Speedup of the Chaplygin Top by Means of Rotors
Doklady Physics, 2019, vol. 64, no. 3, pp. 120-124
Abstract
pdf (486.65 Kb)
In this paper we consider the control of the motion of a dynamically asymmetric unbalanced ball
(Chaplygin top) by means of two perpendicular rotors. We propose a mechanism for control by periodically
changing the gyrostatic momentum of the system, which leads to an unbounded speedup. We then formulate
a general hypothesis of the mechanism for speeding up spherical bodies on a plane by periodically changing
the system parameters.
A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness
Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, pp. 329-352
Abstract
pdf (1.48 Mb)
This paper is a small review devoted to the dynamics of a point on a paraboloid. Specifically, it is concerned with the motion both under the action of a gravitational field and without it. It is assumed that the paraboloid can rotate about a vertical axis with constant angular velocity. The paper includes both well-known results and a number of new results.
We consider the two most widespread friction (resistance) models: dry (Coulomb) friction and viscous friction. It is shown that the addition of external damping (air drag) can lead to stability of equilibrium at the saddle point and hence to preservation of the region of bounded motion
in a neighborhood of the saddle point. Analysis of three-dimensional Poincar´e sections shows that limit cycles can arise in this case in the neighborhood of the saddle point.
Borisov A. V., Kilin A. A., Mamaev I. S., A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness, Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, pp. 329-352
Qualitative Analysis of the Nonholonomic Rolling of a Rubber Wheel with Sharp Edges
Regular and Chaotic Dynamics, 2019, vol. 24, no. 2, pp. 212-233
Abstract
pdf (883.49 Kb)
This paper presents a qualitative analysis of the dynamics in a fixed reference frame of a wheel with sharp edges that rolls on a horizontal plane without slipping at the point of contact and without spinning relative to the vertical. The wheel is a ball that is symmetrically truncated on both sides and has a displaced center of mass. The dynamics of such a system is described by the model of the ball’s motion where the wheel rolls with its spherical part in contact with the supporting plane and the model of the disk’s motion where the contact point lies on the sharp edge of the wheel. A classification is given of possible motions of the wheel depending on whether there are transitions from its spherical part to sharp edges. An analysis is made of the behavior of the point of contact of the wheel with the plane for different values of the system parameters, first integrals and initial conditions. Conditions for boundedness and unboundedness of the wheel’s motion are obtained. Conditions for the fall of the wheel on the
plane of sections are presented.
Keywords:
integrable system, system with discontinuity, nonholonomic constraint, bifurcation diagram, body of revolution, sharp edge, wheel, rubber body model, permanent rotations, dynamics in a fixed reference frame, resonance, quadrature, unbounded motion
Citation:
Kilin A. A., Pivovarova E. N., Qualitative Analysis of the Nonholonomic Rolling of a Rubber Wheel with Sharp Edges, Regular and Chaotic Dynamics, 2019, vol. 24, no. 2, pp. 212-233
Integrability and Chaos in Vortex Lattice Dynamics
Regular and Chaotic Dynamics, 2019, vol. 24, no. 1, pp. 101-113
Abstract
pdf (1.61 Mb)
This paper is concerned with the problem of the interaction of vortex lattices, which
is equivalent to the problem of the motion of point vortices on a torus. It is shown that the
dynamics of a system of two vortices does not depend qualitatively on their strengths. Steadystate
configurations are found and their stability is investigated. For two vortex lattices it is
also shown that, in absolute space, vortices move along closed trajectories except for the case
of a vortex pair. The problems of the motion of three and four vortex lattices with nonzero
total strength are considered. For three vortices, a reduction to the level set of first integrals
is performed. The nonintegrability of this problem is numerically shown. It is demonstrated
that the equations of motion of four vortices on a torus admit an invariant manifold which
corresponds to centrally symmetric vortex configurations. Equations of motion of four vortices
on this invariant manifold and on a fixed level set of first integrals are obtained and their
nonintegrability is numerically proved.
