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# Volume 17, Number 2, 2012

 Roy A.,  Georgakarakos N. Escape Distribution for an Inclined Billiard Abstract Hénon [8] used an inclined billiard to investigate aspects of chaotic scattering which occur in satellite encounters and in other situations. His model consisted of a piecewise mapping which described the motion of a point particle bouncing elastically on two disks. A one parameter family of orbits, named $h$-orbits, was obtained by starting the particle at rest from a given height. We obtain an analytical expression for the escape distribution of the $h$-orbits, which is also compared with results from numerical simulations. Finally, some discussion is made about possible applications of the $h$-orbits in connection with Hill’s problem. Keywords: chaotic scattering, inclined billiards, Hill’s problem Citation: Roy A.,  Georgakarakos N., Escape Distribution for an Inclined Billiard, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 113-121 DOI:10.1134/S1560354712020013
 Golubtsov P. E. Typical Singularities of Polymorphisms Generated by the Problem of Destruction of an Adiabatic Invariant Abstract Polymorphisms are a class of multivalued measure-preserving self-maps of Lebesgue spaces. Specifically, polymorphisms can be used to describe the change in the adiabatic invariant due to separatrix crossing. In this case, it consists of smooth functions mapping the unit interval into itself. In addition, there are some conditions these functions must satisfy in a typical case, namely, that their endpoints form rigid structures that persist under small perturbations. Here we will describe these conditions. Keywords: adiabatic invariant, adiabatic approximation, polymorphisms, typical singularities Citation: Golubtsov P. E., Typical Singularities of Polymorphisms Generated by the Problem of Destruction of an Adiabatic Invariant, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 122-130 DOI:10.1134/S1560354712020025
 Kozlov V. V. On Invariant Manifolds of Nonholonomic Systems Abstract Invariant manifolds of equations governing the dynamics of conservative nonholonomic systems are investigated. These manifolds are assumed to be uniquely projected onto configuration space. The invariance conditions are represented in the form of generalized Lamb’s equations. Conditions are found under which the solutions to these equations admit a hydrodynamical description typical of Hamiltonian systems. As an illustration, nonholonomic systems on Lie groups with a left-invariant metric and left-invariant (right-invariant) constraints are considered. Keywords: invariant manifold, Lamb’s equation, vortex manifold, Bernoulli’s theorem, Helmholtz’ theorem Citation: Kozlov V. V., On Invariant Manifolds of Nonholonomic Systems, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 131-141 DOI:10.1134/S1560354712020037
 Markeev A. P. On a Periodic Motion of a Rigid Body Carrying a Material Point in the Presence of Impacts with a Horizontal Plane Abstract A material system consisting of an outer rigid body (a shell) and an inner body (a material point) is considered. The system moves in a uniform field of gravity over a fixed absolutely smooth horizontal plane. The central ellipsoid of inertia of the shell is an ellipsoid of rotation. The material point moves according to the harmonic law along a straight-line segment rigidly attached to the shell and lying on its axis of dynamical symmetry. During its motion, the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. The periodic motion of the shell is found when its symmetry axis is situated along a fixed vertical, and the shell rotates around this vertical with an arbitrary constant angular velocity. The conditions for existence of this periodic motion are obtained, and its linear stability is studied. Keywords: rigid body dynamics, collision, periodic motion, stability Citation: Markeev A. P., On a Periodic Motion of a Rigid Body Carrying a Material Point in the Presence of Impacts with a Horizontal Plane, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 142-149 DOI:10.1134/S1560354712020049
 Kurakin L. G. On the Stability of Thomson’s Vortex Pentagon Inside a Circular Domain Abstract We investigate the stability problem for stationary rotation of five identical point vortices located at the vertices of a regular pentagon inside a circular domain. The main result is the proof of theorems which have been announced by the author in Doklady Physics (2004, vol. 49, no. 11, pp. 658–661). Keywords: point vortices, stationary motion, stability, resonance Citation: Kurakin L. G., On the Stability of Thomson’s Vortex Pentagon Inside a Circular Domain, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 150-169 DOI:10.1134/S1560354712020050
 Borisov A. V.,  Kilin A. A.,  Mamaev I. S. Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support Abstract We discuss explicit integration and bifurcation analysis of two non-holonomic problems. One of them is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetric ball on a horizontal plane. The other, first posed by Yu.N.Fedorov, deals with the motion of a rigid body in a spherical support. For Chaplygin’s problem we consider in detail the transformation that Chaplygin used to integrate the equations when the constant of areas is zero. We revisit Chaplygin’s approach to clarify the geometry of this very important transformation, because in the original paper the transformation looks a cumbersome collection of highly non-transparent analytic manipulations. Understanding its geometry seriously facilitate the extension of the transformation to the case of a rigid body in a spherical support – the problem where almost no progress has been made since Yu.N. Fedorov posed it in 1988. In this paper we show that extending the transformation to the case of a spherical support allows us to integrate the equations of motion explicitly in terms of quadratures, detect mostly remarkable critical trajectories and study their stability, and perform an exhaustive qualitative analysis of motion. Some of the results may find their application in various technical devices and robot design. We also show that adding a gyrostat with constant angular momentum to the spherical-support system does not affect its integrability. Keywords: nonholonomic mechanics, spherical support, Chaplygin ball, explicit integration, isomorphism, bifurcation analysis Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 170-190 DOI:10.1134/S1560354712020062
 Borisov A. V.,  Mamaev I. S. Two Non-holonomic Integrable Problems Tracing Back to Chaplygin Abstract The paper considers two new integrable systems which go back to Chaplygin. The systems consist of a spherical shell that rolls on a plane; within the shell there is a ball or Lagrange’s gyroscope. All necessary first integrals and an invariant measure are found. The solutions are shown to be expressed in terms of quadratures. Keywords: non-holonomic constraint, integrability, invariant measure, gyroscope, quadrature, coupled rigid bodies Citation: Borisov A. V.,  Mamaev I. S., Two Non-holonomic Integrable Problems Tracing Back to Chaplygin, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 191-198 DOI:10.1134/S1560354712020074
 Chaplygin S. A. On Some Generalization of the Area Theorem with Applications to the Problem of Rolling Balls Abstract This publication contributes to the series of RCD translations of Sergey Alexeevich Chaplygin’s scientific heritage. Earlier we published three of his papers on non-holonomic dynamics (vol. 7, no. 2; vol. 13, no. 4) and two papers on hydrodynamics (vol. 12, nos. 1, 2). The present paper deals with mechanical systems that consist of several spheres and discusses generalized conditions for the existence of integrals of motion (linear in velocities) in such systems. First published in 1897 and awarded by the Gold Medal of Russian Academy of Sciences, this work has not lost its scientific significance and relevance. (In particular, its principal ideas are further developed and extended in the recent article "Two Non-holonomic Integrable Problems, Tracing Back to Chaplygin", published in this issue, see p. 191). Note that non-holonomic models for rolling motion of spherical shells, including the case where the shells contain intricate mechanisms inside, are currently of particular interest in the context of their application in the design of ball-shaped mobile robots. We hope that this classical work will be estimated at its true worth by the English-speaking world. Keywords: nonholonomic systems, integration Citation: Chaplygin S. A., On Some Generalization of the Area Theorem with Applications to the Problem of Rolling Balls, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 199-217 DOI:10.1134/S1560354712020086

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