# Conserved quantities and Hamiltonization of nonholonomic systems

## TL;DR

This paper develops a geometric approach to hamiltonize nonholonomic systems, using symmetries and first integrals to define a global 2-form that transforms the dynamics into an almost symplectic structure, with applications to rolling ball systems.

## Contribution

It introduces a coordinate-free method to Hamiltonize nonholonomic systems via a global 2-form, extending previous local approaches and providing new geometric insights.

## Key findings

- Explicit global 2-form for nonholonomic brackets
- Coordinate-free geometric framework for hamiltonization
- New proof of hamiltonization for rolling homogeneous ball

## Abstract

This paper studies hamiltonization of nonholonomic systems using geometric tools. By making use of symmetries and suitable first integrals of the system, we explicitly define a global 2-form for which the gauge transformed nonholonomic bracket gives rise to a new bracket on the reduced space codifying the nonholonomic dynamics and carrying an almost symplectic foliation (determined by the common level sets of the first integrals). In appropriate coordinates, this 2-form is shown to agree with the one previously introduced locally in [34]. We use our coordinate-free viewpoint to study various geometric features of the reduced brackets. We apply our formulas to obtain a new geometric proof of the hamiltonization of a homogeneous ball rolling without sliding in the interior side of a convex surface of revolution using our formulas.