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.

The onset of adiabatic chaos in rigid body dynamics is considered. A comparison of the analytically calculated diffusion coefficient describing probabilistic effects in the zone of chaos with a numerical experiment is made. An analysis of the splitting of asymptotic surfaces is performed and uncertainty curves are constructed in the Poincaré ? Zhukovsky problem. The application of Hamiltonian methods to nonholonomic systems is discussed. New problem statements are given which are related to the destruction of an adiabatic invariant and to the acceleration of the system (Fermi?s acceleration).

This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found. In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.

In this paper, we consider the dynamics of a heavy homogeneous ball moving under the influence of dry friction on a fixed horizontal plane. We assume the ball to slide without rolling. We demonstrate that the plane may be divided into two regions, each characterized by a distinct coefficient of friction, so that balls with equal initial linear and angular velocity will converge upon the same point from different initial locations along a certain segment. We construct the boundary between the two regions explicitly and discuss possible applications to real physical systems.

In this paper, using the Hojman construction, we give examples of various Poisson brackets which differ from those which are usually analyzed in Hamiltonian mechanics. They possess a nonmaximal rank, and in the general case an invariant measure and Casimir functions can be globally absent for them.

A new integrable problem of nonholonomic mechanics is considered and its mechanical realization is proposed. This problem is a generalization of the well-known problem of À. P. Veselov and L. E. Veselova concerning the rolling motion of the Chaplygin ball in a straight line. Particular cases are found in which integration can be reduced to explicit quadratures.

We consider differential equations with quadratic right-hand sides which admit two quadratic first integrals, one of which is a positive definite quadratic form. We present general conditions under which a linear change of variables reduces this system to some "canonical" form. Under these conditions the system turns out to be nondivergent and is reduced to Hamiltonian form, however, the corresponding linear Lie–Poisson bracket does not always satisfy the Jacobi identity. In the three-dimensional case the equations are reduced to the classical equations of the Euler top, and in the four-dimensional space the system turns out to be superintegrable and coincides with the Euler–Poincare? equations on some Lie algebra. In the five-dimensional case we find a reducing multiplier after multiplication with which the Poisson bracket satisfies the Jacobi identity. In the general case, we prove that there is no reducing multiplier for $n>5$. As an example, we consider a system of Lotka–Volterra type with quadratic right-hand sides, which was studied already by Kovalevskaya from the viewpoint of the conditions for uniqueness of its solutions as functions of complex time.

This is a reply to the comment by V.F. Zhuravlev (see Usp. Fiz. Nauk 185 1337 (2015) [Phys. Usp. 58 (12) (2015)]) on the inadequacy of the nonholonomic model when applied to the rolling of rigid bodies. The model of nonholonomic mechanics is discussed. Using recent results as examples, it is shown that the validity and potential of the nonholonomic model are not inferior to those of other dynamics and friction models.

This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with a stratified density and a steady-state velocity field. As in the classical formulation of the problem, it is assumed that the figures, or their layers, uniformly rotate about an axis fixed in space. It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with a constant angular velocity, is at equilibrium. Expressions are obtained for the gravitational potential, change in the angular velocity and pressure, and the conclusion is drawn that the angular velocity on the outer surface is the same as that of the corresponding Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification. We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis.

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.

This paper is an experimental investigation of a round uniform disk rolling on a horizontal surface. Two methods for experimentally determining the loss of contact of the rolling disk from the horizontal surface before its stop are proposed. Results of experiments for disks having different masses and manufactured from different materials are presented. Causes of ?microlosses of contact? detected in the processes of motion are discussed.

This paper presents an experimental investigation of the influence of rolling friction on the dynamics of a robot wheel. The robot is set in motion by changing the proper gyrostatic momentum using the controlled rotation of a rotor installed in the robot. The problem is considered under the assumption that the center of mass of the system does not coincide with its geometric center. In this paper we derive equations describing the dynamics of the system and give an example of the controlled motion of a wheel by specifying a constant angular acceleration of the rotor. A description of the design of the robot wheel is given and a method for experimentally determining the rolling friction coefficient is proposed. For the verification of the proposed mathematical model, experimental studies of the controlled motion of the robot wheel are carried out. We show that the theoretical results qualitatively agree with the experimental ones, but are quantitatively different.

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.

The paper is devoted to the experimental verification of the Andersen?Pesavento?Wang model describing the falling of a heavy plate through a resisting medium. As a main research method the authors have used video filming of a falling plate with PIV measurement of the velocity of surrounding fluid flows. The trajectories of plates and streamlines were determined and oscillation frequencies were estimated using experimental results. A number of experiments for plates of various densities and sizes were performed. The trajectories of plates made of plastic are qualitatively similar to the trajectories predicted by the Andersen?Pesavento?Wang model. However, measured and computed frequencies of oscillations differ significantly. For a plate made of high carbon steel the results of experiments are quantitatively and qualitatively in disagreement with computational results.

The paper is devoted to the development of a model of an underwater robot actuated by inner rotors. This design has no moving elements interacting with an environment, which minimizes a negative impact on it, and increases noiselessness of the robot motion in a liquid. Despite numerous discussions on the possibility and efficiency of motion by means of internal masses' movement, a large number of works published in recent years confirms a relevance of the research. The paper presents an overview of works aimed at studying the motion by moving internal masses. A design of a screwless underwater robot that moves by the rotation of inner rotors to conduct theoretical and experimental investigations is proposed. In the context of theoretical research a robot model is considered as a hollow ellipsoid with three rotors located inside so that the axes of their rotation are mutually orthogonal. For the proposed model of a screwless underwater robot equations of motion in the form of classical Kirchhoff equations are obtained.

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.

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.

In this article a new model of motif (small ensemble) of neuron-like elements is proposed. It is built with the use of the generalized Lotka?Volterra model with excitatory couplings. The main motivation for this work comes from the problems of neuroscience where excitatory couplings are proved to be the predominant type of interaction between neurons of the brain. In this paper it is shown that there are two modes depending on the type of coupling between the elements: the mode with a stable heteroclinic cycle and the mode with a stable limit cycle. Our second goal is to examine the chaotic dynamics of the generalized three-dimensional Lotka?Volterra model.

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.

In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems. We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.

This paper is a review of the problem of the constructive reduction of nonholonomic systems with symmetries. The connection of reduction with the presence of the simplest tensor invariants (first integrals and symmetry fields) is shown. All theoretical constructions are illustrated by examples encountered in applications. In addition, the paper contains a short historical and critical sketch covering the contribution of various researchers to this problem.

