# Hamiltonisation, measure preservation and first integrals of the multi-dimensional rubber Routh sphere

## TL;DR

This paper extends the analysis of a rolling sphere problem to multiple dimensions, demonstrating measure preservation, Hamiltonian structure, and conserved quantities using recent geometric methods.

## Contribution

It generalizes the multi-dimensional rubber Routh sphere problem and establishes Hamiltonian structure and first integrals using Chaplygin's reducing multiplier method.

## Key findings

- Reduced equations have an invariant measure.
- System can be expressed in Hamiltonian form.
- Existence of conserved quantities for the system.

## Abstract

We consider the multi-dimensional generalisation of the problem of a sphere, with axi-symmetric mass distribution, that rolls without slipping or spinning over a plane. Using recent results from Garc\'ia-Naranjo (arXiv: 1805:06393) and Garc\'ia-Naranjo and Marrero (arXiv: 1812.01422), we show that the reduced equations of motion possess an invariant measure and may be represented in Hamiltonian form by Chaplygin's reducing multiplier method. We also prove a general result on the existence of first integrals for certain Hamiltonisable Chaplygin systems with internal symmetries that is used to determine conserved quantities of the problem.