Identification of parameters of the model of toroidal body motion using experimental data
Russian Journal of Nonlinear Dynamics, 2018, vol. 14, no. 1, pp. 99-121
Abstract
pdf (1.49 Mb)
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/m^{3}). 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, pp. 99-121
Hamilton’s Principle and the Rolling Motion of a Symmetric Ball
Doklady Physics, 2017, vol. 62, no. 6, pp. 314-317
Abstract
pdf (206.42 Kb)
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, pp. 314-317
Retrograde motion of a rolling disk
Physics-Uspekhi, 2017, vol. 60, no. 9, pp. 931-934
Abstract
pdf (364.25 Kb)
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, pp. 931-934
Control of the Motion of a Triaxial Ellipsoid in a Fluid Using Rotors
Mathematical Notes, 2017, vol. 102, no. 4, pp. 455-464
Abstract
pdf (616.14 Kb)
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, pp. 455-464
Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics
Russian Mathematical Surveys, 2017, vol. 72, no. 5, pp. 783-840
Abstract
pdf (1.09 Mb)
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.
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, pp. 783-840
A Chaplygin sleigh with a moving point mass
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2017, vol. 27, no. 4, pp. 583-589
Abstract
pdf (314.6 Kb)
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, pp. 583-589
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, pp. 576-582
Abstract
pdf (312.38 Kb)
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, pp. 576-582
Dynamics of Two Point Vortices in an External Compressible Shear Flow
Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 893–908
Abstract
pdf (3.4 Mb)
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, pp. 893–908
The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration
Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 955–975
Abstract
pdf (1.91 Mb)
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.
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, pp. 955–975
A Family of Models with Blue Sky Catastrophes of Different Classes
Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 551-565
Abstract
pdf (1.61 Mb)
A generalized model with bifurcations associated with blue sky catastrophes is introduced. Depending on an integer index $m$, different kinds of attractors arise, including those associated with quasi-periodic oscillations and with hyperbolic chaos. Verification of the hyperbolicity is provided based on statistical analysis of intersection angles of stable and unstable manifolds.
Kuptsov P. V., Kuznetsov S. P., Stankevich N. V., A Family of Models with Blue Sky Catastrophes of Different Classes, Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 551-565
An Inhomogeneous Chaplygin Sleigh
Regular and Chaotic Dynamics, 2017, vol. 22, no. 4, pp. 435-447
Abstract
pdf (676.55 Kb)
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.
The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane
Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 298-317
Abstract
pdf (399.6 Kb)
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, pp. 298-317
The Inertial Motion of a Roller Racer
Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 239-247
Abstract
pdf (3.26 Mb)
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.
Autonomous Strange Nonchaotic Oscillations in a System of Mechanical Rotators
Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 210-225
Abstract
pdf (2.26 Mb)
We investigate strange nonchaotic self-oscillations in a dissipative system consisting of three mechanical rotators driven by a constant torque applied to one of them. The external driving is nonoscillatory; the incommensurable frequency ratio in vibrational-rotational
dynamics arises due to an irrational ratio of diameters of the rotating elements involved. It is shown that, when losing stable equilibrium, the system can demonstrate two- or three-frequency quasi-periodic, chaotic and strange nonchaotic self-oscillations. The conclusions of the work are confirmed by numerical calculations of Lyapunov exponents, fractal dimensions, spectral analysis, and by special methods of detection of a strange nonchaotic attractor (SNA): phase sensitivity and analysis using rational approximation for the frequency ratio. In particular, SNA possesses a zero value of the largest Lyapunov exponent (and negative values of the other exponents), a capacitive dimension close to 2 and a singular continuous power spectrum. In general, the results of this work shed a new light on the occurrence of strange nonchaotic dynamics.
Jalnine A. Y., Kuznetsov S. P., Autonomous Strange Nonchaotic Oscillations in a System of Mechanical Rotators, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 210-225
The Hess–Appelrot Case and Quantization of the Rotation Number
Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 180-196
Abstract
pdf (991.1 Kb)
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, pp. 180-196
An inhomogeneous Chaplygin sleigh
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, pp. 625–639
Abstract
pdf (518.79 Kb)
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.
Stability analysis of steady motions of a spherical robot of combined type
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, pp. 611–623
Abstract
pdf (1.75 Mb)
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.
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, pp. 611–623
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, pp. 599–609
Abstract
pdf (3.54 Mb)
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, pp. 599–609
Experimental investigation of the fall of helical bodies in a fluid
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, pp. 585–598
Abstract
pdf (853.58 Kb)
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, pp. 585–598
The Hess–Appelrot case and quantization of the rotation number
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 3, pp. 433-452
Abstract
pdf (556.32 Kb)
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, pp. 433-452
Chaos generator with the Smale–Williams attractor based on oscillation death
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 3, pp. 303-315
Abstract
pdf (5.99 Mb)
A nonautonomous system with a uniformly hyperbolic attractor of Smale – Williams type in
a Poincaré cross-section is proposed with generation implemented on the basis of the effect of
oscillation death. The results of a numerical study of the system are presented: iteration diagrams
for phases and portraits of the attractor in the stroboscopic Poincaré cross-section, power density
spectra, Lyapunov exponents and their dependence on parameters, and the atlas of regimes. The
hyperbolicity of the attractor is verified using the criterion of angles.
Doroshenko V. M., Kruglov V. P., Kuznetsov S. P., Chaos generator with the Smale–Williams attractor based on oscillation death, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 3, pp. 303-315
Regular and chaotic dynamics in the rubber model of a Chaplygin top
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 2, pp. 277-297
Abstract
pdf (2.3 Mb)
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, pp. 277-297
Autonomous strange non-chaotic oscillations in a system of mechanical rotators
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 2, pp. 257-275
Abstract
pdf (907.9 Kb)
We investigate strange nonchaotic self-oscillations in a dissipative system consisting of three mechanical rotators driven by a constant torque applied to one of them. The external driving is nonoscillatory; the incommensurable frequency ratio in vibrational-rotational dynamics arises due to an irrational ratio of diameters of the rotating elements involved. It is shown that, when losing stable equilibrium, the system can demonstrate two- or three-frequency quasi-periodic, chaotic and strange nonchaotic self-oscillations. The conclusions of the work are confirmed by numerical calculations of Lyapunov exponents, fractal dimensions, spectral analysis, and by special methods of detection of a strange nonchaotic attractor (SNA): phase sensitivity and analysis using rational approximation for the frequency ratio. In particular, SNA possesses a zero value of the largest Lyapunov exponent (and negative values of the other exponents), a capacitive dimension close to “2” and a singular continuous power spectrum. In general, the results of this work shed a new light on the occurrence of strange nonchaotic dynamics.
Jalnine A. Y., Kuznetsov S. P., Autonomous strange non-chaotic oscillations in a system of mechanical rotators, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 2, pp. 257-275
Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 1, pp. 129-146
Abstract
pdf (2.82 Mb)
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, pp. 129-146
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, pp. 741-759
Abstract
pdf (567.52 Kb)
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, pp. 741-759
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, pp. 435-458
Abstract
pdf (1.49 Mb)
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, pp. 435-458
Optimal control of the motion of a helical body in a liquid using rotors
Russian Journal of Mathematical Physics, 2017, vol. 24, no. 3, pp. 399-411
Abstract
pdf (582.92 Kb)
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, pp. 399-411
Chaplygin sleigh with periodically oscillating internal mass
EPL, 2017, vol. 119, no. 6, 60008, 7 pp.
Abstract
pdf (854.88 Kb)
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, 7 pp.
Experimental investigations of a highly maneuverable mobile omniwheel robot
International Journal of Advanced Robotic Systems, 2017, vol. 14, no. 6, pp. 1-9
Abstract
pdf (654.85 Kb)
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, pp. 1-9
Experimental Investigation of the Dynamics of a Brake Shoe
Doklady Physics, 2016, vol. 61, no. 12, pp. 611-614
Abstract
pdf (413.7 Kb)
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, pp. 611-614
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, pp. 293-297
Abstract
pdf (300.44 Kb)
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, pp. 293-297
Controlled Motion of a Rigid Body with Internal Mechanisms in an Ideal Incompressible Fluid
Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, pp. 302-332
Abstract
pdf (1.36 Mb)
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, pp. 302-332
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, pp. 158-167
Abstract
pdf (366.98 Kb)
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, pp. 158-167
Generalizations of the Kovalevskaya Case and Quaternions
Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, pp. 33-44
Abstract
pdf (190.82 Kb)
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, pp. 33-44
The Hess–Appelrot system and its nonholonomic analogs
Proceedings of the Steklov Institute of Mathematics, 2016, vol. 294, pp. 268-292
Abstract
pdf (1.23 Mb)
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, pp. 268-292
Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics
Sbornik: Mathematics, 2016, vol. 207, no. 10, pp. 1435–1449
Abstract
For integrable systems with two degrees of freedom there are well-known inequalities connecting the Euler characteristic of the configuration space (as a closed two-dimensional surface) with the number of singular points of Newtonian type of the potential energy. On the other hand, there are results on conditions for ergodicity of systems on a two-dimensional torus with short-range potential depending only on the distance from an attracting or repelling centre. In the present paper we consider the problem of conditions for the existence of nontrivial first integrals that are polynomial in the momenta of the problem of motion of a particle on a multi-dimensional Euclidean torus in a force field whose potential has singularity points. These conditions depend only on the order of the singularity, and in the two-dimensional case they are satisfied by potentials with singularities of Newtonian type.
Keywords:
polynomial integrals, potentials with singularities, order of singularity, Poincaré condition
Citation:
Kozlov V. V., Treschev D. V., Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics, Sbornik: Mathematics, 2016, vol. 207, no. 10, pp. 1435–1449
Bifurcation Analysis of the Motion of a Cylinder and a Point Vortex in an Ideal Fluid
Mathematical Notes, 2016, vol. 99, no. 6, pp. 834-839
Abstract
pdf (545.15 Kb)
We consider an integrable Hamiltonian system describing the motion of a circular cylinder and a vortex filament in an ideal fluid. We construct bifurcation diagrams and bifurcation complexes for the case in which the integral manifold is compact and for various topological structures of the symplectic leaf. The types of motions corresponding to the bifurcation curves and their stability are discussed.
Borisov A. V., Ryabov P. E., Sokolov S. V., Bifurcation Analysis of the Motion of a Cylinder and a Point Vortex in an Ideal Fluid, Mathematical Notes, 2016, vol. 99, no. 6, pp. 834-839
On the equations of the hydrodynamic type
Journal of Applied Mathematics and Mechanics, 2016, vol. 80, no. 3, pp. 209–214
Abstract
The conditions of stability of ‘pure’ equilibrium states of finite-dimensional dynamical systems of the hydrodynamic type are indicated. Integrable chains allowing full sets of quadratic first integrals are found. The conditions of existence of invariant Gaussian measures in the infinite-dimensional case are discussed. The existence of such measures makes it possible to establish the properties of recurrence of solutions of infinite-dimensional dynamical systems of the hydrodynamic type.
Citation:
Kozlov V. V., On the equations of the hydrodynamic type, Journal of Applied Mathematics and Mechanics, 2016, vol. 80, no. 3, pp. 209–214
Invariant measures of smooth dynamical systems, generalized functions and summation methods
Izvestiya: Mathematics, 2016, vol. 80, no. 2, pp. 342–358
Abstract
pdf (289.71 Kb)
We discuss conditions for the existence of invariant measures of smooth dynamical systems on compact manifolds. If there is an invariant measure with continuously differentiable density, then the divergence of the vector field along every solution tends to zero in the Cesaro sense as time increases unboundedly. Here the Cesaro convergence may be replaced, for example, by any Riesz summation method, which can be arbitrarily close to ordinary convergence (but does not coincide with it). We give an example of a system whose divergence tends to zero in the ordinary sense but none of its invariant measures is absolutely continuous with respect to the `standard' Lebesgue measure (generated by some Riemannian metric) on the phase space. We give examples of analytic systems of differential equations on analytic phase spaces admitting invariant measures of any prescribed smoothness (including a measure with integrable density), but having no invariant measures with positive continuous densities. We give a new proof of the classical Bogolyubov-Krylov theorem using generalized functions and the Hahn-Banach theorem. The properties of signed invariant measures are also discussed.
Keywords:
invariant measures, generalized functions, summation methods, small denominators, Hahn–Banach theorem
Citation:
Kozlov V. V., Invariant measures of smooth dynamical systems, generalized functions and summation methods, Izvestiya: Mathematics, 2016, vol. 80, no. 2, pp. 342–358
Dynamics of a Painlevé-Appel system
Journal of Applied Mathematics and Mechanics, 2016, vol. 80, no. 1, pp. 7-15
Abstract
pdf (1.78 Mb)
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, pp. 7-15
Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas
Russian Mathematical Surveys, 2016, vol. 71, no. 2, pp. 253–290
Abstract
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The problem of conditions ensuring the existence of first integrals that are polynomials in the momenta (velocities) is considered for certain multidimensional billiard systems which play an important role in non-equilibrium statistical mechanics. These are the Lorentz gas, a particle in a Euclidean space with (not necessarily convex) scattering domains, and the Boltzmann–Gibbs gas, a system of small identical balls in a rectangular box which collide elastically with one another and the walls of the box. The ergodic properties of such systems are only partially understood: some problems are still waiting for solution, and in certain cases (for instance, when the scatterers are non-convex) the system is known not to be ergodic. An approach to showing the absence of a non-trivial polynomial first integral with continuously differentiable coefficients is developed. The known first integrals for integrable problems in dynamics are mostly polynomials in the momenta (or functions of polynomials). The investigation of multidimensional billiards with non-compact configuration space, when there is no hope for ergodic behaviour, is of particular interest. Applications of the general results on the absence of non-trivial polynomial integrals to problems in statistical mechanics are discussed.
