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JOURNALS // Symmetry, Integrability and Geometry: Methods and Applications // Archive

SIGMA, 2007 Volume 3, 044, 15 pp. (Mi sigma170)

This article is cited in 8 papers

A Discretization of the Nonholonomic Chaplygin Sphere Problem

Yuri N. Fedorov

Department de Matemática I, Universitat Politecnica de Catalunya, Barcelona, E-08028, Spain

Abstract: The celebrated problem of a non-homogeneous sphere rolling over a horizontal plane was proved to be integrable and was reduced to quadratures by Chaplygin. Applying the formalism of variational integrators (discrete Lagrangian systems) with nonholonomic constraints and introducing suitable discrete constraints, we construct a discretization of the $n$-dimensional generalization of the Chaplygin sphere problem, which preserves the same first integrals as the continuous model, except the energy. We then study the discretization of the classical 3-dimensional problem for a class of special initial conditions, when an analog of the energy integral does exist and the corresponding map is given by an addition law on elliptic curves. The existence of the invariant measure in this case is also discussed.

Keywords: nonholonomic systems; discretization; integrability.

MSC: 37J60; 37J35; 70H45

Received: December 13, 2006; in final form February 26, 2007; Published online March 12, 2007

Language: English

DOI: 10.3842/SIGMA.2007.044



Bibliographic databases:
ArXiv: nlin.SI/0612037


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