Nonlinear Oscillations

, Volume 8, Issue 4, pp 487–512 | Cite as

Theorem on perturbation of coisotropic invariant tori of locally Hamiltonian systems and its applications

  • Yu. V. Loveikin
  • I. O. Parasyuk
Article
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Abstract

We study the problem of perturbations of quasiperiodic motions on coisotropic invariant tori in a class of locally Hamiltonian systems. We prove a general KAM-theorem on the perturbation of coisotropic invariant tori for locally Hamiltonian systems. As applications of this theorem, we consider the motion of an electron on a two-dimensional torus under the action of an electromagnetic field and extend results concerning the bifurcation of a Cantor set of coisotropic invariant tori to the case of locally Hamiltonian systems.

Keywords

Differential Equation Partial Differential Equation Ordinary Differential Equation Electromagnetic Field Functional Equation 
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References

  1. 1.
    A. N. Kolmogorov, “On the preservation of quasiperiodic motions under a small variation in the Hamilton function,” Dokl. Akad. Nauk SSSR, 98, No. 4, 527–530 (1954).MATHMathSciNetGoogle Scholar
  2. 2.
    B. I. Arnol’d, “Proof of the Kolmogorov theorem on the preservation of quasiperiodic motions under a small variation in the Hamilton function,” Usp. Mat. Nauk, 18, Issue 5, 13–40 (1963).Google Scholar
  3. 3.
    B. I. Arnol’d, “Small denominators and problems of stability of motion in classical and celestial mechanics,” Usp. Mat. Nauk, 18, Issue 6, 91–192 (1963).Google Scholar
  4. 4.
    J. Moser, “A rapidly convergent iteration method and non-linear differential equations. I, II,” Ann. Scuola Norm. Super. Pisa, Ser. III, 20, No. 2, 265–315 (1966), No. 3, 499–535 (1966).MATHGoogle Scholar
  5. 5.
    N. N. Bogolyubov, “On quasiperiodic motions in problems of nonlinear mechanics,” in: Proceedings of the First Summer Mathematical School [in Russian], Naukova Dumka, Kiev (1964), pp. 11–101.Google Scholar
  6. 6.
    J. Moser, “Convergent series expansions for quasi-periodic motions,” Math. Ann., 1969, 136–176 (1967).MATHGoogle Scholar
  7. 7.
    N. N. Bogolyubov, Yu. A. Mitropol’skii, and A. M. Samoilenko, Method of Accelerated Convergence in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1969).Google Scholar
  8. 8.
    J. Pöschel, “Integrability of Hamiltonian systems on Cantor sets,” Commun. Pure Appl. Math., 35, No. 5, 653–695 (1982).MATHGoogle Scholar
  9. 9.
    H. W. Broer, G. B. Huitema, and M. B. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems: Order Admits Chaos, Springer, Berlin (1997).Google Scholar
  10. 10.
    H. Rüssmann, “Non-degeneracy in the perturbation theory of integrable dynamical systems,” London Math. Soc. Lect. Note Ser., No. 134, 15–18 (1989).Google Scholar
  11. 11.
    V. I. Arnol’d, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Éditorial URSS, Moscow (2002).Google Scholar
  12. 12.
    M. B. Sevryuk, “The classical KAM theory at the dawn of the 21st century,” Moscow Math. J., 3, No. 3, 1113–1144 (2003).MATHMathSciNetGoogle Scholar
  13. 13.
    R. de la Llave, Introduction to the KAM theory [Russian translation], Institute of Computer Investigations, Moscow-Izhevsk (2003).Google Scholar
  14. 14.
    I. O. Parasyuk, “On the preservation of many-dimensional invariant tori of Hamiltonian systems,” Ukr. Mat. Zh., 36, No. 4, 467–473 (1984).MATHMathSciNetGoogle Scholar
  15. 15.
    I. O. Parasyuk, “Coisotropic invariant tori of locally Hamiltonian systems,” in: Asymptotic Integration of Differential Equations [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1985), pp. 129–133.Google Scholar
  16. 16.
    I. O. Parasyuk, “On the preservation of coisotropic invariant tori of locally Hamiltonian systems,” in: Some Problems of the Theory of Asymptotic Methods in Nonlinear Mechanics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1986) pp. 150–154.Google Scholar
  17. 17.
    I. O. Parasyuk, “Coisotropic invariant tori of Hamiltonian systems in the quasiclassical theory of motion of a conduction electron,” Ukr. Mat. Zh., 42, No. 3, 346–351 (1990).MATHMathSciNetGoogle Scholar
  18. 18.
    M.-R. Herman, “Exemples de flots hamiltoniens dont aucune perturbation en topologie C ∝ n’a d’orbites périodiques sur un ouvert de surfaces d’énergies,” C. R. Acad. Sci. Paris Sér. I. Math., 312, No. 13, 989–994 (1991).MATHMathSciNetGoogle Scholar
  19. 19.
    M.-R. Herman, “Différentiabilité en contre-exemples á la fermeture en topologie C ∝ des orbites récurrentes de flots hamiltoniens,” C. R. Acad. Sci. Paris Sér. I. Math., 313, No. 1, 49–51 (1991).MATHMathSciNetGoogle Scholar
  20. 20.
    I. O. Parasyuk, “Bifurcation of a Cantor set of coisotropic invariant tori of Hamiltonian systems under perturbation of symplectic structure,” Nelin. Kolyvannya, 1, No. 2, 81–89 (1998).MathSciNetGoogle Scholar
  21. 21.
    I. O. Parasyuk, “Perturbation of degenerate coisotropic invariant tori of Hamiltonian systems,” Ukr. Mat. Zh., 50, No. 1, 72–85 (1998).MATHMathSciNetGoogle Scholar
  22. 22.
    F. Cong and Y. Li, “Existence of higher-dimensional invariant tori for Hamiltonian systems,” J. Math. Anal. Appl., 222, No. 1, 255–267 (1998).MathSciNetGoogle Scholar
  23. 23.
    A. A. Kubichka and I. O. Parasyuk, “Whitney-differentiable family of coisotropic invariant tori of a Hamiltonian system close to a degenerate system,” Visn. Kyiv Univ., Ser. Mat. Mekh., Issue. 4, 20–29 (2000).Google Scholar
  24. 24.
    A. A. Kubichka and I. O. Parasyuk, “Bifurcation of a Whitney-smooth family of coisotropic invariant tori of a Hamiltonian system under small deformations of a symplectic structure,” Ukr. Mat. Zh., 53, No. 5, 610–624 (2001).MathSciNetGoogle Scholar
  25. 25.
    O. I. Bogoyavlenskij, “Extended integrability and bi-Hamiltonian systems,” Commun. Math. Phys., 196, 19–51 (1998).CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    H. Alfven and A. Avez, Cosmical Electrodynamics. Fundamental Principles, Clarendon Press, Oxford (1963).Google Scholar
  27. 27.
    P. C. Clemow and J. P. Dougherty, Electrodynamics of Particles and Plasmas, Addison-Wesley, Reading (1969).Google Scholar
  28. 28.
    V. I. Arnol’d, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1989).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Yu. V. Loveikin
    • 1
  • I. O. Parasyuk
    • 1
  1. 1.Shevchenko Kyiv National UniversityKyiv

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