20537/vm150412 |
Full text: | pdf (4.06 Mb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
Vetchanin E. V., Karavaev Y. L., Kalinkin A. A., Klekovkin A. V., Pivovarova E. N.
The paper is devoted to the development of a model of an underwater robot actuated by inner rotors. This design has no moving elements interacting with an environment, which minimizes a negative impact on it, and increases noiselessness of the robot motion in a liquid. Despite numerous discussions on the possibility and efficiency of motion by means of internal masses' movement, a large number of works published in recent years confirms a relevance of the research. The paper presents an overview of works aimed at studying the motion by moving internal masses. A design of a screwless underwater robot that moves by the rotation of inner rotors to conduct theoretical and experimental investigations is proposed. In the context of theoretical research a robot model is considered as a hollow ellipsoid with three rotors located inside so that the axes of their rotation are mutually orthogonal. For the proposed model of a screwless underwater robot equations of motion in the form of classical Kirchhoff equations are obtained.
Keywords: | mobile robot, screwless underwater robot, movement in ideal fluid |
---|---|
Citation: | Vetchanin E. V., Karavaev Y. L., Kalinkin A. A., Klekovkin A. V., Pivovarova E. N., A model of a screwless underwater robot, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, |
DOI: | 10.20537/vm150411 |
Full text: | pdf (308.24 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
Borisov A. V., Mamaev I. S., Kilin A. A., Bizyaev I. A.
This paper is concerned with the problem of the motion of a wheeled vehicle on a plane in the case where one of the wheel pairs is fixed. In addition, the motion of a wheeled vehicle on a plane in the case of two free wheel pairs is considered. A method for obtaining equations of motion for the vehicle with an arbitrary geometry is presented. Possible kinds of motion of the vehicle with a fixed wheel pair are determined.
Keywords: | nonholonomic constraint, system dynamics, wheeled vehicle, Chaplygin system |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Kilin A. A., Bizyaev I. A., Qualitative Analysis of the Dynamics of a Wheeled Vehicle, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, |
DOI: | 10.1134/S156035471506009X |
Full text: | pdf (445.93 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
In this paper, we develop the results obtained by J.Hadamard and G.Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasi-velocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a one-wheeled vehicle and a wheeled vehicle with two rotating wheel pairs.
Keywords: | nonholonomic constraint, wheeled vehicle, reduction, equations of motion |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., On the Hadamard – Hamel Problem and the Dynamics of Wheeled Vehicles, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, |
DOI: | 10.1134/S1560354715060106 |
Full text: | pdf (265.93 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Kilin A. A., Pivovarova E. N., Ivanova T. B.
This paper is concerned with free and controlled motions of a spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum. Equations of motion for the nonholonomic model are obtained and their first integrals are found. Fixed points of the reduced system are found in the absence of control actions. It is shown that they correspond to the motion of the spherical robot in a straight line and in a circle. A control algorithm for the motion of the spherical robot along an arbitrary trajectory is presented. A set of elementary maneuvers (gaits) is obtained which allow one to transfer the spherical robot from any initial point to any end point.
Keywords: | spherical robot, control, nonholonomic constraint, combined mechanism |
---|---|
Citation: | Kilin A. A., Pivovarova E. N., Ivanova T. B., Spherical Robot of Combined Type: Dynamics and Control, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, |
DOI: | 10.1134/S1560354715060076 |
Full text: | pdf (306.92 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Kazakov A. O.
In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems. We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.
Keywords: | Suslov problem, nonholonomic constraint, reversal, strange attractor |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Kazakov A. O., Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors, Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, |
DOI: | 10.1134/S1560354715050056 |
Full text: | pdf (640.12 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper is a review of the problem of the constructive reduction of nonholonomic systems with symmetries. The connection of reduction with the presence of the simplest tensor invariants (first integrals and symmetry fields) is shown. All theoretical constructions are illustrated by examples encountered in applications. In addition, the paper contains a short historical and critical sketch covering the contribution of various researchers to this problem.
Keywords: | reduction, symmetry, tensor invariant, first integral, symmetry group, symmetry field, nonholonomic constraint, Noether theorem |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Symmetries and Reduction in Nonholonomic Mechanics, Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, |
DOI: | 10.1134/S1560354715050044 |
Full text: | pdf (539.38 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Karavaev Y. L., Mamaev I. S., Erdakova N. N., Ivanova T. B., Tarasov V. V.
In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.
Keywords: | dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law |
---|---|
Citation: | Borisov A. V., Karavaev Y. L., Mamaev I. S., Erdakova N. N., Ivanova T. B., Tarasov V. V., Experimental Investigation of the Motion of a Body with an Axisymmetric Base Sliding on a Rough Plane, Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, |
DOI: | 10.1134/S1560354715050020 |
Full text: | pdf (516.92 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper addresses the dynamics of systems with servoconstraints where the constraints are realized by controlling the inertial properties of the system. Vakonomic systems are a particular case. Special attention is given to the motion on Lie groups with left-invariant kinetic energy and a left-invariant constraint. The presence of symmetries allows the dynamical equations to be reduced to a closed system of differential equations with quadratic right-hand sides. As the main example, we consider the rotation of a rigid body with a left-invariant servoconstraint, which implies that the projection of the body’s angular velocity on some body-fixed direction is zero.
Keywords: | servoconstraints, symmetries, Lie groups, left-invariant constraints, systems with quadratic right-hand sides, vakonomic systems |
---|---|
Citation: | Kozlov V. V., The Dynamics of Systems with Servoconstraints. II, Regular and Chaotic Dynamics, 2015, vol. 20, no. 4, |
DOI: | 10.1134/S1560354715040012 |
Full text: | pdf (861.95 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Bizyaev I. A.
In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.
Keywords: | nonholonomic constraint, Jacobi integral, Chaplygin sleigh, rotating table, Suslov problem |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Bizyaev I. A., The Jacobi Integral in Nonholonomic Mechanics, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, |
DOI: | 10.1134/S1560354715030107 |
Full text: | pdf (990.04 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The paper discusses the dynamics of systems with Béghin’s servoconstraints where the constraints are realized by means of controlled forces. Classical nonholonomic systems are an important particular case. Special attention is given to the study of motion on Lie groups with left-invariant kinetic energy and left-invariant constraints. The presence of symmetries allows one to reduce the dynamic equations to a closed system of differential equations with quadratic right-hand sides on a Lie algebra. Examples are given which include the rotation of a rigid body with a left-invariant servoconstraint — the projection of the angular velocity onto some direction fixed in the body is equal to zero (a generalization of the nonholonomic Suslov problem) — and the motion of the Chaplygin sleigh with servoconstraints of a certain type. The dynamics of systems with Béghin’s servoconstraints is richer and more varied than the more usual dynamics of nonholonomic systems.
Keywords: | servoconstraints, symmetries, Lie groups, left-invariant constraints, systems with quadratic right-hand sides |
---|---|
Citation: | Kozlov V. V., The Dynamics of Systems with Servoconstraints. I, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, |
DOI: | 10.1134/S1560354715030016 |
Full text: | pdf (810.79 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
A nonholonomic model of the dynamics of an omniwheel vehicle on a plane and a sphere is considered. A derivation of equations is presented and the dynamics of a free system are investigated. An explicit motion control algorithm for the omniwheel vehicle moving along an arbitrary trajectory is obtained.
Keywords: | omniwheel, roller-bearing wheel, nonholonomic constraint, dynamical system, invariant measure, integrability, controllability |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Dynamics and Control of an Omniwheel Vehicle, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, |
DOI: | 10.1134/S1560354715020045 |
Full text: | pdf (1.11 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper deals with the problem of a spherical robot propelled by an internal omniwheel platform and rolling without slipping on a plane. The problem of control of spherical robot motion along an arbitrary trajectory is solved within the framework of a kinematic model and a dynamic model. A number of particular cases of motion are identified, and their stability is investigated. An algorithm for constructing elementary maneuvers (gaits) providing the transition from one steady-state motion to another is presented for the dynamic model. A number of experiments have been carried out confirming the adequacy of the proposed kinematic model.
Keywords: | spherical robot, kinematic model, dynamic model, nonholonomic constraint, omniwheel |
---|---|
Citation: | Karavaev Y. L., Kilin A. A., The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, |
DOI: | 10.1134/S1560354715020033 |
Full text: | pdf (1.38 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper is a review of the problem of the constructive reduction of nonholonomic systems with symmetries. The connection of reduction with the presence of the simplest tensor invariants (first integrals and symmetry fields) is shown. All theoretical constructions are illustrated by examples encountered in applications. In addition, the paper contains a short historical and critical sketch covering the contribution of various researchers to this problem.
Keywords: | reduction, symmetry, tensor invariant, first integral, symmetry group, symmetry field, nonholonomic constraint, Noether theorem |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Symmetries and Reduction in Nonholonomic Mechanics, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, |
Full text: | pdf (909.32 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bizyaev I. A., Bolsinov A. V., Borisov A. V., Mamaev I. S.
This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie –Poisson bracket of rank 2. This Lie – Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.
Keywords: | nonholonomic hinge, topology, bifurcation diagram, tensor invariants, Poisson bracket, stability |
---|---|
Citation: | Bizyaev I. A., Bolsinov A. V., Borisov A. V., Mamaev I. S., Topology and Bifurcations in Nonholonomic Mechanics, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, |
Full text: | pdf (561.73 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
This paper presents the results of experimental investigations for the rolling of a spherical robot of combined type actuated by an internal wheeled vehicle with rotor on a horizontal plane. The control of spherical robot based on nonholonomic dynamical by means of gaits. We consider the motion of the spherical robot in case of constant control actions, as well as impulse control. A number of experiments have been carried out confirming the importance of rolling friction.
Keywords: | spherical robot of combined type, dynamic model, control by means of gaits, rolling friction |
---|---|
Citation: | Kilin A. A., Karavaev Y. L., Experimental research of dynamic of spherical robot of combined type, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, |
Full text: | pdf (761.73 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
In this paper we consider the problem of motion of a rigid body in an ideal fluid with two material points moving along circular trajectories. The controllability of this system on the zero level set of first integrals is shown. Elementary “gaits” are presented which allow the realization of the body’s motion from one point to another. The existence of obstacles to a controlled motion of the body along an arbitrary trajectory is pointed out.
Keywords: | ideal fluid, Kirchhoff equations, controllability of gaits |
---|---|
Citation: | Kilin A. A., Vetchanin E. V., The contol of the motion through an ideal fluid of a rigid body by means of two moving masses, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, |
Full text: | pdf (413.58 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
This paper addresses the dynamics of systems with servoconstraints where the constraints are realized by controlling the inertial properties of the system. Vakonomic systems are a particular case. Special attention is given to the motion on Lie groups with left-invariant kinetic energy and a left-invariant constraint. The presence of symmetries allows the dynamical equations to be reduced to a closed system of differential equations with quadratic right-hand sides. As the main example, we consider the rotation of a rigid body with a left-invariant servo-constraint, which implies that the projection of the body’s angular velocity on some body-fixed direction is zero.
Keywords: | servoconstraints, symmetries, Lie groups, left-invariant constraints, systems with quadratic right-hand sides, vakonomic systems |
---|---|
Citation: | Kozlov V. V., The dynamics of systems with servoconstraints. II, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 3, |
Full text: | pdf (560.42 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Karavaev Y. L., Mamaev I. S., Erdakova N. N., Ivanova T. B., Tarasov V. V.
In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.
Keywords: | dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law |
---|---|
Citation: | Borisov A. V., Karavaev Y. L., Mamaev I. S., Erdakova N. N., Ivanova T. B., Tarasov V. V., On the dynamics of a body with an axisymmetric base sliding on a rough plane, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 3, |
Full text: | pdf (2.38 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Bizyaev I. A.
In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.
Keywords: | nonholonomic constraint, Jacobi integral, Chaplygin sleigh, rotating table, Suslov problem |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Bizyaev I. A., The Jacobi Integral in NonholonomicMechanics, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 2, |
Full text: | pdf (1.9 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The paper discusses the dynamics of systems with Béghin’s servoconstraints where the constraints are realized by means of controlled forces. Classical nonholonomic systems are an important particular case. Special attention is given to the study of motion on Lie groups with left-invariant kinetic energy and left-invariant constraints. The presence of symmetries allows one to reduce the dynamic equations to a closed system of differential equations with quadratic right-hand sides on a Lie algebra. Examples are given which include the rotation of a rigid body with a left-invariant servoconstraint — the projection of the angular velocity onto some direction fixed in the body is equal to zero (a generalization of the nonholonomic Suslov problem) — and the motion of the Chaplygin sleigh with servoconstraints of a certain type. The dynamics of systems with Béghin’s servoconstraints is richer and more varied than the more usual dynamics of nonholonomic systems.
Keywords: | servoconstraints, symmetries, Lie groups, left-invariant constraints, systems with quadratic right-hand sides |
---|---|
Citation: | Kozlov V. V., The dynamics of systems with servoconstraints. I, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 2, |
Full text: | pdf (505.17 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The dynamic model for a spherical robot with an internal omniwheel platform is presented. Equations of motion and first integrals according to the non-holonomic model are given. We consider particular solutions and their stability. The algorithm of control of spherical robot for movement along a given trajectory are presented.
Keywords: | spherical robot, dynamical model, non-holonomic constraint, omniwheel, stability |
---|---|
Citation: | Karavaev Y. L., Kilin A. A., The dynamic of a spherical robot with an internal omniwheel platform, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 1, |
Full text: | pdf (530.7 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
It is well known that in the Béghin– Appel theory servo-constraints are realized using controlled external forces. In this paper an expansion of the Béghin–Appel theory is given in the case where servo-constraints are realized using controlled change of the inertial properties of a dynamical system. The analytical mechanics of dynamical systems with servo-constraints of general form is discussed. The key principle of the approach developed is to appropriately determine virtual displacements of systems with constraints.
Keywords: | servo-constraints, d’Alembert–Lagrange principle, virtual displacements, Gauss principle, Noether theorem |
---|---|
Citation: | Kozlov V. V., Principles of dynamics and servo-constraints, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 1, |
Full text: | pdf (316.31 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bolsinov A. V., Borisov A. V., Mamaev I. S.
We develop the reducing multiplier theory for a special class of nonholonomic dynamical systems and show that the non-linear Poisson brackets naturally obtained in the framework of this approach are all isomorphic to the Lie–Poisson e(3)-bracket. As two model examples, we consider the Chaplygin ball problem on the plane and the Veselova system. In particular, we obtain an integrable gyrostatic generalisation of the Veselova system.
Keywords: | nonholonomic system, Chaplygin ball, Hamiltonisation, Poisson bracket |
---|---|
Citation: | Bolsinov A. V., Borisov A. V., Mamaev I. S., Geometrisation of Chaplygin's reducing multiplier theorem, Nonlinearity, 2015, vol. 28, no. 7, |
DOI: | 10.1088/0951-7715/28/7/2307 |
Full text: | pdf (156.41 Kb) |
Impact-factor WoS (2022): | 1.700 (Q2) |
---|---|
ISSN (print): | 0951-7715 |
ISSN (online): | 1361-6544 |
Site: | http://iopscience.iop.org/0951-7715 |
Bizyaev I. A., Bolsinov A. V., Borisov A. V., Mamaev I. S.
This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie–Poisson bracket of rank two. This Lie–Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.
Keywords: | Nonholonomic hinge, topology, bifurcation diagram, tensor invariants, Poisson bracket, stability |
---|---|
Citation: | Bizyaev I. A., Bolsinov A. V., Borisov A. V., Mamaev I. S., Topology and Bifurcations in Nonholonomic Mechanics, International Journal of Bifurcation and Chaos, 2015, vol. 25, no. 10, 1530028, |
DOI: | 10.1142/S0218127415300281 |
Full text: | pdf (645.53 Kb) |
Impact-factor WoS (2022): | 2.200 (Q2) |
---|---|
ISSN (print): | 0218-1274 |
ISSN (online): | 1793-6551 |
Site: | http://www.worldscientific.com/worldscinet/ijbc |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
In this paper, we develop the method of Chaplygin’s reducing multiplier; using this method, we obtain a conformally Hamiltonian representation for three nonholonomic systems, namely, for the nonholonomic oscillator, for the Heisenberg system, and for the Chaplygin sleigh. Furthermore, in the case of oscillator and nonholonomic Chaplygin sleigh, we show that the problem reduces to the study of motion of a mass point (in a potential field) on a plane and, in the case of Heisenberg system, on the sphere. Moreover, we consider an example of a nonholonomic system (suggested by Blackall) to which one cannot apply the method of reducing multiplier.
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., Hamiltonization of Elementary Nonholonomic Systems, Russian Journal of Mathematical Physics, 2015, vol. 22, no. 4, |
---|---|
DOI: | 10.1134/S1061920815040032 |
Full text: | pdf (115.49 Kb) |
Impact-factor WoS (2022): | 1.400 (Q3) |
---|---|
ISSN (print): | 1061-9208 |
ISSN (online): | 1555-6638 |
Site: | http://link.springer.com/journal/11503 |
Citation: | Bizyaev I. A., Nonintegrability and Obstructions to the Hamiltonianization of a Nonholonomic Chaplygin Top, Doklady Mathematics, 2014, vol. 90, no. 2, |
---|---|
DOI: | 10.1134/S1064562414060192 |
Full text: | pdf (203.47 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
Borisov A. V., Mamaev I. S., Tsiganov A. V.
Nonholonomic systems describing the rolling of a rigid body on a plane and their relationship with various Poisson structures are considered. The notion of generalized conformally Hamiltonian representation of dynamical systems is introduced. In contrast to linear Poisson structures defined by Lie algebras and used in rigid-body dynamics, the Poisson structures of nonholonomic systems turn out to be nonlinear. They are also degenerate and the Casimir functions for them can be expressed in terms of complicated transcendental functions or not appear at all.
Keywords: | Poisson bracket, nonholonomic system, Poisson structure, dynamical system, con- formally Hamiltonian representation, Casimir function, Routh sphere, rolling of a Chaplygin ball |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Tsiganov A. V., On the Nonlinear Poisson Bracket Arising in Nonholonomic Mechanics , Mathematical Notes, 2014, vol. 95, no. 3, |
DOI: | 10.1134/S0001434614030031 |
Full text: | pdf (509.63 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,502 |
ISSN (print): | 0025-567X |
ISSN (online): | 2305-2880 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=mzm&option_lang=eng |
Borisov A. V., Mamaev I. S., Tsiganov A. V.
This is a survey of basic facts presently known about non-linear Poisson structures in the analysis of integrable systems in non-holonomic mechanics. It is shown that by using the theory of Poisson deformations it is possible to reduce various non-holonomic systems to dynamical systems on well-understood phase spaces equipped with linear Lie-Poisson brackets. As a result, not only can different non-holonomic systems be compared, but also fairly advanced methods of Poisson geometry and topology can be used for investigating them.
Keywords: | non-holonomic systems, Poisson bracket, Chaplygin ball, Suslov system, Veselova system |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Tsiganov A. V., Non-holonomic dynamics and Poisson geometry , Russian Mathematical Surveys, 2014, vol. 69, no. 3, |
DOI: | 10.1070/RM2014v069n03ABEH004899 |
Full text: | pdf (917.58 Kb) |
Impact-factor WoS (2022): | 0.900 (Q2) |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0036-0279 |
ISSN (online): | 1468-4829 |
Site: | http://www.mathnet.ru/umn |
This paper reviews the results of stability analysis for polygonal configurations of a point vortex system in an annular region depending on the ratio of the inner and outer radii of the annulus. Conditions are found for linear stability of Thomsonʼs configurations for the case N<7. The paper also shows that a system of two vortices between parallel walls is a limiting case of a two-vortex system in an annular region, as the radii of the annulus tend to infinity.
Citation: | Erdakova N. N., Mamaev I. S., On the dynamics of point vortices in an annular region , Fluid Dynamics Research, 2014, vol. 46, no. 3, 031420, |
---|---|
DOI: | 10.1088/0169-5983/46/3/031420 |
Full text: | pdf (205.4 Kb) |
Impact-factor WoS (2022): | 1.500 (Q4) |
---|---|
ISSN (print): | 0169-5983 |
ISSN (online): | 1873-7005 |
Site: | http://iopscience.iop.org/1873-7005/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
In this paper, the integrability of the equations of a system of three vortex sources is shown. A reduced system describing, up to similarity, the evolution of the system’s configurations is obtained. Possible phase portraits and various relative equilibria of the system are presented.
Keywords: | integrability, vortex sources, shape sphere, reduction, homothetic configurations |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., The Dynamics of Three Vortex Sources, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, |
DOI: | 10.1134/S1560354714060070 |
Full text: | pdf (244.27 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Erdakova N. N., Ivanova T. B., Mamaev I. S.
In this paper we investigate the dynamics of a body with a flat base sliding on a horizontal and inclined rough plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. For analysis we use the descriptive function method similar to the methods used in the problems of Hamiltonian dynamics with one degree of freedom and allowing a qualitative analysis of the system to be made without explicit integration of equations of motion. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.
Keywords: | dry friction, linear pressure distribution, planar motion, Coulomb law |
---|---|
Citation: | Borisov A. V., Erdakova N. N., Ivanova T. B., Mamaev I. S., The Dynamics of a Body with an Axisymmetric Base Sliding on a Rough Plane, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, |
DOI: | 10.1134/S1560354714060021 |
Full text: | pdf (965.59 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper is concerned with the problem of first integrals of the equations of geodesics on two-dimensional surfaces that are rational in the velocities (or momenta). The existence of nontrivial rational integrals with given values of the degrees of the numerator and the denominator is proved using the Cauchy–Kovalevskaya theorem.
Keywords: | conformal coordinates, rational integral, irreducible integrals, Cauchy–Kovalevskaya theorem |
---|---|
Citation: | Kozlov V. V., On Rational Integrals of Geodesic Flows, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, |
DOI: | 10.1134/S156035471406001X |
Full text: | pdf (145.78 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
In this paper we consider superintegrable systems which are an immediate generalization of the Kepler and Hook problems, both in two-dimensional spaces — the plane R2 and the sphere S2 — and in three-dimensional spaces R3 and S3. Using the central projection and the reduction procedure proposed in [21], we show an interrelation between the superintegrable systems found previously and show new ones. In all cases the superintegrals are presented in explicit form.
Keywords: | superintegrable systems, Kepler and Hook problems, isomorphism, central projection, reduction, highest degree polynomial superintegrals |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., Superintegrable Generalizations of the Kepler and Hook Problems, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, |
DOI: | 10.1134/S1560354714030095 |
Full text: | pdf (300.95 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
In this paper we investigate two systems consisting of a spherical shell rolling without slipping on a plane and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is attached to the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of a nonholonomic hinge. Equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler–Jacobi–Lie theorem, which is a new integration mechanism in nonholonomic mechanics. We also consider the problem of free motion of a bundle of two bodies connected by means of a nonholonomic hinge. For this system, integrable cases and various tensor invariants are found.
Keywords: | nonholonomic constraint, tensor invariants, isomorphism, nonholonomic hinge |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, |
DOI: | 10.1134/S156035471402004X |
Full text: | pdf (241.48 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension ⩾n. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.
Keywords: | integrability by quadratures, adjoint system, Hamiltonian equations, Euler–Jacobi theorem, Lie theorem, symmetries |
---|---|
Citation: | Kozlov V. V., Remarks on Integrable Systems, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, |
DOI: | 10.1134/S1560354714020014 |
Full text: | pdf (186.79 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Ivanova T. B., Pivovarova E. N.
In this paper we consider the control of a dynamically asymmetric balanced ball on a plane in the case of slipping at the contact point. Necessary conditions under which a control is possible are obtained. Specific algorithms of control along a given trajectory are constructed.
Keywords: | control, dry friction, Chaplygin’s ball, spherical robot |
---|---|
Citation: | Ivanova T. B., Pivovarova E. N., Comments on the Paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev "How to Control the Chaplygin Ball Using Rotors. II", Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, |
DOI: | 10.1134/S1560354714010092 |
Full text: | pdf (170.77 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
In this paper we consider the dynamics of a rigid body with a sharp edge in contact with a rough plane. The body can move so that its contact point is fixed or slips or loses contact with the support. In this paper, the dynamics of the system is considered within three mechanical models which describe different regimes of motion. The boundaries of the domain of definition of each model are given, the possibility of transitions from one regime to another and their consistency with different coefficients of friction on the horizontal and inclined surfaces are discussed.
Keywords: | rod, Painlevé paradox, dry friction, loss of contact, frictional impact |
---|---|
Citation: | Mamaev I. S., Ivanova T. B., The Dynamics of a Rigid Body with a Sharp Edge in Contact with an Inclined Surface in the Presence of Dry Friction, Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, |
DOI: | 10.1134/S1560354714010080 |
Full text: | pdf (735.75 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The kinematic control model for a spherical robot with an internal omniwheel platform is presented. We consider singularities of control of spherical robot with an unbalanced internal omniwheel platform. The general algorithm of control of spherical robot according to the kinematical quasi-static model and controls for simple trajectories (a straight line and in a circle) are presented. Experimental investigations have been carried out for all introduced control algorithms.
Keywords: | spherical robot, kinematic model, nonholonomic constraint, omniwheel, displacement of center of mass |
---|---|
Citation: | Kilin A. A., Karavaev Y. L., The kinematic control model for a spherical robot with an unbalanced internal omniwheel platform, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, |
Full text: | pdf (986.08 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Erdakova N. N., Ivanova T. B., Mamaev I. S.
In this paper we investigate the dynamics of a body with a flat base sliding on a inclined plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. Computer-aided analysis of the system’s dynamics on the inclined plane using phase portraits has allowed us to reveal dynamical effects that have not been found earlier.
Keywords: | dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law |
---|---|
Citation: | Borisov A. V., Erdakova N. N., Ivanova T. B., Mamaev I. S., On the dynamics of a body with an axisymmetric base sliding on a rough plane, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, |
Full text: | pdf (667.35 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The problem of motion of a vehicle in the form of a platform with an arbitrary number of Mecanum wheels fastened on it is considered. The controllability of this vehicle is discussed within the framework of the nonholonomic rolling model. An explicit algorithm is presented for calculating the control torques of the motors required to follow an arbitrary trajectory. Examples of controls for executing the simplest maneuvers are given.
Keywords: | omniwheel, roller bearing wheel, nonholonomic constraint, dynamical system, integrability, controllability |
---|---|
Citation: | Kilin A. A., Bobykin A. D., Control of a Vehicle with Omniwheels on a Plane, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, |
Full text: | pdf (520.34 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
This paper is concerned with the problem of first integrals of the equations of geodesics on twodimensional surfaces that are rational in the velocities (or momenta). The existence of nontrivial rational integrals with given values of the degrees of the numerator and the denominator is proved using the Cauchy–Kovalevskaya theorem.
Keywords: | conformal coordinates, rational integral, irreducible integrals, Cauchy–Kovalevskaya theorem |
---|---|
Citation: | Kozlov V. V., On Rational Integrals of Geodesic Flows, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, |
Full text: | pdf (302.05 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
This paper discusses new unresolved problems of nonholonomic mechanics. Hypotheses of the possibility of Hamiltonization and the existence of an invariant measure for such systems are advanced.
Keywords: | nonholonomic mechanics, tensor invariant, invariant measure, Poisson structure |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Invariant Measure and Hamiltonization of Nonholonomic Systems, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, |
Full text: | pdf (283.75 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Vetchanin E. V., Kazakov A. O.
This paper is concerned with the dynamics of two point vortices of the same intensity which are affected by an acoustic wave. Typical bifurcations of fixed points have been identified by constructing charts of dynamical regimes, and bifurcation diagrams have been plotted.
Keywords: | point vortices, nonintegrability, bifurcations, chart of dynamical regimes |
---|---|
Citation: | Vetchanin E. V., Kazakov A. O., Bifurcations and chaos in the problem of the motion of two point vortices in an acoustic wave, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, |
Full text: | pdf (5.62 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
In this paper, the integrability of the equations of a system of three vortex sources is shown. A reduced system describing, up to similarity, the evolution of the system’s configurations is obtained. Possible phase portraits and various relative equilibria of the system are presented.
Keywords: | integrability, vortex sources, shape sphere, reduction, homothetic configurations |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., The dynamics of three vortex sources, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, |
Full text: | pdf (413.32 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
This paper is concerned with a three-body system on a straight line in a potential field proposed by Tsiganov. The Liouville integrability of this system is shown. Reduction and separation of variables are performed.
Keywords: | Calogero systems, reduction, integrable systems, Jacobi problem |
---|---|
Citation: | Bizyaev I. A., On a generalization of systems of Calogero type, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 2, |
Full text: | pdf (237.84 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Ivanova T. B., Pivovarova E. N.