Keywords:
Birkhoff billiards, Lorentz gas, Boltzmann–Gibbs gas, polynomial integral, topological obstructions to integrability, elastic reflection, KAM theory
Citation:
Kozlov V. V., Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas, Russian Mathematical Surveys, 2016, vol. 71, no. 2, pp. 253–290
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, pp. 886-890
Abstract
pdf (508.12 Kb)
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, pp. 886-890
Regular and Chaotic Motions of a Chaplygin Sleigh under Periodic Pulsed Torque Impacts
Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 792-803
Abstract
pdf (1.65 Mb)
For a Chaplygin sleigh on a plane, which is a paradigmatic system of nonholonomic mechanics, we consider dynamics driven by periodic pulses of supplied torque depending on the instant spatial orientation of the sleigh. Additionally, we assume that a weak viscous force and moment affect the sleigh in time intervals between the pulses to provide sustained modes of the motion associated with attractors in the reduced three-dimensional phase space (velocity, angular velocity, rotation angle). The developed discrete version of the problem of the Chaplygin sleigh is an analog of the classical Chirikov map appropriate for the nonholonomic situation. We demonstrate numerically, discuss and classify dynamical regimes depending on the parameters, including regular motions and diffusive-like random walks associated, respectively, with regular and chaotic attractors in the reduced momentum dynamical equations.
Borisov A. V., Kuznetsov S. P., Regular and Chaotic Motions of a Chaplygin Sleigh under Periodic Pulsed Torque Impacts, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 792-803
Control of the Motion of a Helical Body in a Fluid Using Rotors
Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 874-884
Abstract
pdf (1.23 Mb)
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, pp. 874-884
Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top
Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 885-901
Abstract
pdf (2.21 Mb)
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, pp. 885-901
Experimental Investigations of the Controlled Motion of a Screwless Underwater Robot
Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 918-926
Abstract
pdf (6.91 Mb)
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, pp. 918-926
Influence of Vortex Structures on the Controlled Motion of an Above-water Screwless Robot
Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 927-938
Abstract
pdf (6.5 Mb)
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.
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, pp. 927-938
Spiral Chaos in the Nonholonomic Model of a Chaplygin Top
Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 939-954
Abstract
pdf (2.21 Mb)
This paper presents a numerical study of the chaotic dynamics of a dynamically asymmetric unbalanced ball (Chaplygin top) rolling on a plane. It is well known that the dynamics of such a system reduces to the investigation of a three-dimensional map, which in the general case has no smooth invariant measure. It is shown that homoclinic strange attractors of discrete spiral type (discrete Shilnikov type attractors) arise in this model for certain parameters. From the viewpoint of physical motions, the trace of the contact point of a Chaplygin top on a plane is studied for the case where the phase trajectory sweeps out a discrete spiral attractor. Using the analysis of the trajectory of this trace, a conclusion is drawn
about the influence of “strangeness” of the attractor on the motion pattern of the top.
Borisov A. V., Kazakov A. O., Sataev I. R., Spiral Chaos in the Nonholonomic Model of a Chaplygin Top, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 939-954
On the Extendability of Noether’s Integrals for Orbifolds of Constant Negative Curvature
Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 821-831
Abstract
pdf (630.35 Kb)
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).
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, pp. 821-831
Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups
Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 759-774
Abstract
pdf (3.48 Mb)
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, pp. 759-774
The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity
Regular and Chaotic Dynamics, 2016, vol. 21, no. 5, pp. 556-580
Abstract
pdf (1.45 Mb)
In this paper, we consider in detail the 2-body problem in spaces of constant positive curvature $S^2$ and $S^3$. 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, pp. 556-580
Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period
Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, pp. 455-476
Abstract
pdf (1.87 Mb)
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.
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, pp. 455-476
The Dynamics of Vortex Sources in a Deformation Flow
Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 367-376
Abstract
pdf (1.37 Mb)
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.
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, pp. 367-376
Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics
Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 232-248
Abstract
pdf (941.28 Kb)
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, pp. 232-248
Verification of Hyperbolicity for Attractors of Some Mechanical Systems with Chaotic Dynamics
Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 160-174
Abstract
pdf (1.14 Mb)
Computer verification of hyperbolicity is provided based on statistical analysis of the angles of intersection of stable and unstable manifolds for mechanical systems with hyperbolic attractors of Smale – Williams type: (i) a particle sliding on a plane under periodic kicks, (ii) interacting particles moving on two alternately rotating disks, and (iii) a string with parametric excitation of standing-wave patterns by a modulated pump. The examples are of interest as contributing to filling the hyperbolic theory of dynamical systems with physical content.
Kuznetsov S. P., Kruglov V. P., Verification of Hyperbolicity for Attractors of Some Mechanical Systems with Chaotic Dynamics, Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 160-174
Dynamics of the Chaplygin Sleigh on a Cylinder
Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 136-146
Abstract
pdf (268.54 Kb)
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.
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, pp. 136-146
Dynamics of the Chaplygin sleigh on a cylinder
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 4, pp. 675–687
Abstract
pdf (331.42 Kb)
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.
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, pp. 675–687
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, pp. 663–674
Abstract
pdf (304.78 Kb)
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, pp. 663–674
Historical and critical review of the development of nonholonomic mechanics: the classical period
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 3, pp. 385-411
Abstract
pdf (1.9 Mb)
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.
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, pp. 385-411
Dynamics of the Suslov problem in a gravitational field: reversal and strange attractors
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 2, pp. 263-287
Abstract
pdf (3.54 Mb)
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.
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, pp. 263-287
Pendulum system with an infinite number of equilibrium states and quasiperiodic dynamics
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 2, pp. 223-234
Abstract
pdf (2.38 Mb)
Examples of mechanical systems are discussed, where quasi-periodic motions may occur, caused by an irrational ratio of the radii of rotating elements that constitute the system. For the pendulum system with frictional transmission of rotation between the elements, in the conservative and dissipative cases we note the coexistence of an infinite number of stable fixed points, and in the case of the self-oscillating system the presence of many attractors in the form of limit cycles and of quasi-periodic rotational modes is observed. In the case of quasi-periodic dynamics the frequencies of spectral components depend on the parameters, but the ratio of basic incommensurate frequencies remains constant and is determined by the irrational number characterizing the relative size of the elements.
Kuznetsov A. P., Kuznetsov S. P., Sedova Y. V., Pendulum system with an infinite number of equilibrium states and quasiperiodic dynamics, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 2, pp. 223-234
On the Hadamard–Hamel problem and the dynamics of wheeled vehicles
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 1, pp. 145-163
Abstract
pdf (445.79 Kb)
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, pp. 145-163
Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 1, pp. 121-143
Abstract
pdf (1.96 Mb)
Dynamical equations are formulated and a numerical study is provided for selfoscillatory model systems based on the triple linkage hinge mechanism of Thurston–Weeks–Hunt–MacKay. We consider systems with a holonomic mechanical constraint of three rotators as well as systems, where three rotators interact by potential forces. We present and discuss some quantitative characteristics of the chaotic regimes (Lyapunov exponents, power spectrum). Chaotic dynamics of the models we consider are associated with hyperbolic attractors, at least, at relatively small supercriticality of the self-oscillating modes; that follows from numerical analysis of the distribution for angles of intersection of stable and unstable manifolds of phase trajectories on the attractors. In systems based on rotators with interacting potential the hyperbolicity is violated starting from a certain level of excitation.
Kuznetsov S. P., Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 1, pp. 121-143
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, pp. 99-120
Abstract
pdf (4.44 Mb)
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, pp. 99-120
The Hojman Construction and Hamiltonization of Nonholonomic Systems
Symmetry, Integrability and Geometry: Methods and Applications, 2016, vol. 12, 012, 19 pp.
Abstract
pdf (571.09 Kb)
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.
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, 19 pp.
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, 13 pp.
Abstract
pdf (1.38 Mb)
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, 13 pp.
Rigid Body Dynamics in Non-Euclidean Spaces
Russian Journal of Mathematical Physics, 2016, vol. 23, no. 4, pp. 431-454
Abstract
pdf (704.17 Kb)
In this paper, we focus on the study of two-dimensional plate dynamics on the Lobachevskii plane $L^2$. 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, pp. 431-454
Dynamics of a body sliding on a rough plane and supported at three points
Theoretical and Applied Mechanics, 2016, vol. 43, no. 2, pp. 169-190
Abstract
pdf (1.81 Mb)
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
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, pp. 169-190
The phenomenon of reversal in the Euler–Poincaré–Suslov nonholonomic systems
Journal of Dynamical and Control Systems, 2016, vol. 22, no. 4, pp. 713–724
Abstract
The development of robotics makes it necessary to study the problem of controlling nonholonomic systems (Svinin et al., Regul Chaotic Dyn. 2013; 18(1–2): 126–143, Borisov et al., Regul. Chaotic Dyn. 2013; 18(1–2): 144–158, Ivanova et al., Regul Chaotic Dyn. 2014; 19(1): 140–143). In this paper, the dynamics of nonholonomic systems on Lie groups with a left-invariant kinetic energy and left-invariant constraints are considered. Equations of motion form a closed system of differential equations on the corresponding Lie algebra. In addition, the effect of change in the stability of steady motions of these systems with the direction of motion reversed (the reversal found in rattleback dynamics) is discussed. As an illustration, the rotation of a rigid body with a fixed point and the Suslov nonholonomic constraint as well as the motion of the Chaplygin sleigh is considered.
Kozlov V. V., The phenomenon of reversal in the Euler–Poincaré–Suslov nonholonomic systems, Journal of Dynamical and Control Systems, 2016, vol. 22, no. 4, pp. 713–724
Invariant and quasi-invariant measures on infinite-dimensional spaces
Doklady Mathematics, 2015, vol. 92, no. 3, pp. 743–746
Abstract
Extensions of locally convex topological spaces are considered such that finite cylindrical measures which are not countably additive on their initial domains turn out to be countably additive on the extensions. Extensions of certain transformations of the initial spaces with respect to which the initial measures are invariant or quasi-invariant to the extensions of these spaces are described. Similar questions are considered for differentiable measures. The constructions may find applications in statistical mechanics and quantum field theory.
Citation:
Kozlov V. V., Smolyanov O. G., Invariant and quasi-invariant measures on infinite-dimensional spaces, Doklady Mathematics, 2015, vol. 92, no. 3, pp. 743–746
A New Integrable System of Nonholonomic Mechanics
Doklady Physics, 2015, vol. 60, no. 6, pp. 269-271
Abstract
pdf (255.48 Kb)
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, pp. 269-271
Homogeneous systems with quadratic integrals, Lie–Poisson quasi-brackets, and the Kovalevskaya method
Sbornik: Mathematics, 2015, vol. 206, no. 12, pp. 29–54
Abstract
pdf (481.65 Kb)
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, pp. 29–54
Notes on new friction models and nonholonomic mechanics
Physics-Uspekhi, 2015, vol. 58, no. 12, pp. 1220-1222
Abstract
pdf (262.98 Kb)
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, pp. 1220-1222
Figures of equilibrium of an inhomogeneous self-gravitating fluid
Celestial Mechanics and Dynamical Astronomy, 2015, vol. 122, no. 1, pp. 1-26
Abstract
pdf (651.34 Kb)
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, pp. 1-26
Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: Pro or contra?
Journal of Geometry and Physics, 2015, vol. 87, pp. 61-75
Abstract
pdf (754.64 Kb)
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.
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, pp. 61-75
On the loss of contact of the Euler disk
Nonlinear Dynamics, 2015, vol. 79, no. 4, pp. 2287-2294
Abstract
pdf (829.12 Kb)
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, pp. 2287-2294
Rational integrals of quasi-homogeneous dynamical systems
Journal of Applied Mathematics and Mechanics, 2015, vol. 79, no. 3, pp. 209–216
Abstract
Dynamical systems, described by quasi-homogeneous systems of differential equations with polynomial right-hand sides, are considered. The Euler–Poisson equations from solid-state dynamics, as well as the Euler–Poincaré equations in Lie algebras, which describe the dynamics of systems in Lie groups with a left-invariant kinetic energy, can be pointed out as examples. The conditions for the existence of rational first integrals of quasi-homogeneous systems are found. They include the conditions for the existence of invariant algebraic manifolds. Examples of systems with rational integrals which do not admit of first integrals that are polynomial with respect to the momenta are presented. Results of a general nature are also demonstrated in the example of a Hess–Appel’rot invariant manifold from the dynamics of an asymmetric heavy top.
