Michael Deryabin

We study generalized relativistic billiards, which is the following dynamical system. A particle moves in the interior of a domain under the influence of some force fields. As the particle hits the boundary of the domain, its velocity is transformed as if the particle underwent an elastic collision with a moving wall. Both the motion in the domain and the reflection are considered in the framework of the theory of relativity.
We study the periodic and "monotone" action of the boundary for the particle moving in a parallelepiped and in an arbitrary compact domain respectively, and we also consider an "accelerating" model in an unbounded domain. We prove that under some general conditions an invariant manifold in the velocity phase space of the generalized billiard, where the particle velocity equals the velocity of light, either is an exponential attractor or contains one. Thus for an open set of initial conditions the particle energy tends to infinity.

We consider the non-holonomic system of a $n$-dimensional ball rolling on a $(n – 1)$-dimensional surface. We discuss the structure of the equations of motion, the existence of an invariant measure and some generalizations of the problem.

Chaplygin problem on a heavy rigid body falling in the ideal vortexless fluid resting at infinity. As it is well-known, without "initial impact", for almost all initial condition the rigid body tends to fall with its widest side ahead. The asymptotical expression for the solution of Chaplygin equation for large time t is found. It is numerically proved as well that with the "initial impact" the rigid body tends to fall with widest side ahead.