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, 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.

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 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.

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 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.

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.

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.

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.

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.

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 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.

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.

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.

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.

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.

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 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.

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.

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.

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.