Topology and Bifurcations in Nonholonomic Mechanics

Abstract

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.

Homogeneous systems with quadratic integrals, Lie-Poisson quasibrackets, and Kovalevskaya's method

“…Therefore there remains the open question about the types of integral manifolds and the classification of trajectories on them. Apparently, here the situation is similar to the nonholonomic system that was considered in detail in [18].…”

The Hojman Construction and Hamiltonization of Nonholonomic Systems

“…Indeed, the basic methods of Hamiltonian mechanics have been developed for canonical systems. However, as shown in [3], the representation found by us can be useful for the investigation of stability problems.…”

“…Consider a conformally Hamiltonian system in the form (3). From the point of view of integrability, the case where rank J = 2 is regarded as the simplest case.…”

Finite-dimensional integrable systems: A collection of research problems

“…In classical mechanics there are quite simple examples of Poisson brackets whose symplectic leaves are not separated by global Casimir functions which lead to interesting dynamical phenomena related to Problem 5, see [1].…”