Skip to main content
Log in

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

Regular and Chaotic Dynamics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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].

    Article  MathSciNet  Google Scholar 

  4. Bolsinov, A.V. and Fomenko, A. T., Integrable Hamiltonian Systems: Geometry, Topology and Classification, Boca Raton, FL: CRC Press, 2004.

    MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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].

    MathSciNet  MATH  Google Scholar 

  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).

    MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

  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.

  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.

  14. Kharlamov, M.P., Topological Analysis of Integrable Problems of Rigid Body Dynamics, Leningrad: Leningr. Gos. Univ., 1988.

    Google Scholar 

  15. Koiller, J. and Ehlers, K.M., Rubber Rolling Over a Sphere, Regul. Chaotic Dyn., 2007, vol. 12 no. 2, pp. 127–152.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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].

    Article  MathSciNet  Google Scholar 

  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. 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. Walsh, J. A., The Dynamics of Circle Homeomorphisms: A Hands-On Introduction, Math. Mag., 1999, vol. 72, no. 1, pp. 3–13.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alexey V. Bolsinov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bolsinov, A.V., Borisov, A.V. & Mamaev, I.S. Rolling of a ball without spinning on a plane: the absence of an invariant measure in a system with a complete set of integrals. Regul. Chaot. Dyn. 17, 571–579 (2012). https://doi.org/10.1134/S1560354712060081

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354712060081

MSC2010 numbers

Keywords

Navigation