Keywords:
vortices on a torus, vortex lattices, point vortices, nonintegrability, chaos, invariant manifold, Poincarґe map, topological analysis, numerical analysis, accuracy of calculations, reduction, reduced system
Citation:
Kilin A. A., Artemova E. M., Integrability and Chaos in Vortex Lattice Dynamics, Regular and Chaotic Dynamics, 2019, vol. 24, no. 1, pp. 101-113
Controlled Motion of a Spherical Robot with Feedback. II
Journal of Dynamical and Control Systems, 2019, vol. 25, no. 1, pp. 1-16
Abstract
pdf (803.07 Kb)
In this paper, we develop a model of a controlled spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum, with a feedback that stabilizes given partial solutions for a free system at the final stage of motion. According to the proposed approach, feedback depends on phase variables (current position, velocities) and does not depend on the specific type of trajectory. We present integrals of motion and partial solutions, analyze their stability, and give examples of computer simulations of motion with feedback that demonstrate the efficiency of the proposed model.
Ivanova T. B., Kilin A. A., Pivovarova E. N., Controlled Motion of a Spherical Robot with Feedback. II, Journal of Dynamical and Control Systems, 2019, vol. 25, no. 1, pp. 1-16
Stabilization of the motion of a spherical robot using feedbacks
Applied Mathematical Modelling, 2019, vol. 69, pp. 583-592
Abstract
pdf (1005.01 Kb)
The paper is concerned with the problem of stabilizing a spherical robot of combined type during its motion. The focus is on the application of feedback for stabilization of the robot which is an example of an underactuated system. The robot is set in motion by an inter- nal wheeled platform with a rotor placed inside the sphere. The results of experimental investigations for a prototype of the spherical robot are presented.
Borisov A. V., Kilin A. A., Karavaev Y. L., Klekovkin A. V., Stabilization of the motion of a spherical robot using feedbacks, Applied Mathematical Modelling, 2019, vol. 69, pp. 583-592
Invariant Submanifolds of Genus 5 and a Cantor Staircase in the Nonholonomic Model of a Snakeboard
International Journal of Bifurcation and Chaos, 2019, vol. 29, no. 3, 1930008, 19 pp.
Abstract
pdf (2.79 Mb)
In this paper, we address the free (uncontrolled) dynamics of a snakeboard consisting of two
wheel pairs fastened to a platform. The snakeboard is one of the well-known sports vehicles on
which the sportsman executes necessary body movements. From the theoretical point of view,
this system is a direct generalization of the classical nonholonomic system of the Chaplygin
sleigh. We carry out a topological and qualitative analysis of trajectories of this dynamical
system. An important feature of the problem is that the common level set of first integrals is
a compact two-dimensional surface of genus 5. We specify conditions under which the reaction
forces infinitely increase during motion and the so-called phenomenon of nonholonomic jamming
is observed. In this case, the nonholonomic model ceases to work and it is necessary to use more
complex mechanical models incorporating sliding, elasticity, etc.
Keywords:
Nonholonomic mechanics, snakeboard, qualitative analysis, bifurcations, regularization (blowing up singularities), system on a torus, nonholonomic jamming, bifurcation analysis
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Invariant Submanifolds of Genus 5 and a Cantor Staircase in the Nonholonomic Model of a Snakeboard, International Journal of Bifurcation and Chaos, 2019, vol. 29, no. 3, 1930008, 19 pp.
Nonholonomic rolling of a ball on the surface of a rotating cone
Nonlinear Dynamics, 2019, vol. 97, no. 2, pp. 1635-1648
Abstract
pdf (1.04 Mb)
This paper investigates the rolling without slipping of a homogeneous heavy ball on the surface of a rotating cone in two settings: without dissipation in a nonholonomic setting and with rolling friction torque which is proportional to the angular velocity of the ball. In the nonholonomic setting, the resulting system of five differential equations on the level set of first integrals is reduced to quadratures. A bifurcation analysis of the above system is carried out to determine the possible types of motion. In the second case, it is shown that there are not only trajectories emanating from the lower point of the cone (its vertex), but also trajectories to the vertex of the cone (fall). An analysis of the dependence of the type of terminal motion of the center of mass of the ball on initial conditions is carried out.