In this paper we consider the control of a dynamically asymmetric balanced ball on a plane in the case of slipping at the contact point. Necessary conditions under which a control is possible are obtained. Specific algorithms of control along a given trajectory are constructed.
Keywords: | control, dry friction, Chaplygin’s ball, spherical robot |
---|---|
Citation: | Ivanova T. B., Pivovarova E. N., Comment on the paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev “How to control the Chaplygin ball using rotors. II”, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, |
Full text: | pdf (234.11 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Kilin A. A., Karavaev Y. L., Klekovkin A. V.
In this article a kinematic model of the spherical robot is considered, which is set in motion by the internal platform with omni-wheels. It has been introduced a description of construction, algorithm of trajectory planning according to developed kinematic model, it has been realized experimental research for typical trajectories: moving along a straight line and moving along a circle.
Keywords: | spherorobot, kinematic model, non-holonomic constraint, omni-wheel |
---|---|
Citation: | Kilin A. A., Karavaev Y. L., Klekovkin A. V., Kinematic control of a high manoeuvrable mobile spherical robot with internal omni-wheeled platform, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, |
Full text: | pdf (3.43 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with density stratification and a steady-state velocity field. As in the classical setting, it is assumed that the figure or its layers uniformly rotate about an axis fixed in space. As is well known, when there is no rotation, only a ball can be a figure of equilibrium.
It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with inherent constant angular velocity, is at equilibrium. Expressions are obtained for the gravitational potential, change in the angular velocity and pressure, and the conclusion is drawn that the angular velocity on the outer surface is the same as that of the Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification.
We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis.
Keywords: | self-gravitating fluid, confocal stratification, homothetic stratification, space of constant curvature |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., Figures of equilibrium of an inhomogeneous self-gravitating fluid, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, |
Full text: | pdf (492.78 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Alalykin S. S., Bogatyrev A. V., Ivanova T. B., Pivovarova E. N.
In this paper we describe an inertiameter, which is an experimental facility for determining the inertia tensor components and the position of the center of mass of compound bodies. An algorithm for determining these dynamical properties is presented. Using the algorithm obtained, the displacement of the center of mass and the tensor of inertia are determined experimentally for a spherical robot of combined type.
Keywords: | inertiameter, spherical robot, moment of inertia, center of mass |
---|---|
Citation: | Alalykin S. S., Bogatyrev A. V., Ivanova T. B., Pivovarova E. N., Determination of moments of inertia and the position of the center of mass of robotic devices, Bulletin of Udmurt University. Physics and Chemistry, 2014, no. 4, |
Full text: | pdf (311.48 Kb) |
ISSN (print): | 1810-5505 |
---|---|
ISSN (online): | 1999-8597 |
Site: | http://ru.vestnik.udsu.ru/about |
Borisov A. V., Kilin A. A., Mamaev I. S., Tenenev V. A.
We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible and viscous fluid. Using the numerical simulation of the Navier–Stokes equations, we confirm the existence of leapfrogging of three equal vortex rings and suggest the possibility of detecting it experimentally. We also confirm the existence of leapfrogging of two vortex rings with opposite-signed vorticities in a viscous fluid.
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Tenenev V. A., The dynamics of vortex rings: leapfrogging in an ideal and viscous fluid , Fluid Dynamics Research, 2014, vol. 46, no. 3, 031415, |
---|---|
DOI: | 10.1088/0169-5983/46/3/031415 |
Full text: | pdf (746.69 Kb) |
Impact-factor WoS (2022): | 1.500 (Q4) |
---|---|
ISSN (print): | 0169-5983 |
ISSN (online): | 1873-7005 |
Site: | http://iopscience.iop.org/1873-7005/ |
We discuss an embedding of a vector field for the nonholonomic Routh sphere into a subgroup of commuting Hamiltonian vector fields on six-dimensional phase space. The corresponding Poisson brackets are reduced to the canonical Poisson brackets on the Lie algebra e∗(3). It allows us to relate the nonholonomic Routh system with the Hamiltonian system on a cotangent bundle to the sphere with a canonical Poisson structure.
Citation: | Bizyaev I. A., Tsiganov A. V., On the Routh sphere problem , Journal of Physics A: Mathematical and Theoretical, 2013, vol. 46, no. 8, 085202, |
---|---|
DOI: | 10.1088/1751-8113/46/8/085202 |
Full text: | pdf (169.11 Kb) |
Impact-factor WoS (2022): | 2.100 (Q1) |
---|---|
Impact-factor RSCI (2022): | 0.718 (Q2) |
ISSN (print): | 1751-8113 |
ISSN (online): | 1751-8121 |
Site: | http://iopscience.iop.org/1751-8121/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of the reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point.
Keywords: | nonholonomic constraint, absolute dynamics, bifurcation diagram, bifurcation complex, drift, resonance, invariant torus |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., The Problem of Drift and Recurrence for the Rolling Chaplygin Ball, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, |
DOI: | 10.1134/S1560354713060166 |
Full text: | pdf (702.93 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
We consider the problem of rolling of a ball with an ellipsoidal cavity filled with an ideal fluid, which executes a uniform vortex motion, on an absolutely rough plane. We point out the case of existence of an invariant measure and show that there is a particular case of integrability under conditions of axial symmetry.
Keywords: | vortex motion, nonholonomic constraint, Chaplygin ball, invariant measure, integrability, rigid body, ideal fluid |
---|---|
Citation: | Borisov A. V., Mamaev I. S., The Dynamics of the Chaplygin Ball with a Fluid-filled Cavity, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, |
DOI: | 10.1134/S156035471305002X |
Full text: | pdf (405.38 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
A new integrable system describing the rolling of a rigid body with a spherical cavity on a spherical base is considered. Previously the authors found the separation of variables for this system on the zero level set of a linear (in angular velocity) first integral, whereas in the general case it is not possible to separate the variables. In this paper we show that the foliation into invariant tori in this problem is equivalent to the corresponding foliation in the Clebsch integrable system in rigid body dynamics (for which no real separation of variables has been found either). In particular, a fixed point of focus type is possible for this system, which can serve as a topological obstacle to the real separation of variables.
Keywords: | integrable system, bifurcation diagram, conformally Hamiltonian system, bifurcation, Liouville foliation, critical periodic solution |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere, Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, |
DOI: | 10.1134/S1560354713040035 |
Full text: | pdf (488.37 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper addresses a class of problems associated with the conditions for exact integrability of systems of ordinary differential equations expressed in terms of the properties of tensor invariants. The general theorem of integrability of the system of n differential equations is proved, which admits n−2 independent symmetry fields and an invariant volume n-form (integral invariant). General results are applied to the study of steady motions of a continuum with infinite conductivity.
Keywords: | symmetry field, integral invariant, nilpotent group, magnetic hydrodynamics |
---|---|
Citation: | Kozlov V. V., The Euler–Jacobi–Lie Integrability Theorem, Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, |
DOI: | 10.1134/S1560354713040011 |
Full text: | pdf (377.18 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Bizyaev I. A.
In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body’s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior.
Keywords: | nonholonomic constraint, tensor invariant, first integral, invariant measure, integrability, conformally Hamiltonian system, rubber rolling, reversible, involution |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Bizyaev I. A., The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere, Regular and Chaotic Dynamics, 2013, vol. 18, no. 3, |
DOI: | 10.1134/S1560354713030064 |
Full text: | pdf (2.69 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
In our earlier paper [3] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with no-slip condition at the point of contact. In this paper we investigate the controllability of a ball in the presence of friction. We also study the problem of the existence and stability of singular dissipation-free periodic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed.
Keywords: | non-holonomic constraint, control, dry friction, viscous friction, stability, periodic solutions |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., How to Control the Chaplygin Ball Using Rotors. II, Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, |
DOI: | 10.1134/S1560354713010103 |
Full text: | pdf (1.73 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Vetchanin E. V., Mamaev I. S., Tenenev V. A.
An investigation of the characteristics of motion of a rigid body with variable internal mass distribution in a viscous fluid is carried out on the basis of a joint numerical solution of the Navier–Stokes equations and equations of motion for a rigid body. A nonstationary three-dimensional solution to the problem is found. The motion of a sphere and a drop-shaped body in a viscous fluid in a gravitational field, which is caused by the motion of internal material points, is explored. The possibility of self-propulsion of a body in an arbitrary given direction is shown.
Keywords: | finite-volume numerical method, Navier–Stokes equations, variable internal mass distribution, motion control |
---|---|
Citation: | Vetchanin E. V., Mamaev I. S., Tenenev V. A., The Self-propulsion of a Body with Moving Internal Masses in a Viscous Fluid, Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, |
DOI: | 10.1134/S1560354713010073 |
Full text: | pdf (1.71 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a method for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions where the relative distances remain finite (we call them pseudo-leapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.
Keywords: | ideal fluid, vortex ring, leapfrogging motion of vortex rings, bifurcation complex, periodic solution, integrability, chaotic dynamics |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem, Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, |
DOI: | 10.1134/S1560354713010036 |
Full text: | pdf (857.35 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of a reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point.
Keywords: | nonholonomic constraint, absolute dynamics, bifurcation diagram, bifurcation complex, drift, resonance, invariant torus |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., The problem of drift and recurrence for the rolling Chaplygin ball, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 4, |
Full text: | pdf (875.6 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bolsinov A. V., Borisov A. V., Mamaev I. S.
This paper develops the theory of the reducing multiplier for a special class of nonholonomic dynamical systems, when the resulting nonlinear Poisson structure is reduced to the Lie–Poisson bracket of the algebra e(3). As an illustration, the Chaplygin ball rolling problem and the Veselova system are considered. In addition, an integrable gyrostatic generalization of the Veselova system is obtained.
Keywords: | nonholonomic dynamical system, Poisson bracket, Poisson structure, reducing multiplier, Hamiltonization, conformally Hamiltonian system, Chaplygin ball |
---|---|
Citation: | Bolsinov A. V., Borisov A. V., Mamaev I. S., Geometrization of the Chaplygin reducing-multiplier theorem, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 4, |
Full text: | pdf (373.67 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
In this paper we consider the dynamics of rigid body whose sharp edge is in contact with a rough plane. The body can move so that its contact point does not move or slips or loses touch with the support. In this paper, the dynamics of the system is considered within three mechanical models that describe different modes of motion. The boundaries of definition range of each model are given, the possibility of transitions from one mode to another and their consistency with different coefficients of friction on the horizontal and inclined surfaces is discussed.
Keywords: | rod, Painlevé paradox, dry friction, separation, frictional impact |
---|---|
Citation: | Mamaev I. S., Ivanova T. B., The dynamics of rigid body whose sharp edge is in contact with a inclined surface with dry friction, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, |
Full text: | pdf (886.59 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bizyaev I. A., Borisov A. V., Mamaev I. S.
In this paper we investigate two systems consisting of a spherical shell rolling on a plane without slipping and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is fixed at the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of the nonholonomic hinge. The equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler–Jacobi–Lie theorem, which is a new integration mechanism in nonholonomic mechanics.
Keywords: | nonholonomic constraint, tensor invariants, isomorphism, nonholonomic hinge |
---|---|
Citation: | Bizyaev I. A., Borisov A. V., Mamaev I. S., The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, |
Full text: | pdf (441.83 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
In this paper we investigate the dynamics of a body with a flat base sliding on a horizontal plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model.
For analysis we use the descriptive function method similar to the methods used in the problems of Hamiltonian dynamics with one degree of freedom and allowing a qualitative analysis of the system to be made without explicit integration of equations of motion. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.
Keywords: | dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law |
---|---|
Citation: | Erdakova N. N., Mamaev I. S., On the dynamics of a body with an axisymmetric base sliding on a rough plane, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, |
Full text: | pdf (612.94 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Ivanova T. B., Pivovarova E. N.
This paper investigates the possibility of the motion control of a ball with a pendulum mechanism with non-holonomic constraints using gaits — the simplest motions such as acceleration and deceleration during the motion in a straight line, rotation through a given angle and their combination. Also, the controlled motion of the system along a straight line with a constant acceleration is considered. For this problem the algorithm for calculating the control torques is given and it is shown that the resulting reduced system has the first integral of motion.
Keywords: | non-holonomic constraint, control, spherical shell, integral of motion |
---|---|
Citation: | Ivanova T. B., Pivovarova E. N., Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, |
Full text: | pdf (582.48 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Karavaev Y. L.
The paper presents experimental investigation of a homogeneous circular disk rolling on a horizontal plane. In this paper two methods of experimental determination of the loss of contact between the rolling disk and the horizontal surface before the abrupt halt are proposed. Experimental results for disks of different masses and different materials are presented. The reasons for “micro losses” of contact with surface revealed during the rolling are discussed.
Keywords: | Euler disk, loss of contact, experiment |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Karavaev Y. L., On the loss of contact of the Euler disk, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, |
Full text: | pdf (362.06 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space which permit the algebra of symmetry fields of dimension ⩾n. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momentums in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.
Keywords: | integrability by quadratures, adjoint system, Hamilton equations, Euler–Jacobi theorem, Lie theorem, symmetries |
---|---|
Citation: | Kozlov V. V., Notes on integrable systems, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, |
Full text: | pdf (375.2 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of an ellipsoid on a plane and a sphere. We research these problems using Poincare maps, which investigation helps to discover a new integrable case.
Keywords: | nonholonomic constraint, invariant measure, first integral, Poincare map, integrability and chaos |
---|---|
Citation: | Bizyaev I. A., Kazakov A. O., Integrability and stochastic behavior in some nonholonomic dynamics problems, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, |
Full text: | pdf (2.27 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
This paper addresses a class of problems associated with the conditions for exact integrability of a system of ordinary differential equations expressed in terms of the properties of tensor invariants. The general theorem of integrability of the system of n differential equations is proved, which admits n−2 independent symmetry fields and an invariant volume n-form (integral invariant). General results are applied to the study of steady motions of a continuous medium with infinite conductivity.
Keywords: | symmetry field, integral invariant, nilpotent group, magnetic hydrodynamics |
---|---|
Citation: | Kozlov V. V., The Euler–Jacobi–Lie integrability theorem, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, |
Full text: | pdf (377.18 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bolsinov A. V., Kilin A. A., Kazakov A. O.
The phenomenon of a topological monodromy in integrable Hamiltonian and nonholonomic systems is discussed. An efficient method for computing and visualizing the monodromy is developed. The comparative analysis of the topological monodromy is given for the rolling ellipsoid of revolution problem in two cases, namely, on a smooth and on a rough plane. The first of these systems is Hamiltonian, the second is nonholonomic. We show that, from the viewpoint of monodromy, there is no difference between the two systems, and thus disprove the conjecture by Cushman and Duistermaat stating that the topological monodromy gives a topological obstruction for Hamiltonization of the rolling ellipsoid of revolution on a rough plane.
Keywords: | topological monodromy, integrable systems, nonholonomic systems, Poincaré map, bifurcation analysis, focus-focus singularities |
---|---|
Citation: | Bolsinov A. V., Kilin A. A., Kazakov A. O., Topological monodromy in nonholonomic systems, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, |
Full text: | pdf (890.26 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Bizyaev I. A.
In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body’s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior.
Keywords: | nonholonomic constraint, tensor invariant, first integral, invariant measure, integrability, conformally Hamiltonian system, rubber rolling, reversible, involution |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Bizyaev I. A., The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, |
Full text: | pdf (7.91 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Karavaev Y. L., Trefilov S. A.
The paper deals with deviation based control algorithm for trajectory following of omni-wheeled mobile robot. The kinematic model and the dynamics of the robot actuators are described.
Keywords: | omni-wheeled mobile robot, discrete algorithm, deviation based control, linearization, feedback |
---|---|
Citation: | Karavaev Y. L., Trefilov S. A., Deviation based discrete control algorithm for omni-wheeled mobile robot, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 1, |
Full text: | pdf (434.7 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
In our earlier paper [2] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with no-slip condition at the point of contact. In this paper we investigate the controllability of a ball in the presence of friction. We also study the problem of the existence and stability of singular dissipation-free periodic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed.
Keywords: | non-holonomic constraint, control, dry friction, viscous friction, stability, periodic solutions |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., How to control the Chaplygin ball using rotors. II, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 1, |
Full text: | pdf (2.71 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Vetchanin E. V., Tenenev V. A., Shaura A. S.
We consider the optimal motion control problem for a mobile device with an external rigid shell moving along a prescribed trajectory in a viscous fluid. The mobile robot under consideration possesses the property of self-locomotion. Self-locomotion is implemented due to back-and-forth motion of an internal material point. The optimal motion control is based on the Sugeno fuzzy inference system. An approach based on constructing decision trees using the genetic algorithm for structural and parametric synthesis has been proposed to obtain the base of fuzzy rules.
Keywords: | optimal motion control, self-locomotion, genetic algorithm, structural-parametrical synthesis, decision tree, fuzzy logic |
---|---|
Citation: | Vetchanin E. V., Tenenev V. A., Shaura A. S., Motion control of a rigid body in viscous fluid, Computer Research and Modeling, 2013, vol. 5, no. 4, |
Full text: | pdf (359.11 Kb) |
Impact-factor RSCI (2022): | 0.257 (Q4) |
---|---|
ISSN (print): | 2076-7633 |
ISSN (online): | 2077-6853 |
Site: | http://crm.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Treschev D. V.
In this paper we investigate various kinematic properties of rolling of one rigid body on another both for the classical model of rolling without slipping (the velocities of bodies at the point of contact coincide) and for the model of rubber-rolling (with the additional condition that the spinning of the bodies relative to each other be excluded). Furthermore, in the case where both bodies are bounded by spherical surfaces and one of them is fixed, the equations of motion for a moving ball are represented in the form of the Chaplygin system. When the center of mass of the moving ball coincides with its geometric center, the equations of motion are represented in conformally Hamiltonian form, and in the case where the radii of the moving and fixed spheres coincides, they are written in Hamiltonian form.
Keywords: | Rolling without slipping, Nonholonomic constraint, Chaplygin system, Conformally Hamiltonian system |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Treschev D. V., Rolling of a Rigid Body Without Slipping and Spinning: Kinematics and Dynamics, Journal of Applied Nonlinear Dynamics, 2013, vol. 2, no. 2, |
DOI: | 10.5890/JAND.2013.04.005 |
Full text: | pdf (269.77 Kb) |
ISSN (print): | 2164-6457 |
---|---|
ISSN (online): | 2164-6473 |
Site: | https://lhscientificpublishing.com/journals/JAND-Default.aspx |
Citation: | Kozlov V. V., An extension of the Hamilton-Jacobi method, Doklady Mathematics, 2012, vol. 85, no. 2, |
---|---|
DOI: | 10.1134/S1064562412020305 |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
The invariance conditions of smooth manifolds of Hamilton's equations are represented in the form of multidimensional Lamb's equations from the dynamics of an ideal fluid. In the stationary case these conditions do not depend on the method used to parameterize the invariant manifold. One consequence of Lamb's equations is an equation of a vortex, which is invariant to replacements of the time-dependent variables. A proof of the periodicity conditions of solutions of autonomous Hamilton's equations with n degrees of freedom and compact energy manifolds that admit of 2n – 3 additional first integrals is given as an application of the theory developed.
Citation: | Kozlov V. V., Invariant manifolds of Hamilton's equations, Journal of Applied Mathematics and Mechanics, 2012, vol. 76, no. 4, |
---|---|
DOI: | 10.1016/j.jappmathmech.2012.09.003 |
Full text: | pdf (254.88 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
The statistical mechanics of dynamical systems on which only isotropic viscous friction forces act is developed. A non-stationary analogue of the Gibbs canonical distribution, which enables each such system to be made to correspond to a certain thermodynamic system that satisfies the first and second laws of thermodynamics, is introduced. The evolution of non-Gibbs probability distributions with time is also considered.
Citation: | Kozlov V. V., The statistical mechanics of a class of dissipative systems, Journal of Applied Mathematics and Mechanics, 2012, vol. 76, no. 1, |
---|---|
DOI: | 10.1016/j.jappmathmech.2012.03.002 |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
We develop a new method for solving Hamilton’s canonical differential equations. The method is based on the search for invariant vortex manifolds of special type. In the case of Lagrangian (potential) manifolds, we arrive at the classical Hamilton–Jacobi method.
Keywords: | generalized Lamb’s equations, vortex manifolds, Clebsch potentials, Lagrange brackets |
---|---|
Citation: | Kozlov V. V., An Extended Hamilton–Jacobi Method, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, |
DOI: | 10.1134/S1560354712060093 |
Full text: | pdf (216.54 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Bolsinov A. V., Borisov A. V., Mamaev I. S.
In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.
Keywords: | non-holonomic constraint, Liouville foliation, invariant torus, invariant measure, integrability |
---|---|
Citation: | Bolsinov A. V., Borisov A. V., Mamaev I. S., Rolling of a Ball without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, |
DOI: | 10.1134/S1560354712060081 |
Full text: | pdf (402.12 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Bolsinov A. V., Borisov A. V., Mamaev I. S.
The paper is devoted to the bifurcation analysis and the Conley index in Hamiltonian dynamical systems. We discuss the phenomenon of appearance (disappearance) of equilibrium points under the change of the Morse index of a critical point of a Hamiltonian. As an application of these techniques we find new relative equilibria in the problem of the motion of three point vortices of equal intensity in a circular domain.
Keywords: | Morse index, Conley index, bifurcation analysis, bifurcation diagram, Hamiltonian dynamics, fixed point, relative equilibrium |
---|---|
Citation: | Bolsinov A. V., Borisov A. V., Mamaev I. S., The Bifurcation Analysis and the Conley Index in Mechanics, Regular and Chaotic Dynamics, 2012, vol. 17, no. 5, |
DOI: | 10.1134/S1560354712050073 |
Full text: | pdf (614.32 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
In the paper we study the control of a balanced dynamically non-symmetric sphere with rotors. The no-slip condition at the point of contact is assumed. The algebraic controllability is shown and the control inputs that steer the ball along a given trajectory on the plane are found. For some simple trajectories explicit tracking algorithms are proposed.
Keywords: | non-holonomic constraint, non-holonomic distribution, control, Chow–Rashevsky theorem, drift |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., How to Control Chaplygin’s Sphere Using Rotors, Regular and Chaotic Dynamics, 2012, vol. 17, no. 3-4, |
DOI: | 10.1134/S1560354712030045 |
Full text: | pdf (242.89 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The paper considers two new integrable systems which go back to Chaplygin. The systems consist of a spherical shell that rolls on a plane; within the shell there is a ball or Lagrange’s gyroscope. All necessary first integrals and an invariant measure are found. The solutions are shown to be expressed in terms of quadratures.
Keywords: | non-holonomic constraint, integrability, invariant measure, gyroscope, quadrature, coupled rigid bodies |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Two Non-holonomic Integrable Problems Tracing Back to Chaplygin, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, |
DOI: | 10.1134/S1560354712020074 |
Full text: | pdf (150.32 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
We discuss explicit integration and bifurcation analysis of two non-holonomic problems. One of them is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetric ball on a horizontal plane. The other, first posed by Yu.N.Fedorov, deals with the motion of a rigid body in a spherical support. For Chaplygin’s problem we consider in detail the transformation that Chaplygin used to integrate the equations when the constant of areas is zero. We revisit Chaplygin’s approach to clarify the geometry of this very important transformation, because in the original paper the transformation looks a cumbersome collection of highly non-transparent analytic manipulations. Understanding its geometry seriously facilitate the extension of the transformation to the case of a rigid body in a spherical support – the problem where almost no progress has been made since Yu.N. Fedorov posed it in 1988. In this paper we show that extending the transformation to the case of a spherical support allows us to integrate the equations of motion explicitly in terms of quadratures, detect mostly remarkable critical trajectories and study their stability, and perform an exhaustive qualitative analysis of motion. Some of the results may find their application in various technical devices and robot design. We also show that adding a gyrostat with constant angular momentum to the spherical-support system does not affect its integrability.
Keywords: | nonholonomic mechanics, spherical support, Chaplygin ball, explicit integration, isomorphism, bifurcation analysis |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, |
DOI: | 10.1134/S1560354712020062 |
Full text: | pdf (484.82 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Invariant manifolds of equations governing the dynamics of conservative nonholonomic systems are investigated. These manifolds are assumed to be uniquely projected onto configuration space. The invariance conditions are represented in the form of generalized Lamb’s equations. Conditions are found under which the solutions to these equations admit a hydrodynamical description typical of Hamiltonian systems. As an illustration, nonholonomic systems on Lie groups with a left-invariant metric and left-invariant (right-invariant) constraints are considered.
Keywords: | invariant manifold, Lamb’s equation, vortex manifold, Bernoulli’s theorem, Helmholtz’ theorem |
---|---|
Citation: | Kozlov V. V., On Invariant Manifolds of Nonholonomic Systems, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, |
DOI: | 10.1134/S1560354712020037 |
Full text: | pdf (239.09 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
A new integrable system describing the rolling of a rigid body with a spherical cavity over a spherical base is considered. Previously the authors found the separation of variables for this system at the zero level of a linear (in angular velocity) first integral, whereas in the general case it is not possible to separate the variables. In this paper we show that the foliation into invariant tori in this problem is equivalent to the corresponding foliation in the Clebsch integrable system in rigid body dynamics (for which no real separation of variables has been found either). In particular, a fixed point of focus type is possible for this system, which can serve as a topological obstacle to the real separation of variables.
Keywords: | integrable system, bifurcation diagram, conformally Hamiltonian system, bifurcation, Liouville foliation, critical periodic solution |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Topological analysis of one integrable system related to the rolling of a ball over a sphere, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 5, |
Full text: | pdf (796.84 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Vetchanin E. V., Mamaev I. S., Tenenev V. A.
An investigation of the characteristics of motion of a rigid body with variable internal mass distribution in a viscous fluid is carried out on the basis of a joint numerical solution of the Navier–Stokes equations and equations of motion. A non-stationary three-dimensional solution to the problem is found. The motion of a sphere and a drop-shaped body in a viscous fluid, which is caused by the motion of internal material points, in a gravitational field is explored. The possibility of motion of a body in an arbitrary given direction is shown.
Keywords: | finite-volume numerical method, Navier-Stokes equations, variable internal mass distribution, motion control |
---|---|
Citation: | Vetchanin E. V., Mamaev I. S., Tenenev V. A., The motion of a body with variable mass geometry in a viscous fluid, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, |
Full text: | pdf (15.9 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Treschev D. V.
In this paper we investigate various kinematic properties of rolling of one rigid body on another both for the classical model of rolling without slipping (the velocities of bodies at the point of contact coincide) and for the model of rubber-rolling (with the additional condition that the spinning of the bodies relative to each other be excluded). Furthermore, in the case where both bodies are bounded by spherical surfaces and one of them is fixed, the equations of motion for a moving ball are represented in the form of the Chaplygin system. When the center of mass of the moving ball coincides with its geometric center, the equations of motion are represented in conformally Hamiltonian form, and in the case where the radii of the moving and fixed spheres coincides, they are written in Hamiltonian form.
Keywords: | rolling without slipping, nonholonomic constraint, Chaplygin system, conformally Hamiltonian system |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Treschev D. V., Rolling of a rigid body without slipping and spinning: kinematics and dynamics, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, |
Full text: | pdf (347.06 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bolsinov A. V., Borisov A. V., Mamaev I. S.
In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.
Keywords: | non-holonomic constraint, Liouville foliation, invariant torus, invariant measure, integrability |
---|---|
Citation: | Bolsinov A. V., Borisov A. V., Mamaev I. S., Rolling without spinning of a ball on a plane: absence of an invariant measure in a system with a complete set of first integrals, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, |
Full text: | pdf (328.96 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Treschev D. V., Erdakova N. N., Ivanova T. B.
The problem of a uniform straight cylinder (disc) sliding on a horizontal plane under the action of dry friction forces is considered. The contact patch between the cylinder and the plane coincides with the base of the cylinder. We consider axisymmetric discs, i.e. we assume that the base of the cylinder is symmetric with respect to the axis lying in the plane of the base. The focus is on the qualitative properties of the dynamics of discs whose circular base, triangular base and three points are in contact with a rough plane.
Keywords: | Amontons–Coulomb law, dry friction, disc, final dynamics, stability |
---|---|
Citation: | Treschev D. V., Erdakova N. N., Ivanova T. B., On the final motion of cylindrical solids on a rough plane, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, |
Full text: | pdf (623.55 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
We discuss an embedding of the vector field associated with the nonholonomic Routh sphere in subgroup of the commuting Hamiltonian vector fields associated with this system. We prove that the corresponding Poisson brackets are reduced to canonical ones in the region without of homoclinic trajectories.
