2006/12/31 by Yuri N. Fedorov, Yuri Fedorov
Engineering · Mathematics · Physics and Astronomy · #Applied mathematics #Classical mechanics #Computer science #Control and Dynamics of Mobile Robots #Discretization #Dynamics and Control of Mechanical Systems #Integrable system #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Nonholonomic system #Physics #Vehicle Dynamics and Control Systems #math-ph #math.DS #math.MP #nlin.SI
paper · pdf · doi:10.3842/sigma.2007.044
published as SIGMA 3 (2007), 044, 15 pages · This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/
arxiv created 2007/03/12 · openalex publication_date 2007/03/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.