Citation:
Kozlov V. V., Rational integrals of quasi-homogeneous dynamical systems, Journal of Applied Mathematics and Mechanics, 2015, vol. 79, no. 3, pp. 209–216
Generalized transport-logistic problem as class of dynamical systems
Mathematical Models and Computer Simulations, 2015, vol. 27, no. 12, pp. 65–87
Abstract
pdf (925.4 Kb)
Dynamical systems on network with discrete set of states and discrete time are considered. Sites, channels and particles are forming an abstract model of mass transport, information and so on, on the one hand, and another, they are forming dynamical system of deterministic or stochastic type. State of the system in the following discrete instant of time $S(T+1)$ is defined by transformation of the state at the moment $S(T)$ with given rules $L$, $S(T+1)=L(S(T))$. In this case, $S(T+1)$ does not necessarily belong to the admissible states set $A$. Then "judicial system" is activated, i.e. operator $P$ such that projects $S(T+1)$ to $A$. Thus, $S(T+1)=\{L(S(T))$, if $L(S(T))$ belongs $A$; $PL(S(T))$, if $L(S(T))$ does not belong $A\}$. Properties of these systems are researched, and applications for transport problems are discussed.
Bugaev A. S., Buslaev A. P., Kozlov V. V., Tatashev A. G., Yashina M. V., Generalized transport-logistic problem as class of dynamical systems, Mathematical Models and Computer Simulations, 2015, vol. 27, no. 12, pp. 65–87
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, pp. 583-592
Abstract
pdf (395.52 Kb)
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, pp. 583-592
Experimental determination of the added masses by method of towing
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 568-582
Abstract
pdf (1.88 Mb)
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, pp. 568-582
Optical measurement of a fluid velocity field around a falling plate
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 554-567
Abstract
pdf (4.06 Mb)
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, pp. 554-567
A model of a screwless underwater robot
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 544-553
Abstract
pdf (308.24 Kb)
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, pp. 544-553
Qualitative Analysis of the Dynamics of a Wheeled Vehicle
Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 739-751
Abstract
pdf (445.93 Kb)
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, pp. 739-751
On the Hadamard – Hamel Problem and the Dynamics of Wheeled Vehicles
Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 752-766
Abstract
pdf (265.93 Kb)
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, pp. 752-766
Spherical Robot of Combined Type: Dynamics and Control
Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 716-728
Abstract
pdf (306.92 Kb)
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.
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, pp. 716-728
Hyperbolic Chaos in Self-oscillating Systems Based on Mechanical Triple Linkage: Testing Absence of Tangencies of Stable and Unstable Manifolds for Phase Trajectories
Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 649-666
Abstract
pdf (3.15 Mb)
Dynamical equations are formulated and a numerical study is provided for selfoscillatory model systems based on the triple linkage hinge mechanism of Thurston – Weeks – Hunt – MacKay. We consider systems with a holonomic mechanical constraint of three rotators as well as systems, where three rotators interact by potential forces. We present and discuss some quantitative characteristics of the chaotic regimes (Lyapunov exponents, power spectrum). Chaotic dynamics of the models we consider are associated with hyperbolic attractors, at least, at relatively small supercriticality of the self-oscillating modes; that follows from numerical analysis of the distribution for angles of intersection of stable and unstable manifolds of phase trajectories on the attractors. In systems based on rotators with interacting potential the hyperbolicity is violated starting from a certain level of excitation.
Kuznetsov S. P., Hyperbolic Chaos in Self-oscillating Systems Based on Mechanical Triple Linkage: Testing Absence of Tangencies of Stable and Unstable Manifolds for Phase Trajectories, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 649-666
Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors
Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 605-626
Abstract
pdf (640.12 Kb)
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.
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, pp. 605-626
Symmetries and Reduction in Nonholonomic Mechanics
Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 553-604
Abstract
pdf (539.38 Kb)
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.
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, pp. 518-541
Abstract
pdf (516.92 Kb)
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, pp. 518-541
The Dynamics of Systems with Servoconstraints. II
Regular and Chaotic Dynamics, 2015, vol. 20, no. 4, pp. 401-427
Abstract
pdf (861.95 Kb)
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, pp. 401-427
The Jacobi Integral in Nonholonomic Mechanics
Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 383-400
Abstract
pdf (990.04 Kb)
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, pp. 383-400
The Dynamics of Systems with Servoconstraints. I
Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 205-224
Abstract
pdf (810.79 Kb)
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, pp. 205-224
Plate Falling in a Fluid: Regular and Chaotic Dynamics of Finite-dimensional Models
Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 345-382
Abstract
pdf (1.5 Mb)
Results are reviewed concerning the planar problem of a plate falling in a resisting medium studied with models based on ordinary differential equations for a small number of dynamical variables. A unified model is introduced to conduct a comparative analysis of the dynamical behaviors of models of Kozlov, Tanabe–Kaneko, Belmonte–Eisenberg–Moses and Andersen–Pesavento–Wang using common dimensionless variables and parameters. It is shown that the overall structure of the parameter spaces for the different models manifests certain similarities caused by the same inherent symmetry and by the universal nature of the phenomena involved in nonlinear dynamics (fixed points, limit cycles, attractors, and bifurcations).
Keywords:
body motion in a fluid, oscillations, autorotation, flutter, attractor, bifurcation, chaos, Lyapunov exponent
Citation:
Kuznetsov S. P., Plate Falling in a Fluid: Regular and Chaotic Dynamics of Finite-dimensional Models, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 345-382
Dynamics and Control of an Omniwheel Vehicle
Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 153-172
Abstract
pdf (1.11 Mb)
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.
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, pp. 153-172
The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform
Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 134-152
Abstract
pdf (1.38 Mb)
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.
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, pp. 134-152
Symmetries and Reduction in Nonholonomic Mechanics
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 763–823
Abstract
pdf (909.32 Kb)
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.
Borisov A. V., Mamaev I. S., Symmetries and Reduction in Nonholonomic Mechanics, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 763–823
Topology and Bifurcations in Nonholonomic Mechanics
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 735–762
Abstract
pdf (561.73 Kb)
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.
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, pp. 735–762
Experimental research of dynamic of spherical robot of combined type
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 721–734
Abstract
pdf (761.73 Kb)
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, pp. 721–734
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, pp. 633–645
Abstract
pdf (413.58 Kb)
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, pp. 633–645
The dynamics of systems with servoconstraints. II
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 3, pp. 579-611
Abstract
pdf (560.42 Kb)
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, pp. 579-611
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, pp. 547-577
Abstract
pdf (2.38 Mb)
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, pp. 547-577
The Jacobi Integral in NonholonomicMechanics
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 2, pp. 377-396
Abstract
pdf (1.9 Mb)
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, pp. 377-396
The dynamics of systems with servoconstraints. I
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 2, pp. 353-376
Abstract
pdf (505.17 Kb)
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, pp. 353-376
The dynamic of a spherical robot with an internal omniwheel platform
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 1, pp. 187-204
Abstract
pdf (530.7 Kb)
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.
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, pp. 187-204
Principles of dynamics and servo-constraints
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 1, pp. 169-178
Abstract
pdf (316.31 Kb)
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.
Motion of a falling card in a fluid: Finite-dimensional models, complex phenomena, and nonlinear dynamics
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 1, pp. 3-49
Abstract
pdf (1.39 Mb)
Results are reviewed relating to the planar problem for the falling card in a resisting medium based on models represented by ordinary differential equations for a small number of variables. We introduce a unified model, which gives an opportunity to conduct a comparative analysis of dynamic behaviors of models of Kozlov, Tanabe – Kaneko, Belmonte – Eisenberg – Moses and Andersen – Pesavento – Wang using common dimensionless variables and parameters. It is shown that the overall structure of the parameter spaces for the different models shows certain similarities caused obviously by the same inherent symmetry and by universal nature of the involved phenomena of nonlinear dynamics (fixed points, limit cycles, attractors, bifurcations). In concern of motion of a body of elliptical profile in a viscous medium with imposed circulation of the velocity vector and with the applied constant torque, a presence of the Lorenz-type strange attractor is discovered in the three-dimensional space of generalized velocities.
Keywords:
body motion in fluid, oscillations, autorotation, flutter, attractor, bifurcation, chaos, Lyapunov exponent
Citation:
Kuznetsov S. P., Motion of a falling card in a fluid: Finite-dimensional models, complex phenomena, and nonlinear dynamics, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 1, pp. 3-49
Geometrisation of Chaplygin's reducing multiplier theorem
Nonlinearity, 2015, vol. 28, no. 7, pp. 2307–2318
Abstract
pdf (156.41 Kb)
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.
Bolsinov A. V., Borisov A. V., Mamaev I. S., Geometrisation of Chaplygin's reducing multiplier theorem, Nonlinearity, 2015, vol. 28, no. 7, pp. 2307–2318
Topology and Bifurcations in Nonholonomic Mechanics
International Journal of Bifurcation and Chaos, 2015, vol. 25, no. 10, 1530028, 21 pp.
Abstract
pdf (645.53 Kb)
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.
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, 21 pp.
Hamiltonization of Elementary Nonholonomic Systems
Russian Journal of Mathematical Physics, 2015, vol. 22, no. 4, pp. 444-453
Abstract
pdf (115.49 Kb)
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, pp. 444-453
A dynamical communication system on a network
Journal of Computational and Applied Mathematics, 2015, vol. 275, pp. 247–261
Abstract
A dynamical system is introduced and investigated. The system contains N vertices. The vertices send messages at discrete time instants according to a given rule. A conflict of two vertices takes place if the vertices try to send messages to each other at the same instant. Each vertex sends a message to another vertex at every step if no conflict takes place. In case of a conflict, only one of the two competing vertices sends a message. Deterministic and stochastic conflict resolution rules are considered. We investigate the average number of messages sent by a vertex per a time unit, called the productivity of this vertex, the total productivity of the system and other characteristics. The productivity of vertices depends on the initial state of the system, and the criterion of efficiency is the expected average productivity of vertices provided all possible initial states of the system are equiprobable. An ergodic version of the system is also considered in which any particle moves with approximately equal to 1 probability provided there is no conflict.
Kozlov V. V., Buslaev A. P., Tatashev A. G., A dynamical communication system on a network, Journal of Computational and Applied Mathematics, 2015, vol. 275, pp. 247–261
Monotonic walks on a necklace and a coloured dynamic vector
International Journal of Computer Mathematics, 2015, vol. 92, no. 9, pp. 1910–1920
Abstract
Stochastic and deterministic versions of a discrete dynamical system on a necklace are investigated. This network consists of a sequence of contours NSWE with nodes, i.e. the nodes are common points at W and E. There are two cells and a particle on each contour. Each time instance, the particle occupies a cell and, at every time unit, comes to the next cell in the same direction. The particles of the neighbouring contours move in accordance with rules of stochastic or deterministic type. The behaviour of the model with the rule of the first type is stochastic only at the beginning and after a time interval becomes a pure deterministic system. The system with the second rule comes to a steady mode, which depends on the initial state. The average velocity of particles and characteristics of the system are studied.
Keywords:
dynamical system, networks, monotonic walks of particles, characteristics of dynamical systems, synergy, collapse
Citation:
Kozlov V. V., Buslaev A. P., Tatashev A. G., Monotonic walks on a necklace and a coloured dynamic vector, International Journal of Computer Mathematics, 2015, vol. 92, no. 9, pp. 1910–1920
On real-valued oscillations of a bipendulum
Applied Mathematics Letters, 2015, vol. 46, pp. 44–49
Abstract
Theoretical and computational aspects of special case of logistical-routing problem are considered. Fluctuation of two particles on a grid connected by a channel also is considered. Velocity rate and sufficient conditions of system self-regulation are obtained.
Keywords:
Transport–logistic problem; Dynamical system; Markov’s process
Citation:
Kozlov V. V., Buslaev A. P., Tatashev A. G., On real-valued oscillations of a bipendulum, Applied Mathematics Letters, 2015, vol. 46, pp. 44–49
Coarsening in ergodic theory
Russian Journal of Mathematical Physics, 2015, vol. 22, no. 2, pp. 184–187
Abstract
This paper deals with the coarsening operation in dynamical systems where the phase space with a finite invariant measure is partitioned into measurable pieces and the summable function transferred by the phase flow is averaged over these pieces at each instant of time. Letting the time tend to infinity and then refining the partition, we arrive at a modernization of the von Neumann ergodic theorem, which is useful for the purposes of nonequilibrium statistical mechanics. In particular, for fine-grained partitions, we obtain the law of increment of coarse entropy for systems approaching the state of statistical equilibrium.
