We consider a continuum of interacting particles whose evolution is governed by the Vlasov kinetic equation. An infinite sequence of equations of motion for this medium (in the Eulerian description) is derived and its general properties are explored. An important example is a collisionless gas, which exhibits irreversible behavior. Though individual particles interact via a potential, the dynamics of the continuum bears dissipative features. Applicability of the Vlasov equations to the modeling of small-scale turbulence is discussed.

A new concept of dynamic advection is introduced. The model of dynamic advection deals with the motion of massive particles in a 2D flow of an ideal incompressible liquid. Unlike the standard advection problem, which is widely treated in the modern literature, our equations of motion account not only for particles’ kinematics, governed by the Euler equations, but also for their dynamics (which is obviously neglected if the mass of particles is taken to be zero). A few simple model problems are considered.

In this paper, the system of two vortices in an annular region is shown to be integrable in the sense of Liouville. A few methods for analysis of the dynamics of integrable systems are discussed and these methods are then applied to the study of possible motions of two vortices of equal in magnitude intensities. Using the previously established fact of the existence of relative choreographies, the absolute motions of the vortices are classified in respect to the corresponding regions in the phase portrait of the reduced system.

Recently, Smale horseshoes of new types, the so called half-orientable horseshoes, were found in [1]. Such horseshoes may exist for endomorphisms of the disk and for diffeomorphisms of nonorientable two-dimensional manifolds as well.They have many interesting properties different from those of the classical orientable and non-orientable horseshoes. In particular, half-orientable horseshoes may have boundary points of arbitrary periods. It is shown from this fact that there are infinitely many types of such horseshoes with respect to the local topological congugacy. To prove this and similar results, an effective geometric construction is used.

We prove the absence of an additional meromorphic first integral in the Riemann problem on the motion of a homogeneous liquid ellipsoid with zero angular and vortex momenta in the case of zero self-gravitation.

Classical systems of statistical mechanics — billiards of different geometry the boundaries of which are pertirbed in time — are considered. Dynamics of particles in such billiards and their statistical properties are described. Fermi acceleration which appears in consequence of the boundary oscillations in billiards of arbitrary shapes is investigated. Main attention is given on the analysis of Lorentz gas with stochastically oscillating scatterers and billiards in the form of stadium with periodically perturbed boundary. It is shown that as a result of Fermi acceleration, superdiffusion in the Lorentz gas takes place. It is found that if the shape of the stadium-type billiard is close to rectangular, then the boundary oscillations lead to a new phenomenon — separation of particles by their velocities, when the particle ensemble with high initial velocities is on averaged accelerated, while for particles with relatively low velocities the acceleration is not observed.

Nonlinear problem of motion of two identical pendulums connected by an elastic spring in the neighborhood of their stable vertical equilibrium is investigated. Stiffness of the spring is supposed small, i. e. the case close to resonance 1:1 is considered. The problem of existence and orbital stability of periodical motions of the pendulums arising from the equilibrium is solved. It is indicated existence of motions asymptotic to one of the periodical motions. An analysis of quasi-periodical motions of an approximate system s given in which members up to the forth order inclusively in the normalizing Hamiltonian of the problem are taken into account. Using KAM-theory the question is considered of preservation of these motions in the complete nonlinear system in which members of all orders in the series expansion of Hamiltonian in the sufficiently small neighborhood of the equilibrium are taken account.

In the first part of the article we consider a semiclassical asymptotics for a Cauchy problem for the Schrodinger operator on a metric graph. We discuss the statistical properties of the corresponding classical dynamical system: the behavior of «number of particles» at large times and distribution of «particles» on the graph. We describe the distribution of energy on infinite regular trees. In the second part we describe the asymptotics of the spectrum of the Laplace and Schrodinger operators on a thin torus and on the simplest surfaces with delta-potentials.

We discuss the polynomial bi-Hamiltonian structures for the Kowalewski top in special case of zero square integral. An explicit procedure to find variables of separation and separation relations is considered in detail.