Unimodularity and preservation of volumes in nonholonomic mechanics

TL;DR

This paper investigates conditions under which nonholonomic mechanical systems preserve volume, linking invariant measures to unimodularity of an associated almost Poisson structure, and provides an algorithm for practical analysis.

Contribution

It establishes necessary and sufficient conditions for volume preservation in nonholonomic systems using unimodularity, improving upon previous results and offering a practical algorithm.

Findings

Derived criteria for invariant volume existence in nonholonomic systems.

Unified and enhanced previous conditions for volume preservation.

Applied the algorithm successfully to multiple mechanical examples.

Abstract

The equations of motion of a mechanical system subjected to nonholonomic linear constraints can be formulated in terms of a linear almost Poisson structure in a vector bundle. We study the existence of invariant measures for the system in terms of the unimodularity of this structure. In the presence of symmetries, our approach allows us to give necessary and sufficient conditions for the existence of an invariant volume, that unify and improve results existing in the literature. We present an algorithm to study the existence of a smooth invariant volume for nonholonomic mechanical systems with symmetry and we apply it to several concrete mechanical examples.