In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.

This paper addresses the dynamics of systems with servoconstraints where the constraints are realized by controlling the inertial properties of the system. Vakonomic systems are a particular case. Special attention is given to the motion on Lie groups with left-invariant kinetic energy and a left-invariant constraint. The presence of symmetries allows the dynamical equations to be reduced to a closed system of differential equations with quadratic right-hand sides. As the main example, we consider the rotation of a rigid body with a left-invariant servoconstraint, which implies that the projection of the body?s angular velocity on some body-fixed direction is zero.

In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.

The paper discusses the dynamics of systems with Béghin?s servoconstraints where the constraints are realized by means of controlled forces. Classical nonholonomic systems are an important particular case. Special attention is given to the study of motion on Lie groups with left-invariant kinetic energy and left-invariant constraints. The presence of symmetries allows one to reduce the dynamic equations to a closed system of differential equations with quadratic right-hand sides on a Lie algebra. Examples are given which include the rotation of a rigid body with a left-invariant servoconstraint — the projection of the angular velocity onto some direction fixed in the body is equal to zero (a generalization of the nonholonomic Suslov problem) — and the motion of the Chaplygin sleigh with servoconstraints of a certain type. The dynamics of systems with Béghin?s servoconstraints is richer and more varied than the more usual dynamics of nonholonomic systems.

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.

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.

This paper is a review of the problem of the constructive reduction of nonholonomic systems with symmetries. The connection of reduction with the presence of the simplest tensor invariants (first integrals and symmetry fields) is shown. All theoretical constructions are illustrated by examples encountered in applications. In addition, the paper contains a short historical and critical sketch covering the contribution of various researchers to this problem.

This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie ?Poisson bracket of rank 2. This Lie ? Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.

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.

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.

This paper addresses the dynamics of systems with servoconstraints where the constraints are realized by controlling the inertial properties of the system. Vakonomic systems are a particular case. Special attention is given to the motion on Lie groups with left-invariant kinetic energy and a left-invariant constraint. The presence of symmetries allows the dynamical equations to be reduced to a closed system of differential equations with quadratic right-hand sides. As the main example, we consider the rotation of a rigid body with a left-invariant servo-constraint, which implies that the projection of the body?s angular velocity on some body-fixed direction is zero.

In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.

In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.

The paper discusses the dynamics of systems with Béghin?s servoconstraints where the constraints are realized by means of controlled forces. Classical nonholonomic systems are an important particular case. Special attention is given to the study of motion on Lie groups with left-invariant kinetic energy and left-invariant constraints. The presence of symmetries allows one to reduce the dynamic equations to a closed system of differential equations with quadratic right-hand sides on a Lie algebra. Examples are given which include the rotation of a rigid body with a left-invariant servoconstraint ? the projection of the angular velocity onto some direction fixed in the body is equal to zero (a generalization of the nonholonomic Suslov problem) ? and the motion of the Chaplygin sleigh with servoconstraints of a certain type. The dynamics of systems with Béghin?s servoconstraints is richer and more varied than the more usual dynamics of nonholonomic systems.

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.

It is well known that in the Béghin? Appel theory servo-constraints are realized using controlled external forces. In this paper an expansion of the Béghin?Appel theory is given in the case where servo-constraints are realized using controlled change of the inertial properties of a dynamical system. The analytical mechanics of dynamical systems with servo-constraints of general form is discussed. The key principle of the approach developed is to appropriately determine virtual displacements of systems with constraints.

We develop the reducing multiplier theory for a special class of nonholonomic dynamical systems and show that the non-linear Poisson brackets naturally obtained in the framework of this approach are all isomorphic to the Lie–Poisson $e$(3)-bracket. As two model examples, we consider the Chaplygin ball problem on the plane and the Veselova system. In particular, we obtain an integrable gyrostatic generalisation of the Veselova system.

This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie–Poisson bracket of rank two. This Lie–Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.

In this paper, we develop the method of Chaplygin?s reducing multiplier; using this method, we obtain a conformally Hamiltonian representation for three nonholonomic systems, namely, for the nonholonomic oscillator, for the Heisenberg system, and for the Chaplygin sleigh. Furthermore, in the case of oscillator and nonholonomic Chaplygin sleigh, we show that the problem reduces to the study of motion of a mass point (in a potential field) on a plane and, in the case of Heisenberg system, on the sphere. Moreover, we consider an example of a nonholonomic system (suggested by Blackall) to which one cannot apply the method of reducing multiplier.

This paper investigates the possibility of the motion control of a ball with a pendulum mechanism with non-holonomic constraints using gaits — the simplest motions such as acceleration and deceleration during the motion in a straight line, rotation through a given angle and their combination. Also, the controlled motion of the system along a straight line with a constant acceleration is considered. For this problem the algorithm for calculating the control torques is given and it is shown that the resulting reduced system has the first integral of motion.

For a solid body of convex form moving on a rough horizontal plane that is known as a rattleback, numerical simulations are used to discuss and illustrate dynamical phenomena that are characteristic of the motion due to a nonholonomic nature of the mechanical system; the relevant feature is the nonconservation of the phase volume in the course of the dynamics. In such a system, a local compression of the phase volume can produce behavior features similar to those exhibited by dissipative systems, such as stable equilibrium points corresponding to stationary rotations; limit cycles (rotations with oscillations); and strange attractors. A chart of dynamical regimes is plotted in a plane whose axes are the total mechanical energy and the relative angle between the geometric and dynamic principal axes of the body. The transition to chaos through a sequence of Feigenbaum period doubling bifurcations is demonstrated. A number of strange attractors are considered, for which phase portraits, Lyapunov exponents, and Fourier spectra are presented.

Nonholonomic systems describing the rolling of a rigid body on a plane and their relationship with various Poisson structures are considered. The notion of generalized conformally Hamiltonian representation of dynamical systems is introduced. In contrast to linear Poisson structures defined by Lie algebras and used in rigid-body dynamics, the Poisson structures of nonholonomic systems turn out to be nonlinear. They are also degenerate and the Casimir functions for them can be expressed in terms of complicated transcendental functions or not appear at all.

This is a survey of basic facts presently known about non-linear Poisson structures in the analysis of integrable systems in non-holonomic mechanics. It is shown that by using the theory of Poisson deformations it is possible to reduce various non-holonomic systems to dynamical systems on well-understood phase spaces equipped with linear Lie-Poisson brackets. As a result, not only can different non-holonomic systems be compared, but also fairly advanced methods of Poisson geometry and topology can be used for investigating them.

