Vladimir Vaskin

The motion without bouncing (i.e. in constant contact) of a certain model of a nonhomogeneous ball on a smooth plane is considered. The dependance of the domains of such motion on the shift of the center mass in the space of integrals of motion is analyzed.

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.

With the help of mathematical modelling, we study the dynamics of many point vortices system on the plane. For this system, we consider the following cases:
— vortex rings with outer radius $r = 1$ and variable inner radius $r_0$,
— vortex ellipses with semiaxes $a$, $b$.
The emphasis is on the analysis of the asymptotic $(t → ∞)$ behavior of the system and on the verification of the stability criteria for vorticity continuous distributions.

This study is the continuation of the computer experiment [1] with particles of gas in a one-dimensional tube, described earlier. In this paper we give investigation results for the statistical properties of a relativistic gas in a one-dimensional tube. It is shown that this system reaches the state of thermodynamical equilibrium whose distribution function is determined by the relativistic energy of particles. The system of particles in a one-dimensional tube is described by analogy with the billiards in a polygon.

We analyze domains of an axisymmetric ball with the shifted center mass motion without bouncing (i. e. in constant contact) on a smooth plane. We show that these domains belong to the region of parameters, corresponding to regimes of regular precession (the ball’s axis about axis z). We also give explicit formulas for domain boundaries.

With the help of mathematical modeling, we study the behavior of a gas ($\sim10^6$ particles) in a one-dimensional tube. For this dynamical system, we consider the following cases:
— collisionless gas (with and without gravity) in a tube with both ends closed, the particles of the gas bounce elastically between the ends,
— collisionless gas in a tube with its left end vibrating harmonically in a prescribed manner,
— collisionless gas in a tube with a moving piston, the piston’s mass is comparable to the mass of a particle.
The emphasis is on the analysis of the asymptotic ($t→∞$)) behavior of the system and specifically on the transition to the state of statistical or thermal equilibrium. This analysis allows preliminary conclusions on the nature of relaxation processes.
At the end of the paper the numerical and theoretical results obtained are discussed. It should be noted that not all the results fit well the generally accepted theories and conjectures from the standard texts and modern works on the subject.