Borisov A. V., Ivanova T. B., Kilin A. A., Mamaev I. S., Nonholonomic rolling of a ball on the surface of a rotating cone, Nonlinear Dynamics, 2019, vol. 97, no. 2, pp. 1635-1648
Comment on “Confining rigid balls by mimicking quadrupole ion trapping” [Am. J. Phys. 85, 821 (2017)]
American Journal of Physics, 2019, vol. 87, no. 11, pp. 935-938
Abstract
pdf (2.18 Mb)
This paper discusses two approaches for deriving the equations of motion for a ball that rolls
without slipping on the surface of a rotating hyperbolic paraboloid. We analyze two possible
methods for defining the surface on which the ball rolls, and show the relationship between the
two methods. We describe how the stability of the ball’s rotation at the saddle point depends on
the radius of the ball, in the case where the stability analysis is made in dimensionless parameters.
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Comment on “Confining rigid balls by mimicking quadrupole ion trapping” [Am. J. Phys. 85, 821 (2017)], American Journal of Physics, 2019, vol. 87, no. 11, pp. 935-938
Control of the Rolling Motion of a Spherical Robot on an Inclined Plane
Doklady Physics, 2018, vol. 63, no. 10, pp. 435-440
Abstract
pdf (509.36 Kb)
In this work we consider the controlled motion of a pendulum spherical robot on an inclined plane. The algorithm for determining the control actions for the motion along an arbitrary trajectory and examples of numerical simulation of the controlled motion are given.
Citation:
Ivanova T. B., Kilin A. A., Pivovarova E. N., Control of the Rolling Motion of a Spherical Robot on an Inclined Plane, Doklady Physics, 2018, vol. 63, no. 10, pp. 435-440
Controlled Motion of a Spherical Robot of Pendulum Type on an Inclined Plane
Doklady Physics, 2018, vol. 63, no. 7, pp. 302-306
Abstract
pdf (568.83 Kb)
This paper is concerned with a model of the controlled motion of a spherical robot with an axisymmetric
pendulum actuator on an inclined plane. First integrals of motion and partial solutions are presented
and their stability is analyzed. It is shown that the steady solutions exist only at an inclination angle less than
some critical value and only for constant control action.
Citation:
Ivanova T. B., Kilin A. A., Pivovarova E. N., Controlled Motion of a Spherical Robot of Pendulum Type on an Inclined Plane, Doklady Physics, 2018, vol. 63, no. 7, pp. 302-306
Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges
Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 887-907
Abstract
pdf (2.02 Mb)
This paper is concerned with the dynamics of a wheel with sharp edges moving on a horizontal plane without slipping and rotation about the vertical (nonholonomic rubber model). The wheel is a body of revolution and has the form of a ball symmetrically truncated on both sides. This problem is described by a system of differential equations with a discontinuous
right-hand side. It is shown that this system is integrable and reduces to quadratures. Partial solutions are found which correspond to fixed points of the reduced system. A bifurcation analysis and a classification of possible types of the wheel’s motion depending on the system parameters are presented.
Keywords:
integrable system, system with a discontinuous right-hand side, nonholonomic constraint, bifurcation diagram, body of revolution, sharp edge, wheel, rubber model
Citation:
Kilin A. A., Pivovarova E. N., Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 887-907
A Nonholonomic Model of the Paul Trap
Regular and Chaotic Dynamics, 2018, vol. 23, no. 3, pp. 339-354
Abstract
pdf (6.87 Mb)
In this paper, equations of motion for the problem of a ball rolling without slipping on a rotating hyperbolic paraboloid are obtained. Integrals of motions and an invariant measure are found. A detailed linear stability analysis of the ball’s rotations at the saddle point of the
hyperbolic paraboloid is made. A three-dimensional Poincar´e map generated by the phase flow of the problem is numerically investigated and the existence of a region of bounded trajectories in a neighborhood of the saddle point of the paraboloid is demonstrated. It is shown that a similar problem of a ball rolling on a rotating paraboloid, considered within the framework of the rubber model, can be reduced to a Hamiltonian system which includes the Brower problem as a particular case.
