Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems

TL;DR

This paper explores how gauge transformations can be used to find Hamiltonian structures in nonholonomic systems, revealing that different almost Poisson structures can describe the same system and that twisted Poisson brackets naturally arise.

Contribution

It introduces a geometric framework using gauge transformations to identify Hamiltonizable structures in nonholonomic systems, including examples like the Chaplygin sphere.

Findings

Gauge transformations generate different almost Poisson structures for the same system.

Hamiltonization brackets can be found within gauge-related structures.

Twisted Poisson brackets naturally appear in nonholonomic mechanics.

Abstract

In this paper we study the problem of Hamiltonization of nonholonomic systems from a geometric point of view. We use gauge transformations by 2-forms (in the sense of Severa and Weinstein [29]) to construct different almost Poisson structures describing the same nonholonomic system. In the presence of symmetries, we observe that these almost Poisson structures, although gauge related, may have fundamentally different properties after reduction, and that brackets that Hamiltonize the problem may be found within this family. We illustrate this framework with the example of rigid bodies with generalized rolling constraints, including the Chaplygin sphere rolling problem. We also see how twisted Poisson brackets appear naturally in nonholonomic mechanics through these examples.