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JOURNALS // Regular and Chaotic Dynamics // Archive

Regul. Chaotic Dyn., 2008 Volume 13, Issue 6, Pages 557–571 (Mi rcd601)

This article is cited in 51 papers

JÜRGEN MOSER – 80

Chaplygin ball over a fixed sphere: an explicit integration

A.V. Borisova, Yu.N. Fedorovb, I.S. Mamaeva

a Institute of Computer Science, Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
b Department de Matemática Aplicada I, Universitat Politecnica de Catalunya

Abstract: We consider a nonholonomic system describing the rolling of a dynamically nonsymmetric sphere over a fixed sphere without slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits a separation of variables and reduction to Abel–Jacobi quadratures. The separating variables that we found appear to be a non-trivial generalization of ellipsoidal (spheroconic) coordinates on the Poisson sphere, which can be useful in other integrable problems. Using the quadratures we also perform an explicit integration of the problem in theta-functions of the new time.

Keywords: Chaplygin ball, explicit integration, nonholonomic mechanics.

MSC: 37J60, 37J35, 70H45

Received: 21.07.2008
Accepted: 07.10.2008

Language: English

DOI: 10.1134/S1560354708060063



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