A global version of the Koon-Marsden Jacobiator formula

TL;DR

This paper explores the Jacobiator in nonholonomic systems, providing a global, coordinate-free formula that enhances understanding of the Jacobi identity's failure in almost Poisson brackets.

Contribution

It introduces a global, coordinate-free formula for the Jacobiator in nonholonomic systems, extending previous local results and clarifying their geometric significance.

Findings

Global formula relates to local Koon-Marsden formula

Coordinate-free approach clarifies geometric structure

Example illustrates advantages of the global viewpoint

Abstract

In this paper we study the Jacobiator (the cyclic sum that vanishes when the Jacobi identity holds) of the almost Poisson brackets describing nonholonomic systems. We revisit the local formula for the Jacobiator established by Koon and Marsden in \cite{MarsdenKoon} using suitable local coordinates and explain how it is related to the global formula obtained in \cite{paula}, based on the choice of a complement to the constraint distribution. We use an example to illustrate the benefits of the coordinate-free viewpoint.