The paper develop a new approach to the justification of Gibbs canonical distribution for Hamiltonian systems with finite number of degrees of freedom. It uses the condition of nonintegrability of the ensemble of weak interacting Hamiltonian systems.

New foundations of some aspects of statistical mechanics proposed.

In this paper we present a generalization of the Goryachev–Chaplygin integrable case on a bundle of Poisson brackets, and on Sokolov terms in his new integrable case of Kirchhoff equations. We also present a new analogous integrable case for the quaternion form of rigid body dynamics equations. This form of equations is recently developed and we can use it for the description of rigid body motions in specific force fields, and for the study of different problems of quantum mechanics. In addition we present new invariant relations in the considered problems.

The paper deals with problems of existence of periodic travelling wave solutions with non-small amplitudes of a PDE describing oscillations of an infinite beam, which lies on a non-linearly elastic support. Such solutions are in fact critical points of a functional on a suitable functional space. By means of a minimax variational technique, the authors found a domain in the parameter space for which there exist periodic travelling waves of a certain fixed period $\Sigma$.

The main directions in the development of the nonholonomic dynamics are briefly considered in this paper. The first direction is connected with the general formalizm of the equations of dynamics that differs from the Lagrangian and Hamiltonian methods of the equations of motion's construction. The second direction, substantially more important for dynamics, includes investigations concerning the analysis of the specific nonholonomic problems. We also point out rather promising direction in development of nonholonomic systems that is connected with intensive use of the modern computer-aided methods.

The motion of disks spun on tables has the well-known feature that the associated acoustic signal increases in frequency as the motion tends towards its abrupt halt. Recently, a commercial toy, known as Euler's disk, was designed to maximize the time before this abrupt ending. In this paper, we present and simulate a rigid body model for Euler's disk. Based on the nature of the contact force between the disk and the table revealed by the simulations, we conjecture a new mechanism for the abrupt halt of the disk and the increased acoustic frequency associated with the decline of the disk.

We study a nonholonomically constrained Hamiltonian system with a symmetry group which acts properly and freely on a constraint distribution. We show that the reduced dynamics is described by a generalized distributional Hamiltonian system. The general theory is illustrated by the example of Chaplygin's skate.

In this paper we study integrability of the classical Suslov problem. We prove that in a version of this problem introduced by V.V. Kozlov the problem is integrable only in one known case. We consider also a generalisation of Kozlov version and prove that the system is not integrable. Our proofs are based on the Morales–Ramis theory.

In this review we discuss methods of investigation of steady motions of nonholonomic mechanical systems. General conclusions are illustrated by examples from the rigid bodies dynamics on a absolutely rough horisontal plane.