Ivan A. Bizyaev

TOPOLOGY AND BIFURCATIONS IN NONHOLONOMIC MECHANICS

This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie–Poisson bracket of rank two. This Lie–Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.

The Chaplygin sleigh with friction moving due to periodic oscillations of an internal mass

For a Chaplygin sleigh moving in the presence of weak friction, we present and investigate two mechanisms of arising acceleration due to oscillations of an internal mass. In certain parameter regions, the mechanism induced by small oscillations determines acceleration which is on average one-directional. The role of friction is that the velocity reached in the process of the acceleration is stabilized at a certain level. The second mechanism is due to the effect of the developing oscillatory parametric instability in the motion of the sleigh. It occurs when the internal oscillating particle is comparable in mass with the main platform and the oscillations are of a sufficiently large amplitude. In the nonholonomic model the magnitude of the parametric oscillations and the level of mean energy achieved by the system turn out to be bounded if the line of the oscillations of the moving particle is displaced from the center of mass; the observed sustained motion is in many cases associated with a chaotic attractor. Then, the motion of the sleigh appears to be similar to the process of two-dimensional random walk on the plane.

The Hojman Construction and Hamiltonization of Nonholonomic Systems

In this paper, using the Hojman construction, we give examples of various Pois- son brackets which differ from those which are usually analyzed in Hamiltonian mechanics. They possess a nonmaximal rank, and in the general case an invariant measure and Casimir functions can be globally absent for them.