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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  1. 1.
    Beghin, H., Sur les conditions d’application des équations de Lagrange à un système non holonome, Bulletin de la S.M.F., 1929, vol. 57, pp. 118–124.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Hamiltonisation of Non-Holonomic Systems in the Neighborhood of Invariant Manifolds, Regul. Chaotic Dyn., 2011, vol. 16, no 5, pp. 443–464.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bolsinov, A.V., Matveev, V. S., and Fomenko, A.T., Two-Dimensional Riemannian Metrics with Integrable Geodesic Flows: Local and Global Geometry, Mat. Sb., 1998, vol. 189, no. 10, pp. 5–32 [Sb. Math., 1998, vol. 189, no. 10, pp. 1441–1466].MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bolsinov, A.V. and Fomenko, A. T., Integrable Hamiltonian Systems: Geometry, Topology and Classification, Boca Raton, FL: CRC Press, 2004.zbMATHGoogle Scholar
  5. 5.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., How To Control Chaplygin’s Sphere Using Rotors, Regul. Chaotic Dyn., 2012, vol. 17, nos. 3–4, pp. 258–272.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Borisov A.V., Mamaev I.S. The Rolling Motion of a Rigid Body on a Plane and a Sphere. Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 177–200.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Borisov, A.V. and Mamaev, I. S., Isomorphism and Hamilton Representation of Some Nonholonomic Systems, Sibirsk. Mat. Zh., 2007, vol. 48, no. 1, pp. 33–45 [Siberian Math. J., 2007, vol. 48, no. 1, pp. 26–36].MathSciNetzbMATHGoogle Scholar
  8. 8.
    Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).zbMATHGoogle Scholar
  9. 9.
    Borisov, A. V. and Mamaev, I. S., Rolling of a Non-Homogeneous Ball Over a Sphere without Slipping and Twisting, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 153–159.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Borisov, A.V. and Mamaev, I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443–490.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Borisov, A.V., Mamaev, I. S., and Treschev, D.V., Motion of a Rigid Body without Slipping and Twisting: Kinematics and Dynamics, preprint, 2012.Google Scholar
  12. 12.
    Ehlers, K.M. and Koiller, J., Rubber Rolling: Geometry and Dynamics of 2-3-5 Distributions, Proceedings of the IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence (Moscow, 25–30 August, 2006), A.V. Borisov et al. (Ed.), Springer, 2008, pp. 469–480.Google Scholar
  13. 13.
    Hadamard, J., Sur les mouvements de roulement, Mémoires de la Société des sciences physiques et naturelles de Bordeaux, 4e série, 1895, pp. 397–417.Google Scholar
  14. 14.
    Kharlamov, M.P., Topological Analysis of Integrable Problems of Rigid Body Dynamics, Leningrad: Leningr. Gos. Univ., 1988.Google Scholar
  15. 15.
    Koiller, J. and Ehlers, K.M., Rubber Rolling Over a Sphere, Regul. Chaotic Dyn., 2007, vol. 12 no. 2, pp. 127–152.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lynch, P. and Bustamante, M.D., Precession and Recession of the Rock’n’Roller, J. Phys. A: Math. Theor., 2009, vol. 42, 425203, 25 pp.Google Scholar
  17. 17.
    Nguyen Tien Zung, Polyakova, L. S., and Selivanova, E.N., Topological Classification of Integrable Geodesic Flows on Orientable Two-Dimensional Riemannian Manifolds with Additional Integral Depending on Momenta Linearly or Quadratically, Funktsional. Anal. i Prilozhen., 1993, vol. 27, no. 3, pp. 42–56 [Funct. Anal. Appl., 1993, vol. 27, no. 3, pp. 186–196].MathSciNetCrossRefGoogle Scholar
  18. 18.
    Oshemkov, A.A., Fomenko Invariants for the Main Integrable Cases of the Rigid BodyMotion Equations, in Topological Classification of Integrable Systems, A. T. Fomenko (Ed.), Adv. Soviet Math., vol. 6, Providence, R. I.: AMS, 1991, pp. 67–146.Google Scholar
  19. 19.
    Veselova, L.E., New Cases of the Integrability of the Equations of Motion of a Rigid Body in the Presence of a Nonholonomic Constraint, in Geometry, Differential Equations and Mechanics (Moscow, 1985), Moscow: Moskov. Gos. Univ., Mekh.-Mat. Fak., 1986, pp. 64–68 (Russian).Google Scholar
  20. 20.
    Walsh, J. A., The Dynamics of Circle Homeomorphisms: A Hands-On Introduction, Math. Mag., 1999, vol. 72, no. 1, pp. 3–13.MathSciNetzbMATHGoogle Scholar

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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