Regular and Chaotic Dynamics

, Volume 13, Issue 4, pp 239–249 | Cite as

Stability of steady rotations in the nonholonomic Routh problem

  • A. V. Borisov
  • A. A. Kilin
  • I. S. Mamaev
Nonholonomic Mechanics


We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, subject to gravity, on the surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady rotations of the ball at the upmost, downmost and saddle point.

Key words

nonholonomic constraint stationary rotations stability 

MSC2000 numbers

34D20 70E40 37J35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Routh, E.J., Advanced Dynamics of a System of Rigid Bodies, 6th ed., London: MacMillan Company, 1905. Reprinted by New York: Dover Publications, 1955.Google Scholar
  2. 2.
    Borisov A.V., Mamaev, I.S., and Kilin, A.A., A New Integral in the Problem of Rolling a Ball on an Arbitrary Ellipsoid Dokl.. Phys., 2002, vol. 47, no. 7, pp. 544–547 [Translated from Dokl. Akad. Nauk, 2002, vol. 385, no. 3, 2002, pp. 338–341].CrossRefMathSciNetGoogle Scholar
  3. 3.
    Mamaev, I.S., New Cases when the Invariant Measure and First Integrals Exist in the Problem of a Body Rolling on a Surface, Regul. Chaotic Dyn., 2003, vol. 8, no. 3, pp. 331–335.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Painlevé, P., Leçons sur l’intégration des equations différentielles de la Mécanique et Applications, Paris, A. Hermann, 1895, pp. 291.zbMATHGoogle Scholar
  5. 5.
    Staude, O., Ein Beitrag zur Discussion der Bewegungsleichungen eines Punktes, Math. Ann., vol. 41, no. 2, 1892, pp. 219–259.CrossRefMathSciNetGoogle Scholar
  6. 6.
    Yaroshchuk, V.A., New Cases of the Existence of an Integral Invariant in a Problem on the Rolling of a Rigid Body, Without Slippage, on a Fixed Surface, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1992, no. 6, pp. 26–30 (in Russian).Google Scholar
  7. 7.
    Routh, E.J., A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion (Adams Price Essay), London: MacMillan Company, 1877.Google Scholar
  8. 8.
    Matveev, M.V., Lyapunov Stability of Equilibrium Positions of Reversible Systems, Math. Notes, 1995, vol. 57, no. 1–2, pp. 63–72 [Translated from Mat. Zametki, 1995, vol. 57, no. 1, pp. 90–104].zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Matveyev, M.V., Reversible Systems with First Integrals, Physica D, 1998, vol. 112, pp. 148–157.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Glukhikh, Yu.D., Tkhai, V.N., Chevallier, D.P., On the Stability of Permanent Rotations of a Heavy Homogeneous Ellips oid on an Ideally Rough Plane, in Problems in the Investigation of the Stability and Stabilization of Motion, Part I, Ross. Akad. Nauk, Vychisl. Tsentr im. A. A. Dorodnitsyna, Moscow, 2000, pp. 87–104 (in Russian).Google Scholar
  11. 11.
    Moser, J.K., Lectutes on Hamiltonian Systems, Memoirs Am. Math. Soc., 1968, vol. 81, pp. 1–60.Google Scholar
  12. 12.
    Arnold, V.I., Geometrical Methods in the Theory of Ordinary Differential Equations, New York: Springer-Verlag, 1988.Google Scholar
  13. 13.
    Chaplygin, S.A., On a Paraboloid Pendulum, 1898; reprinted in: Polnoe sobranie sochinenii (Collected Works), Leningrad: Akad. Nauk SSSR, 1933, vol. 1, pp. 194–199.Google Scholar

Copyright information

© MAIK Nauka 2008

Authors and Affiliations

  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia

Personalised recommendations