In memory of A. M. Il’in
Citation:
Kozlov V. V., Coarsening in ergodic theory, Russian Journal of Mathematical Physics, 2015, vol. 22, no. 2, pp. 184–187
Nonintegrability and Obstructions to the Hamiltonianization of a Nonholonomic Chaplygin Top
Doklady Mathematics, 2014, vol. 90, no. 2, pp. 631–634
Abstract
pdf (203.47 Kb)
Citation:
Bizyaev I. A., Nonintegrability and Obstructions to the Hamiltonianization of a Nonholonomic Chaplygin Top, Doklady Mathematics, 2014, vol. 90, no. 2, pp. 631–634
Nonlinear dynamics of the rattleback: a nonholonomic model
Physics-Uspekhi, 2014, vol. 184, no. 5, pp. 453-460
Abstract
pdf (750.09 Kb)
For a solid body of convex form moving on a rough horizontal plane that is known as a rattleback, numerical simulations are used to discuss and illustrate dynamical phenomena that are characteristic of the motion due to a nonholonomic nature of the mechanical system; the relevant feature is the nonconservation of the phase volume in the course of the dynamics. In such a system, a local compression of the phase volume can produce behavior features similar to those exhibited by dissipative systems, such as stable equilibrium points corresponding to stationary rotations; limit cycles (rotations with oscillations); and strange attractors. A chart of dynamical regimes is plotted in a plane whose axes are the total mechanical energy and the relative angle between the geometric and dynamic principal axes of the body. The transition to chaos through a sequence of Feigenbaum period doubling bifurcations is demonstrated. A number of strange attractors are considered, for which phase portraits, Lyapunov exponents, and Fourier spectra are presented.
Citation:
Borisov A. V., Kazakov A. O., Kuznetsov S. P., Nonlinear dynamics of the rattleback: a nonholonomic model , Physics-Uspekhi, 2014, vol. 184, no. 5, pp. 453-460
On the Nonlinear Poisson Bracket Arising in Nonholonomic Mechanics
Mathematical Notes, 2014, vol. 95, no. 3, pp. 308-315
Abstract
pdf (509.63 Kb)
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, pp. 308-315
Liouville's equation as a Schrödinger equation
Izvestiya: Mathematics, 2014, vol. 78, no. 4, pp. 744–757
Abstract
We show that every non-negative solution of Liouville's equation for an
arbitrary (possibly non-Hamiltonian) dynamical system admits a factorization
$\psi\psi^*$, where $\psi$ satisfies a Schrödinger equation of special
form. The corresponding quantum system is obtained by Weyl quantization
of a Hamiltonian system whose Hamiltonian is linear in the momenta.
We discuss the structure of the spectrum of the special Schrödinger
equation on a multidimensional torus and show that the eigenfunctions may
have finite smoothness in the analytic case. Our generalized solutions
of the Schrödinger equation are natural examples of non-selfadjoint
extensions of Hermitian differential operators. We give conditions for
the existence of a smooth invariant measure of a dynamical system. They
are expressed in terms of stability conditions for the conjugate
equations of variations.
The dynamics of systems with large gyroscopic forces and the realization of constraints
Journal of Applied Mathematics and Mechanics, 2014, vol. 78, no. 3, pp. 213–219
Abstract
Lagrangian systems with a large multiplier N on the gyroscopic terms are considered. Simplified equations of motion of general form with holonomic constraints are obtained in the first approximation with respect to the small parameter ɛ = 1/N. The structure of the solutions of the precessional equations is examined.
Citation:
Kozlov V. V., The dynamics of systems with large gyroscopic forces and the realization of constraints, Journal of Applied Mathematics and Mechanics, 2014, vol. 78, no. 3, pp. 213–219
Non-holonomic dynamics and Poisson geometry
Russian Mathematical Surveys, 2014, vol. 69, no. 3, pp. 481-538
Abstract
pdf (917.58 Kb)
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, pp. 481-538
On the dynamics of point vortices in an annular region
Fluid Dynamics Research, 2014, vol. 46, no. 3, 031420, 7 pp.
Abstract
pdf (205.4 Kb)
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, 7 pp.
The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top
Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 718-733
Abstract
pdf (1.29 Mb)
In this paper we consider the motion of a dynamically asymmetric unbalanced ball on a plane in a gravitational field. The point of contact of the ball with the plane is subject to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the nonholonomic reversible system of 6 differential equations. In the case of arbitrary displacement of the center of mass of the ball the system under consideration is a nonintegrable system without an invariant measure. Using qualitative and quantitative analysis we show that the unbalanced ball exhibits reversal (the phenomenon of reversal of the direction of rotation) for some parameter values. Moreover, by constructing charts of Lyaponov exponents we find a few types of strange attractors in the system, including the so-called figure-eight attractor which belongs to the genuine strange attractors of pseudohyperbolic type.
Keywords:
rolling without slipping, reversibility, involution, integrability, reversal, chart of Lyapunov exponents, strange attractor
Citation:
Borisov A. V., Kazakov A. O., Sataev I. R., The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 718-733
The Dynamics of Three Vortex Sources
Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 694-701
Abstract
pdf (244.27 Kb)
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.
The Dynamics of a Body with an Axisymmetric Base Sliding on a Rough Plane
Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 607-634
Abstract
pdf (965.59 Kb)
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, pp. 607-634
On Rational Integrals of Geodesic Flows
Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 601-606
Abstract
pdf (145.78 Kb)
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.
Attractor of Smale–Williams Type in an Autonomous Distributed System
Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 483-494
Abstract
pdf (1.09 Mb)
We consider an autonomous system of partial differential equations for a onedimensional distributed medium with periodic boundary conditions. Dynamics in time consists of alternating birth and death of patterns with spatial phases transformed from one stage of activity to another by the doubly expanding circle map. So, the attractor in the Poincaré section is uniformly hyperbolic, a kind of Smale–Williams solenoid. Finite-dimensional models are derived as ordinary differential equations for amplitudes of spatial Fourier modes (the 5D and 7D models). Correspondence of the reduced models to the original system is demonstrated numerically. Computational verification of the hyperbolicity criterion is performed for the reduced models: the distribution of angles of intersection for stable and unstable manifolds on the attractor is separated from zero, i.e., the touches are excluded. The example considered gives a partial justification for the old hopes that the chaotic behavior of autonomous distributed systems may be associated with uniformly hyperbolic attractors.
Kruglov V. P., Kuznetsov S. P., Pikovsky A., Attractor of Smale–Williams Type in an Autonomous Distributed System, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 483-494
Superintegrable Generalizations of the Kepler and Hook Problems
Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 415-434
Abstract
pdf (300.95 Kb)
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 $\mathbb{R}^2$ and the sphere $S^2$ — and in three-dimensional spaces $\mathbb{R}^3$ and $S^3$. 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, pp. 415-434
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, pp. 198-213
Abstract
pdf (241.48 Kb)
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, pp. 198-213
Remarks on Integrable Systems
Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 145-161
Abstract
pdf (186.79 Kb)
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 $\geqslant n$. 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.
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, pp. 140-143
Abstract
pdf (170.77 Kb)
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.
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, pp. 140-143
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, pp. 116-139
Abstract
pdf (735.75 Kb)
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, pp. 116-139
Paul Painlevé and His Contribution to Science
Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 1-19
Abstract
pdf (1.89 Mb)
The life and career of the great French mathematician and politician Paul Painlevé is described. His contribution to the analytical theory of nonlinear differential equations was significant. The paper outlines the achievements of Paul Painlevé and his students in the investigation of an interesting class of nonlinear second-order equations and new equations defining a completely new class of special functions, now called the Painlevé transcendents. The contribution of Paul Painlevé to the study of algebraic nonintegrability of the $N$-body problem, his remarkable observations in mechanics, in particular, paradoxes arising in the dynamics of systems with friction, his attempt to create the axiomatics of mechanics and his contribution to gravitation theory are discussed.
The kinematic control model for a spherical robot with an unbalanced internal omniwheel platform
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, pp. 497-511
Abstract
pdf (986.08 Kb)
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, pp. 497-511
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, pp. 483-495
Abstract
pdf (667.35 Kb)
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, pp. 483-495
Control of a Vehicle with Omniwheels on a Plane
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, pp. 473-481
Abstract
pdf (520.34 Kb)
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.
On Rational Integrals of Geodesic Flows
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, pp. 439-445
Abstract
pdf (302.05 Kb)
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.
Regular and Chaotic Attractors in the Nonholonomic Model of Chapygin's ball
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 361-380
Abstract
pdf (1.3 Mb)
We study both analytically and numerically the dynamics of an inhomogeneous ball on a rough horizontal plane under the infuence of gravity. A nonholonomic constraint of zero velocity at the point of contact of the ball with the plane is imposed. In the case of an arbitrary displacement of the center of mass of the ball, the system is nonintegrable without the property of phase volume conservation. We show that at certain parameter values the unbalanced ball exhibits the effect of reversal (the direction of the ball rotation reverses). Charts of dynamical regimes on the parameter plane are presented. The system under consideration exhibits diverse chaotic dynamics, in particular, the figure-eight chaotic attractor, which is a special type of pseudohyperbolic chaos.
Keywords:
Chaplygin’s top, rolling without slipping, reversibility, involution, integrability, reverse, chart of dynamical regimes, strange attractor
Citation:
Borisov A. V., Kazakov A. O., Sataev I. R., Regular and Chaotic Attractors in the Nonholonomic Model of Chapygin's ball, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 361-380
Invariant Measure and Hamiltonization of Nonholonomic Systems
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 355-359
Abstract
pdf (283.75 Kb)
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.
Borisov A. V., Mamaev I. S., Invariant Measure and Hamiltonization of Nonholonomic Systems, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 355-359
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, pp. 329-343
Abstract
pdf (5.62 Mb)
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, pp. 329-343
The dynamics of three vortex sources
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 319-327
Abstract
pdf (413.32 Kb)
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.
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, pp. 319-327
Hyperbolic chaos in systems with parametrically excited patterns of standing waves
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 265-277
Abstract
pdf (1018.47 Kb)
We outline a possibility of implementation of Smale–Williams type attractors with different stretching factors for the angular coordinate, namely, $n=3,\,5,\,7,\,9,\,11$, for the maps describing the evolution of parametrically excited standing wave patterns on a nonlinear string over a period of modulation of pump accompanying by alternate excitation of modes with the wavelength ratios of $1:n$.
Kuznetsov S. P., Kuznetsov A. S., Kruglov V. P., Hyperbolic chaos in systems with parametrically excited patterns of standing waves, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 265-277
On a generalization of systems of Calogero type
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 2, pp. 209-212
Abstract
pdf (237.84 Kb)
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, pp. 209-212
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, pp. 127-131
Abstract
pdf (234.11 Kb)
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.
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, pp. 127-131
Kinematic control of a high manoeuvrable mobile spherical robot with internal omni-wheeled platform
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, pp. 113-126
Abstract
pdf (3.43 Mb)
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.
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, pp. 113-126
Figures of equilibrium of an inhomogeneous self-gravitating fluid
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, pp. 73-100
Abstract
pdf (492.78 Kb)
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, pp. 73-100
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, pp. 79-86
Abstract
pdf (311.48 Kb)
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, pp. 79-86
The dynamics of vortex rings: leapfrogging in an ideal and viscous fluid
Fluid Dynamics Research, 2014, vol. 46, no. 3, 031415, 16 pp.
Abstract
pdf (746.69 Kb)
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, 16 pp.
Conservation laws of generalized billiards that are polynomial in momenta
Russian Journal of Mathematical Physics, 2014, vol. 21, no. 2, pp. 226–241
Abstract
This paper deals with dynamics particles moving on a Euclidean n-dimensional torus or in an n-dimensional parallelepiped box in a force field whose potential is proportional to the characteristic function of the region D with a regular boundary. After reaching this region, the trajectory of the particle is refracted according to the law which resembles the Snell -Descartes law from geometrical optics. When the energies are small, the particle does not reach the region D and elastically bounces off its boundary. In this case, we obtain a dynamical system of billiard type (which was intensely studied with respect to strictly convex regions). In addition, the paper discusses the problem of the existence of nontrivial first integrals that are polynomials in momenta with summable coefficients and are functionally independent with the energy integral. Conditions for the geometry of the boundary of the region D under which the problem does not admit nontrivial polynomial first integrals are found. Examples of nonconvex regions are given; for these regions the corresponding dynamical system is obviously nonergodic for fixed energy values (including small ones), however, it does not admit polynomial conservation laws independent of the energy integral.
Citation:
Kozlov V. V., Conservation laws of generalized billiards that are polynomial in momenta, Russian Journal of Mathematical Physics, 2014, vol. 21, no. 2, pp. 226–241
Behavior of pendulums on a regular polygon
Journal of Communication and Computer, 2014, vol. 11, no. 1, pp. 30–38
Abstract
pdf (294.98 Kb)
Stochastic and deterministic versions of a discrete dynamical network system are investigated. This network consists of a sequence of contours NSWE with nodes, which are common points at N, W, S and E. There are four cells and a particle on each contour. At each time instance, the particle occupies a cell, and attempts to occupy the next cell in the same direction. Particles of neighboring contours move in accordance with some rules. Both deterministic and stochastic rules are considered. The behavior of the model with the first rule is stochastic only at the beginning, and after a time interval the system becomes to pure deterministic. The system with the second rule comes to a steady state, which depends on the initial state. The average velocity of particles and other characteristics of the system are studied.
Keywords:
Dynamical system, networks, monotonic walks of particles, characteristics of a dynamical system, synergy
Citation:
Kozlov V. V., Buslaev A. P., Tatashev A. G., Behavior of pendulums on a regular polygon, Journal of Communication and Computer, 2014, vol. 11, no. 1, pp. 30–38
On a system of nonlinear differential equations for the model of totally connected traffic
Journal of Concrete and Applicable Mathematics, 2014, vol. 12, no. 1-2, pp. 86–93
Abstract
pdf (580.72 Kb)
In the paper the qualitative properties solutions of the system nonlinear equations, describing one-way movement of particles chain on a line with follower velocity defined by some function of distance from the leader, are researched. In the case when the given function is the velocity of the first particle (leader) in the chain, the model is called a model of leader. If the given function is the velocity of the last particle (outsider), the model is called a model of “shepherd”.