This paper reviews the results of stability analysis for polygonal configurations of a point vortex system in an annular region depending on the ratio of the inner and outer radii of the annulus. Conditions are found for linear stability of Thomson?s configurations for the case $N<7$. The paper also shows that a system of two vortices between parallel walls is a limiting case of a two-vortex system in an annular region, as the radii of the annulus tend to infinity.

In this paper we consider the motion of a dynamically asymmetric unbalanced ball on a plane in a gravitational field. The point of contact of the ball with the plane is subject to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the nonholonomic reversible system of 6 differential equations. In the case of arbitrary displacement of the center of mass of the ball the system under consideration is a nonintegrable system without an invariant measure. Using qualitative and quantitative analysis we show that the unbalanced ball exhibits reversal (the phenomenon of reversal of the direction of rotation) for some parameter values. Moreover, by constructing charts of Lyaponov exponents we find a few types of strange attractors in the system, including the so-called figure-eight attractor which belongs to the genuine strange attractors of pseudohyperbolic type.

In this paper, the integrability of the equations of a system of three vortex sources is shown. A reduced system describing, up to similarity, the evolution of the system?s configurations is obtained. Possible phase portraits and various relative equilibria of the system are presented.

In this paper we investigate the dynamics of a body with a flat base sliding on a horizontal and inclined rough plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. For analysis we use the descriptive function method similar to the methods used in the problems of Hamiltonian dynamics with one degree of freedom and allowing a qualitative analysis of the system to be made without explicit integration of equations of motion. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.

This paper is concerned with the problem of first integrals of the equations of geodesics on two-dimensional surfaces that are rational in the velocities (or momenta). The existence of nontrivial rational integrals with given values of the degrees of the numerator and the denominator is proved using the Cauchy–Kovalevskaya theorem.

In this paper we consider superintegrable systems which are an immediate generalization of the Kepler and Hook problems, both in two-dimensional spaces — the plane $\mathbb{R}^2$ and the sphere $S^2$ — and in three-dimensional spaces $\mathbb{R}^3$ and $S^3$. Using the central projection and the reduction procedure proposed in [21], we show an interrelation between the superintegrable systems found previously and show new ones. In all cases the superintegrals are presented in explicit form.

In this paper we investigate two systems consisting of a spherical shell rolling without slipping on a plane and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is attached to the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of a nonholonomic hinge. Equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler–Jacobi–Lie theorem, which is a new integration mechanism in nonholonomic mechanics. We also consider the problem of free motion of a bundle of two bodies connected by means of a nonholonomic hinge. For this system, integrable cases and various tensor invariants are found.

The problem of integrability conditions for systems of differential equations is discussed. Darboux?s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension $\geqslant n$. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.

In this paper we consider the control of a dynamically asymmetric balanced ball on a plane in the case of slipping at the contact point. Necessary conditions under which a control is possible are obtained. Specific algorithms of control along a given trajectory are constructed.

In this paper we consider the dynamics of a rigid body with a sharp edge in contact with a rough plane. The body can move so that its contact point is fixed or slips or loses contact with the support. In this paper, the dynamics of the system is considered within three mechanical models which describe different regimes of motion. The boundaries of the domain of definition of each model are given, the possibility of transitions from one regime to another and their consistency with different coefficients of friction on the horizontal and inclined surfaces are discussed.

The life and career of the great French mathematician and politician Paul Painlevé is described. His contribution to the analytical theory of nonlinear differential equations was significant. The paper outlines the achievements of Paul Painlevé and his students in the investigation of an interesting class of nonlinear second-order equations and new equations defining a completely new class of special functions, now called the Painlevé transcendents. The contribution of Paul Painlevé to the study of algebraic nonintegrability of the $N$-body problem, his remarkable observations in mechanics, in particular, paradoxes arising in the dynamics of systems with friction, his attempt to create the axiomatics of mechanics and his contribution to gravitation theory are discussed.

The kinematic control model for a spherical robot with an 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.

In this paper we investigate the dynamics of a body with a flat base sliding on a inclined plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. Computer-aided analysis of the system?s dynamics on the inclined plane using phase portraits has allowed us to reveal dynamical effects that have not been found earlier.

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.

This paper is concerned with the problem of first integrals of the equations of geodesics on twodimensional surfaces that are rational in the velocities (or momenta). The existence of nontrivial rational integrals with given values of the degrees of the numerator and the denominator is proved using the Cauchy?Kovalevskaya theorem.

We study both analytically and numerically the dynamics of an inhomogeneous ball on a rough horizontal plane under the infuence of gravity. A nonholonomic constraint of zero velocity at the point of contact of the ball with the plane is imposed. In the case of an arbitrary displacement of the center of mass of the ball, the system is nonintegrable without the property of phase volume conservation. We show that at certain parameter values the unbalanced ball exhibits the effect of reversal (the direction of the ball rotation reverses). Charts of dynamical regimes on the parameter plane are presented. The system under consideration exhibits diverse chaotic dynamics, in particular, the figure-eight chaotic attractor, which is a special type of pseudohyperbolic chaos.

This paper discusses new unresolved problems of nonholonomic mechanics. Hypotheses of the possibility of Hamiltonization and the existence of an invariant measure for such systems are advanced.

This paper is concerned with the dynamics of two point vortices of the same intensity which are affected by an acoustic wave. Typical bifurcations of fixed points have been identified by constructing charts of dynamical regimes, and bifurcation diagrams have been plotted.

In this paper, the integrability of the equations of a system of three vortex sources is shown. A reduced system describing, up to similarity, the evolution of the system?s configurations is obtained. Possible phase portraits and various relative equilibria of the system are presented.

This paper isšconcerned with ašthree-body system onšašstraight line inšašpotential field proposed byšTsiganov. The Liouville integrability ofšthis system isšshown. Reduction and separation ofšvariables are performed.

Inšthis paper wešconsider the control ofšašdynamically asymmetric balanced ball onšašplane inšthe case ofšslipping atšthe contact point. Necessary conditions under which ašcontrol isšpossible are obtained. Specific algorithms ofšcontrol along ašgiven trajectory are constructed.

Inšthis article aškinematic model ofšthe spherical robot isšconsidered, which isšset inšmotion byšthe internal platform with omni-wheels. Itšhas been introduced ašdescription ofšconstruction, algorithm ofštrajectory planning according tošdeveloped kinematic model, itšhas been realized experimental research for typical trajectories: moving along ašstraight line and moving along ašcircle.

