Generalisation of Chaplygin's Reducing Multiplier Theorem with an application to multi-dimensional nonholonomic dynamics

TL;DR

This paper extends Chaplygin's Reducing Multiplier Theorem to multi-dimensional nonholonomic systems, providing conditions for Hamiltonisation and applying it to a generalized rolling rigid body problem.

Contribution

It introduces a geometric hypothesis for Hamiltonisation of multi-degree nonholonomic systems and demonstrates its applicability to complex multi-dimensional rolling body dynamics.

Findings

Established sufficient conditions for Hamiltonisation in multi-dimensional systems.

Proved Hamiltonisation of a generalized multi-dimensional rolling rigid body.

Provided a systematic approach to verify geometric hypotheses in concrete examples.

Abstract

A generalisation of Chaplygin's Reducing Multiplier Theorem is given by providing sufficient conditions for the Hamiltonisation of Chaplygin nonholonomic systems with an arbitrary number $r$ of degrees of freedom via Chaplygin's multiplier method. The crucial point in the construction is to add an hypothesis of geometric nature that controls the interplay between the kinetic energy metric and the non-integrability of the constraint distribution. Such hypothesis can be systematically examined in concrete examples, and is automatically satisfied in the case $r= 2$ encountered in the original formulation of Chaplygin's theorem. Our results are applied to prove the Hamiltonisation of a multi-dimensional generalisation of the problem of a symmetric rigid body with a flat face that rolls without slipping or spinning over a sphere.