Regular and Chaotic Dynamics

, Volume 17, Issue 6, pp 571–579 | Cite as

Rolling of a ball without spinning on a plane: the absence of an invariant measure in a system with a complete set of integrals

  • Alexey V. Bolsinov
  • Alexey V. Borisov
  • Ivan S. Mamaev


In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.


non-holonomic constraint Liouville foliation invariant torus invariant measure integrability 

MSC2010 numbers

37J60 37J35 70H45 


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Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • Alexey V. Bolsinov
    • 1
    • 2
  • Alexey V. Borisov
    • 2
  • Ivan S. Mamaev
    • 2
  1. 1.School of MathematicsLoughborough UniversityLoughborough, LeicestershireUK
  2. 2.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia

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