Keywords: | nonholonomic mechanics, Routh sphere, Poisson brackets |
---|---|
Citation: | Bizyaev I. A., Tsiganov A. V., On the Routh sphere, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, |
Full text: | pdf (299.77 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
We develop a new method for solving Hamilton’s canonical differential equations. The method is based on the search of invariant vortex manifolds of special type. In the case of Lagrangian (potential) manifolds, we arrive at the classical Hamilton–Jacobi method.
Keywords: | generalized Lamb’s equations, vortex manifolds, Clebsch potentials, Lagrange brackets |
---|---|
Citation: | Kozlov V. V., An extended Hamilton–Jacobi method, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, |
Full text: | pdf (400.68 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
In the paper we study control of a balanced dynamically nonsymmetric sphere with rotors. The no-slip condition at the point of contact is assumed. The algebraic contrability is shown and the control inputs providing motion of the ball along a given trajectory on the plane are found. For some simple trajectories explicit tracking algorithms are proposed.
Keywords: | non-holonomic constraint, non-holonomic distribution, control, Chow–Rashevsky theorem, drift |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., How to control the Chaplygin sphere using rotors, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 2, |
Full text: | pdf (400.44 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
We consider the problem of the motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a method for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions in which the mutual distances remain finite (we call them pseudo-leapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.
Keywords: | ideal fluid, vortex ring, leapfrogging motion of vortex rings, bifurcation complex, periodic solution, integrability, chaotic dynamics |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, |
Full text: | pdf (1.03 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
We consider the problem of rolling of a ball with an ellipsoidal cavity filled with an ideal fluid, which executes a uniform vortex motion, on an absolutely rough plane. We point out the case of existence of an invariant measure and show that there is a particular case of integrability under conditions of axial symmetry.
Keywords: | vortex motion, non-holonomic constraint, Chaplygin ball, invariant measure, integrability, rigid body, ideal fluid |
---|---|
Citation: | Borisov A. V., Mamaev I. S., The dynamics of the Chaplygin ball with a fluid-filled cavity, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, |
Full text: | pdf (305.43 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Invariant manifolds of equations governing the dynamics of conservative nonholonomic systems are investigated. These manifolds are assumed to be uniquely projected onto configuration space. The invariance conditions are represented in the form of generalized Lamb’s equations. Conditions are found under which the solutions to these equations admit a hydrodynamical description typical of Hamiltonian systems. As an illustration, nonholonomic systems on Lie groups with a left-invariant metric and left-invariant (right-invariant) constraints are considered.
Keywords: | invariant manifold, Lamb’s equation, vortex manifold, Bernoulli’s theorem, Helmholtz’ theorem |
---|---|
Citation: | Kozlov V. V., On invariant manifolds of nonholonomic systems, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, |
Full text: | pdf (329.1 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Pivovarova E. N., Ivanova T. B.
In the paper we study the stability of a spherical shell rolling on a horizontal plane with Lagrange’s gyroscope inside. A linear stability analysis is made for the upper and lower position of a top. A bifurcation diagram of the system is constructed. The trajectories of the contact point for different values of the integrals of motion are constructed and analyzed.
Keywords: | rolling motion, stability, Lagrange’s gyroscope, bifurcational diagram |
---|---|
Citation: | Pivovarova E. N., Ivanova T. B., Stability analysis of periodic solutions in the problem of the rolling of a ball with a pendulum, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2012, no. 4, |
DOI: | 10.20537/vm120412 |
Full text: | pdf (688.03 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
Ivanov A. P., Shuvalov N. D., Ivanova T. B.
The classical problem about the motion of a heavy symmetric rigid body (top) with a fixed point on the horizontal plane is discussed. Due to the unilateral nature of the contact, detachments (jumps) are possible under certain conditions. We know two scenarios of detachment related to changing the sign of the normal reaction or the sign of the normal acceleration, and the mismatch of these conditions leads to a paradox. To determine the nature of paradoxes an example of the pendulum (rod) within the limitations of the real coefficient of friction was studied in detail. We showed that in the case of the first type of the paradox (detachment is impossible and contact is impossible) the body begins to slide on the support. In the case of the paradox of the second type (detachment is possible and contact is possible) contact is retained up to the sign change of the normal reaction, and then at the detachment the normal acceleration is non-zero.
Keywords: | friction, Lagrange top, paradox, detachment |
---|---|
Citation: | Ivanov A. P., Shuvalov N. D., Ivanova T. B., On detachment conditions of a top on an absolutely rough support, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2012, no. 3, |
DOI: | 10.20537/vm120310 |
Full text: | pdf (292.26 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
Citation: | Kozlov V. V., Wigner measures on infinite-dimensional spaces and the Bogolyubov equations for quantum systems, Doklady Mathematics, 2011, vol. 84, no. 1, |
---|---|
DOI: | 10.1134/S1064562411050048 |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
Citation: | Kozlov V. V., Friction in the sense of Painlevé and Lagrangian mechanics, Doklady Mathematics, 2011, vol. 56, no. 6, |
---|---|
DOI: | 10.1134/S1028335811060115 |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
Citation: | Kozlov V. V., On the dry-friction mechanism, Doklady Mathematics, 2011, vol. 56, no. 4, |
---|---|
DOI: | 10.1134/S1028335811040124 |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
A linearized problem of stability of simple periodic motions with elastic reflections is considered: a particle moves along a straight-line segment that is orthogonal to the boundary of a billiard at its endpoints. In this problem issues from mechanics (variational principles), linear algebra (spectral properties of products of symmetric operators), and geometry (focal points, caustics, etc.) are naturally intertwined. Multidimensional variants of Hill’s formula, which relates the dynamic and geometric properties of a periodic trajectory, are discussed. Stability conditions are expressed in terms of the geometric properties of the boundary of a billiard. In particular, it turns out that a nondegenerate two-link trajectory of maximum length is always unstable. The degree of instability (the number of multipliers outside the unit disk) is estimated. The estimates are expressed in terms of the geometry of the caustic and the Morse indices of the length function of this trajectory.
Citation: | Kozlov V. V., Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 273, |
---|---|
DOI: | 10.1134/S0081543811040092 |
Impact-factor WoS (2022): | 0.500 (Q4) |
---|---|
Impact-factor RSCI (2022): | 0.276 (Q3) |
ISSN (print): | 0081-5438 |
ISSN (online): | 1531-8605 |
Site: | http://www.maik.ru/ru/journal/trstekl/ |
The equations of motion of a collisionless continuum are derived within an Eulerian approach. They differ from the classical equations of motion of an ideal gas, which take into account heat conduction phenomena. Several problems related to the weak convergence of the solutions of the equations of motion of a continuum when there is an unbounded increase in time are discussed. The problem of the correctness of the operation of truncating the exact infinite chain of equations of a collisionless gas is examined.
Citation: | Kozlov V. V., The equations of motion of a collisionless continuum, Journal of Applied Mathematics and Mechanics, 2011, vol. 75, no. 6, |
---|---|
DOI: | 10.1016/j.jappmathmech.2012.01.001 |
Full text: | pdf (689.16 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
In this paper we develop a new model of non-holonomic billiard that accounts for the intrinsic rotation of the billiard ball. This model is a limit case of the problem of rolling without slipping of a ball without slipping over a quadric surface. The billiards between two parallel walls and inside a circle are studied in detail. Using the three-dimensional-point-map technique, the non-integrability of the non-holonomic billiard within an ellipse is shown.
Keywords: | billiard, impact, point map, nonintegrability, periodic solution, nonholonomic constraint, integral of motion |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., On the Model of Non-holonomic Billiard, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, |
DOI: | 10.1134/S1560354711060062 |
Full text: | pdf (199.9 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
We consider a continuum of interacting particles whose evolution is governed by the Vlasov kinetic equation. An infinite sequence of equations of motion for this medium (in the Eulerian description) is derived and its general properties are explored. An important example is a collisionless gas, which exhibits irreversible behavior. Though individual particles interact via a potential, the dynamics of the continuum bears dissipative features. Applicability of the Vlasov equations to the modeling of small-scale turbulence is discussed.
Keywords: | kinetic Vlasov’s equation, Euler’s equation, continuum, turbulence |
---|---|
Citation: | Kozlov V. V., The Vlasov Kinetic Equation, Dynamics of Continuum and Turbulence, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, |
DOI: | 10.1134/S1560354711060049 |
Full text: | pdf (323.98 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The Kac circular model is a discrete dynamical system which has the property of recurrence and reversibility. Within the framework of this model M.Kac formulated necessary conditions for irreversibility over "short" time intervals to take place and demonstrated Boltzmann’s most important exploration methods and ideas, outlining their advantages and limitations. We study the circular model within the realm of the theory of Gibbs ensembles and offer a new approach to a rigorous proof of the "zeroth" law of thermodynamics based on the analysis of weak convergence of probability distributions.
Keywords: | reversibility, stochastic equilibrium, weak convergence |
---|---|
Citation: | Kozlov V. V., Statistical Irreversibility of the Kac Reversible Circular Model, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, |
DOI: | 10.1134/S1560354711050091 |
Full text: | pdf (202.6 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of "clandestine" linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability.
Keywords: | nonholonomic constraint, rolling motion, Chaplygin ball, integral, invariant measure |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, |
DOI: | 10.1134/S1560354711050042 |
Full text: | pdf (643.15 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Bolsinov A. V., Borisov A. V., Mamaev I. S.
The problem of Hamiltonization of nonholonomic systems, both integrable and non-integrable, is considered. This question is important in the qualitative analysis of such systems and it enables one to determine possible dynamical effects. The first part of the paper is devoted to representing integrable systems in a conformally Hamiltonian form. In the second part, the existence of a conformally Hamiltonian representation in a neighborhood of a periodic solution is proved for an arbitrary (including integrable) system preserving an invariant measure. Throughout the paper, general constructions are illustrated by examples in nonholonomic mechanics.
Keywords: | conformally Hamiltonian system, nonholonomic system, invariant measure, periodic trajectory, invariant torus, integrable system |
---|---|
Citation: | Bolsinov A. V., Borisov A. V., Mamaev I. S., Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, |
DOI: | 10.1134/S1560354711050030 |
Full text: | pdf (425.83 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
The Hamiltonian representation and integrability of the nonholonomic Suslov problem and its generalization suggested by S. A. Chaplygin are considered. This subject is important for understanding the qualitative features of the dynamics of this system, being in particular related to a nontrivial asymptotic behavior (i. e., to a certain scattering problem). A general approach based on studying a hierarchy in the dynamical behavior of nonholonomic systems is developed.
Keywords: | Hamiltonian system, Poisson bracket, nonholonomic constraint, invariant measure, integrability |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Hamiltonicity and integrability of the Suslov problem, Regular and Chaotic Dynamics, 2011, vol. 16, no. 1-2, |
DOI: | 10.1134/S1560354711010035 |
Full text: | pdf (239.81 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
We consider a nonholonomic model of the dynamics of an omni-wheel vehicle on a plane and a sphere. An elementary derivation of equations is presented, the dynamics of a free system is investigated, a relation to control problems is shown.
Keywords: | omni-wheel, roller-bearing wheel, nonholonomic constraint, dynamical system, invariant measure, integrability, controllability |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., An omni-wheel vehicle on a plane and a sphere, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 4, |
Full text: | pdf (726.7 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bolsinov A. V., Borisov A. V., Mamaev I. S.
The paper is concerned with the use of bifurcation analysis and the Conley index in Hamiltonian dynamical systems. We give the proof of the theorem on the appearance (disappearance) of fixed points in the case of the Morse index change. New relative equilibria in the problem of the motion of point vortices of equal intensity in a circle are found.
Keywords: | Morse index, Conley index, bifurcation analysis, bifurcation diagram, Hamiltonian dynamics, fixed point, relative equilibrium |
---|---|
Citation: | Bolsinov A. V., Borisov A. V., Mamaev I. S., The bifurcation analysis and the Conley index in mechanics, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, |
Full text: | pdf (782.35 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The structure of the Lorentz force and the related analogy between electromagnetism and inertia are discussed. The problem of invariant manifolds of the equations of motion for a charge in an electromagnetic field and the conditions for these manifolds to be Lagrangian are considered.
Keywords: | Lorentz force, Maxwell equations, Coriolis force, symplectic structure, Lagrangian manifold |
---|---|
Citation: | Kozlov V. V., The Lorentz force and its generalizations, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, |
Full text: | pdf (359.22 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Kuleshov A. S., Treschev D. V., Ivanova T. B., Naymushina O. S.
The paper considers two two-dimensional dynamical problems for an absolutely rigid cylinder interacting with a deformable flat base (the motion of an absolutely rigid disk on a base which in non-deformed condition is a straight line). The base is a sufficiently stiff viscoelastic medium that creates a normal pressure p(x)=kY(x)+ν˙Y(x), where x is a coordinate on the straight line, Y(x) is a normal displacement of the point x, and k and ν are elasticity and viscosity coefficients (the Kelvin—Voigt medium). We are also of the opinion that during deformation the base generates friction forces, which are subject to Coulomb’s law. We consider the phenomenon of impact that arises during an arbitrary fall of the disk onto the straight line and investigate the disk’s motion «along the straight line» including the stages of sliding and rolling.
Keywords: | Kelvin–Voight medium, impact, viscoelasticity, friction |
---|---|
Citation: | Kuleshov A. S., Treschev D. V., Ivanova T. B., Naymushina O. S., A rigid cylinder on a viscoelastic plane, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, |
Full text: | pdf (499.7 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The paper considers two new integrable systems due to Chaplygin, which describe the rolling of a spherical shell on a plane, with a ball or Lagrange’s gyroscope inside. All necessary first integrals and an invariant measure are found. The reduction to quadratures is given.
Keywords: | non-holonomic constraint, integrability, invariant measure, gyroscope, quadrature, coupled rigid bodies |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Two non-holonomic integrable systems of coupled rigid bodies, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, |
Full text: | pdf (404.52 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Gazizullina L., Mamaev I. S.
This paper has been written for a collection of V.A. Steklov’s selected works, which is being prepared for publication and is entitled «Works on Mechanics 1902–1909: Translations from French». The collection is based on V.A. Steklov’s papers on mechanics published in French journals from 1902 to 1909.
Citation: | Borisov A. V., Gazizullina L., Mamaev I. S., On V.A. Steklov’s legacy in classical mechanics, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 2, |
---|---|
Full text: | pdf (368.21 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
We consider the problem of explicit integration and bifurcation analysis for two systems of nonholonomic mechanics. The first one is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetrical ball on a horizontal plane. The second problem is on the motion of rigid body in a spherical support. We explicitly integrate this problem by generalizing the transformation which Chaplygin applied to the integration of the problem of the rolling ball at a non-zero constant of areas. We consider the geometric interpretation of this transformation from the viewpoint of a trajectory isomorphism between two systems at different levels of the energy integral. Generalization of this transformation for the case of dynamics in a spherical support allows us to integrate the equations of motion explicitly in quadratures and, in addition, to indicate periodic solutions and analyze their stability. We also show that adding a gyrostat does not lead to the loss of integrability.
Keywords: | nonholonomic mechanics, spherical support, Chaplygin ball, explicit integration, isomorphism, bifurcation analysis |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Generalized Chaplygin’s transformation and explicit integration of a system with a spherical support, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 2, |
Full text: | pdf (1.78 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Vaskina A. V.
This paper presents a topological approach to the search and stability analysis of relative equilibria of three point vortices of equal intensities. It is shown that the equations of motion can be reduced by one degree of freedom. We have found two new stationary configurations (isosceles and non-symmetrical collinear) and studied their bifurcations and stability.
Keywords: | point vortex, reduction, bifurcational diagram, relative equilibriums, stability, periodic solutions |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Vaskina A. V., Stability of new relative equilibria of the system of three point vortices in a circular domain, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 1, |
Full text: | pdf (1.2 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The Kac circular model is a discrete dynamical system which has the property of recurrence and reversibility. Within the framework of this model M.Kac formulated necessary conditions for irreversibility over «short» time intervals to take place and demonstrated Boltzmann’s most important exploration methods and ideas, outlining their advantages and limitations. We study the circular model within the realm of the theory of Gibbs ensembles and offer a new approach to a rigorous proof of the «zeroth» law of thermodynamics basing on the analysis of weak convergence of probability distributions.
Keywords: | reversibility, stochastic equilibrium, weak convergence |
---|---|
Citation: | Kozlov V. V., Statistical irreversibility of the Kac reversible circular model, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 1, |
Full text: | pdf (419.07 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Vetchanin E. V., Tenenev V. A.
Statement of a problem of management of movement of a body in a viscous liquid is given. Movement bodies it is induced by moving of internal material points. On a basis the numerical decision of the equations of movement of a body and the hydrodynamic equations approximating dependencies for viscous forces are received. With application approximations the problem of optimum control of body movement dares on the set trajectory with application of hybrid genetic algorithm. Possibility of the directed movement of a body under action is established back and forth motion of an internal point. Optimum control movement direction it is carried out by motion of other internal point on circular trajectory with variable speed
Keywords: | optimum control, the equations of movement, Navier–Stokes equations, numerical methods, fuzzy decision trees, genetic algorithm |
---|---|
Citation: | Vetchanin E. V., Tenenev V. A., Motion control simulating in a viscous liquid of a body with variable geometry of weights, Computer Research and Modeling, 2011, vol. 3, no. 4, |
Full text: | pdf (594.85 Kb) |
Impact-factor RSCI (2022): | 0.257 (Q4) |
---|---|
ISSN (print): | 2076-7633 |
ISSN (online): | 2077-6853 |
Site: | http://crm.ics.org.ru/ |
We consider the inhomogeneous self-gravitating liquid spheroid with confocal stratification which rotates around the minor semiaxis and is in equilibrium. General relationships for pressure, angular velocity and gravitational potential of the spheroid with any density function are obtained. Special cases of piecewise constant and continuous density functions are analyzed.
Keywords: | self-gravitating fluid, confocal stratification, spheroid, Euler equations |
---|---|
Citation: | Bizyaev I. A., Ivanova T. B., Figures of equilibrium of liquid self-gravitating inhomogeneous mass, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2011, no. 3, |
DOI: | 10.20537/vm110313 |
Full text: | pdf (215.68 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
Borisov A. V., Mamaev I. S., Ramodanov S. M.
Citation: | Borisov A. V., Mamaev I. S., Ramodanov S. M., Basic principles and models of dynamic advection, Doklady Physics, 2010, vol. 432, no. 1, |
---|---|
DOI: | 10.1134/S1028335810050058 |
Full text: | pdf (178.99 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
Citation: | Kozlov V. V., Variational principle for periodic orbits of invertible dynamical equations, Doklady Mathematics, 2010, vol. 81, no. 1, |
---|---|
DOI: | 10.1134/S1064562410010382 |
Full text: | pdf (152.97 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
Variational principles, generalizing the classical d’Alembert–Lagrange, Hölder, and Hamilton–Ostrogradskii principles, are established. After the addition of anisotropic dissipative forces and taking the limit, when the coefficient of viscous friction tends to infinity, these variational principles transform into the classical principles, which describe the motion of systems with constraints. New variational relations are established for searching for the periodic trajectories of the reversible equations of dynamics.
Citation: | Kozlov V. V., On the variational principles of mechanics, Journal of Applied Mathematics and Mechanics, 2010, vol. 74, no. 5, |
---|---|
DOI: | 10.1016/j.jappmathmech.2010.11.001 |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Linear systems of differential equations allowing of functions in quadratic forms that do not increase along trajectories with time are considered. The relations between the indices of inertia of these forms and the degrees of instability of equilibrium states are indicated. These assertions generalize known results from the oscillation theory of linear systems with dissipation, and reveal the mechanism of loss of stability when non-increasing quadratic forms lose the property of a minimum.
Citation: | Kozlov V. V., Remarks on the degree of instability, Journal of Applied Mathematics and Mechanics, 2010, vol. 74, no. 1, |
---|---|
DOI: | 10.1016/j.jappmathmech.2010.03.002 |
Full text: | pdf (242.92 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Bolsinov A. V., Borisov A. V., Mamaev I. S.
In this paper a general topological approach is proposed for the study of stability of periodic solutions of integrable dynamical systems with two degrees of freedom. The methods developed are illustrated by examples of several integrable problems related to the classical Euler–Poisson equations, the motion of a rigid body in a fluid, and the dynamics of gaseous expanding ellipsoids. These topological methods also enable one to find non-degenerate periodic solutions of integrable systems, which is especially topical in those cases where no general solution (for example, by separation of variables) is known.
Keywords: | topology, stability, periodic trajectory, critical set, bifurcation set, bifurcation diagram |
---|---|
Citation: | Bolsinov A. V., Borisov A. V., Mamaev I. S., Topology and stability of integrable systems, Russian Mathematical Surveys, 2010, vol. 65, no. 2, |
DOI: | 10.1070/RM2010v065n02ABEH004672 |
Full text: | pdf (1.12 Mb) |
Impact-factor WoS (2022): | 0.900 (Q2) |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0036-0279 |
ISSN (online): | 1468-4829 |
Site: | http://www.mathnet.ru/umn |
Borisov A. V., Mamaev I. S., Ramodanov S. M.
The paper is concerned with a class of problems which involves the dynamical interaction of a rigid body with point vortices on the surface of a two-dimensional sphere. The general approach to the 2D hydrodynamics is further developed. The problem of motion of a dynamically symmetric circular body interacting with a single vortex is shown to be integrable. Mass vortices on S2 are introduced and the related issues (such as equations of motion, integrability, partial solutions, etc.) are discussed. This paper is a natural progression of the author’s previous research on interaction of rigid bodies and point vortices in a plane.
Keywords: | hydrodynamics on a sphere, coupled body-vortex system, mass vortex, equations of motion, integrability |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Ramodanov S. M., Coupled motion of a rigid body and point vortices on a two-dimensional spherical surface, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, |
DOI: | 10.1134/S1560354710040040 |
Full text: | pdf (298.75 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Citation: | Kozlov V. V., Note on dry friction and non-holonomic constraints, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, |
---|---|
Full text: | pdf (183.28 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Citation: | Borisov A. V., Mamaev I. S., Reply to A. T. Fomenko's comments, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, |
---|---|
Full text: | pdf (97.18 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of «clandestine» linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability.
Keywords: | nonholonomic constraint, rolling motion, Chaplygin ball, integral, invariant measure |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Rolling of a homogeneous ball over a dynamically asymmetric sphere, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, |
Full text: | pdf (486.45 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
A generalization of Amantons’ law of dry friction for constrained Lagrangian systems is formulated. Under a change of generalized coordinates the components of the dry-friction force transform according to the covariant rule and the force itself satisfies the Painlevé condition. In particular, the pressure of the system on a constraint is independent of the anisotropic-friction tensor. Such an approach provides an insight into the Painlevé dry-friction paradoxes. As an example, the general formulas for the sliding friction force and torque and the rotation friction torque on a body contacting with a surface are obtained.
Keywords: | Lagrangian system, anisotropic friction, Painlevé condition |
---|---|
Citation: | Kozlov V. V., Lagrangian mechanics and dry friction, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, |
Full text: | pdf (265.11 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Bolsinov A. V., Borisov A. V., Mamaev I. S.
Hamiltonisation problem for non-holonomic systems, both integrable and non-integrable, is considered. This question is important for qualitative analysis of such systems and allows one to determine possible dynamical effects. The first part is devoted to the representation of integrable systems in a conformally Hamiltonian form. In the second part, the existence of a conformally Hamiltonian representation in a neighbourhood of a periodic solution is proved for an arbitrary measure preserving system (including integrable). General consructions are always illustrated by examples from non-holonomic mechanics.
Keywords: | conformally Hamiltonian system, nonholonomic system, invariant measure, periodic trajectory, invariant torus, integrable system |
---|---|
Citation: | Bolsinov A. V., Borisov A. V., Mamaev I. S., Hamiltonisation of non-holonomic systems in the neighborhood of invariant manifolds, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, |
Full text: | pdf (398.78 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Ivanova T. B.
We consider figures of equilibrium and stability of a liquid self-gravitating elliptic cylinder. The flow within the cylinder is assumed to be dew to an elliptic perturbation. A bifurcation diagram is plotted and conditions for steady solutions to exist are indicated.
Keywords: | self-gravitating liquid, elliptic cylinder, bifurcation point, stability, Riemann equations |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Ivanova T. B., Stability of a liquid self-gravitating elliptic cylinder with intrinsic rotation, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, |
Full text: | pdf (902.16 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Ramodanov S. M.
A new concept of dynamic advection is introduced. The model of dynamic advection deals with the motion of massive particles in a 2D flow of an ideal incompressible liquid. Unlike the standard advection problem, which is widely treated in the modern literature, our equations of motion account not only for particles’ kinematics, governed by the Euler equations, but also for their dynamics (which is obviously neglected if the mass of particles is taken to be zero). A few simple model problems are considered.
Keywords: | advection, mixing, point vortex, coarse-grained impurities, bifurcation complex |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Ramodanov S. M., Dynamic advection, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, |
Full text: | pdf (10.3 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
We consider a continuum of interacting particles whose evolution is governed by the Vlasov kinetic equation. An infinite sequence of equations of motion for this medium (in the Eulerian description) is derived and its general properties are explored. An important example is a collisionless gas, which exhibits irreversible behavior. Though individual particles interact via a potential, the dynamics of the continuum bears dissipative features. Applicability of the Vlasov equations to the modeling of small-scale turbulence is discussed.
Keywords: | The Vlasov kinetic equation, dynamics of continuum and turbulence |
---|---|
Citation: | Kozlov V. V., The Vlasov kinetic equation, dynamics of continuum and turbulence, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, |
Full text: | pdf (276.41 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Bolotin S. V., Kilin A. A., Mamaev I. S., Treschev D. V.
Citation: | Borisov A. V., Bolotin S. V., Kilin A. A., Mamaev I. S., Treschev D. V., Valery Vasilievich Kozlov. On his 60th birthday, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, |
---|---|
Full text: | pdf (25.39 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
In this paper we develop a new model of non-holonomic billiard that accounts for the intrinsic rotation of the billiard ball. This model is a limit case of the problem of rolling without slipping of a ball without slipping over a quadric surface. The billiards between two parallel walls and inside a circle are studied in detail. Using the three-dimensional-point-map technique, the non-integrability of the non-holonomic billiard within an ellipse is shown.
Keywords: | billiard, impact, point mapping, nonintegrability, periodic solution, nonholonomic constraint, integral of motion |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., On the model of non-holonomic billiard, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 2, |
Full text: | pdf (237.96 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Vaskin V. V., Vaskina A. V., Mamaev I. S.
With the help of mathematical modelling, we study the dynamics of many point vortices system on the plane. For this system, we consider the following cases:
— vortex rings with outer radius r=1 and variable inner radius r0,
— vortex ellipses with semiaxes a, b.
The emphasis is on the analysis of the asymptotic (t→∞) behavior of the system and on the verification of the stability criteria for vorticity continuous distributions.
Keywords: | vortex dynamics, point vortex, hydrodynamics, asymptotic behavior |
---|---|
Citation: | Vaskin V. V., Vaskina A. V., Mamaev I. S., Problems of stability and asymptotic behavior of vortex patches on the plane, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 2, |
Full text: | pdf (685.19 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
We consider the problems of Hamiltonian representation and integrability of the nonholonomic Suslov system and its generalization suggested by S. A. Chaplygin. These aspects are very important for understanding the dynamics and qualitative analysis of the system. In particular, they are related to the nontrivial asymptotic behaviour (i. e. to some scattering problem). The paper presents a general approach based on the study of the hierarchy of dynamical behaviour of nonholonomic systems.
Keywords: | Hamiltonian system, Poisson bracket, nonholonomic constraint, invariant measure, integrability |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Hamiltonian representation and integrability of the Suslov problem, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 1, |
Full text: | pdf (654.76 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Figures of equilibrium are considered and the stability of liquid self-gravitating elliptic cylinder with an internal flow in a class of elliptic indignations are researched. The bifurcation diagram of given system is constructed, areas of existence of the stationary solutions are specified.
Keywords: | self-gravitating liquid, elliptic cylinder, stability, Riemann equations |
---|---|
Citation: | Ivanova T. B., Construction of bifurcation diagram and analysis of stability of self-gravitating fluid elliptical cylinder with internal flow, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2010, no. 4, |
DOI: | 10.20537/vm100409 |
Full text: | pdf (588.63 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
Borisov A. V., Lutsenko S. G., Mamaev I. S.
The paper deals with the problem of motion of a wheeled carriage on a plane in the case where one of the wheeled pairs is fixed. In addition, the case of motion of a wheeled carriage on a plane in the case of two free wheeled pairs is considered.
