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Regular and Chaotic Dynamics

, Volume 18, Issue 6, pp 832–859 | Cite as

The problem of drift and recurrence for the rolling Chaplygin ball

  • Alexey V. Borisov
  • Alexander A. Kilin
  • Ivan S. Mamaev
Article

Abstract

We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of the reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point.

Keywords

nonholonomic constraint absolute dynamics bifurcation diagram bifurcation complex drift resonance invariant torus 

MSC2010 numbers

70E18 37J60 70K43 37J15 37J20 

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Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • Alexey V. Borisov
    • 1
    • 2
    • 3
  • Alexander A. Kilin
    • 1
    • 2
    • 3
  • Ivan S. Mamaev
    • 1
    • 2
    • 3
  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia
  2. 2.A.A. Blagonravov Mechanical Engineering Research Institute of RASMoscowRussia
  3. 3.Institute of Mathematics and Mechanics of the Ural Branch of RASEkaterinburgRussia

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