Keywords:
Paul trap, stability, nonholonomic system, three-dimensional map, gyroscopic stabilization, noninertial coordinate system, Poincaré map, nonholonomic constraint, rolling without slipping, region of linear stability
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., A Nonholonomic Model of the Paul Trap, Regular and Chaotic Dynamics, 2018, vol. 23, no. 3, pp. 339-354
Controlling the movement of the body using internal masses in a viscous liquid
Computer Research and Modeling, 2018, vol. 10, no. 4, pp. 445-460
Abstract
pdf (523.34 Kb)
This article is devoted to the study of self-propulsion of bodies in a fluid by the action of internal
mechanisms, without changing the external shape of th
e body. The paper presents an overview of theoretical
papers that justify the possibility of this displacement in ideal and viscous liquids.
A special case of self-propulsion of a rigid body along the surface of a liquid is considered due to the
motion of two internal masses along the circles. The paper presents a mathematical model of the motion of
a solid body with moving internal masses in a three-dime
nsional formulation. This model takes into account the
three-dimensional vibrations of the body during motion,
which arise under the action of external forces-gravity
force, Archimedes force and forces acting on the body, from the side of a viscous fluid.
The body is a homogeneous elliptical cylinder with a k
eel located along the larger diagonal. Inside the
cylinder there are two material point masses moving along the circles. The centers of the circles lie on the
smallest diagonal of the ellipse at an equal distance from the center of mass.
Equations of motion of the system (a body with two mater
ial points, placed in a fluid) are represented as
Kirchhoff equations with the addition of external for
ces and moments acting on the body. The phenomenological
model of viscous friction is quadratic in velocity used
to describe the forces of resistance to motion in a fluid.
The coefficients of resistance to movement were determ
ined experimentally. The forces acting on the keel were
determined by numerical modeling of the keel oscillations in a viscous liquid using the Navier – Stokes equations.
In this paper, an experimental verification of the
proposed mathematical model was carried out. Several
series of experiments on self-propulsion of a body in a liquid by means of rotation of internal masses with
different speeds of rotation are presented. The dependence of the average propagation velocity, the amplitude of
the transverse oscillations as a function of the rotational speed of internal masses is investigated. The obtained
experimental data are compared with the results obtai
ned within the framework of the proposed mathematical
model.
Keywords:
motion in a fluid, self-promotion, the equations of movement, above-water screwless robot, Navier – Stokes equations
Citation:
Kilin A. A., Klenov A. I., Tenenev V. A., Controlling the movement of the body using internal masses in a viscous liquid, Computer Research and Modeling, 2018, vol. 10, no. 4, pp. 445-460
Controlled Motion of a Spherical Robot with Feedback. I
Journal of Dynamical and Control Systems, 2018, vol. 24, no. 3, pp. 497-510
Abstract
pdf (606.17 Kb)
In this paper, we develop a model of a controlled spherical robot with an axisymmetric pendulum-type actuator with a feedback system suppressing the pendulum’s oscillations at the final stage of motion. According to the proposed approach, the feedback depends on phase variables (the current position and velocities) and does not depend on the type of trajectory. We present integrals of motion and partial solutions, analyze their stability, and give examples of computer simulation of motion using feedback to illustrate compensation of the pendulum’s oscillations.
