Regular and Chaotic Dynamics

, Volume 13, Issue 6, pp 557–571 | Cite as

Chaplygin ball over a fixed sphere: an explicit integration

  • A. V. Borisov
  • Yu. N. Fedorov
  • I. S. Mamaev
Jürgen Moser - 80


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.

Key words

Chaplygin ball explicit integration nonholonomic mechanics 

MSC2000 numbers

37J60 37J35 70H45 


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Copyright information

© MAIK Nauka 2008

Authors and Affiliations

  • A. V. Borisov
    • 1
  • Yu. N. Fedorov
    • 2
  • I. S. Mamaev
    • 1
  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia
  2. 2.Department de Matemáatica Aplicada IUniversitat Politecnica de CatalunyaBarcelonaSpain

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