A Discretization of the Nonholonomic Chaplygin Sphere Problem

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.