Regular and Chaotic Dynamics

, Volume 22, Issue 2, pp 136–147 | Cite as

Classical perturbation theory and resonances in some rigid body systems

Article

Abstract

We consider the system of a rigid body in a weak gravitational field on the zero level set of the area integral and study its Poincaré sets in integrable and nonintegrable cases. For the integrable cases of Kovalevskaya and Goryachev–Chaplygin we investigate the structure of the Poincaré sets analytically and for nonintegrable cases we study these sets by means of symbolic calculations. Based on these results, we also prove the existence of periodic solutions in the perturbed nonintegrable system. The Chaplygin integrable case of Kirchhoff’s equations is also briefly considered, for which it is shown that its Poincaré sets are similar to the ones of the Kovalevskaya case.

Keywords

Poincaré method Poincaré sets resonances periodic solutions small divisors rigid body Kirchhoff’s equations 

MSC2010 numbers

70E17 70E20 70E40 

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References

  1. 1.
    Kozlov, V.V., Symmetries, Topology and Resonances in Hamiltonian Mechanics, Ergeb. Math. Grenzgeb. (3), vol. 31, Berlin: Springer, 1996.CrossRefGoogle Scholar
  2. 2.
    Poincaré, H., Les méthodes nouvelles de la mécanique céleste: Vol. 1. Solutions périodiques. Non-existence des intégrales uniformes. Solutions asymptotique, Paris: Gauthier-Villars, 1892.MATHGoogle Scholar
  3. 3.
    Born, M., The Mechanics of the Atom, New York: Ungar, 1967.Google Scholar
  4. 4.
    Kozlov, V.V. and Treshchev, D. V., On the Integrability of Hamiltonian Systems with Toral Position Space, Math. USSR-Sb., 1989, vol. 63, no. 1, pp. 121–139; see also: Mat. Sb. (N. S.), 1988, vol. 135(177), no. 1, pp. 119–138, 144.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Golubev, V. V., Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point, Jerusalem: Israel Program for Scientific Translations, 1960.Google Scholar
  6. 6.
    Kozlov, V.V., Topological Obstacles to the Integrability of Natural Mechanical Systems, Sov. Math. Dokl., 1979, vol. 20, pp. 1413–1415; see also: Dokl. Akad. Nauk SSSR, 1979, vol. 249, no. 6, pp. 1299–1302.MATHGoogle Scholar
  7. 7.
    Byalyi, M. L., First Integrals That Are Polynomial in the Momenta for a Mechanical System on the Two-Dimensional Torus, Funct. Anal. Appl., 1987, vol. 21, no. 4, pp. 310–312; see also: Funktsional. Anal. i Prilozhen., 1987, vol. 21, no. 4, pp. 64–65.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Denisova, N.V. and Kozlov, V.V., Polynomial Integrals of Reversible Mechanical Systems with a Two- Dimensional Torus As Configuration Space, Sb. Math., 2000, vol. 191, nos. 1–2, pp. 189–208; see also: Mat. Sb., 2000, vol. 191, no. 2, pp. 43–63.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Mironov, A. E., Polynomial Integrals of a Mechanical System on a Two-Dimensional Torus, Izv. Math., 2010, vol. 74, no. 4, pp. 805–817; see also: Izv. Ross. Akad. Nauk Ser. Mat., 2010, vol. 74, no. 4, pp. 145–.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Denisova, N.V., Kozlov, V.V., and Treshchev, D.V., Remarks on Polynomial Integrals of Higher Degree for Reversible Systems with a Toral Configuration Space, Izv. Math., 2012, vol. 76, no. 5, pp. 907–921; see also: Izv. Ross. Akad. Nauk Ser. Mat., 2012, vol. 76, no. 5, pp. 57–72.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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