The sufficient conditions for the existence of the chain with the given constraints on the velocity and acceleration are obtained.
Keywords:
systems of nonlinear ordinary differential equations; follow-theleader model; interpretation for traffic
Citation:
Buslaev A. P., Kozlov V. V., On a system of nonlinear differential equations for the model of totally connected traffic, Journal of Concrete and Applicable Mathematics, 2014, vol. 12, no. 1-2, pp. 86–93
The classical Bohl argument theorem of a conditionally periodic function is generalized. Conditionally periodic motions on a torus are replaced by the solutions of a nonlinear system of differential equations with invariant measure. Cases in which this system is assumed ergodic or strictly ergodic are considered.
Keywords:
Bohl's argument theorem; conditionally periodic motion on the n-dimensional torus; (strictly) ergodic system of differential equations; uniformly distributed function; Birkhoff-Khinchine ergodic theorem
Citation:
Kozlov V. V., On Bohl's Argument Theorem, Mathematical Notes, 2013, vol. 93, no. 1, pp. 83–89
The behaviour of cyclic variables in integrable systems
Journal of Applied Mathematics and Mechanics, 2013, vol. 77, no. 2, pp. 128–136
Abstract
pdf (246.21 Kb)
A general theorem on the behaviour of the angular variables of integrable dynamical systems as functions of time is established. Problems on the motion of the nodal line of a Kovalevskaya top and of a three-dimensional rigid body in a fluid are considered in integrable cases as examples. This range of topics is discussed for systems of a more general form obtained from completely integrable systems after changing the time.
Citation:
Kozlov V. V., The behaviour of cyclic variables in integrable systems, Journal of Applied Mathematics and Mechanics, 2013, vol. 77, no. 2, pp. 128–136
On the Routh sphere problem
Journal of Physics A: Mathematical and Theoretical, 2013, vol. 46, no. 8, 085202, 11 pp.
Abstract
pdf (169.11 Kb)
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, 11 pp.
Traffic modeling: monotonic total-connected random walk on a network
Mathematical Models and Computer Simulations, 2013, vol. 25, no. 8, pp. 3–21
Abstract
pdf (449.44 Kb)
Monotonic (particles move in the same direction) and total-connected (particles that occupy neighboring cells move synchronized) random ($p<1$) and deterministic ($p=1$) walks on closed networks, which consist of circles, are considered. An algorithm has been developed that allows to calculate the duration of the time interval after that all the particles will be contained in the unique cluster. It is proved that such the interval is finite in the considered model. Some statements are proved that allow to found the velocity of movement if deterministic movement occurs on the follows structures: two rings (two closed sequences of cells) that have a common cell; a closed sequence of rings each of that has two common cells with two the neighboring rings; a two-dimensional network structure in that each cell has common cells with four the neighboring rings; a similar infinite network.
Keywords:
stochastic models; random walk; traffic flows
Citation:
Bugaev A. S., Buslaev A. P., Kozlov V. V., Tatashev A. G., Yashina M. V., Traffic modeling: monotonic total-connected random walk on a network , Mathematical Models and Computer Simulations, 2013, vol. 25, no. 8, pp. 3–21
The Problem of Drift and Recurrence for the Rolling Chaplygin Ball
Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 832-859
Abstract
pdf (702.93 Kb)
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.
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, pp. 832-859
The Dynamics of the Chaplygin Ball with a Fluid-filled Cavity
Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 490-496
Abstract
pdf (405.38 Kb)
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.
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, pp. 490-496
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, pp. 356-371
Abstract
pdf (488.37 Kb)
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.
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, pp. 356-371
The Euler–Jacobi–Lie Integrability Theorem
Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 329-343
Abstract
pdf (377.18 Kb)
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, pp. 329-343
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, pp. 277-328
Abstract
pdf (2.69 Mb)
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.
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, pp. 277-328
How to Control the Chaplygin Ball Using Rotors. II
Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 144-158
Abstract
pdf (1.73 Mb)
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.
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, pp. 144-158
The Self-propulsion of a Body with Moving Internal Masses in a Viscous Fluid
Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 100-117
Abstract
pdf (1.71 Mb)
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, pp. 100-117
The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem
Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 33-62
Abstract
pdf (857.35 Kb)
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.
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, pp. 33-62
The problem of drift and recurrence for the rolling Chaplygin ball
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 4, pp. 721-754
Abstract
pdf (875.6 Kb)
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.
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, pp. 721-754
Geometrization of the Chaplygin reducing-multiplier theorem
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 4, pp. 627-640
Abstract
pdf (373.67 Kb)
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.
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, pp. 627-640
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, pp. 567-594
Abstract
pdf (886.59 Kb)
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.
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, pp. 567-594
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, pp. 547-566
Abstract
pdf (441.83 Kb)
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, pp. 547-566
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, pp. 521-545
Abstract
pdf (612.94 Kb)
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, pp. 521-545
Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 507-520
Abstract
pdf (582.48 Kb)
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, pp. 507-520
On the loss of contact of the Euler disk
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 499-506
Abstract
pdf (362.06 Kb)
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, pp. 499-506
Notes on integrable systems
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 459-478
Abstract
pdf (375.2 Kb)
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 $\geqslant n$. 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, pp. 459-478
On a bifurcation scenario of a birth of attractor of Smale–Williams type
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 267-294
Abstract
pdf (2.91 Mb)
We describe one possible scenario of destruction or of a birth of the hyperbolic attractors considering the Smale—Williams solenoid as an example. The content of the transition observed under variation of the control parameter is the pairwise merge of the orbits belonging to the attractor and to the unstable invariant set on the border of the basin of attraction, in the course of the set of bifurcations of the saddle-node type. The transition is not a single event, but occupies a finite interval on the control parameter axis. In an extended space of the state variables and the control parameter this scenario can be regarded as a mutual transformation of the stable and unstable solenoids one to each other. Several model systems are discussed manifesting this scenario e.g. the specially designed iterative maps and the physically realizable system of coupled alternately activated non-autonomous van der Pol oscillators. Detailed studies of inherent features and of the related statistical and scaling properties of the scenario are provided.
Isaeva O. B., Kuznetsov S. P., Sataev I. R., Pikovsky A., On a bifurcation scenario of a birth of attractor of Smale–Williams type, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 267-294
Integrability and stochastic behavior in some nonholonomic dynamics problems
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 257-265
Abstract
pdf (2.27 Mb)
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, pp. 257-265
The Euler–Jacobi–Lie integrability theorem
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 229-245
Abstract
pdf (377.18 Kb)
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, pp. 229-245
Topological monodromy in nonholonomic systems
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 203-227
Abstract
pdf (890.26 Kb)
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.
Bolsinov A. V., Kilin A. A., Kazakov A. O., Topological monodromy in nonholonomic systems, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 203-227
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, pp. 141-202
Abstract
pdf (7.91 Mb)
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.
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, pp. 141-202
Deviation based discrete control algorithm for omni-wheeled mobile robot
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 1, pp. 91-100
Abstract
pdf (434.7 Kb)
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, pp. 91-100
How to control the Chaplygin ball using rotors. II
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 1, pp. 59-76
Abstract
pdf (2.71 Mb)
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.
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, pp. 59-76
Hyperbolic chaos in parametric oscillations of a string
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 1, pp. 3-10
Abstract
pdf (528.87 Kb)
We outline a possibility of chaotic dynamics associated with a hyperbolic attractor of the Smale–Williams type in mechanical vibrations of a nonhomogeneous string with nonlinear dissipation arising due to parametric excitation of modes at the frequencies $\omega$ and $3\omega$, when the pump is supplied by means of the string tension variations alternately at frequencies of $2\omega$ and $6\omega$.
Isaeva O. B., Kuznetsov A. S., Kuznetsov S. P., Hyperbolic chaos in parametric oscillations of a string, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 1, pp. 3-10
Motion control of a rigid body in viscous fluid
Computer Research and Modeling, 2013, vol. 5, no. 4, pp. 659-675
Abstract
pdf (359.11 Kb)
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.
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, pp. 659-675
Rolling of a Rigid Body Without Slipping and Spinning: Kinematics and Dynamics
Journal of Applied Nonlinear Dynamics, 2013, vol. 2, no. 2, pp. 161-173
Abstract
pdf (269.77 Kb)
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, pp. 161-173
Polynomial in momenta invariants of Hamilton's equations, divergent series and generalized functions
Nonlinearity, 2013, vol. 26, no. 3, pp. 745–756
Abstract
We consider the problem of the motion of two identical interacting particles on a circle in a potential force field. We study conditions for the existence of an additional first integral in the form of a polynomial of degree 4 in momenta with periodic coefficients. These conditions are reduced to the solvability of a certain nonlinear functional-differential equation for the potential of the external force and the interaction potential. If the potentials are smooth, this equation has only trivial solutions: at least one of the potentials is a constant function. We classify generalized solutions of this equation when the interaction potential is a distribution while the external potential is a smooth function. Formal Fourier series for the interaction potential can be summed up by the (C, 2)-Cesaro method to periodic functions with singularities.
Citation:
Kozlov V. V., Polynomial in momenta invariants of Hamilton's equations, divergent series and generalized functions, Nonlinearity, 2013, vol. 26, no. 3, pp. 745–756
Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space
Izvestiya: Mathematics, 2012, vol. 76, no. 5, pp. 907–921
Abstract
pdf (519.88 Kb)
We consider problems related to the well-known conjecture on the degrees of irreducible polynomial integrals of a reversible Hamiltonian system with two degrees of freedom and toral position space. The main object of study is a special system arising in the analysis of irreducible polynomial integrals of degree 4. In a particular case we have the problem of the motion of two interacting particles on a circle in given potential fields. We prove that if the three potentials are smooth non-constant functions, then this problem has no non-trivial polynomial integrals of arbitrarily high degree. We prove the conjecture completely for systems with a polynomial first integral of degree 4 in the momenta.
Keywords:
irreducible integrals, systems with impacts, spectrum of a potential
Citation:
Denisova N. V., Kozlov V. V., Treschev D. V., Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space, Izvestiya: Mathematics, 2012, vol. 76, no. 5, pp. 907–921
Invariant manifolds of Hamilton's equations
Journal of Applied Mathematics and Mechanics, 2012, vol. 76, no. 4, pp. 378–387
Abstract
pdf (254.88 Kb)
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, pp. 378–387
The statistical mechanics of a class of dissipative systems
Journal of Applied Mathematics and Mechanics, 2012, vol. 76, no. 1, pp. 15–24
Abstract
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, pp. 15–24
An Extended Hamilton–Jacobi Method
Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 580-596
Abstract
pdf (216.54 Kb)
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.
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, pp. 571-579
Abstract
pdf (402.12 Kb)
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.
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, pp. 571-579
Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback
Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 512-532
Abstract
pdf (4.23 Mb)
We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.
Borisov A. V., Jalnine A. Y., Kuznetsov S. P., Sataev I. R., Sedova Y. V., Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 512-532
The Bifurcation Analysis and the Conley Index in Mechanics
Regular and Chaotic Dynamics, 2012, vol. 17, no. 5, pp. 457-478
Abstract
pdf (614.32 Kb)
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.
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, pp. 457-478
How to Control Chaplygin’s Sphere Using Rotors
Regular and Chaotic Dynamics, 2012, vol. 17, no. 3-4, pp. 258-272
Abstract
pdf (242.89 Kb)
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.
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, pp. 258-272
Two Non-holonomic Integrable Problems Tracing Back to Chaplygin
Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 191-198
Abstract
pdf (150.32 Kb)
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.
Borisov A. V., Mamaev I. S., Two Non-holonomic Integrable Problems Tracing Back to Chaplygin, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 191-198
Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support
Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 170-190
Abstract
pdf (484.82 Kb)
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.
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, pp. 170-190
On Invariant Manifolds of Nonholonomic Systems
Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 131-141
Abstract
pdf (239.09 Kb)
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.
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, pp. 957-975
Abstract
pdf (796.84 Kb)
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.
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, pp. 957-975
Landau–Hopf scenario in the ensemble of interacting oscillators
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 5, pp. 863-873
Abstract
pdf (578.41 Kb)
The conditions are discussed for which the ensemble of interacting oscillators may demonstrate Landau–Hopf scenario of successive birth of multi-frequency regimes. A model is proposed in the form of a network of five globally coupled oscillators, characterized by varying degree of excitement of individual oscillators. Illustrations are given for the birth of the tori of increasing dimension by successive quasi-periodic Hopf bifurcation.
