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

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.

15 pages. The third version includes minor corrections suggested by the referees of the journal "Regular and Chaotic Dynamics". The paper will be published in the special issue of RCD dedicated to S. A. Chaplygin (vol. 24, issue 5, 2019)