Keywords: | nonholonomic constraint, dynamics of the system, wheeled carriage |
---|---|
Citation: | Borisov A. V., Lutsenko S. G., Mamaev I. S., Dynamics of a wheeled carriage on a plane, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2010, no. 4, |
DOI: | 10.20537/vm100405 |
Full text: | pdf (385.82 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
Borisov A. V., Mamaev I. S., Tsiganov A. V.
Citation: | Borisov A. V., Mamaev I. S., Tsiganov A. V., Lyapunov A.M. Works on theoretical mechanics. From the 1882-1894 handwritten heritage, Izhevsk: Regular and Chaotic Dynamics, 2010, |
---|---|
Full text: | pdf (174.49 Kb) |
Citation: | Kozlov V. V., On the stabilization of unstable equilibria by time-periodic gyroscopic forces, Doklady Mathematics, 2009, vol. 54, no. 12, |
---|---|
DOI: | 10.1134/S1028335809120106 |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
Citation: | Kozlov V. V., Kelvin's instability theorem: Topological meaning and generalizations, Doklady Mathematics, 2009, vol. 79, no. 1, |
---|---|
DOI: | 10.1134/S1064562409010086 |
Full text: | pdf (279.94 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
The problem of the motion of a Chaplygin sleigh on horizontal and inclined surfaces is considered. The possibility of representing the equations of motion in Hamiltonian form and of integration using Liouville’s theorem (with a redundant algebra of integrals) is investigated. The asymptotics for the rectilinear uniformly accelerated sliding of a sleigh along the line of steepest descent are determined in the case of an inclined plane. The zones in the plane of the initial conditions, corresponding to a different behaviour of the sleigh, are constructed using numerical calculations. The boundaries of these domains are of a complex fractal nature, which enables a conclusion to be drawn concerning the probable character from of the dynamic behaviour.
Citation: | Borisov A. V., Mamaev I. S., The dynamics of a Chaplygin sleigh, Journal of Applied Mathematics and Mechanics, 2009, vol. 73, no. 2, |
---|---|
DOI: | 10.1016/j.jappmathmech.2009.04.005 |
Full text: | pdf (263.77 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., The Steklov Mathematical Institute has turned 75 years old, Russian Mathematical Surveys, 2009, vol. 64, no. 4, |
---|---|
DOI: | 10.4213/rm9311 |
Full text: | pdf (724.87 Kb) |
Impact-factor WoS (2022): | 0.900 (Q2) |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0036-0279 |
ISSN (online): | 1468-4829 |
Site: | http://www.mathnet.ru/umn |
We consider linear systems of differential equations admitting functions in the form of quadratic forms that do not increase along trajectories in the course of time. We find new relations between the inertia indices of these forms and the instability degrees of the equilibria. These assertions generalize well-known results in the oscillation theory of linear systems with dissipation and clarify the mechanism of stability loss, whereby nonincreasing quadratic forms lose the property of minimum.
Citation: | Kozlov V. V., On the mechanism of stability loss, Differential Equations, 2009, vol. 45, no. 4, |
---|---|
DOI: | 10.1134/S0012266109040041 |
Full text: | pdf (316.78 Kb) |
Impact-factor WoS (2015): | 0.344 |
---|---|
Impact-factor RSCI (2014): | 0,585 |
ISSN (print): | 0374-0641 |
Site: | http://www.maik.ru/ru/journal/deqrus/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
We consider the motion of a material point on the surface of a sphere in the field of 2n+1 identical Hooke centers (singularities with elastic potential) lying on a great circle. Our main result is that this system is superintegrable. The property of superintegrability for this system has been conjectured by us in [1], where the structure of a superintegral of arbitrarily high odd degree in momemnta was outlined. We also indicate an isomorphism between this system and the one-dimensional N-particle system discussed in the recent paper [2] and show that for the latter system an analogous superintegral can be constructed.
Keywords: | superintegrable systems, systems with a potential, Hooke center |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Superintegrable system on a sphere with the integral of higher degree, Regular and Chaotic Dynamics, 2009, vol. 14, no. 6, |
DOI: | 10.1134/S156035470906001X |
Full text: | pdf (125.27 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The Poincaré model for dynamics of a collisionless gas in a rectangular parallelepiped with mirror walls is considered. The question on smoothing of the density and the temperature of this gas and conditions for the monotone growth of the coarse-grained entropy are discussed. All these effects provide a new insight of the classical paradox of mixing of gases.
Keywords: | collisionless gas, coarse-grained entropy, Gibbs paradox |
---|---|
Citation: | Kozlov V. V., Kinetics of collisionless gas: Equalization of temperature, growth of the coarse-grained entropy and the Gibbs paradox, Regular and Chaotic Dynamics, 2009, vol. 14, no. 4-5, |
DOI: | 10.1134/S1560354709040091 |
Full text: | pdf (172.17 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
We consider several well-known isomorphisms between Jacobi’s geodesic problem and some integrable cases from rigid body dynamics (the cases of Clebsch and Brun). A relationship between these isomorphisms is indicated. The problem of compactification for geodesic flows on noncompact surfaces is stated. This problem is hypothesized to be intimately connected with the property of integrability.
Keywords: | quadric, geodesic flows, integrability, compactification, regularization, isomorphism |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Isomorphisms of geodesic flows on quadrics, Regular and Chaotic Dynamics, 2009, vol. 14, no. 4-5, |
DOI: | 10.1134/S1560354709040030 |
Full text: | pdf (376.58 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Kilin A. A.
The dynamics of self-gravitating liquid and gas ellipsoids is considered. A literary survey and authors’ original results obtained using modern techniques of nonlinear dynamics are presented. Strict Lagrangian and Hamiltonian formulations of the equations of motion are given; in particular, a Hamiltonian formalism based on Lie algebras is described. Problems related to nonintegrability and chaos are formulated and analyzed. All the known integrability cases are classified, and the most natural hypotheses on the nonintegrability of the equations of motion in the general case are presented. The results of numerical simulations are described. They, on the one hand, demonstrate a chaotic behavior of the system and, on the other hand, can in many cases serve as a numerical proof of the nonintegrability (the method of transversally intersecting separatrices).
Keywords: | liquid and gas self-gravitating ellipsoids, integrability, chaotic behavior |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Kilin A. A., The Hamiltonian Dynamics of Self-gravitating Liquid and Gas Ellipsoids, Regular and Chaotic Dynamics, 2009, vol. 14, no. 2, |
DOI: | 10.1134/S1560354709020014 |
Full text: | pdf (885.59 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
Systems of material points interacting both with one another and with an external field are considered in Euclidean space. For the case of arbitrary binary interaction depending solely on the mutual distance between the bodies, new integrals are found, which form a Galilean momentum vector. A corresponding algebra of integrals constituted by the integrals of momentum, angular momentum, and Galilean momentum is presented. Particle systems with a particleinteraction potential homogeneous of degree α=–2 are considered. The most general form of the additional integral of motion, which we term the Jacobi integral, is presented for such systems. A new nonlinear algebra of integrals including the Jacobi integral is found. A systematic description is given to a new reduction procedure and possibilities of applying it to dynamics with the aim of lowering the order of Hamiltonian systems.
Some new integrable and superintegrable systems generalizing the classical ones are also described. Certain generalizations of the Lagrangian identity for systems with a particle interaction potential homogeneous of degree α=–2 are presented. In addition, computational experiments are used to prove the nonintegrability of the Jacobi problem on a plane.
Keywords: | multiparticle systems, Jacobi integral |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Multiparticle Systems. The Algebra of Integrals and Integrable Cases, Regular and Chaotic Dynamics, 2009, vol. 14, no. 1, |
DOI: | 10.1134/S1560354709010043 |
Full text: | pdf (472.45 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
We consider the motion of a material point on the surface of a sphere in the field of 2n+1 identical Hooke centers (singularities with elastic potential) lying on a great circle. Our main result is that this system is superintegrable. The property of superintegrability for this system has been conjectured by us in [3], where the structure of a superintegral of arbitrarily high odd degree in momemnta was outlined. We also indicate an isomorphism between this system and the one-dimensional N-particle system discussed in the recent paper [13] and show that for the latter system an analogous superintegral can be constructed.
Keywords: | superintegrable systems, systems with a potential, Hooke center |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., New superintegrable system on a sphere, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 4, |
Full text: | pdf (214.58 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Vaskin V. V., Erdakova N. N., Mamaev I. S.
With the help of mathematical modeling, we study the behavior of a gas (∼106 particles) in a one-dimensional tube. For this dynamical system, we consider the following cases:
— collisionless gas (with and without gravity) in a tube with both ends closed, the particles of the gas bounce elastically between the ends,
— collisionless gas in a tube with its left end vibrating harmonically in a prescribed manner,
— collisionless gas in a tube with a moving piston, the piston’s mass is comparable to the mass of a particle.
The emphasis is on the analysis of the asymptotic (t→∞)) behavior of the system and specifically on the transition to the state of statistical or thermal equilibrium. This analysis allows preliminary conclusions on the nature of relaxation processes.
At the end of the paper the numerical and theoretical results obtained are discussed. It should be noted that not all the results fit well the generally accepted theories and conjectures from the standard texts and modern works on the subject.
Keywords: | one-dimensional collisionless gas, statistical equilibrium, thermodynamical equilibrium, weak limit |
---|---|
Citation: | Vaskin V. V., Erdakova N. N., Mamaev I. S., Statistical mechanics of nonlinear dynamical systems, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, |
Full text: | pdf (896.55 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The Poincaré model for dynamics of a collisionless gas in a rectangular parallelepiped with mirror walls is considered. The question on smoothing of the density and the temperature of this gas and conditions for the monotone growth of the coarse-grained entropy are discussed. All these effects provide a new insight of the classical paradox of mixing of gases.
Keywords: | collisionless gas, coarse-grained entropy, Gibbs paradox |
---|---|
Citation: | Kozlov V. V., Kinetics of collisionless gas: equalization of temperature, growth of the coarse-grained entropy and the Gibbs paradox, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, |
Full text: | pdf (208.37 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Ramodanov S. M.
The paper is concerned with a class of problems which involves the dynamical interaction of a rigid body with point vortices on the surface of a two-dimensional sphere. The general approach to the 2D hydrodynamics is further developed. The problem of motion of a dynamically symmetric circular body interacting with a single vortex is shown to be integrable. Mass vortices on S2 are introduced and the related issues (such as equations of motion, integrability, partial solutions, etc.) are discussed. This paper is a natural progression of the author’s previous research on interaction of rigid bodies and point vortices in a plane.
Keywords: | hydrodynamics on a sphere, coupled body-vortex system, mass vortex, equations of motion, integrability |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Ramodanov S. M., Coupled motion of a rigid body and point vortices on a sphere, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, |
Full text: | pdf (429.33 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
We consider several well-known isomorphisms between Jacobi’s geodesic problem and some integrable cases from rigid body dynamics (the cases of Clebsch and Brun). A relationship between these isomorphisms is indicated. The problem of compactification for geodesic flows on noncompact surfaces is stated. This problem is hypothesized to be intimately connected with the property of integrability.
Keywords: | quadric, geodesic flows, integrability, compactification, regularization, isomorphism |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Isomorphisms of geodesic flows on quadrics, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 2, |
Full text: | pdf (532.83 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
3-particle systems with a particle-interaction homogeneous potential of degree α=−2 is considered. A constructive procedure of reduction of the system by 2 degrees of freedom is performed. The nonintegrability of the systems is shown using the Poincare mapping.
Keywords: | multiparticle system, potential, Hamiltonian, reduction, integrability |
---|---|
Citation: | Kilin A. A., The Jacobi problem on a plane, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 1, |
Full text: | pdf (137.41 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
Systems of material points interacting both with one another and with an external field are considered in Euclidean space. For the case of arbitrary binary interaction depending solely on the mutual distance between the bodies, new integrals are found, which form a Galilean momentum vector.
A corresponding algebra of integrals constituted by the integrals of momentum, angular momentum, and Galilean momentum is presented. Particle systems with a particle-interaction potential homogeneous of degree α=−2 are considered. The most general form of the additional integral of motion, which we term the Jacobi integral, is presented for such systems. A new nonlinear algebra of integrals including the Jacobi integral is found. A systematic description is given to a new reduction procedure and possibilities of applying it to dynamics with the aim of lowering the order of Hamiltonian systems.
Some new integrable and superintegrable systems generalizing the classical ones are also described. Certain generalizations of the Lagrangian identity for systems with a particle-interaction potential homogeneous of degree α=−2 are presented. In addition, computational experiments are used to prove the nonintegrability of the Jacobi problem on a plane.
Keywords: | multiparticle systems, Jacobi integral |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Multiparticle Systems. The Algebra of Integrals and Integrable Cases, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 1, |
Full text: | pdf (508.81 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The generalized model of formation of a new phase is considered. The basic stages of process of growth are gathered in a model at phase transition of the first sort. The numerical solution of the kinetic equation of Fokker–Planck is received. Dependence of the solution on parametres of system is investigated. Areas of applicability of assumptions made by Zeldovich, Lifshits and Slezov are revealed. Also it is shown, that depending on parametres of system it is possible to reserve both equilibrium distribution, and automodelling distribution of Lifshits–Slezov. At some values of parametres the equation has the oscillatory solution.
Keywords: | Generalized model of kinetics of formation of a new phase |
---|---|
Citation: | Ivanova T. B., Vaskin V. V., Generalized model of kinetics of formation of a new phase, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2009, no. 2, |
DOI: | 10.20537/vm090212 |
Full text: | pdf (724.17 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
Borisov A. V., Mamaev I. S., Marikhin V. G.
Citation: | Borisov A. V., Mamaev I. S., Marikhin V. G., Explicit integration of one problem in nonholonomic mechanics, Doklady Physics, 2008, vol. 53, no. 10, |
---|---|
DOI: | 10.1134/S1028335808100066 |
Full text: | pdf (229.05 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
Citation: | Kozlov V. V., Gyroscopic stabilization of degenerate equilibria and the topology of real algebraic varieties, Doklady Mathematics, 2008, vol. 77, no. 3, |
---|---|
DOI: | 10.1134/S1064562408030253 |
Full text: | pdf (178.12 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
Citation: | Kozlov V. V., Novikov day at the Mathematical Institute, Russian Mathematical Surveys, 2008, vol. 63, no. 6, |
---|---|
DOI: | 10.4213/rm9241 |
Full text: | pdf (251.64 Kb) |
Impact-factor WoS (2022): | 0.900 (Q2) |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0036-0279 |
ISSN (online): | 1468-4829 |
Site: | http://www.mathnet.ru/umn |
Citation: | Kozlov V. V., The generalized Vlasov kinetic equation, Russian Mathematical Surveys, 2008, vol. 63, no. 4, |
---|---|
DOI: | 10.4213/rm9216 |
Full text: | pdf (719.96 Kb) |
Impact-factor WoS (2022): | 0.900 (Q2) |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0036-0279 |
ISSN (online): | 1468-4829 |
Site: | http://www.mathnet.ru/umn |
We study systems of differential equations admitting first integrals with degenerate critical points. We find conditions for the instability of equilibria for the cases in which the first integral loses the minimum property. Results of general nature are used in the proof of the impossibility of gyroscopic stabilization of equilibria in conservative mechanical systems under simple typical bifurcations.
Citation: | Kozlov V. V., On the instability of equilibria of conservative systems under typical degenerations, Differential Equations, 2008, vol. 44, no. 8, |
---|---|
DOI: | 10.1134/S001226610808003X |
Full text: | pdf (285.36 Kb) |
Impact-factor WoS (2015): | 0.344 |
---|---|
Impact-factor RSCI (2014): | 0,585 |
ISSN (print): | 0374-0641 |
Site: | http://www.maik.ru/ru/journal/deqrus/ |
The problem on the existence of an additional first integral of the equations of geodesics on noncompact algebraic surfaces is considered. This problem was discussed as early as by Riemann and Darboux. We indicate coarse obstructions to integrability, which are related to the topology of the real algebraic curve obtained as the line of intersection of such a surface with a sphere of large radius. Some yet unsolved problems are discussed.
Keywords: | geodesic flow, analytic first integral, geodesic convexity, M-curve |
---|---|
Citation: | Kozlov V. V., Topology of Real Algebraic Curves, Functional Analysis and Its Applications, 2008, vol. 42, no. 2, |
DOI: | 10.1007/s10688-008-0015-5 |
Full text: | pdf (117.67 Kb) |
Impact-factor WoS (2015): | 0.486 |
---|---|
Impact-factor RSCI (2014): | 0,565 |
ISSN (print): | 0374-1990 |
ISSN (online): | 2305-2899 |
Site: | http://www.mathnet.ru/faa |
Borisov A. V., Fedorov Y. N., Mamaev I. S.
We consider a nonholonomic system describing the rolling of a dynamically nonsymmetric sphere over a fixed sphere without slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits a separation of variables and reduction to Abel–Jacobi quadratures. The separating variables that we found appear to be a non-trivial generalization of ellipsoidal (spheroconic) coordinates on the Poisson sphere, which can be useful in other integrable problems.
Using the quadratures we also perform an explicit integration of the problem in theta-functions of the new time.
Keywords: | Chaplygin ball, explicit integration, nonholonomic mechanics |
---|---|
Citation: | Borisov A. V., Fedorov Y. N., Mamaev I. S., Chaplygin ball over a fixed sphere: an explicit integration, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, |
DOI: | 10.1134/S1560354708060063 |
Full text: | pdf (282.96 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
This paper can be regarded as a continuation of our previous work [1, 2] on the hierarchy of the dynamical behavior of nonholonomic systems. We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.
Keywords: | nonholonomic systems, implementation of constraints, conservation laws, hierarchy of dynamics, explicit integration |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems , Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, |
DOI: | 10.1134/S1560354708050079 |
Full text: | pdf (508.23 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The paper generalizes the classical Gauss principle for non-constrained dynamical systems. For large anisotropic external forces of viscous friction our statement transforms into the common Gauss principle for systems with constraints.
Keywords: | Gauss principle, constraints, anisotropic friction |
---|---|
Citation: | Kozlov V. V., Gauss Principle and Realization of Constraints, Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, |
DOI: | 10.1134/S1560354708050055 |
Full text: | pdf (144.15 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, subject to gravity, on the surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady rotations of the ball at the upmost, downmost and saddle point.
Keywords: | nonholonomic constraint, stationary rotations, stability |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Stability of Steady Rotations in the Nonholonomic Routh Problem, Regular and Chaotic Dynamics, 2008, vol. 13, no. 4, |
DOI: | 10.1134/S1560354708040011 |
Full text: | pdf (392.42 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
In this paper, we consider the transition to chaos in the phase portrait of a restricted problem of rotation of a rigid body with a fixed point. Two interrelated mechanisms responsible for chaotization are indicated: (1) the growth of the homoclinic structure and (2) the development of cascades of period doubling bifurcations. On the zero level of the area integral, an adiabatic behavior of the system (as the energy tends to zero) is noted. Meander tori induced by the break of the torsion property of the mapping are found.
Keywords: | motion of a rigid body, phase portrait, mechanism of chaotization, bifurcations |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Chaos in a Restricted Problem of Rotation of a Rigid Body with a Fixed Point, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, |
DOI: | 10.1134/S1560354708030076 |
Full text: | pdf (491.62 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
For the classical problem of motion of a rigid body about a fixed point with zero area integral, we present a family of solutions that are periodic in the absolute space. Such solutions are known as choreographies. The family includes the well-known Delone solutions (for the Kovalevskaya case), some particular solutions for the Goryachev–Chaplygin case, and the Steklov solution. The "genealogy" of solutions of the family naturally appearing from the energy continuation and their connection with the Staude rotations are considered. It is shown that if the integral of areas is zero, the solutions are periodic with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies).
Keywords: | rigid-body dynamics, periodic solutions, continuation by a parameter, bifurcation |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Absolute and Relative Choreographies in Rigid Body Dynamics, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, |
DOI: | 10.1134/S1560354708030064 |
Full text: | pdf (447.19 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The paper develops an approach to the proof of the "zeroth" law of thermodynamics. The approach is based on the analysis of weak limits of solutions to the Liouville equation as time grows infinitely. A class of linear oscillating systems is indicated for which the average energy becomes eventually uniformly distributed among the degrees of freedom for any initial probability density functions. An example of such systems are sympathetic pendulums. Conditions are found for nonlinear Hamiltonian systems with finite number of degrees of freedom to converge in a weak sense to the state where the mean energies of the interacting subsystems are the same. Some issues related to statistical models of the thermostat are discussed.
Keywords: | Hamiltonian system, sympathetic oscillators, weak convergence, thermostat |
---|---|
Citation: | Kozlov V. V., Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, |
DOI: | 10.1134/S1560354708030015 |
Full text: | pdf (192.61 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The famous Lagrange identity expresses the second derivative of the moment of inertia of a system of material points through the kinetic energy and homogeneous potential energy. The paper presents various extensions of this brilliant result to the case 1) of constrained mechanical systems, 2) when the potential energy is quasi-homogeneous in coordinates and 3) of continuum of interacting particles governed by the well-known Vlasov kinetic equation.
Keywords: | Lagrange's identity, quasi-homogeneous function, dilations, Vlasov’s equation |
---|---|
Citation: | Kozlov V. V., Lagrange’s Identity and Its Generalizations, Regular and Chaotic Dynamics, 2008, vol. 13, no. 2, |
DOI: | 10.1134/S1560354708020019 |
Full text: | pdf (144.77 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Ramodanov S. M.
The paper develops further the algebraic-reduction method for SO(4)-symmetric systems on the three-dimensional sphere. Canonical variables for the reduced system are constructed both on two-dimensional and three-dimensional spheres. The method is illustrated by applying it to the two-body problem on a sphere (the bodies are assumed to interact with a potential that depends only on the geodesic distance between them) and the three-vortex problem on a two-dimensional sphere.
Keywords: | Poisson structure, Lie algebra, subalgebra, Andoyer variables |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Ramodanov S. M., Algebraic reduction of systems on two- and three-dimensional spheres, Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 4, |
Full text: | pdf (180.6 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Kilin A. A.
The paper contains the review and original results on the dynamics of liquid and gas self-gravitating ellipsoids. Equations of motion are given in Lagrangian and Hamiltonian form, in particular, the Hamiltonian formalism on Lie algebras is presented. Problems of nonintegrability and chaotical behavior of the system are formulated and studied. We also classify all known integrable cases and give some hypotheses about nonintegrability in the general case. Results of numerical modelling are presented, which can be considered as a computer proof of nonintegrability.
Keywords: | liquid and gas self-gravitating ellipsoids, integrability, chaotic behavior |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Kilin A. A., Hamiltonian Dynamics of Liquid and Gas Self-Gravitating Ellipsoids, Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 4, |
Full text: | pdf (994.54 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The paper generalizes the classical Gauss principle for non-constrained dynamical systems. For large anisotropic external forces of viscous friction our statement transforms into the common Gauss principle for systems with constraints.
Keywords: | Gauss principle, constraints, anisotropic friction |
---|---|
Citation: | Kozlov V. V., Gauss Principle and Realization of Constraints, Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 3, |
Full text: | pdf (78.44 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
This paper can be regarded as a continuation of our previous work [70,71] on the hierarchy of the dynamical behavior of nonholonomic systems. We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.
Keywords: | nonholonomic systems, implementation of constraints, conservation laws, hierarchy of dynamics, explicit integration |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 3, |
Full text: | pdf (634.39 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The famous Lagrange identity expresses the second derivative of the moment of inertia of a system of material points through the kinetic energy and homogeneous potential energy. The paper presents various extensions of this brilliant result to the case 1) of constrained mechanical systems, 2) when the potential energy is quasi-homogeneous in coordinates and 3) of continuumof interacting particles governed by the well-known Vlasov kinetic equation.
Keywords: | Lagrange’s identity, quasi-homogeneous function, dilations, Vlasov’s equation |
---|---|
Citation: | Kozlov V. V., Lagrange’s identity and its generalizations, Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 2, |
Full text: | pdf (128.42 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
We discuss system of material points in Euclidean space interacting both with each other and with external field. In particular we consider systems of particles whose interacting is described by homogeneous potential of degree of homogeneity α=−2. Such systems were first considered by Newton and—more systematically—by Jacobi). For such systems there is an extra hidden symmetry, and corresponding first integral of motion which we call Jacobi integral. This integral was given in different papers starting with Jacobi, but we present in general. Furthermore, we construct a new algebra of integrals including Jacobi integral. A series of generalizations of Lagrange's identity for systems with homogeneous potential of degree of homogeneity α=−2 is given. New integrals of motion for these generalizations are found.
Keywords: | Lagrange’s identity, many-particle system, first integral, integrability, algebra of integrals |
---|---|
Citation: | Kilin A. A., Generalization of Lagrange’s identity and new integrals of motion, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2008, no. 3, |
DOI: | 10.20537/vm080308 |
Full text: | pdf (154.14 Kb) |
Impact-factor WoS (2022): | 0.500 |
---|---|
Impact-factor RSCI (2022): | 0.342 (Q3) |
ISSN (print): | 1994-9197 |
ISSN (online): | 2076-5959 |
Site: | http://vst.ics.org.ru |
The motion of two vortex rings on a sphere is considered. This motion generalizes the well-known centrally symmetrical solution of the equations of point vortex dynamics on a plane derived by D.N. Goryachev, N.S. Vasiliev and H. Aref. The equations of motion in this case are shown to be Liouville integrable, and an explicit reduction to a Hamiltonian system with one degree of freedom is described. Two particular cases in which the solutions are periodical are presented. Explicit quadratures are given for these solutions. Phase portraits are described and bifurcation diagrams are shown for centrally symmetrical motion of four vortices on a sphere.
Keywords: | Vortices, Hamiltonian, motion on a sphere, phase portrait |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Dynamics of Two Rings of Vortices on a Sphere, in IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, Springer, 2008, vol. 6, |
DOI: | 10.1007/978-1-4020-6744-0_40 |
Full text: | pdf (7.9 Mb) |
Borisov A. V., Kilin A. A., Mamaev I. S.
The dynamics of an antipodal vortex on a sphere (a point vortex plus its antipode with opposite circulation) is considered. It is shown that the system of n antipodal vortices can be reduced by four dimensions (two degrees of freedom). The cases n=2,3 are explored in greater detail both analytically and numerically. We discuss Thomson, collinear and isosceles configurations of antipodal vortices and study their bifurcations.
Keywords: | Hydrodynamics, ideal fluid, vortex dynamics, point vortex, reduction, bifurcation analysis |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., A New Integrable Problem of Motion of Point Vortices on the Sphere, in IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, Springer, 2008, vol. 6, |
DOI: | 10.1007/978-1-4020-6744-0_4 |
Full text: | pdf (10.34 Mb) |
We discuss some open problems in the theory of dynamical systems, classical and quantum mechanics.
Citation: | Kozlov V. V., Several problems on dynamical systems and mechanics, Nonlinearity, 2008, vol. 21, no. 9, |
---|---|
DOI: | 10.1088/0951-7715/21/9/T01 |
Full text: | pdf (79.46 Kb) |
Impact-factor WoS (2022): | 1.700 (Q2) |
---|---|
ISSN (print): | 0951-7715 |
ISSN (online): | 1361-6544 |
Site: | http://iopscience.iop.org/0951-7715 |
Citation: | Kozlov V. V., Statistical properties of billiards in polytopes, Doklady Mathematics, 2007, vol. 76, no. 2, |
---|---|
DOI: | 10.1134/S1064562407050158 |
Full text: | pdf (169.48 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
This article concerns the life of Leonhard Euler and his achievements in theoretical mechanics. A number of topics are discussed related to the development of Euler’s ideas and methods: divergent series and asymptotics of solutions of non-linear differential equations; the hydrodynamics of a perfect fluid and Hamiltonian systems; vortex theory for systems on Lie groups with left-invariant kinetic energy; energy criteria of stability; Euler’s problem of two gravitating centres in curved spaces.
Citation: | Kozlov V. V., Euler and mathematical methods in mechanics (on the 300th anniversary of the birth of Leonhard Euler), Russian Mathematical Surveys, 2007, vol. 62, no. 4, |
---|---|
DOI: | 10.1070/RM2007v062n04ABEH004427 |
Full text: | pdf (425.6 Kb) |
Impact-factor WoS (2022): | 0.900 (Q2) |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0036-0279 |
ISSN (online): | 1468-4829 |
Site: | http://www.mathnet.ru/umn |
We consider some questions connected with the Hamiltonian form of two problems of nonholonomic mechanics, namely the Chaplygin ball problem and the Veselova problem. For these problems we find representations in the form of the generalized Chaplygin systems that can be integrated by the reducing multiplier method. We give a concrete algebraic form of the Poisson brackets which, together with an appropriate change of time, enable us to write down the equations of motion of the problems under study. Some generalization of these problems are considered and new ways of implementation of nonholonomic constraints are proposed. We list a series of nonholonomic systems possessing an invariant measure and sufficiently many first integrals for which the question about the Hamiltonian form remains open even after change of time. We prove a theorem on isomorphism of the dynamics of the Chaplygin ball and the motion of a body in a fluid in the Clebsch case.