This paper isšconcerned with the figures ofšequilibrium ofšašself-gravitating ideal fluid with density stratification and ašsteady-state velocity field. Asšinšthe classical setting, itšisšassumed that the figure oršits layers uniformly rotate about anšaxis fixed inšspace. Asšisšwell known, when there isšnošrotation, only ašball can bešašfigure ofšequilibrium.

Itšisšshown that the ellipsoid ofšrevolution (spheroid) with confocal stratification, inšwhich each layer rotates with inherent constant angular velocity, isšatšequilibrium. Expressions are obtained for the gravitational potential, change inšthe angular velocity and pressure, and the conclusion isšdrawn that the angular velocity onšthe outer surface isšthe same asšthat ofšthe Maclaurin spheroid. Wešnote that the solution found generalizes ašpreviously known solution for piecewise constant density distribution. For comparison, wešalso present ašsolution, due tošChaplygin, for ašhomothetic density stratification.

Wešconclude byšconsidering ašhomogeneous spheroid inšthe space ofšconstant positive curvature. Wešshow that inšthis case the spheroid cannot rotate asšašrigid body, since the angular velocity distribution ofšfluid particles depends onšthe distance tošthe symmetry axis.

In this paper we describe an inertiameter, which is an experimental facility for determining the inertia tensor components and the position of the center of mass of compound bodies. An algorithm for determining these dynamical properties is presented. Using the algorithm obtained, the displacement of the center of mass and the tensor of inertia are determined experimentally for a spherical robot of combined type.

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.

We discuss an embedding of a vector field for the nonholonomic Routh sphere into a subgroup of commuting Hamiltonian vector fields on six-dimensional phase space. The corresponding Poisson brackets are reduced to the canonical Poisson brackets on the Lie algebra $e^*$(3). It allows us to relate the nonholonomic Routh system with the Hamiltonian system on a cotangent bundle to the sphere with a canonical Poisson structure.

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.

We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.

We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.

We consider the problem of rolling of a ball with an ellipsoidal cavity filled with an ideal fluid, which executes a uniform vortex motion, on an absolutely rough plane. We point out the case of existence of an invariant measure and show that there is a particular case of integrability under conditions of axial symmetry.

A new integrable system describing the rolling of a rigid body with a spherical cavity on a spherical base is considered. Previously the authors found the separation of variables for this system on the zero level set of a linear (in angular velocity) first integral, whereas in the general case it is not possible to separate the variables. In this paper we show that the foliation into invariant tori in this problem is equivalent to the corresponding foliation in the Clebsch integrable system in rigid body dynamics (for which no real separation of variables has been found either). In particular, a fixed point of focus type is possible for this system, which can serve as a topological obstacle to the real separation of variables.

This paper addresses a class of problems associated with the conditions for exact integrability of systems of ordinary differential equations expressed in terms of the properties of tensor invariants. The general theorem of integrability of the system of $n$ differential equations is proved, which admits $n?2$ independent symmetry fields and an invariant volume $n$-form (integral invariant). General results are applied to the study of steady motions of a continuum with infinite conductivity.

In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body?s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior.

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.

An investigation of the characteristics of motion of a rigid body with variable internal mass distribution in a viscous fluid is carried out on the basis of a joint numerical solution of the Navier–Stokes equations and equations of motion for a rigid body. A nonstationary three-dimensional solution to the problem is found. The motion of a sphere and a drop-shaped body in a viscous fluid in a gravitational field, which is caused by the motion of internal material points, is explored. The possibility of self-propulsion of a body in an arbitrary given direction is shown.

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.

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.

This paper develops the theory ofšthe reducing multiplier for ašspecial class ofšnonholonomic dynamical systems, when the resulting nonlinear Poisson structure isšreduced tošthe Lie?Poisson bracket ofšthe algebra $e(3)$. Asšanšillustration, the Chaplygin ball rolling problem and the Veselova system are considered. Inšaddition, anšintegrable gyrostatic generalization ofšthe Veselova system isšobtained.

Inšthis paper wešconsider the dynamics ofšrigid body whose sharp edge isšinšcontact with ašrough plane. The body can move sošthat its contact point does not move oršslips oršloses touch with the support. Inšthis paper, the dynamics ofšthe system isšconsidered within three mechanical models that describe different modes ofšmotion. The boundaries ofšdefinition range ofšeach model are given, the possibility ofštransitions from one mode tošanother and their consistency with different coefficients ofšfriction onšthe horizontal and inclined surfaces isšdiscussed.

Inšthis paper wešinvestigate two systems consisting ofšašspherical shell rolling onšašplane without slipping and ašmoving rigid body fixed inside the shell byšmeans ofštwo different mechanisms. Inšthe former case the rigid body isšfixed atšthe center ofšthe ball onšašspherical hinge. Wešshow anšisomorphism between the equations ofšmotion for the inner body with those for the ball moving onšašsmooth plane. Inšthe latter case the rigid body isšfixed byšmeans ofšthe nonholonomic hinge. The equations ofšmotion for this system have been obtained and new integrable cases found. Ašspecial feature ofšthe set ofštensor invariants ofšthis system isšthat itšleads tošthe Euler?Jacobi?Lie theorem, which isšašnew integration mechanism inšnonholonomic mechanics.

Inšthis paper wešinvestigate the dynamics ofšašbody with ašflat base sliding onšašhorizontal plane under the assumption ofšlinear pressure distribution ofšthe body onšthe plane asšthe simplest dynamically consistent friction model.

For analysis wešuse the descriptive function method similar tošthe methods used inšthe problems ofšHamiltonian dynamics with one degree ofšfreedom and allowing ašqualitative analysis ofšthe system tošbešmade without explicit integration ofšequations ofšmotion. Inšaddition, wešgive ašsystematic review ofšthe well-known experimental and theoretical results inšthis area.

This paper investigates the possibility ofšthe motion control ofšašball with ašpendulum mechanism with non-holonomic constraints using gaitsš? the simplest motions such asšacceleration and deceleration during the motion inšašstraight line, rotation through ašgiven angle and their combination. Also, the controlled motion ofšthe system along ašstraight line with ašconstant acceleration isšconsidered. For this problem the algorithm for calculating the control torques isšgiven and itšisšshown that the resulting reduced system has the first integral ofšmotion.

The paper presents experimental investigation of a homogeneous circular disk rolling on a horizontal plane. In this paper two methods of experimental determination of the loss of contact between the rolling disk and the horizontal surface before the abrupt halt are proposed. Experimental results for disks of different masses and different materials are presented. The reasons for ?micro losses? of contact with surface revealed during the rolling are discussed.

