Gauge momenta as Casimir functions of nonholonomic systems

TL;DR

This paper introduces a systematic method to modify the almost Poisson structure in symmetric nonholonomic systems, turning certain linear velocity integrals into Casimir functions, which aids in Hamiltonization and understanding equilibria.

Contribution

It provides a new approach to Hamiltonize nonholonomic systems by making specific integrals Casimir functions through structure modification.

Findings

The method explains recent Hamiltonization results.

It clarifies the role of first integrals in nonholonomic dynamics.

Implications for analyzing relative equilibria.

Abstract

We consider nonholonomic systems with symmetry possessing a certain type of first integrals that are linear in the velocities. We develop a systematic method for modifying the standard nonholonomic almost Poisson structure that describes the dynamics so that these integrals become Casimir functions after reduction. This explains a number of recent results on Hamiltonization of nonholonomic systems, and has consequences for the study of relative equilibria in such systems.