Regular and Chaotic Dynamics

, Volume 17, Issue 2, pp 131–141 | Cite as

On invariant manifolds of nonholonomic systems

Article

Abstract

Invariant manifolds of equations governing the dynamics of conservative nonholonomic systems are investigated. These manifolds are assumed to be uniquely projected onto configuration space. The invariance conditions are represented in the form of generalized Lamb’s equations. Conditions are found under which the solutions to these equations admit a hydrodynamical description typical of Hamiltonian systems. As an illustration, nonholonomic systems on Lie groups with a left-invariant metric and left-invariant (right-invariant) constraints are considered.

Keywords

invariant manifold Lamb’s equation vortex manifold Bernoulli’s theorem Helmholtz’ theorem 

MSC2010 numbers

70Hxx 37J60 

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References

  1. 1.
    Arzhanykh, I. S., Momentum Fields, Tashkent: Nauka, 1965 (Russian).Google Scholar
  2. 2.
    Kozlov, V.V., General Theory of Vortices, Encyclopaedia Math. Sci., vol. 67, Berlin-Heidelberg: Springer, 2003.MATHGoogle Scholar
  3. 3.
    Kozlov, V.V., On Invariant Manifolds of Hamilton’s Equations, J. Appl. Math. Mech., 2012, in press. (Russian).Google Scholar
  4. 4.
    Cartan, É. J., Leçons sur les invariants intégraux, Paris: Hermann et fils, 1922.MATHGoogle Scholar
  5. 5.
    Godbillon, C., Géométrie différentielle et méchanique analytique, Paris: Hermann, 1969.MATHGoogle Scholar
  6. 6.
    Kozlov, V.V., Notes on Steady Vortex Motions of Continuous Medium, Prikl. Mat. Mekh., 1983, vol. 47, no. 2, pp. 341–342 [J. Appl. Math. Mech., 1983, vol. 47, no. 2, pp. 288–289].MathSciNetGoogle Scholar
  7. 7.
    Arnold V.I. Kozlov V.V. and Neishtadt A.I. Mathematical Aspects of Classical and Celestial Mechanics Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 1993, pp. 1–291MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kozlov, V.V., On the Existence of an Integral Invariant of a Smooth Dynamic System, Prikl. Mat. Mekh., 1987, vol. 51, no. 4, pp. 538–545 [J. Appl. Math. Mech., 1987, vol. 51, no. 4, pp. 420–426].MathSciNetGoogle Scholar
  9. 9.
    Fedorov, Yu.N. and Kozlov, V.V., Various Aspects of N-dimensional Rigid Body Dynamics, Dynamical Systems in Classical Mechanics, Amer. Math. Soc. Transl. Ser. 2, vol. 168, Providence, RI: AMS, 1995, pp. 141–171.Google Scholar
  10. 10.
    Suslov, G. K., Theoretical Mechanics, Moscow-Leningrad: Gostekhizdat, 1951 (Russian).Google Scholar
  11. 11.
    Veselov, A.P. and Veselova, L. E., Flows on Lie Groups with a Nonholonomic Constraint and Integrable Non-Hamiltonian Systems, Funktsional. Anal. i Prilozhen., 1986, vol. 20, no. 4, pp. 65–66 [Funct. Anal. Appl., 1986, vol. 20, no. 5, pp. 308–309].MathSciNetGoogle Scholar
  12. 12.
    Kozlov, V.V. and Kolesnikov, N.N., On Theorems of Dynamics, Prikl. Mat. Mekh., 1978, vol. 42, no. 1, pp. 28–33 [J. Appl. Math. Mech., 1978, vol. 42, no. 1, pp. 26–31].MathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.V.A. Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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