The problem ofšintegrability conditions for systems ofšdifferential equations isšdiscussed. Darboux?s classical results onšthe integrability ofšlinear non-autonomous systems with anšincomplete set ofšparticular solutions are generalized. Special attention isšpaid tošlinear Hamiltonian systems. The paper discusses the general problem ofšintegrability ofšthe systems ofšautonomous differential equations inšanš$n$-dimensional space which permit the algebra ofšsymmetry fields ofšdimension $\geqslant n$. Using ašmethod due tošLiouville, this problem isšreduced tošinvestigating the integrability conditions for Hamiltonian systems with Hamiltonians linear inšthe momentums inšphase space ofšdimension that isštwice asšlarge. Inšconclusion, the integrability ofšanšautonomous system inšthree-dimensional space with two independent non-trivial symmetry fields isšproved. Itšshould bešemphasized that nošadditional conditions are imposed onšthese fields.

Inšthis paper wešstudy ašproblem ofšrolling ofšthe dynamically asymmetric ball with displacement center ofšgravity onšašplane without slipping and vertical rotating. Itšisšshown that the dynamics ofšthe ball isšsignificantly affected byšthe type ofšreversibility. Depending onšthe type ofšthe reversibility wešfound two different types ofšdynamical chaos: strange attractors and mixed chaotic dynamics. Inšthis paper wešdescribe ašstrange attractor development, and then its basic properties. Ašset ofšcriteria byšwhich inšnumerical experiments mixed dynamics may bešdistinguished from other types ofšdynamical chaos are given.

Inšthis paper, wešinvestigate the dynamics ofšsystems describing the rolling without slipping and spinning (rubber rolling) ofšanšellipsoid onšašplane and ašsphere. Wešresearch these problems using Poincare maps, which investigation helps tošdiscover ašnew integrable case.

This paper addresses ašclass ofšproblems associated with the conditions for exact integrability ofšašsystem ofšordinary differential equations expressed inšterms ofšthe properties ofštensor invariants. The general theorem ofšintegrability ofšthe system ofš$n$šdifferential equations isšproved, which admits $n ? 2$šindependent symmetry fields and anšinvariant volume $n$-form (integral invariant). General results are applied tošthe study ofšsteady motions ofšašcontinuous medium with infinite conductivity.

The phenomenon ofšaštopological monodromy inšintegrable Hamiltonian and nonholonomic systems isšdiscussed. Anšefficient method for computing and visualizing the monodromy isšdeveloped. The comparative analysis ofšthe topological monodromy isšgiven for the rolling ellipsoid ofšrevolution problem inštwo cases, namely, onšašsmooth and onšašrough plane. The first ofšthese systems isšHamiltonian, the second isšnonholonomic. Wešshow that, from the viewpoint ofšmonodromy, there isšnošdifference between the two systems, and thus disprove the conjecture byšCushman and Duistermaat stating that the topological monodromy gives aštopological obstruction for Hamiltonization ofšthe rolling ellipsoid ofšrevolution onšašrough plane.

Inšthis paper, wešinvestigate the dynamics ofšsystems describing the rolling without slipping and spinning (rubber rolling) ofšvarious rigid bodies onšašplane and ašsphere. Itšisšshown that ašhierarchy ofšpossible types ofšdynamical behavior arises depending onšthe body?s surface geometry and mass distribution. New integrable cases and cases ofšexistence ofšanšinvariant measure are found. Inšaddition, these systems are used tošillustrate that the existence ofšseveral nontrivial involutions inšreversible dissipative systems leads tošquasi-Hamiltonian behavior.

The paper deals with deviation based control algorithm for trajectory following ofšomni-wheeled mobile robot. The kinematic model and the dynamics ofšthe robot actuators are described.

Inšour earlier paper [2] wešexamined the problem ofšcontrol ofšašbalanced dynamically nonsymmetric sphere with rotors with no-slip condition atšthe point ofšcontact. Inšthis paper wešinvestigate the controllability ofšašball inšthe presence ofšfriction. Wešalso study the problem ofšthe existence and stability ofšsingular dissipation-free periodic solutions for ašfree ball inšthe presence ofšfriction forces. The issues ofšconstructive realization ofšthe proposed algorithms are discussed.

We consider the optimal motion control problem for a mobile device with an external rigid shell moving along a prescribed trajectory in a viscous fluid. The mobile robot under consideration possesses the property of self-locomotion. Self-locomotion is implemented due to back-and-forth motion of an internal material point. The optimal motion control is based on the Sugeno fuzzy inference system. An approach based on constructing decision trees using the genetic algorithm for structural and parametric synthesis has been proposed to obtain the base of fuzzy rules.

An algorithm for the investigation of a mathematical model of the impact of a double pendulum against an obstacle is constructed and realized by computer. This algorithm allows calculation of impact loads and coefficients of restitution at the contact point and hinges. The main goal of this investigation is to minimize negative impact conditions in the hingings.

We develop a new method for solving Hamilton?s canonical differential equations. The method is based on the search for invariant vortex manifolds of special type. In the case of Lagrangian (potential) manifolds, we arrive at the classical Hamilton–Jacobi method.

In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.

We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.

The paper is devoted to the bifurcation analysis and the Conley index in Hamiltonian dynamical systems. We discuss the phenomenon of appearance (disappearance) of equilibrium points under the change of the Morse index of a critical point of a Hamiltonian. As an application of these techniques we find new relative equilibria in the problem of the motion of three point vortices of equal intensity in a circular domain.

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.

The paper considers two new integrable systems which go back to Chaplygin. The systems consist of a spherical shell that rolls on a plane; within the shell there is a ball or Lagrange?s gyroscope. All necessary first integrals and an invariant measure are found. The solutions are shown to be expressed in terms of quadratures.

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.

Invariant manifolds of equations governing the dynamics of conservative nonholonomic systems are investigated. These manifolds are assumed to be uniquely projected onto configuration space. The invariance conditions are represented in the form of generalized Lamb?s equations. Conditions are found under which the solutions to these equations admit a hydrodynamical description typical of Hamiltonian systems. As an illustration, nonholonomic systems on Lie groups with a left-invariant metric and left-invariant (right-invariant) constraints are considered.