Ivanova T. B., Kilin A. A., Pivovarova E. N., Controlled Motion of a Spherical Robot with Feedback. I, Journal of Dynamical and Control Systems, 2018, vol. 24, no. 3, pp. 497-510
Chaplygin Top with a Periodic Gyrostatic Moment
Russian Journal of Mathematical Physics, 2018, vol. 25, no. 4, pp. 509-524
Abstract
pdf (1.14 Mb)
In the paper, a study of rolling of a dynamically asymmetrical unbalanced ball (Chaplygin top) on a horizontal plane under the action of periodic gyrostatic moment is carried out. The problem is considered in the framework of the model of a rubber body, i.e., under the assumption that there is no slipping and spinning at the point of contact. It is shown that, for certain values of the parameters of the system and certain dependence of the gyrostatic moment on time, an acceleration of the system, i.e., an unbounded growth of the energy of the system, is observed. Investigations of the dependence of the presence of acceleration on the parameters of the system and on the initial conditions are carried out. On the basis of the investigations of the dynamics of the frozen system, a conjecture concerning the general mechanism of acceleration at the expense to periodic impacts in
nonholonomic systems is expressed.
Citation:
Kilin A. A., Pivovarova E. N., Chaplygin Top with a Periodic Gyrostatic Moment, Russian Journal of Mathematical Physics, 2018, vol. 25, no. 4, pp. 509-524
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
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 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
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
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
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
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
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
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
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
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
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
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
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
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
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 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
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
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.
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
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.
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
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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Stability of steady rotations in the non-holonomic Routh problem
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 3, pp. 333-345
Abstract
pdf (398.23 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 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
New integral in nonholonomic Painleve-Chaplygin problem
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 2, pp. 193-198
Abstract
pdf (132.6 Kb)
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.
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
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
pdf (473.55 Kb)
We offer a new method of reduction for a system of point vortices on a plane and a sphere. This method is similar to the classical node elimination procedure. However, as applied to the vortex dynamics, it requires substantial modification. Reduction of four vortices on a sphere is given in more detail. We also use the Poincare surface-of-section technique to perform the reduction a four-vortex system on a sphere.
Keywords:
reduction, point vortex, equations of motion, Poincare map
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Reduction and chaotic behavior of point vortices on a plane and a sphere, Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 2, pp. 233-246
Chaos in a restricted problem of rotation of a rigid body with a fixed point
Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 2, pp. 191-207
Abstract
pdf (650.7 Kb)
The paper deals with a transition to chaos in the phase-plane portrait of a restricted problem of rotation of a rigid body with a fixed point. Two interrelated mechanisms responsible for chaotisation have been indicated: 1) growth of the homoclinic structure and 2) development of cascades of period doubling bifurcations. On the zero level of the integral of areas, an adiabatic behavior of the system (as the energy tends to zero) has been noticed. Meander tori induced by the breakdown of the torsion property of the mapping have been found.
Keywords:
motion of a rigid body, phase-plane portrait, mechanism of chaotisation, bifurcations
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Chaos in a restricted problem of rotation of a rigid body with a fixed point, Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 2, pp. 191-207
Absolute and relative choreographies in rigid body dynamics
Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 1, pp. 123-141
Abstract
pdf (401.63 Kb)
For the classical problem of motion of a rigid body about a fixed point with zero integral of areas, the paper presents a family of solutions which are periodic in the absolute space. Such solutions are known as choreographies. The family includes the famous Delaunay solution in the case of Kovalevskaya, some particular solutions in the Goryachev-Chaplygin case and Steklov’s solution. The «genealogy» of the solutions of the family, arising naturally from the energy continuation, and their connection with the Staude rotations are considered.
It is shown that if the integral of areas is zero, the solutions are periodic but with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies).
Keywords:
rigid body dynamics, periodic solutions, continuation by a parameter, bifurcation
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Absolute and relative choreographies in rigid body dynamics, Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 1, pp. 123-141
Reduction and chaotic behavior of point vortices on a plane and a sphere
Discrete and Continuous Dynamical Systems - Series B (Supplement Volume devoted to the 5th AIMS International Conference on Dynamical Systems and Differential Equations (Pomona, California, USA, June 2004)), 2005, pp. 100-109
Abstract
pdf (360.5 Kb)
We offer a new method of reduction for a system of point vortices on a plane and a sphere. This method is similar to the classical node elimination procedure. However, as applied to the vortex dynamics, it requires substantial modification. Reduction of four vortices on a sphere is given in more detail. We also use the Poincare surface-of-section technique to perform the reduction a four-vortex system on a sphere.
