Alexey Vladimirovich Borisov

Топология и устойчивость интегрируемых систем@@@Topology and stability of integrable systems

В работе предложен общий топологический подход к исследованию устойчивости периодических решений интегрируемых динамических систем с двумя степенями свободы. Развиваемые методы проиллюстрированы на примерах нескольких интегрируемых задач, связанных с классическими уравнениями Эйлера–Пуассона, движением твердого тела в жидкости, а также динамикой газообразных расширяющихся эллипсоидов. Данные топологические методы позволяют также отыскивать невырожденные периодические решения интегрируемых систем, что является особенно актуальным в тех случаях, когда общее решение, например, при помощи разделения переменных, неизвестно. Библиография: 82 названия.

Geometrisation of Chaplygin's reducing multiplier theorem

We develop the reducing multiplier theory for a special class of nonholonomic dynamical systems and show that the non-linear Poisson brackets naturally obtained in the framework of this approach are all isomorphic to the Lie-Poisson

Superintegrable generalizations of the Kepler and Hook problems

e(3)

Figures of equilibrium of an inhomogeneous self-gravitating fluid

-bracket. As two model examples, we consider the Chaplygin ball problem on the plane and the Veselova system. In particular, we obtain an integrable gyrostatic generalisation of the Veselova system.

Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups

In this paper we consider superintegrable systems which are an immediate generalization of the Kepler and Hook problems, both in two-dimensional spaces — the plane ℝ2 and the sphere S2 — and in three-dimensional spaces ℝ3 and S3. Using the central projection and the reduction procedure proposed in [21], we show an interrelation between the superintegrable systems found previously and show new ones. In all cases the superintegrals are presented in explicit form.

Dynamics of rolling disk

This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with a stratified density and a steady-state velocity field. As in the classical formulation of the problem, it is assumed that the figures, or their layers, uniformly rotate about an axis fixed in space. It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with a constant angular velocity, is at equilibrium. Expressions are obtained for the gravitational potential, change in the angular velocity and pressure, and the conclusion is drawn that the angular velocity on the outer surface is the same as that of the corresponding Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification. We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis.

ABSOLUTE AND RELATIVE CHOREOGRAPHIES IN THE PROBLEM OF POINT VORTICES MOVING ON A PLANE

This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector (3, 6, 14), the other is defined by two generatrices and growth vector (2, 3, 5, 8). Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.