Ašnew integrable system describing the rolling ofšašrigid body with ašspherical cavity over ašspherical base isšconsidered. Previously the authors found the separation ofšvariables for this system atšthe zero level ofšašlinear (inšangular velocity) first integral, whereas inšthe general case itšisšnot possible tošseparate the variables. Inšthis paper wešshow that the foliation into invariant tori inšthis problem isšequivalent tošthe corresponding foliation inšthe Clebsch integrable system inšrigid body dynamics (for which nošreal separation ofšvariables has been found either). Inšparticular, ašfixed point ofšfocus type isšpossible for this system, which can serve asšaštopological obstacle tošthe real separation ofšvariables.

Anšinvestigation ofšthe characteristics ofšmotion ofšašrigid body with variable internal mass distribution inšašviscous fluid isšcarried out onšthe basis ofšašjoint numerical solution ofšthe Navier?Stokes equations and equations ofšmotion. Ašnon-stationary three-dimensional solution tošthe problem isšfound. The motion ofšašsphere and ašdrop-shaped body inšašviscous fluid, which isšcaused byšthe motion ofšinternal material points, inšašgravitational field isšexplored. The possibility ofšmotion ofšašbody inšanšarbitrary given direction isšshown.

Inšthis paper wešinvestigate various kinematic properties ofšrolling ofšone rigid body onšanother both for the classical model ofšrolling without slipping (the velocities ofšbodies atšthe point ofšcontact coincide) and for the model ofšrubber-rolling (with the additional condition that the spinning ofšthe bodies relative tošeach other bešexcluded). Furthermore, inšthe case where both bodies are bounded byšspherical surfaces and one ofšthem isšfixed, the equations ofšmotion for ašmoving ball are represented inšthe form ofšthe Chaplygin system. When the center ofšmass ofšthe moving ball coincides with its geometric center, the equations ofšmotion are represented inšconformally Hamiltonian form, and inšthe case where the radii ofšthe moving and fixed spheres coincides, they are written inšHamiltonian form.

The problem ofšdynamics ofšheavy uniform ball moving onšthe fixed rough plane under its own inertia and forces ofšdry friction isšconsidered. Assuming that friction coefficient isšvariable, the switching curve for change the value ofšfriction coefficient isšconstructed. Using this curve tošchange the value ofšfriction coefficient wešhave shown that the bundle ofšequal balls starting from one interval with equal linear and angular velocities should gather atšone point.

Inšthe paper wešconsider ašsystem ofšašball that rolls without slipping onšašplane. The ball isšassumed tošbešinhomogeneous and its center ofšmass does not necessarily coincide with its geometric center. Wešhave proved that the governing equations can bešrecast into ašsystem ofšsix ODEs that admits four integrals ofšmotion. Thus, the phase space ofšthe system isšfoliated byšinvariant 2-tori; moreover, this foliation isšequivalent tošthe Liouville foliation encountered inšthe case ofšEuler ofšthe rigid body dynamics. However, the system cannot bešsolved inšterms ofšquadratures because there isšnošinvariant measure which wešproved byšfinding limit cycles.

The problem ofšašuniform straight cylinder (disc) sliding onšašhorizontal plane under the action ofšdry friction forces isšconsidered. The contact patch between the cylinder and the plane coincides with the base ofšthe cylinder. Wešconsider axisymmetric discs, i.e. wešassume that the base ofšthe cylinder isšsymmetric with respect tošthe axis lying inšthe plane ofšthe base. The focus isšonšthe qualitative properties ofšthe dynamics ofšdiscs whose circular base, triangular base and three points are inšcontact with ašrough plane.

Wešdiscuss anšembedding ofšthe vector field associated with the nonholonomic Routh sphere inšsubgroup ofšthe commuting Hamiltonian vector fields associated with this system. Wešprove that the corresponding Poisson brackets are reduced tošcanonical ones inšthe region without ofšhomoclinic trajectories.

Wešdevelop ašnew method for solving Hamilton?s canonical differential equations. The method isšbased onšthe search ofšinvariant vortex manifolds ofšspecial type. Inšthe case ofšLagrangian (potential) manifolds, wešarrive atšthe classical Hamilton?Jacobi method.

Wešstudy chaotic dynamics ofšašnonholonomic model ofšceltic stone movement onšthe plane. Scenarious ofšappearance and development ofšchaos are investigated.

Inšthe paper wešstudy control ofšašbalanced dynamically nonsymmetric sphere with rotors. The no-slip condition atšthe point ofšcontact isšassumed. The algebraic contrability isšshown and the control inputs providing motion ofšthe ball along ašgiven trajectory onšthe plane are found. For some simple trajectories explicit tracking algorithms are proposed.

Wešconsider the problem ofšthe motion ofšaxisymmetric vortex rings inšanšideal incompressible fluid. Using the topological approach, wešpresent ašmethod for complete qualitative analysis ofšthe dynamics ofšašsystem ofštwo vortex rings. Inšparticular, wešcompletely solve the problem ofšdescribing the conditions for the onset ofšleapfrogging motion ofšvortex rings. Inšaddition, for the system ofštwo vortex rings wešfind new families ofšmotions inšwhich the mutual distances remain finite (wešcall them pseudo-leapfrogging). Wešalso find solutions for the problem ofšthree vortex rings, which describe both the regular and chaotic leapfrogging motion ofšvortex rings.

Wešconsider the problem ofšrolling ofšašball with anšellipsoidal cavity filled with anšideal fluid, which executes ašuniform vortex motion, onšanšabsolutely rough plane. Wešpoint out the case ofšexistence ofšanšinvariant measure and show that there isšašparticular case ofšintegrability under conditions ofšaxial symmetry.

Invariant manifolds ofšequations governing the dynamics ofšconservative nonholonomic systems are investigated. These manifolds are assumed tošbešuniquely projected onto configuration space. The invariance conditions are represented inšthe form ofšgeneralized Lamb?s equations. Conditions are found under which the solutions tošthese equations admit ašhydrodynamical description typical ofšHamiltonian systems. Asšanšillustration, nonholonomic systems onšLie groups with ašleft-invariant metric and left-invariant (right-invariant) constraints are considered.

In this paper we investigate various kinematic properties of rolling of one rigid body on another both for the classical model of rolling without slipping (the velocities of bodies at the point of contact coincide) and for the model of rubber-rolling (with the additional condition that the spinning of the bodies relative to each other be excluded). Furthermore, in the case where both bodies are bounded by spherical surfaces and one of them is fixed, the equations of motion for a moving ball are represented in the form of the Chaplygin system. When the center of mass of the moving ball coincides with its geometric center, the equations of motion are represented in conformally Hamiltonian form, and in the case where the radii of the moving and fixed spheres coincides, they are written in Hamiltonian form.