Keywords:
Vortex dynamics, reduction, Poincaré map, point vortices
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Reduction and chaotic behavior of point vortices on a plane and a sphere, Discrete and Continuous Dynamical Systems - Series B (Supplement Volume devoted to the 5th AIMS International Conference on Dynamical Systems and Differential Equations (Pomona, California, USA, June 2004)), 2005, pp. 100-109
New periodic solutions for three or four identical vortices on a plane and a sphere
Discrete and Continuous Dynamical Systems - Series B (Supplement Volume devoted to the 5th AIMS International Conference on Dynamical Systems and Differential Equations (Pomona, California, USA, June 2004)), 2005, pp. 110-120
Abstract
pdf (194.7 Kb)
In this paper we describe new classes of periodic solutions for point vortices on a plane and a sphere. They correspond to similar solutions (so-called choreographies) in celestial mechanics.
Borisov A. V., Mamaev I. S., Kilin A. A., New periodic solutions for three or four identical vortices on a plane and a sphere, Discrete and Continuous Dynamical Systems - Series B (Supplement Volume devoted to the 5th AIMS International Conference on Dynamical Systems and Differential Equations (Pomona, California, USA, June 2004)), 2005, pp. 110-120
Two-body problem on a sphere. Reduction, stochasticity, periodic orbits
Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 265-279
Abstract
pdf (13.58 Mb)
We consider the problem of two interacting particles on a sphere. The potential of the interaction depends on the distance between the particles. The case of Newtonian-type potentials is studied in most detail. We reduce this system to a system with two degrees of freedom and give a number of remarkable periodic orbits. We also discuss integrability and stochastization of the motion.
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., Two-body problem on a sphere. Reduction, stochasticity, periodic orbits, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 265-279
Absolute and relative choreographies in the problem of point vortices moving on a plane
Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 101-111
Abstract
pdf (389.11 Kb)
We obtained new periodic solutions in the problems of three and four point vortices moving on a plane. In the case of three vortices, the system is reduced to a Hamiltonian system with one degree of freedom, and it is integrable. In the case of four vortices, the order is reduced to two degrees of freedom, and the system is not integrable. We present relative and absolute choreographies of three and four vortices of the same intensity which are periodic motions of vortices in some rotating and fixed frame of reference, where all the vortices move along the same closed curve. Similar choreographies have been recently obtained by C. Moore, A. Chenciner, and C. Simo for the $n$-body problem in celestial mechanics [6, 7, 17]. Nevertheless, the choreographies that appear in vortex dynamics have a number of distinct features.
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., Absolute and relative choreographies in the problem of point vortices moving on a plane, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 101-111
Dynamics of rolling disk
Regular and Chaotic Dynamics, 2003, vol. 8, no. 2, pp. 201-212
Abstract
pdf (648.4 Kb)
In the paper we present the qualitative analysis of rolling motion without slipping of a homogeneous round disk on a horisontal plane. The problem was studied by S.A. Chaplygin, P. Appel and D. Korteweg who showed its integrability. The behavior of the point of contact on a plane is investigated and conditions under which its trajectory is finit are obtained. The bifurcation diagrams are constructed.
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., Dynamics of rolling disk, Regular and Chaotic Dynamics, 2003, vol. 8, no. 2, pp. 201-212
A New Integral in the Problem of Rolling a Ball on an Arbitrary Ellipsoid
Doklady Physics, 2002, vol. 47, no. 7, pp. 544-547
Abstract
pdf (85.97 Kb)
The problem of rolling motion without slipping of an unbalanced ball on 1) an arbitrary ellipsoid and 2) an ellipsoid of revolution is considered. In his famous treatise E. Routh showed that the problem of rolling motion of a body on a surface of revolution even in the presence of axisymmetrical potential fields is integrable. In case 1, we present a new integral of motion. New solutions expressed in elementary functions are found in case 2.