Kuznetsov A. P., Turukina L. V., Kuznetsov S. P., Sataev I. R., Landau–Hopf scenario in the ensemble of interacting oscillators, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 5, pp. 863-873
The motion of a body with variable mass geometry in a viscous fluid
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, pp. 815-836
Abstract
pdf (15.9 Mb)
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, pp. 815-836
Rolling of a rigid body without slipping and spinning: kinematics and dynamics
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, pp. 783-797
Abstract
pdf (347.06 Kb)
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, pp. 783-797
Phenomena of nonlinear dynamics of dissipative systems in nonholonomic mechanics of the rattleback
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, pp. 735-762
Abstract
pdf (1.42 Mb)
We perform a numerical study of the motion of the rattleback, a rigid body with a convex surface on a rough horizontal plane in dependence on the parameters, applying the methods used previously for the treatment of dissipative dynamical systems, and adapted for the nonholonomic model. Charts of dynamical regimes are presented on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body. Presence of characteristic structures in the parameter space, previously observed only for dissipative systems, is demonstrated. A method of calculating for the full spectrum of Lyapunov exponents is developed and implemented. It is shown that analysis of the Lyapunov exponents of chaotic regimes of the nonholonomic model reveals two classes, one of which is typical for relatively high energies, and the second for the relatively small energies. For the model reduced to a three-dimensional map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of the quasiconservative type, with close in magnitude positive and negative Lyapunov exponents, and the rest one about zero. The transition to chaos through a sequence of period-doubling bifurcations is illustrated, and the observed scaling corresponds to that intrinsic to the dissipative systems. A study of strange attractors is provided, in particularly, phase portraits are presented as well as the Lyapunov exponents, the Fourier spectra, the results of calculating the fractal dimensions.
Kuznetsov S. P., Jalnine A. Y., Sataev I. R., Sedova Y. V., Phenomena of nonlinear dynamics of dissipative systems in nonholonomic mechanics of the rattleback, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, pp. 735-762
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, pp. 605-616
Abstract
pdf (328.96 Kb)
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.
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, pp. 605-616
On the final motion of cylindrical solids on a rough plane
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 585-603
Abstract
pdf (623.55 Kb)
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, pp. 585-603
On the Routh sphere
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 569-583
Abstract
pdf (299.77 Kb)
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.
An extended Hamilton–Jacobi method
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 549-568
Abstract
pdf (400.68 Kb)
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.
Universal two-dimensional map and its radiophysical realization
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 461-471
Abstract
pdf (637.32 Kb)
We suggest a simple two-dimensional map, parameters of which are the trace and Jacobian of the perturbation matrix of the fixed point. On the parameters plane it demonstrates the main universal bifurcation scenarios: the threshold to chaos via period-doublings, the situation of quasiperiodic oscillations and Arnold tongues. We demonstrate the possibility of implementation of such map in radiophysical device.
Keywords:
maps, bifurcations, phenomena of quasiperiodicity
Citation:
Kuznetsov A. P., Kuznetsov S. P., Pozdnyakov M. V., Sedova Y. V., Universal two-dimensional map and its radiophysical realization, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 461-471
How to control the Chaplygin sphere using rotors
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 2, pp. 289-307
Abstract
pdf (400.44 Kb)
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.
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, pp. 289-307
Viatcheslav Vladimirovich Meleshko (07.10.1951–14.11.2011)
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, pp. 179-182
Abstract
pdf (1.73 Mb)
Citation:
Grinchenko V. T., Krasnopolskaya T. S., Borisov A. V., van Heijst G. J., Viatcheslav Vladimirovich Meleshko (07.10.1951–14.11.2011), Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, pp. 179-182
The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, pp. 113-147
Abstract
pdf (1.03 Mb)
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.
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, pp. 113-147
The dynamics of the Chaplygin ball with a fluid-filled cavity
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, pp. 103-111
Abstract
pdf (305.43 Kb)
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.
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, pp. 103-111
On invariant manifolds of nonholonomic systems
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, pp. 57-69
Abstract
pdf (329.1 Kb)
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.
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, pp. 146-155
Abstract
pdf (688.03 Kb)
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, pp. 146-155
On detachment conditions of a top on an absolutely rough support
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2012, no. 3, pp. 103-113
Abstract
pdf (292.26 Kb)
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, pp. 103-113
Metropolis traffic modeling: from intelligent monitoring through physical representation to mathematical problems
Proc. International Conference on Computational and Mathematical Methods in Science and Engineering, 2012, vol. 1, pp. 750–756
Abstract
Citation:
Kozlov V. V., Buslaev A. P., Metropolis traffic modeling: from intelligent monitoring through physical representation to mathematical problems, Proc. International Conference on Computational and Mathematical Methods in Science and Engineering, 2012, vol. 1, pp. 750–756
Wigner measures on infinite-dimensional spaces and the Bogolyubov equations for quantum systems
Doklady Mathematics, 2011, vol. 84, no. 1, pp. 571–575
Abstract
Citation:
Kozlov V. V., Wigner measures on infinite-dimensional spaces and the Bogolyubov equations for quantum systems, Doklady Mathematics, 2011, vol. 84, no. 1, pp. 571–575
Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard
Proceedings of the Steklov Institute of Mathematics, 2011, vol. 273, pp. 196–213
Abstract
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, pp. 196–213
The equations of motion of a collisionless continuum
Journal of Applied Mathematics and Mechanics, 2011, vol. 75, no. 6, pp. 619–630
Abstract
pdf (689.16 Kb)
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, pp. 619–630
Hassan Aref (1950–2011)
Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 671-684
Abstract
pdf (777.79 Kb)
Citation:
Borisov A. V., Meleshko V. V., Stremler M. A., van Heijst G. J., Hassan Aref (1950–2011), Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 671-684
On the Model of Non-holonomic Billiard
Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 653-662
Abstract
pdf (199.9 Kb)
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, pp. 653-662
The Vlasov Kinetic Equation, Dynamics of Continuum and Turbulence
Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 602-622
Abstract
pdf (323.98 Kb)
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.
Hassan Aref (1950–2011)
Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 671-684
Abstract
pdf (777.79 Kb)
Citation:
Borisov A. V., Meleshko V. V., Stremler M. A., van Heijst G. J., Hassan Aref (1950–2011), Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 671-684
Statistical Irreversibility of the Kac Reversible Circular Model
Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 536-549
Abstract
pdf (202.6 Kb)
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.
Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere
Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 465-483
Abstract
pdf (643.15 Kb)
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, pp. 465-483
Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds
Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 443-464
Abstract
pdf (425.83 Kb)
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.
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, pp. 443-464
Hamiltonicity and integrability of the Suslov problem
Regular and Chaotic Dynamics, 2011, vol. 16, no. 1-2, pp. 104-116
Abstract
pdf (239.81 Kb)
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.
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, pp. 104-116
An omni-wheel vehicle on a plane and a sphere
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 4, pp. 785-801
Abstract
pdf (726.7 Kb)
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.
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, pp. 785-801
The bifurcation analysis and the Conley index in mechanics
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, pp. 649-681
Abstract
pdf (782.35 Kb)
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.
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, pp. 649-681
The Lorentz force and its generalizations
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, pp. 627-634
Abstract
pdf (359.22 Kb)
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.
A rigid cylinder on a viscoelastic plane
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, pp. 601-625
Abstract
pdf (499.7 Kb)
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 $p(x) = kY(x)+ν\dot{Y}(x)$, 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.
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, pp. 601-625
Two non-holonomic integrable systems of coupled rigid bodies
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, pp. 559-568
Abstract
pdf (404.52 Kb)
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.
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, pp. 559-568
On V.A. Steklov’s legacy in classical mechanics
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 2, pp. 389-403
Abstract
pdf (368.21 Kb)
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, pp. 389-403
Generalized Chaplygin’s transformation and explicit integration of a system with a spherical support
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 2, pp. 313-338
Abstract
pdf (1.78 Mb)
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.
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, pp. 313-338
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, pp. 119-138
Abstract
pdf (1.2 Mb)
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, pp. 119-138
Statistical irreversibility of the Kac reversible circular model
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 1, pp. 101-117
Abstract
pdf (419.07 Kb)
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.
Kozlov V. V., Statistical irreversibility of the Kac reversible circular model, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 1, pp. 101-117
Motion control simulating in a viscous liquid of a body with variable geometry of weights
Computer Research and Modeling, 2011, vol. 3, no. 4, pp. 371-381
Abstract
pdf (594.85 Kb)
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, pp. 371-381
Some mathematical and information aspects of traffic modeling
T-Comm: Telecommunications and Transport, 2011, no. 4, pp. 29–31
Abstract
pdf (809.09 Kb)
Transport, communications and mathematics (in wider understanding - natural sciences) it becomes more and more obvious to despite different "age" of these three components. If the mathematics in Russia (USSR) was always, at least, since that moment as to Russia were invited L.Eyler (1976) and D. Bernulli (1725) that a traffic (traffic) as the appreciable phenomenon and an acute problem, appeared in the early nineties after opening of borders and a mass import of cars. With transition to market economy and small business mobility of the population, need for multipurpose cars and load of a street road network sharply increased.
Citation:
Bugaev A. S., Buslaev A. P., Kozlov V. V., Yashina M. V., Some mathematical and information aspects of traffic modeling, T-Comm: Telecommunications and Transport, 2011, no. 4, pp. 29–31
Figures of equilibrium of liquid self-gravitating inhomogeneous mass
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2011, no. 3, pp. 142-153
Abstract
pdf (215.68 Kb)
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.
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, pp. 142-153
Bogoliubov type equations via infinite-dimensional equations for measures
Quantum bio-informatics IV, QP–PQ: Quantum Probability and White Noise Analysis, 2011, vol. 28, pp. 321–337
Abstract
The following sections are included:
Introduction
Symplectic locally convex spaces and Hamilton's equations.
Liouville's equations with respect to measures.
Systems of equations with respect to finite-dimensional distributions of probabilities.
Bogolyubov's systems of equations.
Wigner measures.
Generalization of Poincaré's model.
References
Citation:
Kozlov V. V., Smolyanov O. G., Bogoliubov type equations via infinite-dimensional equations for measures, Quantum bio-informatics IV, QP–PQ: Quantum Probability and White Noise Analysis, 2011, vol. 28, pp. 321–337
Distributed problems of monitoring and modern approaches to traffic modeling
2011 14th IEEE International IEEE Conference on Intelligent Transportation Systems (ITSC), 14th International IEEE Conference on Intelligent Transportation Systems-ITSC, 2011, pp. 477–481
Abstract
The paper discusses some mathematical models of traffic flow. We have introduced the concept of a stationary r-connected traffic flow on k-lane road as a development of the hydrodynamic approach and cellular automata method. A client-server based software “SSSR”-system, using smart-phone programming, for evaluating a distance of safety in continuous traffic was developed. A series of experiments were carried out using the SSSR-system, the results showing good agreement with those obtained by Greenshields in 1933. Other problems of traffic monitoring and control by the programmed SSSR-system are discussed. We also introduce a few open problems.
Citation:
Bugaev A. S., Buslaev A. P., Kozlov V. V., Yashina M. V., Distributed problems of monitoring and modern approaches to traffic modeling, 2011 14th IEEE International IEEE Conference on Intelligent Transportation Systems (ITSC), 14th International IEEE Conference on Intelligent Transportation Systems-ITSC, 2011, pp. 477–481
Infinite-dimensional Liouville equations with respect to measures
Doklady Mathematics, 2010, vol. 81, no. 3, pp. 476–480
Abstract
pdf (335.38 Kb)
Citation:
Kozlov V. V., Smolyanov O. G., Infinite-dimensional Liouville equations with respect to measures, Doklady Mathematics, 2010, vol. 81, no. 3, pp. 476–480
On the variational principles of mechanics
Journal of Applied Mathematics and Mechanics, 2010, vol. 74, no. 5, pp. 505–512
Abstract
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, pp. 505–512
Remarks on the degree of instability
Journal of Applied Mathematics and Mechanics, 2010, vol. 74, no. 1, pp. 10–12
Abstract
pdf (242.92 Kb)
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, pp. 10–12
Topology and stability of integrable systems
Russian Mathematical Surveys, 2010, vol. 65, no. 2, pp. 259–318
Abstract
pdf (1.12 Mb)
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.
Bolsinov A. V., Borisov A. V., Mamaev I. S., Topology and stability of integrable systems, Russian Mathematical Surveys, 2010, vol. 65, no. 2, pp. 259–318
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, pp. 440-461
Abstract
pdf (298.75 Kb)
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 $S^2$ 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, pp. 440-461
Example of blue sky catastrophe accompanied by a birth of Smale–Williams attractor
Regular and Chaotic Dynamics, 2010, vol. 15, no. 2-3, pp. 348-353
Abstract
pdf (272.33 Kb)
A model system is proposed, which manifests a blue sky catastrophe giving rise to a hyperbolic attractor of Smale–Williams type in accordance with theory of Shilnikov and Turaev. Some essential features of the transition are demonstrated in computations, including Bernoulli-type discrete-step evolution of the angular variable, inverse square root dependence of the first return time on the bifurcation parameter, certain type of dependence of Lyapunov exponents on control parameter for the differential equations and for the Poincaré map.
Kuznetsov S. P., Example of blue sky catastrophe accompanied by a birth of Smale–Williams attractor, Regular and Chaotic Dynamics, 2010, vol. 15, no. 2-3, pp. 348-353
Rolling of a homogeneous ball over a dynamically asymmetric sphere
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, pp. 869-889
Abstract
pdf (486.45 Kb)
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, pp. 869-889
Lagrangian mechanics and dry friction
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, pp. 855-868
Abstract
pdf (265.11 Kb)
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.
