The Jacobiator of nonholonomic systems and the geometry of reduced nonholonomic brackets

TL;DR

This paper investigates the geometric structure of reduced nonholonomic systems, deriving formulas for their Jacobiators, and explores conditions under which these systems admit genuine Poisson structures, with applications to classical mechanics examples.

Contribution

It provides global formulas for nonholonomic Jacobiators and clarifies their relationship with twisted Poisson structures, extending understanding of reduced brackets in nonholonomic mechanics.

Findings

Derived formulas for nonholonomic Jacobiators before and after reduction.

Identified conditions for reduced brackets to be genuine Poisson structures.

Applied results to classical mechanical examples, including Chaplygin systems.

Abstract

In this paper, we consider the hamiltonian formulation of nonholonomic systems with symmetries and study several aspects of the geometry of their reduced almost Poisson brackets, including the integrability of their characteristic distributions. Our starting point is establishing global formulas for the nonholonomic Jacobiators, before and after reduction, which are used to clarify the relationship between reduced nonholonomic brackets and twisted Poisson structures. For certain types of symmetries (generalizing the Chaplygin case), we obtain genuine Poisson structures on the reduced spaces and analyze situations in which the reduced nonholonomic brackets arise by applying a gauge transformation to these Poisson structures. We illustrate our results with mechanical examples, and in particular show how to recover several well-known facts in the special case of Chaplygin symmetries.