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

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 García-Naranjo (arXiv: 1805:06393) and García-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.

23 pages, 1 figure. Submitted to the special issue of Theor. Appl. Mech. in honour of Chaplygin's 150th anniversary

References in corpus (1)

• Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces

Cited by in corpus (2)

• Demchenko's nonholonomic case of gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis
• Spherical and planar ball bearings -- nonholonomic systems with invariant measures