Existence of invariant volumes in nonholonomic systems subject to nonlinear constraints

TL;DR

This paper establishes conditions under which nonholonomic systems with nonlinear constraints preserve volume forms, linking volume invariance to geometric properties like the exactness of a 1-form and torsion integrability.

Contribution

It provides new criteria for volume preservation in nonholonomic systems with nonlinear constraints, extending to affine and geodesic flows, and relates invariance to torsion properties.

Findings

Invariant volume exists if a specific 1-form is exact.

Volume preservation relates to the vanishing of a certain function.

Torsion integrability is key for volume invariance in geodesic flows.

Abstract

We derive conditions for a nonholonomic system subject to nonlinear constraints (obeying Chetaev's rule) to preserve a smooth volume form. When applied to affine constraints, these conditions dictate that a basic invariant density exists if and only if a certain 1-form is exact and a certain function vanishes (this function automatically vanishes for linear constraints). Moreover, this result can be extended to geodesic flows for arbitrary metric connections and the sufficient condition manifests as integrability of the torsion. As a consequence, volume-preservation of a nonholonomic system is closely related to the torsion of the nonholonomic connection. Examples of nonlinear/affine/linear constraints are considered.