Skip to main content
SpringerLink
Log in

Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable Hamiltonian systems. II

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville—Arnold integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytical method for the investigation of the corresponding mapping of imbedding of an invariant torus into the phase space. This enables one to describe analytically the structure of quasiperiodic solutions of the Hamiltonian system under consideration. We also consider the problem of existence of adiabatic invariants associated with a slowly perturbed Hamiltonian system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. Arnol’d,Additional Chapters of the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  2. J. J. Duistermaat, “On global action-angles,”Comm. Pure Appl. Math.,33, 687–706 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  3. N. N. Nekhoroshev, “Action-angle variables and their generalizations,”Tr. Mosk. Mat. Obshch.,26, 181–198 (1972).

    MATH  Google Scholar 

  4. V. I. Arnol’d, V. V. Kozlov, and A. N. Nekhoroshev, “Mathematical aspects of classical and celestial mechanics,” in:VINITI Series in Contemporary Problems in Mathematics [in Russian], Vol. 3, VINITI, Moscow (1985).

    Google Scholar 

  5. V. I. Arnol’d,Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  6. C. Godbillon,Differential Geometry and Analytic Mechanics [Russian translation], Mir, Moscow (1960).

    Google Scholar 

  7. A. K. Prykarpats’kyi, M. M. Prytula, and I. V. Mykytyuk,Elements of the Theory of Differential-Geometric Structures and Dynamical Systems [in Ukrainian], UMK VO, Kiev (1988).

    Google Scholar 

  8. M. I. Kopych, Ya. A. Prykarpats’kyi, and R. V. Samulyak, “Adiabatic invariants of the generalized Henon—Heiles Hamiltonian system and the structure of the chaotic action,”Dopov. Akad. Nauk. Ukrainy, No. 2, 32–36 (1997).

    Google Scholar 

  9. N. Ercolani and E. D. Siggia, “Painleve property and geometry,”Physica D. 34, 303–346 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Blaszak and S. Wojciechowski-Rauch, “A generalized Henon-Heiles system and related integrable Newton equations,”J. Math. Phys.,35, No. 4, 1693–1709 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  11. A. M. Samoilenko and Ya. A. Prykarpats’kyi, “Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable Hamiltonian systems. I,”Ukr. Mat. Zh.,51, No. 10, 1379–1390 (1999).

    Article  Google Scholar 

  12. 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 

  13. Yu. A. Mitropol’skii,Averaging Method in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1971).

    Google Scholar 

  14. A. M. Samoilenko, “Asymptotic method for investigation ofm-frequency oscillating systems,”Ukr. Mat. Zh.,50, No. 10, 1366–1387 (1998).

    Article  MATH  Google Scholar 

  15. A. M. Samoilenko,Elements of the Mathematical Theory of Multifrequency Oscillations [in Russian], Nauka, Moscow (1987).

    Google Scholar 

  16. H. Cartan,Differential Calculus. Differential Forms [Russian translation], Mir, Moscow (1971).

    MATH  Google Scholar 

  17. M. