Rolling of a homogeneous ball over a dynamically asymmetric sphere

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


We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of “clandestine” linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability.


nonholonomic constraint rolling motion Chaplygin ball integral invariant measure 

MSC2010 numbers



  1. 1.
    Chaplygin, S. A., On the Ball Rolling on a Horizontal Plane, in Collected works, vol. 1, pp. 76–101, Moscow-Leningrad: Gostekhizdat, 1948 (Russian).Google Scholar
  2. 2.
    Chaplygin, S. A., On some generalization of the area theorem with applications to the problem of rolling balls, in Collected works, vol. 1, pp. 26–56, Moscow-Leningrad: Gostekhizdat, 1948 (Russian).Google Scholar
  3. 3.
    Fedorov, Yu.N., The Motion of a Rigid Body in a Spherical Support, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1988, no. 5, pp. 91–93 (Russian).Google Scholar
  4. 4.
    Markeev, A.P., Integrability of the Problem of Rolling of a Sphere with a Multiply Connected Cavity Filled with an Ideal Fluid, Izv. Akad. Nauk SSSR. Mekhanika tverdogo tela, 1986, vol. 21, no 1. pp. 64–65 (Russian).Google Scholar
  5. 5.
    Borisov, A.V. and Fedorov, Y.N., On Two Modified Integrable Problems of Dynamics, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1995, no. 6, pp. 102–105 (Russian); English transl.:
  6. 6.
    Borisov, A. V., Fedorov, Yu. N., and Mamaev, I. S., Chaplygin Ball over a Fixed Sphere: An Explicit Integration, Regul. Chaotic Dyn., 2008, vol. 13, no. 6, pp. 557–571.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Borisov, A. V., Mamaev, I. S., and Marikhin, V.G., Explicit Integration of one Problem in Nonholonomic Mechanics, Dokl. Akad. Nauk, 2008, vol. 422, no. 4, pp. 475–478 [Dokl. Phys., 2008, vol. 53, no. 10, pp. 525–528].MathSciNetGoogle Scholar
  8. 8.
    Veselov, A.P. and Veselova, L.E., Integrable Nonholonomic Systems on Lie Groups, Mat. Zametki, 1988, vol. 44, no. 5, pp. 604–619 [Math. Notes, 1988, vol. 44, no. 5, pp. 810–819].MathSciNetzbMATHGoogle Scholar
  9. 9.
    Borisov, A. V., Mamaev, I. S., and Kilin, A.A., The Rolling Motion of a Ball on a Surface: New Integrals and Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 201–219.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fedorov Yu. N., Multidimensional Integrable Generalizations of the Nonholonomic Chaplygin Sphere Problem, Report, Nov. 29, 2006, 13 p. See also:
  12. 12.
    Jovanović, B., Hamiltonization and Integrability of the Chaplygin Sphere in ℝn, J. Nonlinear Sci., 2010, vol. 20, no. 5, pp. 569–593.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Kilin, A.A., The Dynamics of Chaplygin Ball: The Qualitative and Computer Analysis, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 291–306.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Duistermaat, J. J., Chaplygin’s Sphere, in Chaplygin and the Geometry of Nonholonomically Constrained Systems, R. Cushman, J. J. Duistermaat, J. Sniatycki (Eds.), Singapore: World Sci. Publ., 2010.Google Scholar
  15. 15.
    Shen, J., Schneider, D.A., and Bloch, A.M., Controllability and Motion Planning of a Multibody Chaplygin’s Sphere and Chaplygin’s Top, Int. J. Robust Nonlinear Control, 2008, vol. 18, no. 9, pp. 905–945.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Alves, J. and Dias, J., Design and Control of a Spherical Mobile Robot, J. Systems Control Engrg., 2003, vol. 217, no. 6, pp. 457–467.Google Scholar
  17. 17.
    Bhattacharya, S. and Agrawal, S.K., Design, Experiments and Motion Planning of a Spherical Rolling Robot, Proc. IEEE Intern. Conference on Robotics & Automation (San Francisco, CA, April 2000), pp. 1208–1212.Google Scholar
  18. 18.
    Crossley, V.A., A Literature Review on the Design of Spherical Rolling Robots, Pittsburgh, PA, 2006.Google Scholar
  19. 19.
    Behar, A., Matthews, J., Carsey, F., and Jones, J., NASA/JPL Tumbleweed Polar Rover, Proc. IEEE Aerospace Conference (Big Sky, MO, March 2004), pp.1–8.Google Scholar
  20. 20.
    Routh, E. J., Dynamics of a System of Rigid Bodies: 2 vols, New York: Dover, 1905.Google Scholar
  21. 21.
    Kozlov, V.V., On the Theory of Integration of the Equations of Nonholonomic Mechanics, Adv. in Mech., 1985, vol. 8, no. 3, pp. 85–107 (Russian).MathSciNetGoogle Scholar
  22. 22.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds, Regul. Chaotic Dyn., 2011, vol. 16, pp. ?–?.Google Scholar
  23. 23.
    Kozlov, V.V., Tensor Invariants of Quasihomogeneous Systems of Differential Equations, and the Asymptotic Kovalevskaya-Lyapunov Method, Mat. Zametki, 1992, vol. 51, no. 2, pp. 46–52 [Math. Notes, 1992, vol. 51, no. 2, pp. 138–142].Google Scholar
  24. 24.
    Kozlov, V.V., Symmetries and Regular Behavior of Hamiltonian Systems, Chaos, 1996, vol. 6, no. 1, pp. 1–5.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kozlov, V.V., Integral Invariants after Poincaré and Cartan, in Cartan, É., Integral Invariants, Moscow: URSS, 1998, pp. 215–260 (Russian).Google Scholar
  26. 26.
    Kozlov, V.V. and Ramodanov, S. M., Motion of a Variable Body in an Ideal Liquid, J. Appl. Math. Mech., 2001, vol. 65, no. 4, pp. 579–587.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Borisov, A. V. and Mamaev, I. S., Rolling of a Rigid Body on a Plane and Sphere: Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 177–200.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Arnol’d, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1989.zbMATHGoogle Scholar
  29. 29.
    Arnol’d, V. I., Kozlov, V.V., and Neïshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 1993.zbMATHGoogle Scholar
  30. 30.
    Kozlov, V.V., Symmetries, Topology and Resonances in Hamiltonian Mechanics, Ergeb. Math. Grenzgeb. (3), vol. 31, Berlin: Springer, 1996.Google Scholar
  31. 31.
    Borisov, A.V. and Mamaev, I. S., Modern Methods of the Theory of Integrable Systems, Moscow-Izhevsk: Institute of Computer Science, 2003 (Russian).zbMATHGoogle Scholar
  32. 32.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., On the Model of Non-Holonomic Billiard, Rus. J. Nonlin. Dyn., 2010, vol. 6, no. 2, pp. 373–385 (Russian).Google Scholar
  33. 33.
    Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Moscow-Izhevsk: Institute of Computer Science, 2005 (Russian).zbMATHGoogle Scholar
  34. 34.
    Borisov, A. V., Mamaev, I. S., and Kilin, A.A., Dynamics of Rolling Disk, Regul. Chaotic Dyn., 2003, vol. 8, no. 2, pp. 201–212.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  • Alexey V. Borisov
    • 1
  • Alexander A. Kilin
    • 1
  • Ivan S. Mamaev
    • 1
  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia

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