Nonholonomic momentum map reduction and a Chaplygin-type foliation

TL;DR

This paper develops a framework for reducing nonholonomic mechanical systems with symmetries using momentum maps, resulting in a foliation of almost symplectic leaves akin to Chaplygin systems, enhancing understanding of their geometric structure.

Contribution

It extends momentum map reduction techniques for nonholonomic systems, introducing a foliation by Chaplygin-type leaves with almost symplectic structures, based on a nonholonomic momentum bundle map.

Findings

Reduced manifolds carry an almost symplectic form.

Each leaf is isomorphic to a cotangent bundle with a magnetic term.

The reduction generalizes previous constructions in nonholonomic mechanics.

Abstract

This paper presents a set-up for momentum map reduction of nonholonomic systems with symmetries, extending previous constructions in [3,25], based on the existence of certain conserved quantities and making essential use of the nonholonomic momentum bundle map of [10]. We show that the reductions of the momentum level sets carry an almost symplectic form codifying the reduced dynamics. These reduced manifolds are the leaves of the foliation associated with an almost Poisson bracket obtained by a (dynamically compatible) gauge transformation of the nonholonomic bracket. We show that each leaf is a "Chaplygin-type leaf", in the sense that it is isomorphic to a cotangent bundle with the canonical symplectic form plus a "magnetic term".