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., A New Integral in the Problem of Rolling a Ball on an Arbitrary Ellipsoid, Doklady Physics, 2002, vol. 47, no. 7, pp. 544-547
The rolling motion of a ball on a surface. New integrals and hierarchy of dynamics
Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 201-219
Abstract
pdf (628.29 Kb)
The paper is concerned with the problem on rolling of a homogeneous ball on an arbitrary surface. New cases when the problem is solved by quadratures are presented. The paper also indicates a special case when an additional integral and invariant measure exist. Using this case, we obtain a nonholonomic generalization of the Jacobi problem for the inertial motion of a point on an ellipsoid. For a ball rolling, it is also shown that on an arbitrary cylinder in the gravity field the ball's motion is bounded and, on the average, it does not move downwards. All the results of the paper considerably expand the results obtained by E. Routh in XIX century.
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., The rolling motion of a ball on a surface. New integrals and hierarchy of dynamics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 201-219
The Dynamics of Chaplygin Ball: the Qualitative and Computer Analysis
Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 291-306
Abstract
pdf (1.16 Mb)
The motion of Chaplygin ball with and without gyroscope in the absolute space is analyzed. In particular, the trajectories of the point of contact are studied in detail. We discuss the motions in the absolute space, that correspond to the different types of motion in the moving frame of reference related to the body. The existence of the bounded trajectories of the ball's motion is shown by means of numerical methods in the case when the problem is reduced to a certain Hamiltonian system.
Citation:
Kilin A. A., The Dynamics of Chaplygin Ball: the Qualitative and Computer Analysis, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 291-306
First Integral in the Problem of Motion of a Circular Cylinder and a Point Vortex in Unbounded Ideal Fluid
Regular and Chaotic Dynamics, 2001, vol. 6, no. 2, pp. 233-234
Abstract
pdf (139.78 Kb)
In the paper Motion of a circular cylinder and a vortex in an ideal fluid (Reg. & Chaot. Dyn. V. 6. 2001. No 1. P. 33-38) Ramodanov S.M. showed the integrability of the problem of motion of a circular cylinder and a point vortex in unbounded ideal fluid. In the present paper we find additional first integral and invariant measure of motion equations.
Citation:
Kilin A. A., First Integral in the Problem of Motion of a Circular Cylinder and a Point Vortex in Unbounded Ideal Fluid, Regular and Chaotic Dynamics, 2001, vol. 6, no. 2, pp. 233-234
Stability of Thomson's Configurations of Vortices on a Sphere
Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 189-200
Abstract
pdf (284.39 Kb)
In this work stability of polygonal configurations on a plane and sphere is investigated. The conditions of linear stability are obtained. A nonlinear analysis of the problem is made with the help of Birkhoff normalization. Some problems are also formulated.
Citation:
Borisov A. V., Kilin A. A., Stability of Thomson's Configurations of Vortices on a Sphere, Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 189-200
Libration points in spaces $S^2$ and $L^2$
Regular and Chaotic Dynamics, 1999, vol. 4, no. 1, pp. 91-103
Abstract
pdf (455.22 Kb)
We consider two-body problem and restricted three-body problem in spaces $S^2$ and $L^2$. For two-body problem we have showed the absence of exponential instability of partiбular solutions relevant to roundabout motion on the plane. New libration points are found, and the dependence of their positions on parameters of a system is explored. The regions of existence of libration points in space of parameters were constructed. Basing on a examination of the Hill's regions we found the qualitative estimation of stability of libration points was produced.
Citation:
Kilin A. A., Libration points in spaces $S^2$ and $L^2$, Regular and Chaotic Dynamics, 1999, vol. 4, no. 1, pp. 91-103
Libration points in a bounded problem of three bodies on $S^2$
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 1998, no. 1, pp. 61-66
Abstract
pdf (3.96 Mb)
Citation:
Kilin A. A., Mamaev I. S., Libration points in a bounded problem of three bodies on $S^2$, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 1998, no. 1, pp. 61-66