In the paper we study the stability of a spherical shell rolling on a horizontal plane with Lagrange?s gyroscope inside. A linear stability analysis is made for the upper and lower position of a top. A bifurcation diagram of the system is constructed. The trajectories of the contact point for di?erent values of the integrals of motion are constructed and analyzed.

The classical problem about the motion of a heavy symmetric rigid body (top) with a fixed point on the horizontal plane is discussed. Due to the unilateral nature of the contact, detachments (jumps) are possible under certain conditions. We know two scenarios of detachment related to changing the sign of the normal reaction or the sign of the normal acceleration, and the mismatch of these conditions leads to a paradox. To determine the nature of paradoxes an example of the pendulum (rod) within the limitations of the real coefficient of friction was studied in detail. We showed that in the case of the first type of the paradox (detachment is impossible and contact is impossible) the body begins to slide on the support. In the case of the paradox of the second type (detachment is possible and contact is possible) contact is retained up to the sign change of the normal reaction, and then at the detachment the normal acceleration is non-zero.

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.

We consider a continuum of interacting particles whose evolution is governed by the Vlasov kinetic equation. An infinite sequence of equations of motion for this medium (in the Eulerian description) is derived and its general properties are explored. An important example is a collisionless gas, which exhibits irreversible behavior. Though individual particles interact via a potential, the dynamics of the continuum bears dissipative features. Applicability of the Vlasov equations to the modeling of small-scale turbulence is discussed.

The Kac circular model is a discrete dynamical system which has the property of recurrence and reversibility. Within the framework of this model M.Kac formulated necessary conditions for irreversibility over "short" time intervals to take place and demonstrated Boltzmann?s most important exploration methods and ideas, outlining their advantages and limitations. We study the circular model within the realm of the theory of Gibbs ensembles and offer a new approach to a rigorous proof of the "zeroth" law of thermodynamics based on the analysis of weak convergence of probability distributions.

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.

The problem of Hamiltonization of nonholonomic systems, both integrable and non-integrable, is considered. This question is important in the qualitative analysis of such systems and it enables one to determine possible dynamical effects. The first part of the paper is devoted to representing integrable systems in a conformally Hamiltonian form. In the second part, the existence of a conformally Hamiltonian representation in a neighborhood of a periodic solution is proved for an arbitrary (including integrable) system preserving an invariant measure. Throughout the paper, general constructions are illustrated by examples in nonholonomic mechanics.

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.

Wešconsider ašnonholonomic model ofšthe dynamics ofšanšomni-wheel vehicle onšašplane and ašsphere. Anšelementary derivation ofšequations isšpresented, the dynamics ofšašfree system isšinvestigated, ašrelation tošcontrol problems isšshown.

The paper isšconcerned with the use ofšbifurcation analysis and the Conley index inšHamiltonian dynamical systems. Wešgive the proof ofšthe theorem onšthe appearance (disappearance) ofšfixed points inšthe case ofšthe Morse index change. New relative equilibria inšthe problem ofšthe motion ofšpoint vortices ofšequal intensity inšašcircle are found.

The structure ofšthe Lorentz force and the related analogy between electromagnetism and inertia are discussed. The problem ofšinvariant manifolds ofšthe equations ofšmotion for ašcharge inšanšelectromagnetic field and the conditions for these manifolds tošbešLagrangian are considered.

The paper considers two two-dimensional dynamical problems for anšabsolutely rigid cylinder interacting with ašdeformable flat base (the motion ofšanšabsolutely rigid disk onšašbase which inšnon-deformed condition isšašstraight line). The base isšašsufficiently stiff viscoelastic medium that creates ašnormal pressure $p(x) = kY(x)+?\dot{Y}(x)$, where $x$šisšašcoordinate onšthe straight line, $Y(x)$ isšašnormal displacement ofšthe pointš$x$, and $k$šand $?$šare elasticity and viscosity coefficients (the Kelvin?Voigt medium). Wešare also ofšthe opinion that during deformation the base generates friction forces, which are subject tošCoulomb?s law. Wešconsider the phenomenon ofšimpact that arises during anšarbitrary fall ofšthe disk onto the straight line and investigate the disk?s motion ?along the straight line? including the stages ofšsliding and rolling.

The paper considers two new integrable systems due tošChaplygin, which describe the rolling ofšašspherical shell onšašplane, with ašball oršLagrange?s gyroscope inside. All necessary first integrals and anšinvariant measure are found. The reduction tošquadratures isšgiven.

This paper has been written for ašcollection ofšV.A. Steklov?s selected works, which isšbeing prepared for publication and isšentitled ?Works onšMechanics 1902?1909: Translations from French?. The collection isšbased onšV.A. Steklov?s papers onšmechanics published inšFrench journals from 1902 toš1909.

Wešconsider the problem ofšexplicit integration and bifurcation analysis for two systems ofšnonholonomic mechanics. The first one isšthe Chaplygin?s problem onšno-slip rolling ofšašbalanced dynamically non-symmetrical ball onšašhorizontal plane. The second problem isšonšthe motion ofšrigid body inšašspherical support. Wešexplicitly integrate this problem byšgeneralizing the transformation which Chaplygin applied tošthe integration ofšthe problem ofšthe rolling ball atšašnon-zero constant ofšareas. Wešconsider the geometric interpretation ofšthis transformation from the viewpoint ofšaštrajectory isomorphism between two systems atšdifferent levels ofšthe energy integral. Generalization ofšthis transformation for the case ofšdynamics inšašspherical support allowsšus tošintegrate the equations ofšmotion explicitly inšquadratures and, inšaddition, tošindicate periodic solutions and analyze their stability. Wešalso show that adding ašgyrostat does not lead tošthe loss ofšintegrability.

This paper presents aštopological approach tošthe search and stability analysis ofšrelative equilibria ofšthree point vortices ofšequal intensities. Itšisšshown that the equations ofšmotion can bešreduced byšone degree ofšfreedom. Wešhave found two new stationary configurations (isosceles and non-symmetrical collinear) and studied their bifurcations and stability.

The Kac circular model isšašdiscrete dynamical system which has the property ofšrecurrence and reversibility. Within the framework ofšthis model M.Kac formulated necessary conditions for irreversibility over ?short? time intervals toštake place and demonstrated Boltzmann?s most important exploration methods and ideas, outlining their advantages and limitations. Wešstudy the circular model within the realm ofšthe theory ofšGibbs ensembles and offer ašnew approach tošašrigorous proof ofšthe ?zeroth? law ofšthermodynamics basing onšthe analysis ofšweak convergence ofšprobability distributions.