Hamiltonisation of non-holonomic systems in the neighborhood of invariant manifolds
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, pp. 829-854
Abstract
pdf (398.78 Kb)
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.
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, pp. 829-854
Stability of a liquid self-gravitating elliptic cylinder with intrinsic rotation
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, pp. 807-822
Abstract
pdf (902.16 Kb)
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, pp. 807-822
Dynamic advection
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, pp. 521-530
Abstract
pdf (10.3 Mb)
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, pp. 521-530
The Vlasov kinetic equation, dynamics of continuum and turbulence
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, pp. 489-512
Abstract
pdf (276.41 Kb)
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, pp. 489-512
Valery Vasilievich Kozlov. On his 60th birthday
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, pp. 461-488
Abstract
pdf (25.39 Mb)
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, pp. 461-488
On the model of non-holonomic billiard
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 2, pp. 373-385
Abstract
pdf (237.96 Kb)
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, pp. 373-385
Problems of stability and asymptotic behavior of vortex patches on the plane
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 2, pp. 327-343
Abstract
pdf (685.19 Kb)
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 $r_0$,
— 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, pp. 327-343
Hamiltonian representation and integrability of the Suslov problem
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 1, pp. 127-142
Abstract
pdf (654.76 Kb)
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.
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, pp. 127-142
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, pp. 77-86
Abstract
pdf (588.63 Kb)
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.
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, pp. 77-86
Dynamics of a wheeled carriage on a plane
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2010, no. 4, pp. 39-48
Abstract
pdf (385.82 Kb)
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, pp. 39-48
Lyapunov A.M. Works on theoretical mechanics. From the 1882-1894 handwritten heritage
Izhevsk: Regular and Chaotic Dynamics, 2010, pp. 6-20
Abstract
pdf (174.49 Kb)
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, pp. 6-20
The dynamics of a Chaplygin sleigh
Journal of Applied Mathematics and Mechanics, 2009, vol. 73, no. 2, pp. 156-161
Abstract
pdf (263.77 Kb)
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, pp. 156-161
On the mechanism of stability loss
Differential Equations, 2009, vol. 45, no. 4, pp. 510–519
Abstract
pdf (316.78 Kb)
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, pp. 510–519
Superintegrable system on a sphere with the integral of higher degree
Regular and Chaotic Dynamics, 2009, vol. 14, no. 6, pp. 615-620
Abstract
pdf (125.27 Kb)
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, pp. 615-620
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, pp. 535-540
Abstract
pdf (172.17 Kb)
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.
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, pp. 535-540
Isomorphisms of geodesic flows on quadrics
Regular and Chaotic Dynamics, 2009, vol. 14, no. 4-5, pp. 455-465
Abstract
pdf (376.58 Kb)
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.
The Hamiltonian Dynamics of Self-gravitating Liquid and Gas Ellipsoids
Regular and Chaotic Dynamics, 2009, vol. 14, no. 2, pp. 179-217
Abstract
pdf (885.59 Kb)
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, pp. 179-217
Multiparticle Systems. The Algebra of Integrals and Integrable Cases
Regular and Chaotic Dynamics, 2009, vol. 14, no. 1, pp. 18-41
Abstract
pdf (472.45 Kb)
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 $\alpha = –2$ 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 $\alpha = –2$ 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, pp. 18-41
New superintegrable system on a sphere
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 4, pp. 455-462
Abstract
pdf (214.58 Kb)
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, pp. 455-462
An example of a non-autonomous continuous-time system with attractor of Plykin type in the Poincare map
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, pp. 403-424
Abstract
pdf (2.26 Mb)
A non-autonomous flow system is introduced, which may serve as a base for elaboration of real systems and devices demonstrating the structurally stable chaotic dynamics. The starting point is the map of the sphere composed of four stages of sequential continuous geometrically evident transformations. The computations indicate that in a certain parameter range the map posseses an attractor of Plykin type. Accounting the structural stability intrinsic to this attractor, a modification of the model is undertaken, which includes a variable change with passage to representation of instantaneous states on the plane. As a result, a set of two non-autonomous differential equations of the first order with smooth coefficients is obtained explicitly, which has the Plykin type attractor in the plane in the Poincaré cross-section. Results of computations are presented for the sphere map and for the flow system including portraits of attractors, Lyapunov exponents, dimension estimates. Substantiation of the hyperbolic nature of the attractors for the sphere map and for the flow system is based on a computer procedure of verification of the so-called cone criterion; in this context, some hints are applied, which may be useful in similar computations for some other systems.
Kuznetsov S. P., An example of a non-autonomous continuous-time system with attractor of Plykin type in the Poincare map, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, pp. 403-424
Statistical mechanics of nonlinear dynamical systems
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, pp. 385-402
Abstract
pdf (896.55 Kb)
With the help of mathematical modeling, we study the behavior of a gas ($\sim10^6$ 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.
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, pp. 385-402
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, pp. 377-383
Abstract
pdf (208.37 Kb)
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.
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, pp. 377-383
Coupled motion of a rigid body and point vortices on a sphere
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, pp. 319-343
Abstract
pdf (429.33 Kb)
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 $S^2$ 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, pp. 319-343
Isomorphisms of geodesic flows on quadrics
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 2, pp. 145-158
Abstract
pdf (532.83 Kb)
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.
The Jacobi problem on a plane
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 1, pp. 83-86
Abstract
pdf (137.41 Kb)
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.
Multiparticle Systems. The Algebra of Integrals and Integrable Cases
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 1, pp. 53-82
Abstract
pdf (508.81 Kb)
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, pp. 53-82
Generalized model of kinetics of formation of a new phase
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2009, no. 2, pp. 110-117
Abstract
pdf (724.17 Kb)
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, pp. 110-117
Explicit integration of one problem in nonholonomic mechanics
Doklady Physics, 2008, vol. 53, no. 10, pp. 525-528
Abstract
pdf (229.05 Kb)
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, pp. 525-528
Gyroscopic stabilization of degenerate equilibria and the topology of real algebraic varieties
Doklady Mathematics, 2008, vol. 77, no. 3, pp. 412–415
Abstract
pdf (178.12 Kb)
Citation:
Kozlov V. V., Gyroscopic stabilization of degenerate equilibria and the topology of real algebraic varieties, Doklady Mathematics, 2008, vol. 77, no. 3, pp. 412–415
On the instability of equilibria of conservative systems under typical degenerations
Differential Equations, 2008, vol. 44, no. 8, pp. 1064–1071
Abstract
pdf (285.36 Kb)
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, pp. 1064–1071
Topology of Real Algebraic Curves
Functional Analysis and Its Applications, 2008, vol. 42, no. 2, pp. 98–102
Abstract
pdf (117.67 Kb)
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, pp. 98–102
Chaplygin ball over a fixed sphere: an explicit integration
Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 557-571
Abstract
pdf (282.96 Kb)
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.
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, pp. 557-571
Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems
Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 443-490
Abstract
pdf (508.23 Kb)
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, pp. 443-490
Gauss Principle and Realization of Constraints
Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 431-434
Abstract
pdf (144.15 Kb)
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.
Stability of Steady Rotations in the Nonholonomic Routh Problem
Regular and Chaotic Dynamics, 2008, vol. 13, no. 4, pp. 239-249
Abstract
pdf (392.42 Kb)
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.
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, pp. 239-249
Chaos in a Restricted Problem of Rotation of a Rigid Body with a Fixed Point
Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 221-233
Abstract
pdf (491.62 Kb)
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, pp. 221-233
Absolute and Relative Choreographies in Rigid Body Dynamics
Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 204-220
Abstract
pdf (447.19 Kb)
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, pp. 204-220
Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat
Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 141-154
Abstract
pdf (192.61 Kb)
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.
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, pp. 141-154
Lagrange’s Identity and Its Generalizations
Regular and Chaotic Dynamics, 2008, vol. 13, no. 2, pp. 71-80
Abstract
pdf (144.77 Kb)
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.
E. Zermelo Habilitationsschrift on the vortex hydrodynamics on a sphere
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 4, pp. 497-513
Abstract
pdf (286.99 Kb)
Citation:
Borisov A. V., Gazizullina L., Ramodanov S. M., E. Zermelo Habilitationsschrift on the vortex hydrodynamics on a sphere, Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 4, pp. 497-513
Algebraic reduction of systems on two- and three-dimensional spheres
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 4, pp. 407-416
Abstract
pdf (180.6 Kb)
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.
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, pp. 407-416
Hamiltonian Dynamics of Liquid and Gas Self-Gravitating Ellipsoids
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 4, pp. 363-406
Abstract
pdf (994.54 Kb)
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, pp. 363-406
Gauss Principle and Realization of Constraints
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 3, pp. 281-285
Abstract
pdf (78.44 Kb)
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.
Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 3, pp. 223-280
Abstract
pdf (634.39 Kb)
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, pp. 223-280
Lagrange’s identity and its generalizations
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 2, pp. 157-168
Abstract
pdf (128.42 Kb)
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.
Critical point of accumulation of fold-flip bifurcation points and critical quasi-attractor (the review and new results)
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 2, pp. 113-132
Abstract
pdf (1.02 Mb)
In paper we suggest an example of system which dynamics is answered to conception of a «critical quasi-attractor». Besides the brief review of earlier obtained results the new results are presented, namely the illustrations of scaling for basins of attraction of elements of critical quasi-attractor, the renormalization group approach in the presence of additive uncorrelated noise, the calculation of universal constant responsible for the scaling regularities of the noise effect, the illustrations of transitions initialized by noise that are realized between coexisted attractors.
Keywords:
quasi-attractor, renormalization group method, type of criticality, bifurcation, scaling, noise
Citation:
Kuznetsov A. P., Kuznetsov S. P., Sataev I. R., Sedova Y. V., Critical point of accumulation of fold-flip bifurcation points and critical quasi-attractor (the review and new results), Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 2, pp. 113-132
Generalization of Lagrange’s identity and new integrals of motion
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2008, no. 3, pp. 69-74
Abstract
pdf (154.14 Kb)
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 $\alpha=-2$. 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 $\alpha=-2$ 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, pp. 69-74
Dynamics of Two Rings of Vortices on a Sphere
in IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, Springer, 2008, vol. 6, pp. 445–458
Abstract
pdf (7.9 Mb)
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, pp. 445–458
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, pp. 39–53
Abstract
pdf (10.34 Mb)
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, pp. 39–53
Fine-grained and coarse-grained entropy in problems of statistical mechanics
Theoretical and Mathematical Physics, 2007, vol. 151, no. 1, pp. 539–555
Abstract
pdf (502.32 Kb)
We consider dynamical systems with a phase space Γ that preserve a measure μ. A partition of Γ into parts of finite μ-measure generates the coarse-grained entropy, a functional that is defined on the space of probability measures on Γ and generalizes the usual (ordinary or fine-grained) Gibbs entropy. We study the approximation properties of the coarse-grained entropy under refinement of the partition and also the properties of the coarse-grained entropy as a function of time.
Kozlov V. V., Treschev D. V., Fine-grained and coarse-grained entropy in problems of statistical mechanics, Theoretical and Mathematical Physics, 2007, vol. 151, no. 1, pp. 539–555
Euler and mathematical methods in mechanics (on the 300th anniversary of the birth of Leonhard Euler)
Russian Mathematical Surveys, 2007, vol. 62, no. 4, pp. 639–661
Abstract
pdf (425.6 Kb)
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, pp. 639–661
Isomorphism and Hamilton representation of some nonholonomic systems
Siberian Mathematical Journal, 2007, vol. 48, no. 1, pp. 26-36
Abstract
pdf (154.1 Kb)
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.
Borisov A. V., Mamaev I. S., Isomorphism and Hamilton representation of some nonholonomic systems, Siberian Mathematical Journal, 2007, vol. 48, no. 1, pp. 26-36
Asymptotic stability and associated problems of dynamics of falling rigid body
Regular and Chaotic Dynamics, 2007, vol. 12, no. 5, pp. 531-565
Abstract
pdf (1.81 Mb)
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, pp. 531-565
Rolling of a Non-homogeneous Ball Over a Sphere Without Slipping and Twisting
Regular and Chaotic Dynamics, 2007, vol. 12, no. 2, pp. 153-159
Abstract
pdf (189.47 Kb)
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.
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, pp. 153-159
Interaction between Kirchhoff vortices and point vortices in an ideal fluid
Regular and Chaotic Dynamics, 2007, vol. 12, no. 1, pp. 68-80
Abstract
pdf (358.32 Kb)
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, pp. 68-80
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, pp. 411-422
Abstract
pdf (182.63 Kb)
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, pp. 411-422
Asymptotic stability and associated problems of dynamics of falling rigid body
Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 3, pp. 255-296
Abstract
pdf (1.62 Mb)
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, pp. 255-296
A New Integrable Problem of Motion of Point Vortices on the Sphere
Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 2, pp. 211-223
Abstract
pdf (298.41 Kb)
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, pp. 211-223
Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat
Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 2, pp. 123-140
Abstract
pdf (263.66 Kb)
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.
