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SIGMA 3 (2007), 044, 15 pages nlin.SI/0612037
https://doi.org/10.3842/SIGMA.2007.044
Contribution to the Vadim Kuznetsov Memorial Issue
A Discretization of the Nonholonomic Chaplygin Sphere Problem
Yuri N. Fedorov
Department de Matemática I, Universitat Politecnica de Catalunya, Barcelona, E-08028, Spain
Received December 13, 2006, in final form February 26, 2007; Published online March 12, 2007
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.
Key words:
nonholonomic systems; discretization; integrability.
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