Keywords: | nonholonomic system, reducing multiplier, Hamiltonization, isomorphism |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Isomorphism and Hamilton representation of some nonholonomic systems, Siberian Mathematical Journal, 2007, vol. 48, no. 1, |
DOI: | 10.1007/s11202-007-0004-6 |
Full text: | pdf (154.1 Kb) |
Impact-factor WoS (2022): | 0.500 (Q4) |
---|---|
Impact-factor RSCI (2022): | (Q2) |
ISSN (print): | 0037-4466 |
ISSN (online): | 1573-9260 |
Site: | http://a-server.math.nsc.ru/publishing/smz/index.php |
Borisov A. V., Kozlov V. V., Mamaev I. S.
We consider two problems from the rigid body dynamics and use new methods of stability and asymptotic behavior analysis for their solution. The first problem deals with motion of a rigid body in an unbounded volume of ideal fluid with zero vorticity. The second problem, having similar asymptotic behavior, is concerned with motion of a sleigh on an inclined plane. The equations of motion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. A comprehensive survey of references is given and new problems connected with falling motion of heavy bodies in fluid are proposed.
Keywords: | rigid body, ideal fluid, non-holonomic mechanics |
---|---|
Citation: | Borisov A. V., Kozlov V. V., Mamaev I. S., Asymptotic stability and associated problems of dynamics of falling rigid body, Regular and Chaotic Dynamics, 2007, vol. 12, no. 5, |
DOI: | 10.1134/S1560354707050061 |
Full text: | pdf (1.81 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, J. Koiller and K. Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.
Keywords: | nonholonomic mechanics, reducing multiplier, hamiltonization, isomorphism |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Rolling of a Non-homogeneous Ball Over a Sphere Without Slipping and Twisting, Regular and Chaotic Dynamics, 2007, vol. 12, no. 2, |
DOI: | 10.1134/S1560354707010030 |
Full text: | pdf (189.47 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
We consider the interaction of two vortex patches (elliptic Kirchhoff vortices) which move in an unbounded volume of an ideal incompressible fluid. A moment second-order model is used to describe the interaction. The case of integrability of a Kirchhoff vortex and a point vortex is qualitatively analyzed. A new case of integrability of two Kirchhoff vortices is found by the variable separation method . A reduced form of equations for two Kirchhoff vortices is proposed and used to analyze their regular and chaotic behavior.
Keywords: | vortex patch, point vortex, integrability |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Interaction between Kirchhoff vortices and point vortices in an ideal fluid, Regular and Chaotic Dynamics, 2007, vol. 12, no. 1, |
DOI: | 10.1134/S1560354707010066 |
Full text: | pdf (358.32 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Ramodanov S. M.
The paper deals with the derivation of the equations of motion for two spheres in an unbounded volume of ideal and incompressible fluid in 3D Euclidean space. Reduction of order, based on the use of new variables that form a Lie algebra, is offered. A trivial case of integrability is indicated.
Keywords: | motion of two spheres, ideal fluid, reduction, integrability |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Ramodanov S. M., Motion of two spheres in ideal fluid. I. Equations o motions in the Euclidean space. First integrals and reduction, Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 4, |
Full text: | pdf (182.63 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Kozlov V. V., Mamaev I. S.
We consider two problems from the rigid body dynamics and use new methods of stability and asymptotic behavior analysis for their solution. The first problem deals with motion of a rigid body in an unbounded volume of ideal fluid with zero vorticity. The second problem, having similar asymptotic behavior, is concerned with motion of a sleigh on an inclined plane. The equations of motion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. A comprehensive survey of references is given and new problems connected with falling motion of heavy bodies in fluid are proposed.
Keywords: | nonholonomic mechanics, rigid body, ideal fluid, resisting medium |
---|---|
Citation: | Borisov A. V., Kozlov V. V., Mamaev I. S., Asymptotic stability and associated problems of dynamics of falling rigid body, Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 3, |
Full text: | pdf (1.62 Mb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
The dynamics of an antipodal vortex on a sphere (a point vortex plus its antipode with opposite circulation) is considered. It is shown that the system of n antipodal vortices can be reduced by four dimensions (two degrees of freedom). The cases n=2,3 are explored in greater detail both analytically and numerically. We discuss Thomson, collinear and isosceles configurations of antipodal vortices and study their bifurcations.
Keywords: | hydrodynamics, ideal fluid, vortex dynamics, point vortex, reduction, bifurcation analysis |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., A New Integrable Problem of Motion of Point Vortices on the Sphere, Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 2, |
Full text: | pdf (298.41 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The paper develops an approach to the proof of the «zeroth» law of thermodynamics. The approach is based on the analysis of weak limits of solutions to the Liouville equation as time grows infinitely. A class of linear oscillating systems is indicated for which the average energy becomes eventually uniformly distributed among the degrees of freedom for any initial probability density functions. An example of such systems are sympathetic pendulums. Conditions are found for nonlinear Hamiltonian systems with finite number of degrees of freedom to converge in a weak sense to the state where the average energies of the interacting subsystems are the same. Some issues related to statistical models of the thermostat are discussed.
Keywords: | Hamiltonian system, sympathetic oscillators, weak convergence, thermostat |
---|---|
Citation: | Kozlov V. V., Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat, Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 2, |
Full text: | pdf (263.66 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
We consider trajectory isomorphisms between various integrable systems on an n-dimensional sphere Sn and a Euclidean space Rn. Some of the systems are classical integrable problems of Celestial Mechanics in plane and curved spaces. All the systems under consideration have an additional first integral quadratic in momentum and can be integrated analytically by using the separation of variables. We show that some integrable problems in constant curvature spaces are not essentially new from the viewpoint of the theory of integration, and they can be analyzed using known results of classical Celestial Mechanics.
Keywords: | integrable systems, two-center problem, isomorphisms |
---|---|
Citation: | Borisov A. V., Mamaev I. S., On isomorphisms of some integrable systems on a plane and a sphere, Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 1, |
Full text: | pdf (166.52 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Ramodanov S. M.
In this paper we consider the system of an arbitrary two-dimensional cylinder interacting with point vortices in a perfect fluid. We present the equations of motion and discuss their integrability. Simulations show that the system of an elliptic cylinder (with nonzero eccentricity) and a single point vortex already exhibits chaotic features and the equations of motion are nonintegrable. We suggest a Hamiltonian form of the equations. The problem we study here, namely, the equations of motion, the Hamiltonian structure for the interacting system of a cylinder of arbitrary cross-section shape, with zero circulation around it, and N vortices, has been addressed by Shashikanth [Regular Chaotic Dyn. 10, 1 (2005)]. We slightly generalize the work by Shashikanth by allowing for nonzero circulation around the cylinder and offer a different approach than that by Shashikanth by using classical complex variable theory.
Citation: | Borisov A. V., Mamaev I. S., Ramodanov S. M., Dynamic Interaction of Point Vortices and a Two-Dimensional Cylinder, Journal of Mathematical Physics, 2007, vol. 48, no. 6, 065403, |
---|---|
DOI: | 10.1063/1.2425100 |
Full text: | pdf (126.99 Kb) |
Impact-factor WoS (2022): | 1.300 (Q3) |
---|---|
Impact-factor RSCI (2022): | 0.599 (Q2) |
ISSN (print): | 0022-2488 |
ISSN (online): | 1089-7658 |
Site: | http://scitation.aip.org/content/aip/journal/jmp |
We consider trajectory isomorphisms between various integrable systems on an n-dimensional sphere Sn and a Euclidean space Rn. Some of the systems are classical integrable problems of Celestial Mechanics in plane and curved spaces. All the systems under consideration have an additional first integral quadratic in momentum and can be integrated analytically by using the separation of variables. We show that some integrable problems in constant curvature spaces are not essentially new from the viewpoint of the theory of integration, and they can be analyzed using known results of classical Celestial Mechanics.
Keywords: | Integrable systems, Euclidean spaces |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Relations between Integrable Systems in Plane and Curved Spaces, Celestial Mechanics and Dynamical Astronomy, 2007, vol. 99, no. 4, |
DOI: | 10.1007/s10569-007-9098-1 |
Full text: | pdf (151.07 Kb) |
Impact-factor WoS (2022): | 1.600 (Q3) |
---|---|
ISSN (print): | 0923-2958 |
ISSN (online): | 1572-9478 |
Site: | http://link.springer.com/journal/10569 |
Borisov A. V., Mamaev I. S., Ramodanov S. M.
In this paper we consider the system of two 2D rigid circular cylinders immersed in an unbounded volume of inviscid perfect fluid. The circulations around the cylinders are assumed to be equal in magnitude and opposite in sign. We also explore some special cases of this system assuming that the cylinders move along the line through their centers and the circulation around each cylinder is zero. A similar system of two interacting spheres was originally considered in the classical works of Carl and Vilhelm Bjerknes, H. Lamb and N.E. Joukowski. By making the radii of the cylinders infinitesimally small, we have obtained a new mechanical system which consists of two regular point vortices but with non-zero masses. The study of this system can be reduced to the study of the motion of a particle subject to potential and gyroscopic forces. A new integrable case is found. The Hamiltonian equations of motion for this system have been generalized to the case of an arbitrary number of mass vortices with arbitrary intensities. Some first integrals have been obtained. These equations expand upon the classical Kirchhoff equations of motion for n point vortices.
Keywords: | Dimension theory, Poincar\'e recurrences, multifractal analysis |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Ramodanov S. M., Dynamics of Two Interacting Circular Cylinders in Perfect Fluid, Discrete and Continuous Dynamical Systems - Series A, 2007, vol. 19, no. 2, |
DOI: | 10.3934/dcds.2007.19.235 |
Full text: | pdf (350.69 Kb) |
Impact-factor WoS (2015): | 1.127 |
---|---|
ISSN (print): | 1078-0947 |
ISSN (online): | 1553-5231 |
Site: | https://www.aimsciences.org/journals/home.jsp?journalID=1 |
Borisov A. V., Kilin A. A., Mamaev I. S.
The paper considers the dynamics of a rattleback as a model of a heavy balanced ellipsoid of revolution rolling without slippage on a fixed horizontal plane. Central ellipsoid of inertia is an ellipsoid of revolution as well. In presence of the angular displacement between two ellipsoids, there occur dynamical effects somewhat similar to the reverse fenomena in earlier models. However, unlike a customary rattleback model (a truncated biaxial paraboloid) our system allows the motions which are superposition of the reverse motion (reverse of the direction of spinning) and the turn over (change of the axis of rotation). With appropriate values of energies and mass distribution, this effect (reverse + turn over) can occur more than once. Such motions as repeated reverse or repeated turn over are also possible.
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., New effects in dynamics of rattlebacks, Doklady Physics, 2006, vol. 408, no. 2, |
---|---|
DOI: | 10.1134/S1028335806050107 |
Full text: | pdf (214.19 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
The paper considers the process of transition to chaos in the problem of four point vortices on a plane. A new method for constructive reduction of the order for a system of vortices on a plane is presented. Existence of the cascade of period doubling bifurcations in the given problem is indicated.
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Transition to chaos in dynamics of four point vortices on a plane, Doklady Physics, 2006, vol. 51, no. 5, |
---|---|
DOI: | 10.1134/S1028335806050089 |
Full text: | pdf (249.72 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
The rolling motion of a dynamically nonsymmetric balanced ball (Chaplygin ball) on an inclined plane is studied. For the case of a horizontal plane, Chaplygin demonstrated this problem to be integrable. For a nonzero slope, the system is integrable only if the motion starts from a state of rest (E.N. Kharlamova). It is shown that, in the general case, the system exhibits a rather simple asymptotic behavior.
Citation: | Borisov A. V., Mamaev I. S., Motion of Chaplygin ball on an inclined plane, Doklady Physics, 2006, vol. 51, no. 2, |
---|---|
DOI: | 10.1134/S1028335806020078 |
Full text: | pdf (212.42 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
The paper considers a general case of rolling motion of a rigid body with sharp edge on an icy sphere in a field of gravity. Cases of integrability are indicated and probability of a body fall is analyzed.
Keywords: | integrable system, nonholonomic constraint, Schwarzschild–Littlewood theorem |
---|---|
Citation: | Borisov A. V., Mamaev I. S., An Integrable System with a Nonintegrable Constraint, Mathematical Notes, 2006, vol. 80, no. 1, |
DOI: | 10.1007/s11006-006-0116-5 |
Full text: | pdf (98.56 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,502 |
ISSN (print): | 0025-567X |
ISSN (online): | 2305-2880 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=mzm&option_lang=eng |
Borisov A. V., Kilin A. A., Mamaev I. S.
Rolling (without slipping) of a homogeneous ball on an oblique cylinder in different potential fields and the integrability of the equations of motion are considered. We examine also if the equations can be reduced to a Hamiltonian form. We prove the theorem stated that if there is a gravity (and the cylinder is oblique), the ball moves without any vertical shift, on the average.
Keywords: | nonholonomic dynamics, rolling motion without slipping, nonholonomic constraints, quasiperiodic oscillations |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., On a Nonholonomic Dynamical Problem, Mathematical Notes, 2006, vol. 79, no. 5, |
DOI: | 10.1007/s11006-006-0085-8 |
Full text: | pdf (189.22 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,502 |
ISSN (print): | 0025-567X |
ISSN (online): | 2305-2880 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=mzm&option_lang=eng |
Equations of motion of vortex sources (examined earlier by Fridman and Polubarinova) are studied, and the problems of their being Hamiltonian and integrable are discussed. A system of two vortex sources and three sources-sinks was examined. Their behavior was found to be regular. Qualitative analysis of this system was made, and the class of Liouville integrable systems is considered. Particular solutions analogous to the homothetic configurations in celestial mechanics are given.
Keywords: | vortex sources, integrability, Hamiltonian, point vortex |
---|---|
Citation: | Borisov A. V., Mamaev I. S., On the problem of motion of vortex sources on a plane , Regular and Chaotic Dynamics, 2006, vol. 11, no. 4, |
DOI: | 10.1070/RD2006v011n04ABEH000363 |
Full text: | pdf (377.53 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, Koiller and Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.
Keywords: | Chaplygin ball, rolling model, Hamiltonian structure |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Rolling of a heterotgeneous ball over a sphere without sliding and spinning, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 4, |
Full text: | pdf (162.92 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The issues discussed in this paper relate to the description of developed two-dimensional turbulence, when the mean values of characteristics of steady flow stabilize. More exactly, the problem of a weak limit of vortex distribution in two-dimensional flow of an ideal fluid at time tending to infinity is considered. Relations between the vorticity equation and the well-known Vlasov equation are discussed.
Keywords: | vortex motion equation, vorticity, Vlasov equation |
---|---|
Citation: | Kozlov V. V., Vorticity equation of 2D-hydrodynamics, Vlasov steady-state kinetic equation and developed turbulence, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 4, |
Full text: | pdf (152.1 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, subject to gravity, on the surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady rotations of the ball in the upmost, downmost and saddle point.
Keywords: | nonholonomic constraint, stationary rotations, stability |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Stability of steady rotations in the non-holonomic Routh problem, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 3, |
Full text: | pdf (398.23 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
We present a reduction-of-order procedure in the problem of motion of two bodies on the Lobatchevsky plane H2. The bodies interact with a potential that depends only on the distance between the bodies (this holds for an analog of the Newtonian potential). In earlier works, this reduction procedure was used to analyze the motion of two bodies on the sphere
Keywords: | Lobatchevsky plane, first integral, reduction-of-order procedure, potential of interaction |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Reduction in the two-body problem on the Lobatchevsky plane, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 3, |
Full text: | pdf (148.66 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
We consider the interaction of two vortex patches (elliptic Kirchhoff vortices) which move in an unbounded volume of an ideal incompressible fluid. A moment second-order model is used to describe the interaction. The case of integrability of a Kirchhoff vortex and a point vortex by the variable separation method is qualitatively analyzed. A new case of integrability of two Kirchhoff vortices is found. A reduced form of equations for two Kirchhoff vortices is proposed and used to analyze their regular and chaotic behavior.
Keywords: | Kirchhoff vortices, integrability, Hamiltonian, stability, point vortex |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Interaction between Kirchhoff vortices and point vortices in an ideal fluid, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 2, |
Full text: | pdf (535.61 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
In the paper we present a new integral of motion in the problem of rolling motion of a heavy symmetric sphere on the surface of a paraboloid. We use this integral to study the Lyapunov stability of some trivial steady rotations.
Keywords: | dynamical sysytem, non-holonomic constraint, integral, periodic solution, Lyapunov stability |
---|---|
Citation: | Kilin A. A., New integral in nonholonomic Painleve-Chaplygin problem, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 2, |
Full text: | pdf (132.6 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
The motion of two vortex rings on a sphere is considered. This motion generalizes the well-known centrally symmetrical solution of the equations of point vortex dynamics on a plane derived by D.N. Goryachev and H. Aref. The equations of motion in this case are shown to be Liouville integrable, and an explicit reduction to a Hamiltonian system with one degree of freedom is described. Two particular cases in which the solutions are periodical are presented. Explicit quadratures are given for these solutions. Phase portraits are described and bifurcation diagrams are shown for centrally symmetrical motion of four vortices on a sphere.
Keywords: | vortex, Hamiltonian, motion on a sphere, phase portrait |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Dynamics of two vortex rings on a sphere, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 2, |
Full text: | pdf (321.21 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
We consider Chaplygin's equations [Izd. Akad. Nauk SSSR 3(3), 1933] describing the planar motion of a rigid body in an unbounded volume of an ideal fluid while circulation around the body is not zero. Hamiltonian structures and new integrable cases are revealed; certain remarkable partial solutions are found and their stability is examined. The nonintegrability of the system describing the motion of a body in the field of gravity is proved and the chaotic behavior of the system is illustrated.
Citation: | Borisov A. V., Mamaev I. S., On the motion of a heavy rigid body in an ideal fluid with circulation, Chaos, 2006, vol. 16, no. 1, 013118, |
---|---|
DOI: | 10.1063/1.2166530 |
Full text: | pdf (303.93 Kb) |
Impact-factor WoS (2022): | 2.900 (Q1) |
---|---|
ISSN (print): | 1054-1500 |
ISSN (online): | 1089-7682 |
Site: | http://scitation.aip.org/content/aip/journal/chaos |
The bifurcation analysis of the Kepler problem on S3 and H3 is performed. An analogue of the Delaunay variables is introduced and the motion of a point mass in the field of the Newtonian center moving along a geodesic on S2 and H2 (the restricted two-body problem) is investigated. When the curvature is small, the pericenter shift is computed using the perturbation theory. We also present the results of the numerical analysis based on the analogy with the motion of rigid body.
Keywords: | Kepler problem, Bifurcation analysis, Perihelion shift, Delaunay variables, Restricted two-body problem |
---|---|
Citation: | Borisov A. V., Mamaev I. S., The restricted two-body problem in constant curvature spaces, Celestial Mechanics and Dynamical Astronomy, 2006, vol. 96, no. 1, |
DOI: | 10.1007/s10569-006-9012-2 |
Full text: | pdf (479.18 Kb) |
Impact-factor WoS (2022): | 1.600 (Q3) |
---|---|
ISSN (print): | 0923-2958 |
ISSN (online): | 1572-9478 |
Site: | http://link.springer.com/journal/10569 |
This paper presents the most complete classification of Birkhoff-integrable generalized Toda lattices and considers new integrable lattices.
Citation: | Borisov A. V., Mamaev I. S., Classification of Birkhoff-Integrable generalized Toda lattices, in "Topological Methods in the Theory of Integrable Systems", Cambridge: Cambridge Scientific Publishers Ltd., 2006, |
---|---|
Full text: | pdf (560.25 Kb) |
Citation: | Kozlov V. V., To the piston problem, Doklady Mathematics, 2005, vol. 72, no. 1, |
---|---|
Full text: | pdf (188.72 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
Citation: | Kozlov V. V., Remarks on eigenvalues of real matrices, Doklady Mathematics, 2005, vol. 72, no. 1, |
---|---|
Full text: | pdf (186.42 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
Citation: | Kozlov V. V., Topological obstructions to the existence of quantum conservation laws, Doklady Mathematics, 2005, vol. 71, no. 2, |
---|---|
Full text: | pdf (138.71 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
Borisov A. V., Kilin A. A., Mamaev I. S.
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Absolute and Relative Choreographies in the Problem of the Motion of Point Vortices in a Plane, Doklady Physics, 2005, vol. 71, no. 1, |
---|---|
Full text: | pdf (168.54 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
We strengthen the well-known Oxtoby theorem for strictly ergodic transformations by replacing the standard Cesaro convergence by the weaker Riesz or Voronoi convergence with monotonically increasing or decreasing weight coefficients. This general result allows, in particular, to strengthen the classical Weyl theorem on the uniform distribution of fractional parts of polynomials with irrational coefficients.
Keywords: | Weyl theorem, Cesaro and Voronoi convergences, Borel measure, Oxtoby's theorem, strictly ergodic transformation, Riesz summation method |
---|---|
Citation: | Kozlov V. V., Weighted Means, Strict Ergodicity, and Uniform Distributions, Mathematical Notes, 2005, vol. 78, no. 3, |
DOI: | 10.1007/s11006-005-0132-x |
Full text: | pdf (155.58 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,502 |
ISSN (print): | 0025-567X |
ISSN (online): | 2305-2880 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=mzm&option_lang=eng |
This note presents new conditions for nonexistance of an invariant measure for an inhomogeneous ellipsoid with the special mass distribution rolling on an absolutely rough plane. This work supplements results on the nonexistence of the measure in the rolling of a rattleback.
Keywords: | invariant measure, rolling ellipsoid, Liouville equation, Celtic stone |
---|---|
Citation: | Borisov A. V., Mamaev I. S., The Nonexistence of an Invariant Measure for an Inhomogeneous Ellipsoid Rolling on a Plane, Mathematical Notes, 2005, vol. 77, no. 6, |
DOI: | 10.1007/s11006-005-0085-0 |
Full text: | pdf (74.38 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,502 |
ISSN (print): | 0025-567X |
ISSN (online): | 2305-2880 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=mzm&option_lang=eng |
A circle of problems related to the application of the Riesz and Voronoi summation methods in ergodic theory, number theory, and probability theory is considered. The first digit paradox is discussed, strengthenings of the classical result of Weyl on the uniform distribution of the fractional parts of the values of a polynomial are indicated, and the possibility of sharpening the Birkhoff–Khinchin ergodic theorem is considered. In conclusion, some unsolved problems are listed.
Citation: | Kozlov V. V., Weighted averages, uniform distribution, and strict ergodicity, Russian Mathematical Surveys, 2005, vol. 60, no. 6, |
---|---|
DOI: | 10.1070/RM2005v060n06ABEH004284 |
Full text: | pdf (293.33 Kb) |
Impact-factor WoS (2022): | 0.900 (Q2) |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0036-0279 |
ISSN (online): | 1468-4829 |
Site: | http://www.mathnet.ru/umn |
Citation: | Kozlov V. V., Remarks on a Lie theorem on the exact integrability of differential equations, Differential Equations, 2005, vol. 41, no. 4, |
---|---|
DOI: | 10.1007/s10625-005-0193-3 |
Full text: | pdf (85.01 Kb) |
Impact-factor WoS (2015): | 0.344 |
---|---|
Impact-factor RSCI (2014): | 0,585 |
ISSN (print): | 0374-0641 |
Site: | http://www.maik.ru/ru/journal/deqrus/ |
We discuss the symplectic geometry of linear Hamiltonian systems with nondegenerate Hamiltonians. These systems can be reduced to linear second-order differential equations characteristic of linear oscillation theory. This reduction is related to the problem on the signatures of restrictions of quadratic forms to Lagrangian planes. We study vortex symplectic planes invariant with respect to linear Hamiltonian systems. These planes are determined by the solutions of quadratic matrix equations of a special form. New conditions for gyroscopic stabilization are found.
Keywords: | Hamiltonian function, symplectic structure, quadratic form, Williamson normal form, vortex plane |
---|---|
Citation: | Kozlov V. V., Restrictions of Quadratic Forms to Lagrangian Planes, Quadratic Matrix Equations, and Gyroscopic Stabilization, Functional Analysis and Its Applications, 2005, vol. 39, no. 4, |
DOI: | 10.1007/s10688-005-0048-y |
Full text: | pdf (170.93 Kb) |
Impact-factor WoS (2015): | 0.486 |
---|---|
Impact-factor RSCI (2014): | 0,565 |
ISSN (print): | 0374-1990 |
ISSN (online): | 2305-2899 |
Site: | http://www.mathnet.ru/faa |
We consider various generalizations of the Kepler problem to three-dimensional sphere S3, (a compact space of constant curvature). In particular, these generalizations include addition of a spherical analogue of the magnetic monopole (the Poincaré–Appell system) and addition of a more complicated field which is a generalization of the MICZ-system. The mentioned systems are integrable superintegrable, and there exists the vector integral which is analogous to the Laplace–Runge–Lenz vector. We offer a classification of the motions and consider a trajectory isomorphism between planar and spatial motions. The presented results can be easily extended to Lobachevsky space L3.
Keywords: | spaces of constant curvature, Kepler problem, integrability |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Superintegrable systems on a sphere , Regular and Chaotic Dynamics, 2005, vol. 10, no. 3, |
DOI: | 10.1070/RD2005v010n03ABEH000314 |
Full text: | pdf (312.81 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
We offer a new method of reduction for a system of point vortices on a plane and a sphere. This method is similar to the classical node elimination procedure. However, as applied to the vortex dynamics, it requires substantial modification. Reduction of four vortices on a sphere is given in more detail. We also use the Poincare surface-of-section technique to perform the reduction a four-vortex system on a sphere.
Keywords: | reduction, point vortex, equations of motion, Poincare map |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Reduction and chaotic behavior of point vortices on a plane and a sphere, Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 2, |
Full text: | pdf (473.55 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
The paper deals with a transition to chaos in the phase-plane portrait of a restricted problem of rotation of a rigid body with a fixed point. Two interrelated mechanisms responsible for chaotisation have been indicated: 1) growth of the homoclinic structure and 2) development of cascades of period doubling bifurcations. On the zero level of the integral of areas, an adiabatic behavior of the system (as the energy tends to zero) has been noticed. Meander tori induced by the breakdown of the torsion property of the mapping have been found.
Keywords: | motion of a rigid body, phase-plane portrait, mechanism of chaotisation, bifurcations |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Chaos in a restricted problem of rotation of a rigid body with a fixed point, Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 2, |
Full text: | pdf (650.7 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Kilin A. A., Mamaev I. S.
For the classical problem of motion of a rigid body about a fixed point with zero integral of areas, the paper presents a family of solutions which are periodic in the absolute space. Such solutions are known as choreographies. The family includes the famous Delaunay solution in the case of Kovalevskaya, some particular solutions in the Goryachev-Chaplygin case and Steklov’s solution. The «genealogy» of the solutions of the family, arising naturally from the energy continuation, and their connection with the Staude rotations are considered.
It is shown that if the integral of areas is zero, the solutions are periodic but with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies).
Keywords: | rigid body dynamics, periodic solutions, continuation by a parameter, bifurcation |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Absolute and relative choreographies in rigid body dynamics, Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 1, |
Full text: | pdf (401.63 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Ramodanov S. M.
In this paper we consider the system of two 2D rigid circular cylinders immersed in an unbounded volume of inviscid perfect fluid. The circulations around the cylinders are assumed to be equal in magnitude and opposite in sign. Special cases of this system (the cylinders move along the line through their centers and the circulation around each cylinder is zero) are considered. A similar system of two interacting spheres was originally considered in classical works of Carl and Vilhelm Bjerknes, G. Lamb and N.E. Joukowski.
By making the radii of the cylinders infinitesimally small, we have obtained a new mechanical system which consists of two regular point vortices but with non-zero masses. The study of this system can be reduced to the study of the motion of a particle subject to potential and gyroscopic forces. A new integrable case is found. The Hamiltonian equations of motion for this system have been generalized to the case of an arbitrary number of mass vortices with arbitrary intensities. Some first integrals have been obtained. These equations expand upon the classical Kirchhoff equations of motion for n point vortices.
Keywords: | perfect fluid, circulation, rigid body, qualitative analysis |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Ramodanov S. M., Interaction of two circular cylinders in a perfect fluid, Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 1, |
Full text: | pdf (401.14 Kb) |
Impact-factor RSCI (2022): | 0.259 (Q3) |
---|---|
ISSN (print): | 2658-5324 |
ISSN (online): | 2658-5316 |
Site: | http://nd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Ramodanov S. M.
