We consider a system obtained by coupling two Euler–Poinsot systems. The motivation to consider such a system can be traced back to the Riemann Ellipsoids problem. We deal with the problems of integrability and existence of region of chaotic motions.

Counting collisions in a simple dynamical system with two billiard balls can be used to estimate $\pi$ to any accuracy.

This paper is a brief survey of solving the problem of the recurrence for planar Lorentz process. There are two different ways to do this.
1. Using Lai-Sang Young's construction [27] one proves the local central limit theorem from which Pólya's theorem is then deduced — this is the method of D.Szász and T.Varjú [25].
2. Klaus Schmidt [21] and J.-P.Conze [8] proved that the recurrence of the planar Lorentz process follows from the global central limit theorem, established by Bunimovich and Sinai [7].
The history of the problem and the main ingredients of the proofs are given. The details of K.Schmidt's method are analysed in the Appendix written by V.Bognár.

We consider a restricted problem of two bodies in constant curvature spaces. The Newton and Hooke interactions between bodies are considered. For both types of interactions, we prove the non-integrability of this problem in spaces with constant non-zero curvature. Our proof is based on the Morales–Ramis theory.

In this paper, we present results of simulations of a model of the Galton board for various degrees of elasticity of the ball-to-nail collision.

A relationship between the three metrics — Billiard, Euclidean, and Lobachevskian (Hyperbolic) — is established in the article. This relationship is applied to a billiard problem on generalized diagonals of a Euclidean multidimensional convex polyhedron.

The paper studies the system of a rigid body interacting dynamically with point vortices in a perfect fluid. For arbitrary value of vortex strengths and circulation around the cylinder the system is shown to be Hamiltonian (the corresponding Poisson bracket structure is rather complicated). We also reduced the number of degrees of freedom of the system by two using the reduction by symmetry technique and performed a thorough qualitative analysis of the integrable system of a cylinder interacting with one vortex.

In the present note we introduce a full isomorphism between the problems of motion of a triaxial rigid body and motion of a particle on a triaxial ellipsoid. Using this isomorphism, we can obtain a very special case of integrability in the last problem from an integrable case of the first problem. The new case is time irreversible and is not separable in any configurational variables for arbitrary initial conditions. It can be interpreted as a motion of an electrically charged particle under potential and Lorentz forces. The result is extended to cases of motion on one-sheeted and two-sheeted hyperboloid.