Authors: Gajić, Borislav 
Jovanović, Božidar 
Title: Two integrable cases of a ball rolling over a sphere in Rn
Journal: Russian Journal of Nonlinear Dynamics
Volume: 15
Issue: 4
First page: 457
Last page: 475
Issue Date: 1-Jan-2019
ISSN: 2658-5324
DOI: 10.20537/ND190405
Abstract: 
We consider the nonholonomic problem of rolling without slipping and twisting of a balanced ball over a fixed sphere in Rn. By relating the system to a modified LR system, we prove that the problem always has an invariant measure. Moreover, this is a SO(n)-Chaplygin system that reduces to the cotangent bundle T∗Sn-1. We present two integrable cases. The first one is obtained for a special inertia operator that allows the Chaplygin Hamiltonization of the reduced system. In the second case, we consider the rigid body inertia operator Iω = Iω + ωI, I = diag(I1, . , In) with a symmetry I1 = I2 = . = Ir ≠= Ir+1 = Ir+2 = . = In. It is shown that general trajectories are quasi-periodic, while for r ≠= 1, n-1 the Chaplygin reducing multiplier method does not apply.
Keywords: Integrability | Invariant measure | Nonholonomic Chaplygin systems
Publisher: Institute of Computer Science Izhevsk
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 

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