The paper studies the system of a rigid body interacting dynamically with point vortices in a perfect fluid. For arbitrary value of vortex strengths and circulation around the cylinder the system is shown to be Hamiltonian (the corresponding Poisson bracket structure is rather complicated). We also reduced the number of degrees of freedom of the system by two using the reduction by symmetry technique and performed a thorough qualitative analysis of the integrable system of a cylinder interacting with one vortex.
Citation: | Borisov A. V., Mamaev I. S., Ramodanov S. M., Dynamics of a circular cylinder interacting with point vortices, Discrete and Continuous Dynamical Systems - Series B, 2005, vol. 5, no. 1, |
---|---|
Full text: | pdf (210.01 Kb) |
Impact-factor WoS (2015): | 1.257 |
---|---|
ISSN (print): | 1531-3492 |
ISSN (online): | 1553-524X |
Site: | http://aimsciences.org/journals/home.jsp?journalID=2 |
We consider integrable spherical analogue of the Darboux potential, which appear in the problem (and its generalizations) of the planar motion of a particle in the field of two and four fixed Newtonian centers. The obtained results can be useful when constructing a theory of motion of satellites in the field of an oblate spheroid in constant curvature spaces.
Keywords: | spherical two (and four) centers problem, Newtonian potential, sphero-conical coordinates, separation of variables |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Generalized problem of two and four Newtonian centers, Celestial Mechanics and Dynamical Astronomy, 2005, vol. 92, no. 4, |
DOI: | 10.1007/s10569-005-1557-y |
Full text: | pdf (291.25 Kb) |
Impact-factor WoS (2022): | 1.600 (Q3) |
---|---|
ISSN (print): | 0923-2958 |
ISSN (online): | 1572-9478 |
Site: | http://link.springer.com/journal/10569 |
Borisov A. V., Kilin A. A., Mamaev I. S.
We offer a new method of reduction for a system of point vortices on a plane and a sphere. This method is similar to the classical node elimination procedure. However, as applied to the vortex dynamics, it requires substantial modification. Reduction of four vortices on a sphere is given in more detail. We also use the Poincare surface-of-section technique to perform the reduction a four-vortex system on a sphere.
Keywords: | Vortex dynamics, reduction, Poincaré map, point vortices |
---|---|
Citation: | Borisov A. V., Kilin A. A., Mamaev I. S., Reduction and chaotic behavior of point vortices on a plane and a sphere, Discrete and Continuous Dynamical Systems - Series B (Supplement Volume devoted to the 5th AIMS International Conference on Dynamical Systems and Differential Equations (Pomona, California, USA, June 2004)), 2005, |
Full text: | pdf (360.5 Kb) |
Impact-factor WoS (2015): | 1.227 |
---|---|
ISSN (print): | 1531-3492 |
ISSN (online): | 1553-524X |
Site: | http://aimsciences.org/journals/home.jsp?journalID=2 |
Borisov A. V., Mamaev I. S., Kilin A. A.
In this paper we describe new classes of periodic solutions for point vortices on a plane and a sphere. They correspond to similar solutions (so-called choreographies) in celestial mechanics.
Keywords: | periodic solutions, choreographies , vortex dynamics, reduction |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Kilin A. A., New periodic solutions for three or four identical vortices on a plane and a sphere, Discrete and Continuous Dynamical Systems - Series B (Supplement Volume devoted to the 5th AIMS International Conference on Dynamical Systems and Differential Equations (Pomona, California, USA, June 2004)), 2005, |
Full text: | pdf (194.7 Kb) |
Impact-factor WoS (2015): | 1.227 |
---|---|
ISSN (print): | 1531-3492 |
ISSN (online): | 1553-524X |
Site: | http://aimsciences.org/journals/home.jsp?journalID=2 |
Citation: | Kozlov V. V., Gibbs and Poincaré statistical equilibria in systems with slowly varying parameters, Doklady Mathematics, 2004, vol. 69, no. 2, |
---|---|
Full text: | pdf (182.91 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
We present rather complete classification of the Birkhoff integrable generalized Toda lattices and consider new cases of integrable lattices.
Citation: | Borisov A. V., Mamaev I. S., Necessary and Sufficient Conditions for the Polynomial Integrability of Generalized Toda Chains, Doklady Physics, 2004, vol. 69, no. 1, |
---|---|
Full text: | pdf (280.72 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
In this paper, we obtain a nonlinear Poisson structure and two first integrals in the problem of the plane motion of a circular cylinder and n point vortices in an ideal fluid. This problem is a priori not Hamiltonian; specifically, in the case n=1 (i.e., in the problem of the interaction of a cylinder with a vortex) it is integrable.
Keywords: | ideal fluid, motion of a circular cylinder in an ideal fluid, point vortices, Poisson structure, Poisson bracket, Casimir function |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Integrability of the Problem of the Motion of a Cylinder and a Vortex in an Ideal Fluid, Mathematical Notes, 2004, vol. 75, no. 1, |
DOI: | 10.1023/B:MATN.0000015018.63296.ce |
Full text: | pdf (79.87 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,502 |
ISSN (print): | 0025-567X |
ISSN (online): | 2305-2880 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=mzm&option_lang=eng |
Mechanical systems whose configuration space is a Lie group and whose Lagrangian is invariant to left translations on that group are considered. It is assumed that the mass geometry of the system may change under the action of only internal forces. The equations of motion admit of a complete set of Noether integrals which are linear in the velocities. For fixed values of these integrals, the equations of motion reduce to a non-autonomous system of first-order differential equations on the Lie group. Conditions under which the system can be brought from any initial position to another preassigned position by changing its mass geometry are discussed. The “falling cat” problem and the problem of the motion of a body of variable shape in an unlimited volume of ideal fluid are considered as examples.
Citation: | Kozlov V. V., Dynamics of variable systems, and Lie groups, Journal of Applied Mathematics and Mechanics, 2004, vol. 68, no. 6, |
---|---|
DOI: | 10.1016/j.jappmathmech.2004.11.001 |
Full text: | pdf (478.49 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
New relations are established between the spectrum of a linear system and the indices of inertia of its quadratic integral. A detailed investigation is made of the case in which the positive and negative indices of inertia of the quadratic integral are identical. Conditions are found under which the singular planes will be Lagrangian relative to some natural symplectic structure. They are closely related to the conditions for strong stability of a linear system. The general results are applied to the classical problem of gyroscopic stabilization.
Citation: | Kozlov V. V., Linear systems with a quadratic integral and the symplectic geometry of Artin spaces, Journal of Applied Mathematics and Mechanics, 2004, vol. 68, no. 3, |
---|---|
DOI: | 10.1016/S0021-8928(04)00047-4 |
Full text: | pdf (648.21 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Borisov A. V., Mamaev I. S., Kilin A. A.
We consider the problem of two interacting particles on a sphere. The potential of the interaction depends on the distance between the particles. The case of Newtonian-type potentials is studied in most detail. We reduce this system to a system with two degrees of freedom and give a number of remarkable periodic orbits. We also discuss integrability and stochastization of the motion.
Citation: | Borisov A. V., Mamaev I. S., Kilin A. A., Two-body problem on a sphere. Reduction, stochasticity, periodic orbits, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, |
---|---|
DOI: | 10.1070/RD2004v009n03ABEH000280 |
Full text: | pdf (13.58 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Kilin A. A.
We obtained new periodic solutions in the problems of three and four point vortices moving on a plane. In the case of three vortices, the system is reduced to a Hamiltonian system with one degree of freedom, and it is integrable. In the case of four vortices, the order is reduced to two degrees of freedom, and the system is not integrable. We present relative and absolute choreographies of three and four vortices of the same intensity which are periodic motions of vortices in some rotating and fixed frame of reference, where all the vortices move along the same closed curve. Similar choreographies have been recently obtained by C. Moore, A. Chenciner, and C. Simo for the n-body problem in celestial mechanics [6, 7, 17]. Nevertheless, the choreographies that appear in vortex dynamics have a number of distinct features.
Citation: | Borisov A. V., Mamaev I. S., Kilin A. A., Absolute and relative choreographies in the problem of point vortices moving on a plane, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, |
---|---|
DOI: | 10.1070/RD2004v009n02ABEH000269 |
Full text: | pdf (389.11 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The kinetics of collisionless continuous medium is studied in a bounded region on a curved manifold. We have assumed that in statistical equilibrium, the probability distribution density depends only on the total energy. It is shown that in this case, all the fundamental relations for a multi-dimensional ideal gas in thermal equilibrium hold true.
Citation: | Kozlov V. V., Billiards, invariant measures, and equilibrium thermodynamics. II, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, |
---|---|
DOI: | 10.1070/RD2004v009n02ABEH000268 |
Full text: | pdf (283.7 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
A collisionless continuous medium in Euclidean space is discussed, i.e. a continuum of free particles moving inertially, without interacting with each other. It is shown that the distribution density of such medium is weakly converging to zero as time increases indefinitely. In the case of Maxwell's velocity distribution of particles, this density satisfies the well-known diffusion equation, the diffusion coefficient increasing linearly with time.
Citation: | Kozlov V. V., Notes on diffusion in collisionless medium, Regular and Chaotic Dynamics, 2004, vol. 9, no. 1, |
---|---|
DOI: | 10.1070/RD2004v009n01ABEH000262 |
Full text: | pdf (148.96 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Citation: | Kozlov V. V., On a uniform distribution on a torus, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2004, vol. 59, no. 2, |
---|---|
Full text: | pdf (582.1 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., The spectrum of a linear Hamiltonian system and symplectic geometry of a complex Artin space, Doklady Mathematics, 2003, vol. 68, no. 3, |
---|---|
Full text: | pdf (133.43 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
Citation: | Kozlov V. V., Weak limits of probability distributions in systems with nonstationary perturbations, Doklady Mathematics, 2003, vol. 67, no. 2, |
---|---|
Full text: | pdf (148.42 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
In their paper "Kepler's problem in constant curvature spaces" (Cel. Mech. And Dyn. Ast., v. 54, 1992. p. 393-399) Kozlov V.V. and Harin А.О. showed that the Euler problem of planar motion of a particle attracted by two fixed Newtonian centers has an integrable analogue on a two-dimensional sphere S2. The integrability was shown using the separation of variables. In the book "Poisson structures and Lie algebras in Hamiltonian mechanics" by Borisov A.V. and Mamaev I.S. the integrability of a spatial analogue of the problem was proved. The proof is based on the reduction by using a cyclic variable. As a result of this reduction, the 'spatial' problem on S3 becomes a 'plane' one on S2. In this case, however, an additional Hook's center appears at the pole on the perpendicular to an equatorial plane of the two former centers. In this paper we present algebraic integrals for a more general situation when a particle is attracted by two Newtonian centers and three mutually orthogonal Hook's centers of which two together with the Newtonian centers lie in a plane, and the third one is on the perpendicular to the plane.
Citation: | Mamaev I. S., Two Integrable System on a Two-dimensional Sphere, Doklady Physics, 2003, vol. 389, no. 3, |
---|---|
Full text: | pdf (160.12 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
This review is dedicated to the dynamics of the rattleback, a phenomenon with curious physical properties that is studied in nonholonomic mechanics. All known analytical results are collected here, and some results of our numerical simulation are presented. In particular, three-dimensional Poincare maps associated with dynamical systems are systematically investigated for the first time. It is shown that the loss of stability of periodic and quasiperiodic solutions, which gives rise to strange attractors, is typical of the three-dimensional maps related to rattleback dynamics. This explains some newly discovered properties of the rattleback related to the transition from regular to chaotic solutions at certain values of the physical parameters.
Citation: | Borisov A. V., Mamaev I. S., Strange Attractors in Rattleback Dynamics, Physics-Uspekhi, 2003, vol. 46, no. 4, |
---|---|
DOI: | 10.1070/PU2003v046n04ABEH001306 |
Full text: | pdf (484.78 Kb) |
Impact-factor WoS (2022): | 2.700 (Q2) |
---|---|
Impact-factor RSCI (2014): | 1.496 |
ISSN (print): | 0042-1294 |
ISSN (online): | 1996-6652 |
Site: | http://ufn.ru/ |
In the paper we consider modifications of the Hess integral suitable for various forms of the equations of motion for rigid body. This integral occurs due to some additional symmetry properties of the equations of motion. Moreover, we discuss the general conditions under which the integral exists. Assuming that these conditions are satisfied, we discuss the reduction of the order of the equations, their explicit integration and a qualitative analysis of motion. For the first time, the paper indicates new counterparts of the Hess case for the problem of a gyroscope fixed in a gimbal suspension and for Chaplygin's equations describing the motion of a heavy rigid body in an ideal fluid.
Citation: | Borisov A. V., Mamaev I. S., The Hess case in the dynamics of a rigid body, Journal of Applied Mathematics and Mechanics, 2003, vol. 67, no. 2, |
---|---|
Full text: | pdf (605.38 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Borisov A. V., Mamaev I. S., Ramodanov S. M.
The paper studies the system of a rigid body interacting dynamically with point vortices in a perfect fluid. For arbitrary value of vortex strengths and circulation around the cylinder the system is shown to be Hamiltonian (the corresponding Poisson bracket structure is rather complicated). We also reduced the number of degrees of freedom of the system by two using the reduction by symmetry technique and performed a thorough qualitative analysis of the integrable system of a cylinder interacting with one vortex.
Citation: | Borisov A. V., Mamaev I. S., Ramodanov S. M., Motion of a circular cylinder and n point vortices in a perfect fluid, Regular and Chaotic Dynamics, 2003, vol. 8, no. 4, |
---|---|
DOI: | 10.1070/RD2003v008n04ABEH000257 |
Full text: | pdf (585.43 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Kozlov V. V., Mitrofanova M. Y.
In this paper, we present results of simulations of a model of the Galton board for various degrees of elasticity of the ball-to-nail collision.
Citation: | Kozlov V. V., Mitrofanova M. Y., Galton board, Regular and Chaotic Dynamics, 2003, vol. 8, no. 4, |
---|---|
DOI: | 10.1070/RD2003v008n04ABEH000255 |
Full text: | pdf (1.99 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Some new cases when the invariant measure and an additional first integral exist in the problem of a rigid body rolling on a sphere and on an ellipsoid are discussed in the paper. These cases generalize the results obtained previously by V.A.Yaroshchuk and A.V.Borisov, I.S.Mamaev, A.A.Kilin.
Citation: | Mamaev I. S., New cases when the invariant measure and first integrals exist in the problem of a body rolling on a surface, Regular and Chaotic Dynamics, 2003, vol. 8, no. 3, |
---|---|
DOI: | 10.1070/RD2003v008n03ABEH000249 |
Full text: | pdf (148.37 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Kilin A. A.
In the paper we present the qualitative analysis of rolling motion without slipping of a homogeneous round disk on a horisontal plane. The problem was studied by S.A. Chaplygin, P. Appel and D. Korteweg who showed its integrability. The behavior of the point of contact on a plane is investigated and conditions under which its trajectory is finit are obtained. The bifurcation diagrams are constructed.
Citation: | Borisov A. V., Mamaev I. S., Kilin A. A., Dynamics of rolling disk, Regular and Chaotic Dynamics, 2003, vol. 8, no. 2, |
---|---|
DOI: | 10.1070/RD2003v008n02ABEH000237 |
Full text: | pdf (648.4 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
In this paper we present the nonlinear Poisson structure and two first integrals in the problem on plane motion of circular cylinder and N point vortices in the ideal fluid. A priori this problem is not Hamiltonian. The particular case N=1, i.e. the problem on interaction of cylinder and vortex, is integrable.
Citation: | Borisov A. V., Mamaev I. S., An Integrability of the Problem on Motion of Cylinder and Vortex in the Ideal Fluid, Regular and Chaotic Dynamics, 2003, vol. 8, no. 2, |
---|---|
DOI: | 10.1070/RD2003v008n02ABEH000235 |
Full text: | pdf (90.36 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Citation: | Kozlov V. V., Rationality conditions for the ratio of elliptic integrals and the great Poncelet theorem, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2003, vol. 58, no. 4, |
---|---|
Full text: | pdf (508.66 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., On the stochastization of plane-parallel flows of an ideal fluid, Fundamental and applied problems in vortex theory, Ser. Komp'yut. Mat. Fiz. Biol., Izhevsk: Institute of Computer Science, 2003, |
---|---|
Full text: | pdf (144.69 Kb) |
In this article questions on the possibility of sharpening classic ergodic theorems is considered. To sharpen these theorems the author uses methods of summation of divergent sequences and series. The main topic is connected with the individual ergodic Birkhoff–Khinchin theorem. The theorem is studied in connection with the Riesz and Voronoi summation methods. These methods are weaker than those of the Cesaro method of arithmetic means. It is shown that already for the Bernoulli transformation of the unit interval, meaningful problems arise. These problems are interesting in connection with the possibility of extension of the strong law of large numbers. The questions of suitable summation factors and of the solution of homological equations by means of divergent series is also discussed.
Citation: | Kozlov V. V., Summation of divergent series and ergodic theorems, Journal of Mathematical Sciences, 2003, vol. 114, no. 4, |
---|---|
DOI: | 10.1023/A:1022257013220 |
Full text: | pdf (406.3 Kb) |
Impact-factor WoS (2015): | 0.34 |
---|---|
ISSN (print): | 1072-3374 |
ISSN (online): | 1573-8795 |
Site: | http://www.springer.com/mathematics/journal/10958 |
In the paper, classical Chaplygin's problem on an unbalanced ball that is rolling without slipping on a plane is considered. Using numerical simulations, we have shown the possibility of mixing integrable non-holonomic systems on invariant tori. Therein lies the obstacle for this system to be Hamiltonian. It should be noted, that, nevertheless, in such systems there is an invariant measure and conservation of energy.
Citation: | Borisov A. V., Mamaev I. S., Obstacle to the Reduction of Nonholonomic Systems to the Hamiltonian Form, Doklady Physics, 2002, vol. 47, no. 12, |
---|---|
DOI: | 10.1134/1.1536224 |
Full text: | pdf (36.81 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
Borisov A. V., Mamaev I. S., Kilin A. A.
The problem of rolling motion without slipping of an unbalanced ball on 1) an arbitrary ellipsoid and 2) an ellipsoid of revolution is considered. In his famous treatise E. Routh showed that the problem of rolling motion of a body on a surface of revolution even in the presence of axisymmetrical potential fields is integrable. In case 1, we present a new integral of motion. New solutions expressed in elementary functions are found in case 2.
Citation: | Borisov A. V., Mamaev I. S., Kilin A. A., A New Integral in the Problem of Rolling a Ball on an Arbitrary Ellipsoid, Doklady Physics, 2002, vol. 47, no. 7, |
---|---|
DOI: | 10.1134/1.1499198 |
Full text: | pdf (85.97 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
Citation: | Kozlov V. V., Thermal equilibrium in the sense of Gibbs and Poincaré, Doklady Mathematics, 2002, vol. 65, no. 1, |
---|---|
Full text: | pdf (304.83 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
In this article questions on the possibility of sharpening classic ergodic theorems is considered. To sharpen these theorems the author uses methods of summation of divergent sequences and series. The main topic is connected with the individual ergodic Birkhoff–Khinchin theorem. The theorem is studied in connection with the Riesz and Voronoi summation methods. These methods are weaker than those of the Cesaro method of arithmetic means. It is shown that already for the Bernoulli transformation of the unit interval, meaningful problems arise. These problems are interesting in connection with the possibility of extension of the strong law of large numbers. The questions of suitable summation factors and of the solution of homological equations by means of divergent series is also discussed.
Citation: | Kozlov V. V., Summation of divergent series and ergodic theorems, Trudy Seminara imeni I. G. Petrovskogo, 2002, vol. 22, |
---|---|
Full text: | pdf (972.52 Kb) |
ISSN (print): | 0321-2971 |
---|---|
Site: | https://istina.msu.ru/journals/378112/ |
Borisov A. V., Mamaev I. S., Kilin A. A.
The paper is concerned with the problem on rolling of a homogeneous ball on an arbitrary surface. New cases when the problem is solved by quadratures are presented. The paper also indicates a special case when an additional integral and invariant measure exist. Using this case, we obtain a nonholonomic generalization of the Jacobi problem for the inertial motion of a point on an ellipsoid. For a ball rolling, it is also shown that on an arbitrary cylinder in the gravity field the ball's motion is bounded and, on the average, it does not move downwards. All the results of the paper considerably expand the results obtained by E. Routh in XIX century.
Citation: | Borisov A. V., Mamaev I. S., Kilin A. A., The rolling motion of a ball on a surface. New integrals and hierarchy of dynamics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, |
---|---|
DOI: | 10.1070/RD2002v007n02ABEH000205 |
Full text: | pdf (628.29 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
In this paper we study the cases of existence of an invariant measure, additional first integrals, and a Poisson structure in the problem of rigid body's rolling without sliding on a plane and a sphere. The problem of rigid body's motion on a plane was studied by S.A. Chaplygin, P. Appel, D. Korteweg. They showed that the equations of motion are reduced to a second-order linear differential equation in the case when the surface of the dynamically symmetrical body is a surface of revolution. These results were partially generalized by P. Woronetz, who studied the motion of a body of revolution and the motion of round disk with sharp edge on a sphere. In both cases the systems are Euler–Jacobi integrable and have additional integrals and invariant measure. It can be shown that by an appropriate change of time (determined by reducing multiplier), the reduced system is a Hamiltonian one. Here we consider some particular cases when the integrals and the invariant measure can be presented as finite algebraic expressions. We also consider a generalized problem of rolling of a dynamically nonsymmetric Chaplygin ball. The results of investigations are summarized in tables to illustrate the hierarchy of existence of various tensor invariants: invariant measure, integrals, and Poisson structure.
Citation: | Borisov A. V., Mamaev I. S., The rolling motion of a rigid body on a plane and a sphere. Hierarchy of dynamics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, |
---|---|
DOI: | 10.1070/RD2002v007n02ABEH000204 |
Full text: | pdf (784.15 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The paper deals with the problem of integration of equations of motion in nonholonomic systems. By means of well-known theory of the differential equations with an invariant measure the new integrable systems are discovered. Among them there are the generalization of Chaplygin's problem of rolling nonsymmetric ball in the plane and the Suslov problem of rotation of rigid body with a fixed point. The structure of dynamics of systems on the invariant manifold in the integrable problems is shown. Some new ideas in the theory of integration of the equations in nonholonomic mechanics are suggested. The first of them consists in using known integrals as the constraints. The second is the use of resolvable groups of symmetries in nonholonomic systems. The existence conditions of invariant measure with analytical density for the differential equations of nonholonomic mechanics is given.
Citation: | Kozlov V. V., On the Integration Theory of Equations of Nonholonomic Mechanics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, |
---|---|
DOI: | 10.1070/RD2002v007n02ABEH000203 |
Full text: | pdf (456.43 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The main directions in the development of the nonholonomic dynamics are briefly considered in this paper. The first direction is connected with the general formalizm of the equations of dynamics that differs from the Lagrangian and Hamiltonian methods of the equations of motion's construction. The second direction, substantially more important for dynamics, includes investigations concerning the analysis of the specific nonholonomic problems. We also point out rather promising direction in development of nonholonomic systems that is connected with intensive use of the modern computer-aided methods.
Citation: | Borisov A. V., Mamaev I. S., On the History of the Development of the Nonholonomic Dynamics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, |
---|---|
DOI: | 10.1070/RD2002v007n01ABEH000194 |
Full text: | pdf (183.63 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
In this paper we present a generalization of the Goryachev–Chaplygin integrable case on a bundle of Poisson brackets, and on Sokolov terms in his new integrable case of Kirchhoff equations. We also present a new analogous integrable case for the quaternion form of rigid body dynamics equations. This form of equations is recently developed and we can use it for the description of rigid body motions in specific force fields, and for the study of different problems of quantum mechanics. In addition we present new invariant relations in the considered problems.
Citation: | Borisov A. V., Mamaev I. S., Generalization of the Goryachev–Chaplygin Case, Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, |
---|---|
DOI: | 10.1070/RD2002v007n01ABEH000192 |
Full text: | pdf (289.22 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The paper develop a new approach to the justification of Gibbs canonical distribution for Hamiltonian systems with finite number of degrees of freedom. It uses the condition of nonintegrability of the ensemble of weak interacting Hamiltonian systems.
Citation: | Kozlov V. V., On Justification of Gibbs Distribution, Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, |
---|---|
DOI: | 10.1070/RD2002v007n01ABEH000190 |
Full text: | pdf (324.27 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Citation: | Kozlov V. V., On the theory of integration of the equations of nonholonomic mechanics, Nonholonomic dynamical systems, Ser. Komp'yut. Mat. Fiz. Biol., Izhevsk: Institute of Computer Science, 2002, |
---|
Borisov A. V., Mamaev I. S., Sokolov V. V.
In his paper "New integrable Case of Kirchoff's equations" (Theor. and Math. Physics. 2001) V.V. Sokolov proposed a new integrable case with additional integral of fourth degree. We have shown that the integral can be written in a more natural form and consider its generalization to a bundle of Poisson brackets.
Citation: | Borisov A. V., Mamaev I. S., Sokolov V. V., A New Integrable Case on so(4), Doklady Physics, 2001, vol. 46, no. 12, |
---|---|
DOI: | 10.1134/1.1433537 |
Full text: | pdf (28.12 Kb) |
Impact-factor WoS (2022): | 0.800 (Q4) |
---|---|
ISSN (print): | 1028-3358 |
ISSN (online): | 1562-6903 |
Site: | http://www.maik.ru/ru/journal/dan/ |
Citation: | Kozlov V. V., Diffusion in systems with an integral invariant on a torus, Doklady Mathematics, 2001, vol. 64, no. 3, |
---|---|
Full text: | pdf (150.96 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
Citation: | Kozlov V. V., Nonexponential atmosphere and noncanonical probability distributions, Doklady Mathematics, 2001, vol. 46, no. 9, |
---|---|
DOI: | 10.1134/1.1409002 |
Full text: | pdf (277.34 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
In this paper we introduce a new nonlinear Poisson bracket in the problem of rolling motion of a Chaplygin's ball. Thus, upon some change of time, the equations of motion become Hamiltonian. We have also established that the trajectories of this system are isomorphic to the trajectories of the Braden system which describes the motion of a point on a two-dimensional sphere in a potential field; using the isomorphism, we have shown that in the Chaplygin problem the variables are separable.
Keywords: | Chaplygin's ball rolling problem, potential force field, Poisson bracket, Euler\,--\,Jacobi theorem |
---|---|
Citation: | Borisov A. V., Mamaev I. S., Chaplygin's Ball Rolling Problem Is Hamiltonian, Mathematical Notes, 2001, vol. 70, no. 5, |
DOI: | 10.1023/A:1012995330780 |
Full text: | pdf (339.65 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,502 |
ISSN (print): | 0025-567X |
ISSN (online): | 2305-2880 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=mzm&option_lang=eng |
The problem of the conditions for the gyroscopic stabilization of unstable equilibria using gyroscopic forces with a degenerate matrix is considered. Systems with an odd number of degrees of freedom are an important example. The gyroscopic forces can generally be removed using a non-autonomous orthogonal transformation. The equations of motion then become a system of Sturm—Liouville type equations with a time-dependent potential. The conditions imposed on the skew-symmetric matrix of the gyroscopic forces for which the new potential depends periodically on time are indicated. These conditions are necessarily satisfied for non-zero matrices of the gyroscopic forces of minimum rank equal to two. Hence, the problem of gyroscopic stabilization reduces, in a number of cases, to investigating the stability of the equilibrium positions of systems with a periodic potential. The use of parametric-resonance theory enables new constructive conditions to be obtained for the stability of the equilibria of mechanical systems acted upon by additional degenerate gyroscopic forces. These conditions have the form of the conditions for an extremum of certain functions which depend solely on the position of the system. Particular attention is devoted to the stability conditions for large gyroscopic forces. It is shown, using examples, that the conditions of gyroscopic stabilization obtained are only sufficient. However, if the potential energy in the equilibrium position has a maximum and the matrix of the gyroscopic forces are non-degenerate, they are close to the necessary stability conditions.
Citation: | Kozlov V. V., Gyroscopic stabilization and parametric resonance, Journal of Applied Mathematics and Mechanics, 2001, vol. 65, no. 5, |
---|---|
DOI: | 10.1016/S0021-8928(01)00077-6 |
Full text: | pdf (438.42 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
The motion of Chaplygin ball with and without gyroscope in the absolute space is analyzed. In particular, the trajectories of the point of contact are studied in detail. We discuss the motions in the absolute space, that correspond to the different types of motion in the moving frame of reference related to the body. The existence of the bounded trajectories of the ball's motion is shown by means of numerical methods in the case when the problem is reduced to a certain Hamiltonian system.
