Geometrisation of Chaplygin's reducing multiplier theorem

TL;DR

This paper extends the reducing multiplier theory to certain nonholonomic systems, demonstrating their Poisson structures are isomorphic to the Lie-Poisson e(3)-bracket, with applications to the Chaplygin ball and Veselova systems.

Contribution

It develops a new theoretical framework for nonholonomic systems, showing their Poisson brackets are isomorphic to a well-known Lie-Poisson structure, and introduces an integrable gyrostatic Veselova system.

Findings

Poisson brackets are isomorphic to Lie-Poisson e(3)

Applied theory to Chaplygin ball and Veselova system

Derived an integrable gyrostatic Veselova system

Abstract

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 $e(3)$-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.