A Generalization of Chaplygin's Reducibility Theorem

TL;DR

This paper extends Chaplygin's Reducibility Theorem to a broader class of nonholonomic systems with symmetry, including systems without invariant measures, providing new Hamiltonization conditions and illustrating with classical examples.

Contribution

It generalizes the theorem to nonholonomic systems with arbitrary degrees of freedom and nonabelian symmetry groups, including systems lacking invariant measures.

Findings

Extended the theorem to nonholonomic systems with nonabelian symmetry groups.

Provided Hamiltonization conditions for systems without invariant measures.

Illustrated results with classical nonholonomic system examples.

Abstract

In this paper we study Chaplygin's Reducibility Theorem and extend its applicability to nonholonomic systems with symmetry described by the Hamilton-Poincare-d'Alembert equations in arbitrary degrees of freedom. As special cases we extract the extension of the Theorem to nonholonomic Chaplygin systems with nonabelian symmetry groups as well as Euler-Poincare-Suslov systems in arbitrary degrees of freedom. In the latter case, we also extend the Hamiltonization Theorem to nonholonomic systems which do not possess an invariant measure. Lastly, we extend previous work on conditionally variational systems using the results above. We illustrate the results through various examples of well-known nonholonomic systems.