Citation: | Kilin A. A., The Dynamics of Chaplygin Ball: the Qualitative and Computer Analysis, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, |
---|---|
DOI: | 10.1070/RD2001v006n03ABEH000178 |
Full text: | pdf (1.16 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
In this paper we propose a new approach to the study of integrable cases based on intensive computer methods' application. We make a new investigation of Kovalevskaya and Goryachev–Chaplygin cases of Euler–Poisson equations and obtain many new results in rigid body dynamics in absolute space. Also we present the visualization of some special particular solutions.
Citation: | Borisov A. V., Mamaev I. S., Euler–Poisson Equations and Integrable Cases, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, |
---|---|
DOI: | 10.1070/RD2001v006n03ABEH000176 |
Full text: | pdf (1.41 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
In this article we develop Poincaré ideas about a heat balance of ideal gas considered as a collisionless continuous medium. We obtain the theorems on diffusion in nondegenerate completely integrable systems. As a corollary we show that for any initial distribution the gas will be eventually irreversibly and uniformly distributed over all volume, although every particle during this process approaches arbitrarily close to the initial position indefinitely many times. However, such individual returnability is not uniform, which results in diffusion in a reversible and conservative system. Balancing of pressure and internal energy of ideal gas is proved, the formulas for limit values of these quantities are given and the classical law for ideal gas in a heat balance is deduced. It is shown that the increase of entropy of gas under the adiabatic extension follows from the law of motion of a collisionless continuous medium.
Citation: | Kozlov V. V., Kinetics of Collisionless Continuous Medium, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, |
---|---|
DOI: | 10.1070/RD2001v006n03ABEH000175 |
Full text: | pdf (468.67 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
In the paper Motion of a circular cylinder and a vortex in an ideal fluid (Reg. & Chaot. Dyn. V. 6. 2001. No 1. P. 33-38) Ramodanov S.M. showed the integrability of the problem of motion of a circular cylinder and a point vortex in unbounded ideal fluid. In the present paper we find additional first integral and invariant measure of motion equations.
Citation: | Kilin A. A., First Integral in the Problem of Motion of a Circular Cylinder and a Point Vortex in Unbounded Ideal Fluid, Regular and Chaotic Dynamics, 2001, vol. 6, no. 2, |
---|---|
DOI: | 10.1070/RD2001v006n02ABEH000174 |
Full text: | pdf (139.78 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Borisov A. V., Mamaev I. S., Kholmskaya A. G.
Generalizations of the Kovalevskaya, Chaplygin, Goryachev–Chaplygin and Bogoyavlensky systems on a bundle are considered in this paper. Moreover, a method of introduction of separating variables and action-angle variables is described. Another integration method for the Kovalevskaya top on the bundle is found. This method uses a coordinate transformation that reduces the Kovalevskaya system to the Neumann system. The Kolosov analogy is considered. A generalization of a recent Gaffet system to the bundle of Poisson brackets is obtained at the end of the paper.
Citation: | Borisov A. V., Mamaev I. S., Kholmskaya A. G., Kovalevskaya top and generalizations of integrable systems, Regular and Chaotic Dynamics, 2001, vol. 6, no. 1, |
---|---|
DOI: | 10.1070/RD2001v006n01ABEH000161 |
Full text: | pdf (286.68 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Citation: | Kozlov V. V., On uniform distribution, Bulletin of High education schools. North-Kavkaz region. Natural Sciences, 2001, |
---|
Impact-factor RSCI (2014): | 0,482 |
---|---|
ISSN (print): | 0321-3005 |
Site: | http://izvestiya.sfedu.ru/est-sci/index.html |
Citation: | Kozlov V. V., Statistical dynamics of a system of coupled pendulums, Doklady Mathematics, 2000, vol. 62, no. 1, |
---|---|
Full text: | pdf (258.57 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
Citation: | Kozlov V. V., Thermodynamics of Hamiltonian systems and the Gibbs distribution, Doklady Mathematics, 2000, vol. 61, no. 1, |
---|---|
Full text: | pdf (240.15 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,813 |
ISSN (print): | 1064-5624 |
ISSN (online): | 1531-8362 |
Site: | http://link.springer.com/journal/11472 |
Two-link periodic trajectories of a plane convex billiard, when a point mass moves along a segment which is orthogonal to the boundary of the billiard at its end points, are considered. It is established that, if the caustic of the boundary lies within the billiard, then, in a typical situation, there is an even number of two-link trajectories and half of them are hyperbolic (and, consequently, unstable) and the other half are of elliptic type. An example is given of a billiard for which the caustic intersects the boundary and all of the two-link trajectories are hyperbolic. The analysis of the stability is based on an analysis of the extremum of a function of the length of a segment of a convex billiard which is orthogonal to the boundary at one of its ends.
Citation: | Kozlov V. V., Two-link billiard trajectories: extremal properties and stability, Journal of Applied Mathematics and Mechanics, 2000, vol. 64, no. 6, |
---|---|
DOI: | 10.1016/S0021-8928(00)00121-0 |
Full text: | pdf (342.81 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., Sofya Kovalevskaya: a mathematician and a person, Russian Mathematical Surveys, 2000, vol. 55, no. 6, |
---|---|
DOI: | 10.1070/RM2000v055n06ABEH000353 |
Full text: | pdf (192.45 Kb) |
Impact-factor WoS (2022): | 0.900 (Q2) |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0036-0279 |
ISSN (online): | 1468-4829 |
Site: | http://www.mathnet.ru/umn |
In this work stability of polygonal configurations on a plane and sphere is investigated. The conditions of linear stability are obtained. A nonlinear analysis of the problem is made with the help of Birkhoff normalization. Some problems are also formulated.
Citation: | Borisov A. V., Kilin A. A., Stability of Thomson's Configurations of Vortices on a Sphere, Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, |
---|---|
DOI: | 10.1070/RD2000v005n02ABEH000141 |
Full text: | pdf (284.39 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
The questions of justification of the Gibbs canonical distribution for systems with elastic impacts are discussed. A special attention is paid to the description of probability measures with densities depending on the system energy.
Citation: | Kozlov V. V., Billiards, Invariant Measures, and Equilibrium Thermodynamics, Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, |
---|---|
DOI: | 10.1070/RD2000v005n02ABEH000136 |
Full text: | pdf (207.72 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Citation: | Borisov A. V., Mamaev I. S., Some comments to the paper by A.M.Perelomov "A note on geodesics on ellipsoid" RCD 2000 5(1) 89-91, Regular and Chaotic Dynamics, 2000, vol. 5, no. 1, |
---|---|
Full text: | pdf (178.43 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Citation: | Kozlov V. V., The Newton and Ivory theorems of attraction in spaces of constant curvature, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2000, vol. 55, no. 5, |
---|---|
Full text: | pdf (343.59 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Borisov A. V., Mamaev I. S., Kholmskaya A. G.
Citation: | Borisov A. V., Mamaev I. S., Kholmskaya A. G., The Kovalevskaya case and new integrable systems of dynamics, Vestnik molodyh uchenyh. "Prikladnaya matematika i mehanika", 2000, no. 4, |
---|---|
Full text: | pdf (910.69 Kb) |
Keywords: | adiabatic equilibrium process; integral invariants; variational principle |
---|---|
Citation: | Kozlov V. V., The vortex theory of adiabatic equilibrium processes, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2000, vol. 55, no. 2, |
Full text: | pdf (436.48 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
The famous Gauss principle states that an actual motion is the one among conceivable motions that deviates least from the released motion. Herz based his forceless dynamics on this principle [1]. Gauss called the deviations of the conceivable motions from the released one the constraint. An explicit expression of the constraint in generalized coordinates was obtained first by Lipshitz [2]. In this paper, two new theorems are pointed out.
Citation: | Kozlov V. V., Integrable analogue of the Gauss principle, Facta Universitatis. Series Mechanics, Automatic Control and Robotics, 2000, vol. 2, no. 10, |
---|---|
Full text: | pdf (223.44 Kb) |
ISSN (print): | 0354-2009 |
---|---|
Site: | http://facta.junis.ni.ac.rs/macar/macar.html |
A classical theorem of Helmholtz states that vortex lines are frozen into a flow of barotropic ideal fluid in a potential force field. This result leads to the following general problem: it is required to find conditions under which a given dynamical system admits of a direction field frozen into its phase flow. By the rectification theorem for trajectories, a whole family of frozen direction fields always exists locally. It turns out that the problem of the existence of non-trivial frozen direction fields defined in the whole phase space is closely related to the well-known problem of small denominators. Results of a general nature are applied to Hamiltonian systems, and also to steady flows of a viscous fluid.
Citation: | Kozlov V. V., A condition for the freezing of a direction field, small denominators, and the chaotization of steady flows of a viscous fluid, Journal of Applied Mathematics and Mechanics, 1999, vol. 63, no. 2, |
---|---|
DOI: | 10.1016/S0021-8928(99)00031-3 |
Full text: | pdf (544.53 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Chernoivan V. A., Mamaev I. S.
In this work we carry out the bifurcation analysis of the Kepler problem on S3 and L3, and construct the analogues of Delaunau variables. We consider the problem of motion of a mass point in the field of moving Newtonian center on S2 and L2. The perihelion deviation is derived by the method of perturbation theory under the small curvature, and a numerical investigation is made, using anology of this problem with rigid body dynamics.
Citation: | Chernoivan V. A., Mamaev I. S., The restricted two-body problem and the kepler problem in the constant curvature spaces, Regular and Chaotic Dynamics, 1999, vol. 4, no. 2, |
---|---|
DOI: | 10.1070/RD1999v004n02ABEH000107 |
Full text: | pdf (807.85 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Traditional derivation of Gibbs canonical distribution and the justification of thermodynamics are based on the assumption concerning an isoenergetic ergodicity of a system of n weakly interacting identical subsystems and passage to the limit n→∞. In the presented work we develop another approach to these problems assuming that n is fixed and n⩾2. The ergodic hypothesis (which frequently is not valid due to known results of the KAM-theory) is substituted by a weaker assumption that the perturbed system does not have additional first integrals independent of the energy integral. The proof of nonintegrability of perturbed Hamiltonian systems is based on the Poincare method. Moreover, we use the natural Gibbs assumption concerning a thermodynamic equilibrium of bsystems at vanishing interaction. The general results are applied to the system of the weakly connected pendula. The averaging with respect to the Gibbs measure allows to pass from usual dynamics of mechanical systems to the classical thermodynamic model.
Citation: | Kozlov V. V., Canonical Gibbs distribution and thermodynamics of mechanical systems with a finite number of degrees of freedom, Regular and Chaotic Dynamics, 1999, vol. 4, no. 2, |
---|---|
DOI: | 10.1070/RD1999v004n02ABEH000106 |
Full text: | pdf (242 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
We consider two-body problem and restricted three-body problem in spaces S2 and L2. For two-body problem we have showed the absence of exponential instability of partiбular solutions relevant to roundabout motion on the plane. New libration points are found, and the dependence of their positions on parameters of a system is explored. The regions of existence of libration points in space of parameters were constructed. Basing on a examination of the Hill's regions we found the qualitative estimation of stability of libration points was produced.
Citation: | Kilin A. A., Libration points in spaces S2 and L2, Regular and Chaotic Dynamics, 1999, vol. 4, no. 1, |
---|---|
DOI: | 10.1070/RD1999v004n01ABEH000101 |
Full text: | pdf (455.22 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Bolsinov A. V., Borisov A. V., Mamaev I. S.
1.Classificaton of the algebra of n vortices on a plane
2.Solvable problems of vortex dynamics
3.Algebraization and reduction in a three-body problem
The work [13] introduces a naive description of dynamics of point vortices on a plane in terms of variables of distances and areas which generate Lie–Poisson structure. Using this approach a qualitative description of dynamics of point vortices on a plane and a sphere is obtained in the works [14,15]. In this paper we consider more formal constructions of the general problem of n vortices on a plane and a sphere. The developed methods of algebraization are also applied to the classical problem of the reduction in the three-body problem.
Citation: | Bolsinov A. V., Borisov A. V., Mamaev I. S., Lie algebras in vortex dynamics and celestial mechanics — IV, Regular and Chaotic Dynamics, 1999, vol. 4, no. 1, |
---|---|
DOI: | 10.1070/RD1999v004n01ABEH000097 |
Full text: | pdf (1.16 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Finite-dimensional systems with viscous friction are studied in the case when the Rayleigh function modeling this friction is proportional to the kinetic energy. Hydrodynamic analogies are presented for these systems.
Citation: | Kozlov V. V., Hydrodynamic Theory of a Class of Finite-Dimensional Dissipative Systems, Proceedings of the Steklov Institute of Mathematics, 1998, vol. 223, |
---|---|
Full text: | pdf (231.63 Kb) |
Impact-factor WoS (2022): | 0.500 (Q4) |
---|---|
Impact-factor RSCI (2022): | 0.276 (Q3) |
ISSN (print): | 0081-5438 |
ISSN (online): | 1531-8605 |
Site: | http://www.maik.ru/ru/journal/trstekl/ |
Differential equations with quadratic right-hand sides and additional constant terms are considered. Important examples are self-driven gyroscopes and the problem of the motion of a rigid body in an unbounded volume of ideal fluid subject to a force and a torque which are constant in an attached frame of reference. Under certain simple conditions, these equations have solutions that increase linearly with time. In problems of dynamics they describe uniformly accelerated motions of mechanical systems. The stability of such motions is investigated in the first approximation and using bundles of integrals. The general results are used to investigate the stability of uniformly accelerated screw motions of a rigid body in a fluid.
Citation: | Kozlov V. V., The stability of uniformly accelerated motions, Journal of Applied Mathematics and Mechanics, 1998, vol. 62, no. 5, |
---|---|
DOI: | 10.1016/S0021-8928(98)00086-0 |
Full text: | pdf (353.15 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
The conditions for branching of the solutions of the equations of motion of natural mechanical systems in the complex time plane are obtained. The relation between the structure of the branching and the number of independent momentum-polynomial first integrals is investigated. Results of a general form are illustrated by examples from dynamics.
Citation: | Kozlov V. V., Branching of solutions and polynomial integrals of the equations of dynamics, Journal of Applied Mathematics and Mechanics, 1998, vol. 62, no. 1, |
---|---|
DOI: | 10.1016/S0021-8928(98)00020-3 |
Full text: | pdf (620.19 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kilin A. A., Mamaev I. S., Libration points in a bounded problem of three bodies on S2, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 1998, no. 1, |
---|---|
Full text: | pdf (3.96 Mb) |
Impact-factor WoS (2022): | 0.400 |
---|---|
Impact-factor RSCI (2022): | 0.318 (Q3) |
ISSN (print): | 2226-3594 |
ISSN (online): | 2410-1737 |
Site: | http://journals.udsu.ru/mathematics |
Citation: | Mamaev I. S., Integrability of the Problems of Motion of a Particle in Constant Curvature Spaces in the Presence of Magnetic Monopole and in the Presence of Two Fixed Newtonian Centers, Proceedings of IX Seminar "Gravitational Energy and Gravity Waves", 1998, |
---|---|
Full text: | pdf (4.6 Mb) |
Citation: | Kozlov V. V., On periodic solutions of Duffing's equations, Proceedings of Scientific Seminar at A. A. Blagonravov Institute of Engineering, 1998, |
---|---|
Full text: | pdf (283.81 Kb) |
Citation: | Borisov A. V., Mamaev I. S., A degenerate Poisson Structure and Lie algebras in the Two Problems of Hamiltonian Dynamics, Proceedings of IX Seminar "Gravitational Energy and Gravity Waves", 1998, |
---|---|
Full text: | pdf (3.81 Mb) |
The problem of stabilizing unstable (by Earnshaw's theorem) equilibria of a free charge in an electrostatic field by adding a steady magnetic field is considered. The additional Lorentz force that thereby arises has a gyroscopic form. An example of the possibility of stabilization in a rigorous relativistic formulation of the problem is given. Criteria for the stabilization of unstable equilibria of linearized systems are obtained. The conditions for charge stability in intense magnetic fields are investigated and estimates of the stabilization probability are given. Some multidimensional analogues of these results are presented. In particular, the problem of gyroscopic stabilization when the matrix of the gyroscopic forces is degenerate is considered. Some extremal criteria of the stability of the equilibrium positions are given.
Citation: | Kozlov V. V., Stabilization of the unstable equilibria of charges by intense magnetic fields, Journal of Applied Mathematics and Mechanics, 1997, vol. 61, no. 3, |
---|---|
DOI: | 10.1016/S0021-8928(97)00048-8 |
Full text: | pdf (576.42 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
In the paper the equations of motion of a rigid body in the Hamiltonian form on the subalgebra of algebra e(4) are written. With the help of the algebraic methods a number of new isomorphisms in dynamics is established. We consider the lowering of the order as the process of decreasing rank of the Poisson structure with the algebraic point of view and indicate the possibility of arising the nonlinear Poisson brackets at this reduction as well.
Citation: | Borisov A. V., Mamaev I. S., Non-linear Poisson brackets and isomorphisms in dynamics, Regular and Chaotic Dynamics, 1997, vol. 2, no. 3-4, |
---|---|
DOI: | 10.1070/RD1997v002n03ABEH000049 |
Full text: | pdf (1.69 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
We analyse the operation of averaging of smooth functions along exact trajectories of dynamic systems in a neighborhood of stable nonresonance invariant tori. It is shown that there exists the first integral after the averaging; however in the typical situation the mean value is discontinuous or even not everywhere defind. If the temporal mean were a smooth function it would take its stationary values in the points of nondegenerate invariant tori. We demonstrate that this result can be properly derived if we change the operations of averaging and differentiating with respect to the initial data by their places. However, in general case for nonstable tori this property is no longer preserved. We also discuss the role of the reducibility condition of the invariant tori and the possibility of the generalization for the case of arbitrary compact invariant manifolds on which the initial dynamic system is ergodic.
Citation: | Kozlov V. V., Averaging in a neighborhood of stable invariant tori, Regular and Chaotic Dynamics, 1997, vol. 2, no. 3-4, |
---|---|
DOI: | 10.1070/RD1997v002n03ABEH000046 |
Full text: | pdf (564.5 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
We consider arising of adiabatic chaos in rigid body dynamics. The comparison of analytical diffusion coefficient describing probable effects in the chaos zone with numerical experiment is carried out. The analysis of split of asymptotic surfaces is carried out the curves of indfenition in the Poincare-Zhukovsky problem.
Citation: | Borisov A. V., Mamaev I. S., Adiabatic Chaos in Rigid Body Dynamics, Regular and Chaotic Dynamics, 1997, vol. 2, no. 2, |
---|---|
DOI: | 10.1070/RD1997v002n02ABEH000037 |
Full text: | pdf (1.71 Mb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
We study motion of a charged particle on the two dimensional torus in a constant direction magnetic field. This analysis can be applied to the description of electron dynamics in metals, which admit a 2-dimensional translation group (Bravais crystal lattice). We found the threshold magnetic value, starting from which there exist three closed Larmor orbits of a given energy. We demonstrate that if there are n lattice atoms in a primitive Bravais cell then there are 4+n different Larmor orbits in the nondegenerate case. If the magnetic field is absent the electron dynamics turns out to be chaotic, dynamical systems on the corresponding energy shells possess positive entropy in the case that the total energy is positive.
Citation: | Kozlov V. V., Closed Orbits and Chaotic Dynamics of a Charged Particle in a Periodic Electromagnetic Field, Regular and Chaotic Dynamics, 1997, vol. 2, no. 1, |
---|---|
DOI: | 10.1070/RD1997v002n01ABEH000021 |
Full text: | pdf (811.01 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
A method of solving the canonical Hamilton equations, based on a search for invariant manifolds, which are uniquely projected onto position space, is proposed. These manifolds are specified by covector fields, which satisfy a system of first-order partial differential equations, similar in their properties to Lamb's equations in the dynamic of an ideal fluid. If the complete integral of Lamb's equations is known, then, with certain additional assumptions, one can integrate the initial Hamilton equations explicitly. This method reduces to the well-known Hamilton-Jacobi method for gradient fields. Some new conditions for Hamilton's equations to be accurately integrable are indicated. The general results are applied to the problem of the motion of a variable body.
Citation: | Kozlov V. V., An extension of the Hamilton-Jacobi method, Journal of Applied Mathematics and Mechanics, 1996, vol. 60, no. 6, |
---|---|
DOI: | 10.1016/S0021-8928(96)00113-X |
Full text: | pdf (585.47 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
The paper discusses relationship between regular behavior of Hamilton's systems and the existence a sufficient number of fields of symmetry. Some properties of quite regular schemes and their relationship with various characteristics of stochastic behavior are studied.
Citation: | Kozlov V. V., Symmetries and Regular Behavior of Hamilton's Systems, Regular and Chaotic Dynamics, 1996, vol. 1, no. 1, |
---|---|
DOI: | 10.1070/RD1996v001n01ABEH000001 |
Full text: | pdf (708.49 Kb) |
Impact-factor WoS (2022): | 1.400 (Q2) |
---|---|
Impact-factor RSCI (2022): | 0.399 (Q2) |
ISSN (print): | 1560-3547 |
ISSN (online): | 1468-4845 |
Site: | http://rcd.ics.org.ru/ |
Citation: | Kozlov V. V., Motion of a disk on an inclined plane, Mechanics of Solids, 1996, no. 5, |
---|---|
Full text: | pdf (327.68 Kb) |
Impact-factor WoS (2015): | 0.233 |
---|---|
Impact-factor RSCI (2014): | 0,659 |
ISSN (print): | 0572-3299 |
Site: | http://mtt.ipmnet.ru/ru/ |
The behavior of the phase trajectories of the Hamilton equations is commonly classified as regular and chaotic. Regularity is usually related to the condition for complete integrability, i.e., a Hamiltonian system with n degrees of freedom has n independent integrals in involution. If at the same time the simultaneous integral manifolds are compact, the solutions of the Hamilton equations are quasiperiodic. In particular, the entropy of the Hamiltonian phase flow of a completely integrable system is zero. It is found that there is a broader class of Hamiltonian systems that do not show signs of chaotic behavior. These are systems that allow n commuting ‘‘Lagrangian’’ vector fields, i.e., the symplectic 2‐form on each pair of such fields is zero. They include, in particular, Hamiltonian systems with multivalued integrals.
Citation: | Kozlov V. V., Symmetries and regular behavior of Hamiltonian systems, Chaos, 1996, vol. 6, no. 1, |
---|---|
DOI: | 10.1063/1.166153 |
Full text: | pdf (453.44 Kb) |
Impact-factor WoS (2022): | 2.900 (Q1) |
---|---|
ISSN (print): | 1054-1500 |
ISSN (online): | 1089-7682 |
Site: | http://scitation.aip.org/content/aip/journal/chaos |
The problem of a point moving on the surface of an n-dimensional ellipsoid in a conservative field of force is considered. It is shown that if the potential energy terms are inversely proportional to the squares of the distances to the (n−1)-dimensional planes of symmetry of the ellipsoid, the problem can be explicitly integrated by using separation variables in elliptic Jacobi coordinates. It has n independent commuting integrals that are quadratic functions of the momenta. If n=2, an additional integral can be found explicitly by using redundant coordinates. In the limit, when the least semi-axis approaches zero, one obtains a new integrable billiards problem inside the ellipse. Extensions of these results to a space of constant non-zero curvature are discussed.
Citation: | Kozlov V. V., Some integrable generalizations of the Jacobi problem on geodesics on an ellipsoid, Journal of Applied Mathematics and Mechanics, 1995, vol. 59, no. 1, |
---|---|
DOI: | 10.1016/0021-8928(95)00001-6 |
Full text: | pdf (432.65 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Conditions are found for the existence of integral invariants of Hamiltonian systems. For two-degrees-of-freedom systems these conditions are intimately related to the existence of nontrivial symmetry fields and multivalued integrals. Any integral invariant of a geodesic flow on an analytic surface of genus greater than 1 is shown to be a constant multiple of the Poincaré-Cartan invariant. Poincaré's conjecture that there are no additional integral invariants in the restricted three-body problem is proved.
Citation: | Kozlov V. V., Integral invariants of the Hamilton equations, Mathematical Notes, 1995, vol. 58, no. 3, |
---|---|
DOI: | 10.1007/BF02304771 |
Full text: | pdf (599.62 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,502 |
ISSN (print): | 0025-567X |
ISSN (online): | 2305-2880 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=mzm&option_lang=eng |
Citation: | Kozlov V. V., Raboty P. L. Chebysheva po prikladnoi mekhanike, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1995, no. 6, |
---|
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., On multivalued integrals of Hamiltonian equations, Problems of nonlinear analysis in engineering systems, 1995, no. 1, |
---|---|
Full text: | pdf (284.07 Kb) |
ISSN (print): | 1727-687Х |
---|---|
Site: | http://kpfu.ru/science/journals/ansj/pnaes |
Citation: | Kozlov V. V., Problemata nova, ad quorum solutionem mathematici invitantur, Dynamical systems in classical mechanics, Amer. Math. Soc. Transl. Ser. 2, 1995, vol. 168, |
---|---|
Full text: | pdf (826.77 Kb) |
Citation: | Kozlov V. V., Hydrodynamics of noncommutative integration of Hamiltonian systems, Dynamical systems in classical mechanics, Amer. Math. Soc. Transl. Ser. 2, 1995, vol. 168, |
---|---|
Full text: | pdf (578.71 Kb) |
Citation: | Kozlov V. V., The asymptotic motions of systems with dissipation, Journal of Applied Mathematics and Mechanics, 1994, vol. 58, no. 5, |
---|---|
Full text: | pdf (311.84 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
We consider dynamical systems on compact three-dimensional manifolds which have an invariant volume form. An important example is given by Hamilton equations of a system with two degrees of freedom restricted to three-dimensional lever surface of the energy integral. In this system we study the existence of tensor invariants (a first integral, a symmetry field, an invariant form) and give conditions of integrability by quadratures under the existence of a tensor invariant. We show that the infinite number of nondegenerate periodic trajectories and splitting of separatices obstruct the existence of nontrivial integral invariants analytical on the three-dimensional manifold.
Keywords: | dynamical systems; compact three-dimensional manifolds; invariant volume form; Hamilton equations; tensor invariants; integrability; periodic trajectories |
---|---|
Citation: | Kozlov V. V., On tensor invariants of dynamical systems on three-dimensional manifolds, Teorijska i Primenjena Mehanika, 1994, no. 20, |
Full text: | pdf (476.53 Kb) |
ISSN (print): | 0353-8249; 0350-2708 |
---|---|
Site: | http://www.ssm.org.rs/WebTAM/journal.html |
Citation: | Kozlov V. V., On equilibria of nonholonomic systems, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1994, vol. 49, no. 2-3, |
---|---|
Full text: | pdf (279.02 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., Dynamics in spaces of constant curvature, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1994, vol. 49, no. 2, |
---|---|
Full text: | pdf (391.2 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
This volume offers a state-of-the-art overview of Hamiltonian mechanics and its applications. Its contents include papers by distinguished Russian scientists not previously published in English. Topics covered in the refereed contributions include: Non-integrability criterion of Hamiltonian systems based on Ziglin’s theorem and its relation to the singular point analysis; Natural boundaries of normalizing transformations; The structure of chaos; Successive elimination of harmonics: a way to explore the resonant structure of a Hamiltonian system; The tendency toward ergodicity with increasing number of degrees of freedom in Hamiltonian systems; Numerical integration of Hamiltonian systems in the presence of additional integrals: application of the observer method.
Keywords: | numerical integration; Ziglin’s theorem; singular point analysis; normalizing transformations; resonant structure; ergodicity |
---|---|
Citation: | Kozlov V. V., Symmetries and topology of dynamical systems with two degrees of freedom, Hamiltonian mechanics. Integrability and chaotic behavior. NATO ASI Series. Series B. Physics, 1994, vol. 331, |
DOI: | 10.1007/978-1-4899-0964-0_13 |
ISSN (print): | 0258-1221 |
---|---|
Site: | http://link.springer.com/bookseries/5657 |
The degree of instability of an equilibrium position in an autonomous dynamical system is defined as the number of eigenvalues of its linearization that lie in the right half-plane. Dissipative systems with Morse functions that do not increase along their trajectories are considered. The critical points of such functions are precisely the equilibrium positions. It will be shown that the degree of instability of a non-degenerate equilibrium position has the same parity as the index of the Morse function at that point. In particular, if the index is odd, the equilibrium is unstable. This result carries over to compact invariant manifolds of a dynamical system, provided they are non-degenerate, reducible and ergodic. An example is the problem of the stability of the steady motion of a heavy cylindrical rigid body in an unbounded volume of ideal liquid with non-zero circulation.