Statement of a problem of management of movement of a body in a viscous liquid is given. Movement bodies it is induced by moving of internal material points. On a basis the numerical decision of the equations of movement of a body and the hydrodynamic equations approximating dependencies for viscous forces are received. With application approximations the problem of optimum control of body movement dares on the set trajectory with application of hybrid genetic algorithm. Possibility of the directed movement of a body under action is established back and forth motion of an internal point. Optimum control movement direction it is carried out by motion of other internal point on circular trajectory with variable speed

We consider the inhomogeneous self-gravitating liquid spheroid with confocal stratification which rotates around the minor semiaxis and is in equilibrium. General relationships for pressure, angular velocity and gravitational potential of the spheroid with any density function are obtained. Special cases of piecewise constant and continuous density functions are analyzed.

In this paper a general topological approach is proposed for the study of stability of periodic solutions of integrable dynamical systems with two degrees of freedom. The methods developed are illustrated by examples of several integrable problems related to the classical Euler–Poisson equations, the motion of a rigid body in a fluid, and the dynamics of gaseous expanding ellipsoids. These topological methods also enable one to find non-degenerate periodic solutions of integrable systems, which is especially topical in those cases where no general solution (for example, by separation of variables) is known.

The paper is concerned with a class of problems which involves the dynamical interaction of a rigid body with point vortices on the surface of a two-dimensional sphere. The general approach to the 2D hydrodynamics is further developed. The problem of motion of a dynamically symmetric circular body interacting with a single vortex is shown to be integrable. Mass vortices on $S^2$ are introduced and the related issues (such as equations of motion, integrability, partial solutions, etc.) are discussed. This paper is a natural progression of the author?s previous research on interaction of rigid bodies and point vortices in a plane.

Wešconsider ašnovel mechanical system consisting ofštwo spherical bodies rolling over each other, which isšašnatural extension ofšthe famous Chaplygin problem ofšrolling motion ofšašball onšašplane. Inšcontrast tošthe previously explored non-holonomic systems, this one has ašhigher dimension and isšconsiderably more complicated. One remarkable property ofšour system isšthe existence ofš?clandestine? linear inšmomenta first integrals. For ašmore trivial integrable system, their counterparts were discovered byšChaplygin. Wešhave also found ašfew cases ofšintegrability.

Ašgeneralization ofšAmantons? law ofšdry friction for constrained Lagrangian systems isšformulated. Under ašchange ofšgeneralized coordinates the components ofšthe dry-friction force transform according tošthe covariant rule and the force itself satisfies the Painlev? condition. Inšparticular, the pressure ofšthe system onšašconstraint isšindependent ofšthe anisotropic-friction tensor. Such anšapproach provides anšinsight into the Painlev? dry-friction paradoxes. Asšanšexample, the general formulas for the sliding friction force and torque and the rotation friction torque onšašbody contacting with ašsurface are obtained.

Hamiltonisation problem for non-holonomic systems, both integrable and non-integrable, isšconsidered. This question isšimportant for qualitative analysis ofšsuch systems and allows one tošdetermine possible dynamical effects. The first part isšdevoted tošthe representation ofšintegrable systems inšašconformally Hamiltonian form. Inšthe second part, the existence ofšašconformally Hamiltonian representation inšašneighbourhood ofšašperiodic solution isšproved for anšarbitrary measure preserving system (including integrable). General consructions are always illustrated byšexamples from non-holonomic mechanics.

Wešconsider figures ofšequilibrium and stability ofšašliquid self-gravitating elliptic cylinder. The flow within the cylinder isšassumed tošbešdew tošanšelliptic perturbation. Ašbifurcation diagram isšplotted and conditions for steady solutions tošexist are indicated.

Inšthis paper, the system ofštwo vortices inšanšannular region isšshown tošbešintegrable inšthe sense ofšLiouville. Ašfew methods for analysis ofšthe dynamics ofšintegrable systems are discussed and these methods are then applied tošthe study ofšpossible motions ofštwo vortices ofšequal inšmagnitude intensities. Using the previously established fact ofšthe existence ofšrelative choreographies, the absolute motions ofšthe vortices are classified inšrespect tošthe corresponding regions inšthe phase portrait ofšthe reduced system.

Ašnew concept ofšdynamic advection isšintroduced. The model ofšdynamic advection deals with the motion ofšmassive particles inšaš2Dšflow ofšanšideal incompressible liquid. Unlike the standard advection problem, which isšwidely treated inšthe modern literature, our equations ofšmotion account not only for particles? kinematics, governed byšthe Euler equations, but also for their dynamics (which isšobviously neglected ifšthe mass ofšparticles isštaken tošbešzero). Ašfew simple model problems are considered.

Wešconsider ašcontinuum ofšinteracting particles whose evolution isšgoverned byšthe Vlasov kinetic equation. Anšinfinite sequence ofšequations ofšmotion for this medium (inšthe Eulerian description) isšderived and its general properties are explored. Anšimportant example isšašcollisionless gas, which exhibits irreversible behavior. Though individual particles interact via ašpotential, the dynamics ofšthe continuum bears dissipative features. Applicability ofšthe Vlasov equations tošthe modeling ofšsmall-scale turbulence isšdiscussed.

Inšthis paper wešdevelop ašnew model ofšnon-holonomic billiard that accounts for the intrinsic rotation ofšthe billiard ball. This model isšašlimit case ofšthe problem ofšrolling without slipping ofšašball without slipping over ašquadric surface. The billiards between two parallel walls and inside ašcircle are studied inšdetail. Using the three-dimensional-point-map technique, the non-integrability ofšthe non-holonomic billiard within anšellipse isšshown.

With the help ofšmathematical modelling, wešstudy the dynamics ofšmany point vortices system onšthe plane. For this system, wešconsider the following cases:
?švortex rings with outer radius $r = 1$šand variable inner radius $r_0$,
?švortex ellipses with semiaxesš$a$, $b$.
The emphasis isšonšthe analysis ofšthe asymptotic $(t ? ?)$ behavior ofšthe system and onšthe verification ofšthe stability criteria for vorticity continuous distributions.

Wešconsider the problems ofšHamiltonian representation and integrability ofšthe nonholonomic Suslov system and its generalization suggested byšS.šA.šChaplygin. These aspects are very important for understanding the dynamics and qualitative analysis ofšthe system. Inšparticular, they are related tošthe nontrivial asymptotic behaviour (i.še. tošsome scattering problem). The paper presents ašgeneral approach based onšthe study ofšthe hierarchy ofšdynamical behaviour ofšnonholonomic systems.