Reduction of Almost Poisson brackets and Hamiltonization of the Chaplygin Sphere

TL;DR

This paper introduces new almost Poisson brackets for nonholonomic systems, applies them to the Chaplygin sphere, and offers a geometric explanation for its Hamiltonization, advancing the understanding of nonholonomic mechanics.

Contribution

The authors develop novel almost Poisson brackets based on non-canonical two-forms and demonstrate their application to the Hamiltonization of the Chaplygin sphere.

Findings

New almost Poisson brackets constructed for nonholonomic systems

Application of brackets to reduce and analyze the Chaplygin sphere

Geometric explanation for the Hamiltonization of the Chaplygin sphere

Abstract

We construct different almost Poisson brackets for nonholonomic systems than those existing in the literature and study their reduction. Such brackets are built by considering non-canonical two-forms on the cotangent bundle of configuration space and then carrying out a projection onto the constraint space that encodes the Lagrange-D'Alembert principle. We justify the need for this type of brackets by working out the reduction of the celebrated Chaplygin sphere rolling problem. Our construction provides a geometric explanation of the Hamiltonization of the problem given by A. V. Borisov and I. S. Mamaev.