# Integrability of the $n$-dimensional axially symmetric Chaplygin sphere

## TL;DR

This paper investigates the integrability of an n-dimensional axially symmetric Chaplygin sphere, demonstrating quasi-periodic dynamics under certain conditions and proving integrability for the 4-dimensional case via symmetry reduction.

## Contribution

It establishes quasi-periodic behavior for specific initial conditions and proves integrability of the 4D system through symmetry reduction and classical integrability theorems.

## Key findings

- Dynamics are quasi-periodic for vertical angular momentum initial conditions.
- The 4D reduced system is integrable by Euler-Jacobi theorem.
- Provides new insights into high-dimensional nonholonomic integrable systems.

## Abstract

We consider the $n$-dimensional Chaplygin sphere under the assumption that the mass distribution of the sphere is axisymmetric. We prove that for initial conditions whose angular momentum about the contact point is vertical, the dynamics is quasi-periodic. For $n=4$ we perform the reduction by the associated $SO(3)$ symmetry and show that the reduced system is integrable by the Euler-Jacobi theorem.