Citation: | Kozlov V. V., On the degree of instability, Journal of Applied Mathematics and Mechanics, 1993, vol. 57, no. 5, |
---|---|
DOI: | 10.1016/0021-8928(93)90141-8 |
Full text: | pdf (331.54 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., The Liouville property of invariant measures of completely integrable systems and the Monge–Ampère equation, Mathematical Notes, 1993, vol. 53, no. 4, |
---|---|
DOI: | 10.1007/BF01210221 |
Full text: | pdf (301.55 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,502 |
ISSN (print): | 0025-567X |
ISSN (online): | 2305-2880 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=mzm&option_lang=eng |
Citation: | Kozlov V. V., On a heavy cylindrical body falling in a fluid, Mechanics of Solids, 1993, no. 4, |
---|---|
Full text: | pdf (277.03 Kb) |
Impact-factor WoS (2015): | 0.233 |
---|---|
Impact-factor RSCI (2014): | 0,659 |
ISSN (print): | 0572-3299 |
Site: | http://mtt.ipmnet.ru/ru/ |
Time-independent differential equations describing the velocity field of a stationary flow of a viscous incompressible liquid in three-dimensional Euclidean space are considered. It is proved that, in the general case, this system does not admit any nontrivial first integrals, symmetry fields, and linear integral invariants. Certain examples of stationary flows with chaotic properties are given.
Keywords: | Navier-Stokes equations; stationary flow; viscous incompressible liquid |
---|---|
Citation: | Kozlov V. V., Dynamical systems determined by the Navier–Stokes equations, Russian Journal of Mathematical Physics, 1993, vol. 1, no. 1, |
Full text: | pdf (6.88 Mb) |
Impact-factor WoS (2022): | 1.400 (Q3) |
---|---|
ISSN (print): | 1061-9208 |
ISSN (online): | 1555-6638 |
Site: | http://link.springer.com/journal/11503 |
It is shown that a linear system of n differential equations with constant coefficients, at least one of whose integrals is a non-degenerate quadratic form, may be reduced to a canonical system of Hamiltonian equations. In particular, n is even and the phase flow preserves the standard measure; if the index of the quadratic integral is odd, the trivial solution is unstable, and so on. For the case n=4 the stability conditions are given a geometrical form. The general results are used to investigate small oscillations of non-holonomic systems, and also the problem of the stability of invariant manifolds of non-linear systems that have Morse functions as integrals.
Citation: | Kozlov V. V., Linear-systems with a quadratic integral, Journal of Applied Mathematics and Mechanics, 1992, vol. 56, no. 6, |
---|---|
DOI: | 10.1016/0021-8928(92)90114-N |
Full text: | pdf (334.92 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Problems associated with the limiting transition in the second-order Lagrange equations, when the coefficients of rigidity and viscosity and added masses tend to infinity are considered. Under certain conditions, the solutions of the initial equations approach those of the limiting problem with constraints. For integrable constraints, the limiting equations are identical with the usual equations with constraint multipliers. In the case of non-integrable constraints, the solutions depends closely on the way in which they are realized. The generalized models of the dynamics of systems with non-integrable constraints and the properties of the limiting equations of motion are discussed.
Citation: | Kozlov V. V., The problem of realizing constraints in dynamics, Journal of Applied Mathematics and Mechanics, 1992, vol. 56, no. 4, |
---|---|
DOI: | 10.1016/0021-8928(92)90017-3 |
Full text: | pdf (475.25 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., Tensor invariants of quasihomogeneous systems of differential equations, and the Kovalevskaya–Lyapunov asymptotic method, Mathematical Notes, 1992, vol. 51, no. 2, |
---|---|
DOI: | 10.1007/BF02102118 |
Full text: | pdf (341.6 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,502 |
ISSN (print): | 0025-567X |
ISSN (online): | 2305-2880 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=mzm&option_lang=eng |
The stability of equilibrium positions is investigated for mechanical systems in force fields with potentials of the form p(t)V, where V is a function of the generalized coordinates. Systems of this form are frequently encountered in applications. It is shown that if the factor p(t) increases monotonically to +∞ as t→+∞, then stability conditions for equilibria can be formulated in the form of extremal properties of the function V. The general results are applied to the problem of the motion of a rigid body in an infinite volume of an ideal fluid.
Citation: | Kozlov V. V., The stability of equilibrium positions in a nonstationary force field, Journal of Applied Mathematics and Mechanics, 1991, vol. 55, no. 1, |
---|---|
DOI: | 10.1016/0021-8928(91)90054-X |
Full text: | pdf (521.17 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Keywords: | Chebyshev polynomials; asymptotic solution; Birkhoff billards; stability; mechanical systems; works of Chebyshev and Lyapunov |
---|---|
Citation: | Kozlov V. V., Stability of periodic trajectories and Chebyshev polynomials, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1991, vol. 46, no. 5, |
Full text: | pdf (366.41 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
A sinusoidal perturbation of constant direction on a stationary velocity field has been shown to give rise to chaotic motion of an ideal fluid. The perturbed motion due to a pair of vortices of opposite intensities has been numerically computed.
Keywords: | sinusoidal perturbation; chaotic motion; pair of vortices of opposite intensities |
---|---|
Citation: | Kozlov V. V., On randomization of plane parallel flow of an ideal fluid, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1991, vol. 46, no. 1, |
Full text: | pdf (237.57 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., Groups of symmetries of geodesic flows on closed surfaces, Mathematical Notes, 1990, vol. 48, no. 5, |
---|---|
DOI: | 10.1007/BF01236297 |
Full text: | pdf (283.9 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,502 |
ISSN (print): | 0025-567X |
ISSN (online): | 2305-2880 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=mzm&option_lang=eng |
Citation: | Kozlov V. V., Eddy theory of the top, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1990, vol. 45, no. 4, |
---|---|
Full text: | pdf (305.41 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
A simplified mathematical model of motion of a rigid body in a resisting medium is considered, in which viscous friction forces defined by the Rayleigh dissipative function are taken into account along with the associated mass effect. Using the small parameter method, the existence of a stable auto-rotation is established, where, on the average, the center of mass of the body comes down with a constant velocity along an inclined line. The stability of stationary vertical descent of the rigid body is analyzed.
Keywords: | motion of a rigid body; resisting medium; viscous friction forces; Rayleigh dissipative function; stable auto-rotation |
---|---|
Citation: | Kozlov V. V., On the problem of fall of a rigid body in a resisting medium, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1990, vol. 45, no. 1, |
Full text: | pdf (417.86 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., Relyativistskaya zadacha mnogikh tel i ee kvantovanie, v kn.: E. M. Nikishin, Izbrannye voprosy matematicheskogo analiza, Moskva–Tula, 1990, |
---|---|
Full text: | pdf (141.82 Kb) |
Citation: | Kozlov V. V., Constructive approach to proof of the dynamics of systems with constraints, Theoretical Mechanics, Collection of scientific and methodological papers, no. 20, MPI, Moscow, 1990, |
---|---|
Full text: | pdf (382.68 Kb) |
A problem of the stability of equilibrium of a system of interacting particles distributed within a bounded volume of Euclidean space is considered. Sufficient conditions for the instability and existence of the motions approaching the position of equilibrium without bounds, containing the Kelvin theorem /1/ as a special case, are obtained. The results are based on the general theory of instability of equilibrium in a force field with a subharmonic force function.
Citation: | Kozlov V. V., A problem of Kelvin, Journal of Applied Mathematics and Mechanics, 1989, vol. 53, no. 1, |
---|---|
DOI: | 10.1016/0021-8928(89)90145-7 |
Full text: | pdf (208.35 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., Polynomial integrals of dynamical systems with one-and-a-half degrees of freedom, Mathematical Notes, 1989, vol. 45, no. 4, |
---|---|
DOI: | 10.1007/BF01158890 |
Full text: | pdf (304.73 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,502 |
ISSN (print): | 0025-567X |
ISSN (online): | 2305-2880 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=mzm&option_lang=eng |
Citation: | Kozlov V. V., On impact with friction, Izv. Akad. Nauk SSSR Mekh. Tverd. Tela, 1989, no. 6, |
---|---|
Full text: | pdf (463.51 Kb) |
ISSN (print): | 0572-3299 |
---|---|
Site: | http://mtt.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., Principles of dynamics and servoconstraints, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1989, no. 5, |
---|---|
Full text: | pdf (382.7 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., On falling of a heavy rigid body in an ideal fluid, Izv. Akad. Nauk SSSR Mekh. Tverd. Tela, 1989, no. 5, |
---|---|
Full text: | pdf (584.35 Kb) |
ISSN (print): | 0572-3299 |
---|---|
Site: | http://mtt.ipmnet.ru/ru/ |
Keywords: | interaction of three identical particles; complete integrability; existence of an additional integral; Moser-Calogero potentials; existence of an infinite number of short-periodic solutions; existence of a polynomial integral |
---|---|
Citation: | Kozlov V. V., Polynomial integrals of a system of interacting particles, Soviet Mathematics. Doklady, 1988, vol. 38, no. 1, |
Full text: | pdf (188.83 Kb) |
Site: | www.maik.ru/ru/journal/dan/ |
---|
The formal axiomatic approach to establishing the validity of the theory of constrained systems has obvious disadvantages: the source of the initial axioms (such as the Befreiungsprinzip and the conditions for constraints to be ideal) remains unclear. A constructive method is proposed for establishing the validity of the main principles of the dynamics of unilaterally constrained systems (including systems with collisions). The idea of the method is related to the analysis of physical methods for realising constraints (stiff systems, anisotropic viscosity, and apparent additional masses). This approach yields simple equations of motion, suitable for the entire time interval and more accurately incorporating the actual dynamics. Several problems of the mechanics of oscillatory systems with collisions are solved by the method. In particular, conditions are determined for the stability of periodic oscillatory modes and a study is made of the evolution of motion with inelastic collisions when the coefficient of restitution is close to unity. Total integrability is established and a qualitative analysis is presented of the problem of parabolic billiards in a uniform force field.
Keywords: | formal axiomatic approach; theory of constrained systems; Befreiungsprinzip; systems with collisions |
---|---|
Citation: | Kozlov V. V., A constructive method for justifying the theory of systems with nonretaining constraints, Journal of Applied Mathematics and Mechanics, 1988, vol. 52, no. 6, |
DOI: | 10.1016/0021-8928(88)90001-9 |
Full text: | pdf (820.22 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
The existence of vector fields which commute with the vector field of the initial system and are defined in the entire phase space is discussed. The phase fluxes of these fields are well-known to be symmetry groups of a dynamic system, since they map the set of all its solutions into itself. Obstacles to the existence of non-trivial symmetry groups are the generation of a large number of non-degenerate periodic solutions, and the transversal intersection of asymptotic surfaces. The symmetry groups of systems of a “normal” type, which play an important part in perturbation theory, are examined in detail. The general results are applied, in particular, to Hamiltonian systems. It is shown that the equations of rotation of a heavy asymmetric rigid body with a fixed point do not have a non-trivial symmetry group if the centre of mass of the body is not the same as the point of suspension. In particular, there is no supplementary many-valued analytic integral which is independent of the classical energy and area integrals.
Keywords: | existence of vector fields; phase space; phase fluxes; symmetry groups; dynamic system; non-degenerate periodic solutions; asymptotic surfaces; Hamiltonian systems |
---|---|
Citation: | Kozlov V. V., Symmetry groups of dynamical systems, Journal of Applied Mathematics and Mechanics, 1988, vol. 52, no. 4, |
DOI: | 10.1016/0021-8928(88)90026-3 |
Full text: | pdf (703.66 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., Solution branching and polynomial integrals in an invertible system on a torus, Mathematical Notes, 1988, vol. 44, no. 1, |
---|---|
Full text: | pdf (225.22 Kb) |
Impact-factor WoS (2022): | 0.600 (Q3) |
---|---|
Impact-factor RSCI (2014): | 0,502 |
ISSN (print): | 0025-567X |
ISSN (online): | 2305-2880 |
Site: | http://www.mathnet.ru/php/journal.phtml?jrnid=mzm&option_lang=eng |
Keywords: | integral invariant; Euler-Poincaré equations; invariant measure |
---|---|
Citation: | Kozlov V. V., Invariant measures of Euler–Poincaré equations on Lie algebras, Functional Analysis and Its Applications, 1988, vol. 22, no. 1, |
DOI: | 10.1007/BF01077727 |
Full text: | pdf (166.93 Kb) |
Impact-factor WoS (2015): | 0.486 |
---|---|
Impact-factor RSCI (2014): | 0,565 |
ISSN (print): | 0374-1990 |
ISSN (online): | 2305-2899 |
Site: | http://www.mathnet.ru/faa |
Citation: | Kozlov V. V., Soudarenie tel, Kvant, 1988, no. 9, |
---|---|
Full text: | pdf (5.41 Mb) |
Site: | http://kvant.mccme.ru/index.htm |
---|
Generalized mathematical models are considered of the motion of mechanical systems with nonintegrable constraints produced by a limiting process approaching infinity in anisotropic visous friction coefficient and attached mass. Provided an appropriate definition, the variations of kinematically admissible paths of such systems turn out to be extrema of the action functional. The validity is shown of the generalized principle of releasability and the idealness of constraints in an integral form. Constraint reactions in this case are no longer functions of the system states but are functionals of the system motions.
Keywords: | motion of mechanical systems with nonintegrable constraints; anisotropic visous friction; attached mass; generalized principle of releasability |
---|---|
Citation: | Kozlov V. V., Dynamics of systems with nonintegrable constraints. V: Freedom principle and ideal constraints condition, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1988, vol. 43, no. 6, |
Full text: | pdf (176.21 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
The analytical aspects are discussed of the classical perturbation theory of Hamiltonian systems for the case where the perturbation function is the sum of real exponents. The necessary conditions are formulated of complete integrability of disturbed Hamiltonian systems. General results are applied to a study of the integrability of generalized Toda chains.
Keywords: | Liouville theorem; classical perturbation theory of Hamiltonian systems; complete integrability; integrability of generalized Toda chains |
---|---|
Citation: | Kozlov V. V., Perturbation theory of Hamiltonian systems with noncompact invariant surfaces, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1988, vol. 43, no. 2, |
Full text: | pdf (364.79 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
The existence of an integral invariant with a smooth density for a dynamic system in a cylindrical phase space is considered. The well-known Krylov-Bogolyubov theorem guarantees the existence of an invariant measure for any system in a compact space (for a discussion of these topics see /1, 2/). But this measure is often concentrated in invariant sets of small dimensionality and in general is not an integral invariant with a summable density. For useful applications of ergodic theory, and in the theory of the Euler-Jacobi integrating factor, an invariant measure in the form of an integral invariant with smooth density is useful. Effective criteria for the existence of such measures in smooth dynamic systems are described. The general results are illustrated by examples from non-holonomic mechanics.
Keywords: | existence of an integral invariant; cylindrical phase space; Krylov- Bogolyubov theorem; invariant measure; ergodic theory; Euler-Jacobi integrating factor; non-holonomic mechanics |
---|---|
Citation: | Kozlov V. V., On the existence of an integral invariant of a smooth dynamic system, Journal of Applied Mathematics and Mechanics, 1987, vol. 51, no. 4, |
DOI: | 0.1016/0021-8928(87)90078-5 |
Full text: | pdf (542.86 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Various mathematical models of the motion of mechanical systems with nonintegrable constraints are known. Their validity is demonstrated by limiting processes in systems with no constraints. The general Hamilton-Ostrogradski principle, combined with the technique of limiting process, makes it possible to derive integral principles in the various models of motion of “unfree” systems. They are similar formally to Hamilton’s principle, but use a different method of variation of motions. The laws of conservation and the conditions of existence of an invariant measure are discussed.
Keywords: | motion of mechanical systems; nonintegrable constraints; limiting processes; Hamilton-Ostrogradski principle; integral principles; Hamilton’s principle; existence of an invariant measure |
---|---|
Citation: | Kozlov V. V., Dynamics of systems with nonintegrable constraints. IV. Integral principles, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1987, vol. 42, no. 5, |
Full text: | pdf (365.45 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Keywords: | integrability; nonintegrability; dynamical system |
---|---|
Citation: | Kozlov V. V., Phenomena of nonintegrability in Hamiltonian systems, Proceedings of the International Congress of Mathematicians (Berkeley, CA, 1986), vol. 2, Amer. Math. Soc., Providence, RI, 1987, |
Full text: | pdf (556.38 Kb) |
Citation: | Kozlov V. V., On the stability of equilibria of nonholonomic systems, Soviet Mathematics. Doklady, 1986, vol. 33, no. 2, |
---|---|
Full text: | pdf (153.51 Kb) |
Site: | www.maik.ru/ru/journal/dan/ |
---|
Keywords: | motions of natural mechanical systems; equilibrium position; frequencies of small oscillations; existence theorem; asymptotic trajectories; Maclaurin series; potential energy |
---|---|
Citation: | Kozlov V. V., Asymptotic motions and the inversion of the Lagrange–Dirichlet theorem, Journal of Applied Mathematics and Mechanics, 1986, vol. 50, no. 6, |
DOI: | 10.1016/0021-8928(86)90079-1 |
Full text: | pdf (664.8 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., The splitting of separatrices and the generation of isolated periodic solutions in Hamiltonian systems with one-and-a-half degrees of freedom, Russian Mathematical Surveys, 1986, vol. 41, no. 5, |
---|---|
DOI: | 10.1070/RM1986v041n05ABEH003439 |
Full text: | pdf (100.7 Kb) |
Impact-factor WoS (2022): | 0.900 (Q2) |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0036-0279 |
ISSN (online): | 1468-4829 |
Site: | http://www.mathnet.ru/umn |
Citation: | Kozlov V. V., Some aspects of the theory of dynamical systems, Geometry, differential equations and mechanics (Moscow, 1985), eds. V. V. Kozlov, A. T. Fomenko, Moskov. Gos. Univ., Mekh.-Mat. Fak., Moscow, 1986, |
---|---|
Full text: | pdf (519.56 Kb) |
Citation: | Kozlov V. V., Calculus of variations in the large and classical mechanics, Russian Mathematical Surveys, 1985, vol. 40, no. 2, |
---|---|
DOI: | 10.1070/RM1985v040n02ABEH003557 |
Full text: | pdf (1.7 Mb) |
Impact-factor WoS (2022): | 0.900 (Q2) |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0036-0279 |
ISSN (online): | 1468-4829 |
Site: | http://www.mathnet.ru/umn |
Citation: | Kozlov V. V., On the theory of integration of the equations of nonholonomic mechanics, Advances in Mechanics (Uspekhi Mekhaniki), 1985, vol. 8, no. 3, |
---|---|
Full text: | pdf (325.42 Kb) |
Citation: | Kozlov V. V., On the theory of integration of the equations of nonholonomic mechanics, Izv. Akad. Nauk SSSR Mekh. Tverd. Tela, 1985, no. 6, |
---|---|
Full text: | pdf (199.79 Kb) |
ISSN (print): | 0572-3299 |
---|---|
Site: | http://mtt.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., Integrable cases of the problem of motion of a point over a three- dimensional sphere in a force field with fourth-degree potential, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1985, vol. 40, no. 3, |
---|---|
Full text: | pdf (80.24 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., Realization of nonintegrable constraints in classical mechanics, Soviet Mathematics. Doklady, 1983, vol. 28, no. 3, |
---|---|
Full text: | pdf (153.5 Kb) |
Site: | www.maik.ru/ru/journal/dan/ |
---|
The general equation for variation of solenoidal vector field which is satisfied by the magnetic field intensity in magnetohydrodynamics and the vector of vortex barotropic flows of ideal fluid, with the equation of continuity taken into account, is reduced to the “Euler equation for the change of momentum”. This note is used for investigating of the topology of steady barotropic flows of inviscid compressible fluid.
Citation: | Kozlov V. V., Notes on steady vortex motions of continuous medium, Journal of Applied Mathematics and Mechanics, 1983, vol. 47, no. 2, |
---|---|
DOI: | 10.1016/0021-8928(83)90020-5 |
Full text: | pdf (171.18 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., Integrability and non-integrability in Hamiltonian mechanics, Russian Mathematical Surveys, 1983, vol. 38, no. 1, |
---|---|
DOI: | 10.1070/RM1983v038n01ABEH003330 |
Full text: | pdf (3.69 Mb) |
Impact-factor WoS (2022): | 0.900 (Q2) |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0036-0279 |
ISSN (online): | 1468-4829 |
Site: | http://www.mathnet.ru/umn |
Citation: | Kozlov V. V., The hydrodynamics of Hamiltonian systems, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1983, vol. 38, no. 6, |
---|---|
Full text: | pdf (325.77 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., Dynamics of systems with nonintegrable constraints. III”, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1983, vol. 38, no. 3, |
---|---|
Full text: | pdf (318.53 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., Averaging in the neighborhood of stable periodic motions, Soviet Mathematics. Doklady, 1982, vol. 27, no. 3, |
---|---|
Full text: | pdf (135.43 Kb) |
Site: | www.maik.ru/ru/journal/dan/ |
---|
Motions of natural mechanical systems that approach their equilibrium positions with unlimited increase of time are considered.
Citation: | Kozlov V. V., Asymptotic solutions of equations of classical mechanics, Journal of Applied Mathematics and Mechanics, 1982, vol. 46, no. 4, |
---|---|
DOI: | 10.1016/0021-8928(82)90029-6 |
Full text: | pdf (286 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., A conjecture on the existence of asymptotic motions in classical mechanics, Functional Analysis and Its Applications, 1982, vol. 16, no. 4, |
---|---|
Full text: | pdf (152.44 Kb) |
Impact-factor WoS (2015): | 0.486 |
---|---|
Impact-factor RSCI (2014): | 0,565 |
ISSN (print): | 0374-1990 |
ISSN (online): | 2305-2899 |
Site: | http://www.mathnet.ru/faa |
Citation: | Kozlov V. V., Hamiltonian equation in a problem of motion of a rigid body with an unmovable point in redundant coordinates, Teor. Primen. Meh., 1982, vol. 8, |
---|---|
Full text: | pdf (189.3 Kb) |
Citation: | Kozlov V. V., Dynamics of systems with nonintegrable constraints. II, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1982, vol. 37, no. 3-4, |
---|---|
Full text: | pdf (371.05 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., Dynamics of systems with nonintegrable constraints. I, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1982, vol. 37, no. 3-4, |
---|---|
Full text: | pdf (388.56 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., Equilibrium stability in a potential field, taking account of viscous friction, Journal of Applied Mathematics and Mechanics, 1981, vol. 45, no. 3, |
---|---|
DOI: | 10.1016/0021-8928(81)90077-0 |
Full text: | pdf (179.39 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., On the instability of equilibrium in a potential field, Russian Mathematical Surveys, 1981, vol. 36, no. 3, |
---|---|
DOI: | 10.1070/RM1981v036n03ABEH004261 |
Full text: | pdf (170.8 Kb) |
Impact-factor WoS (2022): | 0.900 (Q2) |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0036-0279 |
ISSN (online): | 1468-4829 |
Site: | http://www.mathnet.ru/umn |
Citation: | Kozlov V. V., Instability of equilibrium in a potential field, Russian Mathematical Surveys, 1981, vol. 36, no. 1, |
---|---|
Full text: | pdf (112.67 Kb) |
Impact-factor WoS (2022): | 0.900 (Q2) |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0036-0279 |
ISSN (online): | 1468-4829 |
Site: | http://www.mathnet.ru/umn |
Citation: | Kozlov V. V., Two integrable problems of classical dynamics, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1981, vol. 36, no. 3-4, |
---|---|
Full text: | pdf (128.9 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., Periodic oscillations of a composite pendulum, Journal of Applied Mathematics and Mechanics, 1980, vol. 44, no. 2, |
---|---|
DOI: | 10.1016/0021-8928(80)90142-2 |
Full text: | pdf (262.09 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., Oscillations of one-dimensional systems with periodic potential, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1980, vol. 35, no. 5-6, |
---|---|
Full text: | pdf (144.12 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., Topological obstructions to the integrability of natural mechanical systems, Soviet Mathematics. Doklady, 1979, vol. 20, no. 6, |
---|---|
Full text: | pdf (161.56 Kb) |
Site: | www.maik.ru/ru/journal/dan/ |
---|
Citation: | Kozlov V. V., Nonexistence of univalued integrals and branching of solutions in rigid body dynamics, Journal of Applied Mathematics and Mechanics, 1978, vol. 42, no. 3, |
---|---|
DOI: | 10.1016/0021-8928(78)90109-0 |
Full text: | pdf (490.39 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., On integrals of quasiperiodic functions, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1978, vol. 33, no. 1-2, |
---|---|
Full text: | pdf (266.47 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Qualitative properties of typical rotations of a heavy solid body are analyzed in the case of the Goriachev-Chaplygin problem in which the first integrals of equations of motion are independent. Gyration numbers of tangent vector fields on two-dimensional invariant tori are determined. It is shown that the nutation of a solid body is a quasi-periodic motion, and spin and precession have a pcincipal motion. If the gyration number is irrational, then in the case of a solid body the principal motion of node lines is zero.
Citation: | Kozlov V. V., On the qualitative analysis of motion of a solid body in the Goriachev–Chaplygin problem, Journal of Applied Mathematics and Mechanics, 1977, vol. 41, no. 2, |
---|---|
DOI: | 10.1016/0021-8928(77)90005-3 |
Full text: | pdf (649.41 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., The geometry of domains of possible motions with boundary, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1977, no. 5, |
---|---|
Full text: | pdf (121.39 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., New periodic solutions in the problem of motion of a heavy rigid body about a fixed point, Issledovaniya po mekhanike zhidkikh i tverdykh tel, Moscow: Izdatel'stvo Moskovskogo Universiteta, 1977, |
---|---|
Full text: | pdf (263.68 Kb) |
Citation: | Kozlov V. V., On a structure of additional integrals in the problem of rotation of a rigid body around a fixed point, Issledovaniya po mekhanike zhidkikh i tverdykh tel, Moscow: Izdatel'stvo Moskovskogo Universiteta, 1977, |
---|---|
Full text: | pdf (332.25 Kb) |
Citation: | Kozlov V. V., The principle of least action and periodic solutions in problems of classical mechanics, Journal of Applied Mathematics and Mechanics, 1976, vol. 40, no. 3, |
---|---|
DOI: | 10.1016/0021-8928(76)90027-7 |
Full text: | pdf (669.87 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., On a problem of Poincaré, Journal of Applied Mathematics and Mechanics, 1976, vol. 40, no. 2, |
---|---|
DOI: | 10.1016/0021-8928(76)90070-8 |
Full text: | pdf (256.75 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., Splitting of the separatrices in the perturbed Euler–Poinsot problem, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1976, vol. 31, no. 6, |
---|---|
Full text: | pdf (184.74 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., Nonexistence of analytic integrals near equilibrium positions of Hamiltonian systems, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1976, vol. 31, no. 1, |
---|---|
Full text: | pdf (186.68 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
The theory of generation of periodic solutions in canonic systems of near-integrable differential equations was developed by Poincaré for the purposes of celestial mechanics. In this paper we establish the applicability of these results to the classical problem of the motion of a heavy solid body with a fixed point. By the same token we have succeeded in essentially widening the class of periodic solutions appearing in this problem.
Citation: | Kozlov V. V., New periodic solutions for the problem of motion of a heavy solid body around a fixed point, Journal of Applied Mathematics and Mechanics, 1975, vol. 39, no. 3, |
---|---|
DOI: | 10.1016/0021-8928(75)90003-9 |
Full text: | pdf (626.19 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., Dynamic systems arising on the invariant tori of the Kowalewska problem, Journal of Applied Mathematics and Mechanics, 1975, vol. 39, no. 1, |
---|---|
DOI: | 10.1016/0021-8928(75)90030-1 |
Full text: | pdf (522.86 Kb) |
Impact-factor WoS (2017): | 0.461 |
---|---|
Impact-factor RSCI (2014): | 0,996 |
ISSN (print): | 0032-8235 |
Site: | http://pmm.ipmnet.ru/ru/ |
Citation: | Kozlov V. V., The nonexistence of an additional analytic integral in the problem of the motion of a nonsymmetric heavy solid around a fixed point, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1975, vol. 30, no. 1, |
---|---|
Full text: | pdf (212.32 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., The geometry of the “action-angle” variables in the Euler–Poinsot problem, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1974, vol. 29, no. 5, |
---|---|
Full text: | pdf (163.57 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., The nonexistence of analytic integrals of canonical systems that are nearly integrable, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1974, vol. 29, no. 2, |
---|---|
Full text: | pdf (219.42 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|
Citation: | Kozlov V. V., Certain properties of particular integrals of the canonical equations, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1973, vol. 28, no. 1, |
---|---|
Full text: | pdf (131.24 Kb) |
Site: | http://new.math.msu.su/vestnik/ |
---|