Existence of invariant volumes in nonholonomic systems subject to nonlinear constraints

William Clark; Anthony Bloch; William Clark; Anthony Bloch

doi:10.3934/jgm.2023011

Journal of Geometric Mechanics

2023, Volume 15, Issue 1: 256-286. doi: 10.3934/jgm.2023011

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Research article

Existence of invariant volumes in nonholonomic systems subject to nonlinear constraints

William Clark ^{1
,
,},
Anthony Bloch ²

1.
Department of Mathematics, Cornell University, Ithaca, NY 14850, USA
2.
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

Received: 23 September 2020 Revised: 01 November 2022 Accepted: 23 January 2023 Published: 16 February 2023
70F25, 37C40, 70G45

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.
- geometric mechanics,
- nonholonomic systems,
- invariant volumes
Citation: William Clark, Anthony Bloch. Existence of invariant volumes in nonholonomic systems subject to nonlinear constraints[J]. Journal of Geometric Mechanics, 2023, 15(1): 256-286. doi: 10.3934/jgm.2023011

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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.

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