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, pp. 123-140
On isomorphisms of some integrable systems on a plane and a sphere
Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 1, pp. 49-56
Abstract
pdf (166.52 Kb)
We consider
trajectory isomorphisms between various integrable
systems on an $n$-dimensional sphere $S^n$ and a Euclidean space $R^n$.
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.
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, pp. 49-56
Dynamic Interaction of Point Vortices and a Two-Dimensional Cylinder
Journal of Mathematical Physics, 2007, vol. 48, no. 6, 065403, 9 pp.
Abstract
pdf (126.99 Kb)
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, 9 pp.
Relations between Integrable Systems in Plane and Curved Spaces
Celestial Mechanics and Dynamical Astronomy, 2007, vol. 99, no. 4, pp. 253–260
Abstract
pdf (151.07 Kb)
We consider trajectory isomorphisms between various integrable systems on an $n$-dimensional sphere $S^n$ and a Euclidean space $\mathbb{R}^n$. 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, pp. 253–260
Dynamics of Two Interacting Circular Cylinders in Perfect Fluid
Discrete and Continuous Dynamical Systems - Series A, 2007, vol. 19, no. 2, pp. 235-253
Abstract
pdf (350.69 Kb)
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.
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, pp. 235-253
Information entropy in problems of classical and quantum statistical mechanics
Doklady Mathematics, 2006, vol. 74, no. 3, pp. 910–913
Abstract
pdf (199.38 Kb)
Citation:
Kozlov V. V., Smolyanov O. G., Information entropy in problems of classical and quantum statistical mechanics, Doklady Mathematics, 2006, vol. 74, no. 3, pp. 910–913
Square integrable solutions to the Klein-Gordon equation on a manifold
Doklady Mathematics, 2006, vol. 73, no. 3, pp. 441–444
Abstract
pdf (292.17 Kb)
Citation:
Volovich I. V., Kozlov V. V., Square integrable solutions to the Klein-Gordon equation on a manifold, Doklady Mathematics, 2006, vol. 73, no. 3, pp. 441–444
New effects in dynamics of rattlebacks
Doklady Physics, 2006, vol. 408, no. 2, pp. 192-195
Abstract
pdf (214.19 Kb)
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, pp. 192-195
Transition to chaos in dynamics of four point vortices on a plane
Doklady Physics, 2006, vol. 51, no. 5, pp. 262-267
Abstract
pdf (249.72 Kb)
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, pp. 262-267
Motion of Chaplygin ball on an inclined plane
Doklady Physics, 2006, vol. 51, no. 2, pp. 73-76
Abstract
pdf (212.42 Kb)
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, pp. 73-76
An Integrable System with a Nonintegrable Constraint
Mathematical Notes, 2006, vol. 80, no. 1, pp. 127-130
Abstract
pdf (98.56 Kb)
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.
On a Nonholonomic Dynamical Problem
Mathematical Notes, 2006, vol. 79, no. 5, pp. 734-740
Abstract
pdf (189.22 Kb)
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, pp. 734-740
Wigner function and diffusion in collisionfree media of quantum particles
Theory of Probability and its Applications, 2006, vol. 51, no. 1, pp. 168–181
Abstract
pdf (952.66 Kb)
A quantum Poincaré model (realizing behavior of ideal gas of noninteracting quantum Bolztman particles) is introduced. We use the fact that the evolution of the Wigner function corresponding to a quantum system with a quadratic Hamiltonian coincides with the evolution of a probability distribution on a phase space of the Hamiltonian system, the quantization of which gives the quantum system under consideration.
Kozlov V. V., Smolyanov O. G., Wigner function and diffusion in collisionfree media of quantum particles, Theory of Probability and its Applications, 2006, vol. 51, no. 1, pp. 168–181
On the problem of motion of vortex sources on a plane
Regular and Chaotic Dynamics, 2006, vol. 11, no. 4, pp. 455-466
Abstract
pdf (377.53 Kb)
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, pp. 455-466
Dynamics of small perturbations of orbits on a torus in a quasiperiodically forced 2D dissipative map
Regular and Chaotic Dynamics, 2006, vol. 11, no. 1, pp. 19-30
Abstract
pdf (520.95 Kb)
We consider the dynamics of small perturbations of stable two-frequency quasiperiodic orbits on an attracting torus in the quasiperiodically forced Hénon map. Such dynamics consists in an exponential decay of the radial component and in a complex behaviour of the angle component. This behaviour may be two- or three-frequency quasiperiodicity, or it may be irregular. In the latter case a graphic image of the dynamics of the perturbation angle is a fractal object, namely a strange nonchaotic attractor, which appears in auxiliary map for the angle component. Therefore, we claim that stable trajectories may approach the attracting torus either in a regular or in an irregular way. We show that the transition from quasiperiodic dynamics to chaos in the model system is preceded by the appearance of an irregular behaviour in the approach of the perturbed quasiperiodic trajectories to the smooth attracting torus. We also demonstrate a link between the evolution operator of the perturbation angle and a quasiperiodically forced circle mapping of a special form and with a Harper equation with quasiperiodic potential.
Jalnine A. Y., Kuznetsov S. P., Osbaldestin A. H., Dynamics of small perturbations of orbits on a torus in a quasiperiodically forced 2D dissipative map , Regular and Chaotic Dynamics, 2006, vol. 11, no. 1, pp. 19-30
Rolling of a heterotgeneous ball over a sphere without sliding and spinning
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 4, pp. 445-452
Abstract
pdf (162.92 Kb)
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, pp. 445-452
Vorticity equation of 2D-hydrodynamics, Vlasov steady-state kinetic equation and developed turbulence
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 4, pp. 425-434
Abstract
pdf (152.1 Kb)
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.
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, pp. 425-434
Stability of steady rotations in the non-holonomic Routh problem
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 3, pp. 333-345
Abstract
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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.
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, pp. 333-345
Transition to a synchronous chaos regime in a system of coupled non-autonomous oscillators presented in terms of amplitude equations
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 3, pp. 307-331
Abstract
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Amplitude equations are obtained for a system of two coupled van der Pol oscillators that has been recently suggested as a simple system with hyperbolic chaotic attractor allowing physical realization. We demonstrate that an approximate model based on the amplitude equations preserves basic features of a hyperbolic dynamics of the initial system. For two coupled amplitude equations models having the hyperbolic attractors a transition to synchronous chaos is studied. Phenomena typically accompanying this transition, as riddling and bubbling, are shown to manifest themselves in a specific way and can be observed only in a small vicinity of a critical point. Also, a structure of many-dimensional attractor of the system is described in a region below the synchronization point.
Kuptsov P. V., Kuznetsov S. P., Transition to a synchronous chaos regime in a system of coupled non-autonomous oscillators presented in terms of amplitude equations, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 3, pp. 307-331
Reduction in the two-body problem on the Lobatchevsky plane
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 3, pp. 279-285
Abstract
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We present a reduction-of-order procedure in the problem of motion of two bodies on the Lobatchevsky plane $H^2$. 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, pp. 279-285
Interaction between Kirchhoff vortices and point vortices in an ideal fluid
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 2, pp. 199-213
Abstract
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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, pp. 199-213
New integral in nonholonomic Painleve-Chaplygin problem
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 2, pp. 193-198
Abstract
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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.
Dynamics of two vortex rings on a sphere
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 2, pp. 181-192
Abstract
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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, pp. 181-192
On the motion of a heavy rigid body in an ideal fluid with circulation
Chaos, 2006, vol. 16, no. 1, 013118, 7 pp.
Abstract
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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, 7 pp.
The restricted two-body problem in constant curvature spaces
Celestial Mechanics and Dynamical Astronomy, 2006, vol. 96, no. 1, pp. 1-17
Abstract
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The bifurcation analysis of the Kepler problem on $\mathbb{S}^3$ and $\mathbb{H}^3$ 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 $\mathbb{S}^2$ and $\mathbb{H}^2$ (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.
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, pp. 1-17
Classification of Birkhoff-Integrable generalized Toda lattices
in "Topological Methods in the Theory of Integrable Systems", Cambridge: Cambridge Scientific Publishers Ltd., 2006, pp. 69-79
Abstract
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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, pp. 69-79
Finite action Klein–Gordon solutions on Lorentzian manifolds
International Journal of Geometric Methods in Modern Physics, 2006, vol. 3, no. 7, pp. 1349–1357
Abstract
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The eigenvalue problem for the square integrable solutions is studied usually for elliptic equations. In this paper we consider such a problem for the hyperbolic Klein–Gordon equation on Lorentzian manifolds. The investigation could help to answer the question why elementary particles have a discrete mass spectrum. An infinite family of square integrable solutions for the Klein–Gordon equation on the Friedman type manifolds is constructed. These solutions have a discrete mass spectrum and a finite action. In particular the solutions on de Sitter space are investigated.
Citation:
Kozlov V. V., Volovich I. V., Finite action Klein–Gordon solutions on Lorentzian manifolds, International Journal of Geometric Methods in Modern Physics, 2006, vol. 3, no. 7, pp. 1349–1357
Absolute and Relative Choreographies in the Problem of the Motion of Point Vortices in a Plane
Doklady Physics, 2005, vol. 71, no. 1, pp. 139-144
Abstract
pdf (168.54 Kb)
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, pp. 139-144
Uniform distribution and Voronoi convergence
Sbornik: Mathematics, 2005, vol. 196, no. 10, pp. 1495–1502
Abstract
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There is a broad generalization of a uniformly distributed sequence according to Weyl where the frequency of elements of this sequence falling into an interval is defined by using a matrix summation method of a general form. In the present paper conditions for uniform distribution are found in the case where a regular Voronoi method is chosen as the summation method. The proofs are based on estimates of trigonometric sums of a certain special type. It is shown that the sequence of the fractional parts of the logarithms of positive integers is not uniformly distributed for any choice of a regular Voronoi method.
Citation:
Kozlov V. V., Madsen T., Uniform distribution and Voronoi convergence, Sbornik: Mathematics, 2005, vol. 196, no. 10, pp. 1495–1502
Weighted Means, Strict Ergodicity, and Uniform Distributions
Mathematical Notes, 2005, vol. 78, no. 3, pp. 329–337
Abstract
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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.
The Nonexistence of an Invariant Measure for an Inhomogeneous Ellipsoid Rolling on a Plane
Mathematical Notes, 2005, vol. 77, no. 6, pp. 855-857
Abstract
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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, pp. 855-857
On periodic trajectories of a billiard in a magnetic field
Journal of Applied Mathematics and Mechanics, 2005, vol. 69, no. 6, pp. 844–851
Abstract
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The problem of the existence of periodic trajectories of a charged particle in a magnetic field, when the particle moves inside a closed convex region and is elastically reflected from its boundary, is considered. The presence of an infinite number of different periodic trajectories at low magnetic field strengths is established using Poincaré's geometric theorem. The conditions for two-link trajectories to be stable in the case of a uniform magnetic field are obtained.
Citation:
Kozlov V. V., Polikarpov S. A., On periodic trajectories of a billiard in a magnetic field, Journal of Applied Mathematics and Mechanics, 2005, vol. 69, no. 6, pp. 844–851
Weighted averages, uniform distribution, and strict ergodicity
Russian Mathematical Surveys, 2005, vol. 60, no. 6, pp. 1121–1146
Abstract
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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, pp. 1121–1146
Remarks on a Lie theorem on the exact integrability of differential equations
Differential Equations, 2005, vol. 41, no. 4, pp. 588–590
Abstract
pdf (85.01 Kb)
Citation:
Kozlov V. V., Remarks on a Lie theorem on the exact integrability of differential equations, Differential Equations, 2005, vol. 41, no. 4, pp. 588–590
Restrictions of Quadratic Forms to Lagrangian Planes, Quadratic Matrix Equations, and Gyroscopic Stabilization
Functional Analysis and Its Applications, 2005, vol. 39, no. 4, pp. 271–283
Abstract
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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.
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, pp. 271–283
Superintegrable systems on a sphere
Regular and Chaotic Dynamics, 2005, vol. 10, no. 3, pp. 257-266
Abstract
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We consider various generalizations of the Kepler problem to three-dimensional sphere $S^3$, (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 $L^3$.
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, pp. 257-266
Generalized dimensions of the golden-mean quasiperiodic orbit from renormalization-group functional equation
Regular and Chaotic Dynamics, 2005, vol. 10, no. 1, pp. 33-38
Abstract
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A method is suggested for computation of the generalized dimensions for a fractal attractor associated with the quasiperiodic transition to chaos at the golden-mean rotation number. The approach is based on an eigenvalue problem formulated in terms of functional equations with coeficients expressed via the universal fixed-point function of Feigenbaum–Kadanoff–Shenker. The accuracy of the results is determined only by precision of representation of the universal function.
Keywords:
circle map, golden mean, renormalization, dimension, generalized dimensions
Citation:
Kuznetsov S. P., Generalized dimensions of the golden-mean quasiperiodic orbit from renormalization-group functional equation, Regular and Chaotic Dynamics, 2005, vol. 10, no. 1, pp. 33-38
Reduction and chaotic behavior of point vortices on a plane and a sphere
Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 2, pp. 233-246
Abstract
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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, Poinca