Authors: Dragović, Vladimir 
Gajić, Borislav 
Jovanović, Božidar 
Affiliations: Mechanics 
Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Spherical and Planar Ball Bearings — Nonholonomic Systems with Invariant Measures
Journal: Regular and Chaotic Dynamics
First page: 424
Last page: 442
Issue Date: 1-Jul-2022
Rank: ~M22
ISSN: 1560-3547
DOI: 10.1134/S1560354722040037
Abstract: 
We first construct nonholonomic systems of n homogeneous balls (Formula presented.) with centers (Formula presented.) and with the same radius r that are rolling without slipping around a fixed sphere S0 with center O and radius R. In addition, it is assumed that a dynamically nonsymmetric sphere S of radius R + 2r and the center that coincides with the center O of the fixed sphere S0 rolls without slipping over the moving balls B1, ..., Bn. We prove that these systems possess an invariant measure. As the second task, we consider the limit, when the radius R tends to infinity. We obtain a corresponding planar problem consisting of n homogeneous balls B1, ..., Bn with centers B1, ..., Bn and the same radius r that are rolling without slippingover a fixed plane Σ0, and a moving plane Σ that moves without slipping over the homogeneous balls. We prove that this system possesses an invariant measure and that it is integrable in quadratures according to the Euler – Jacobi theorem.
Keywords: integrability | invariant measure | nonholonimic dynamics | rolling without slipping; Mathematical Physics; Mathematical Physics; Mathematics - Dynamical Systems; Mathematics - Mathematical Physics; Nonlinear Sciences - Exactly Solvable and Integrable Systems; 37J60, 37J35, 70E40, 70F25
Publisher: Springer Link
Project: Integrability and Extremal Problems in Mechanics, Geometry and Combinatorics

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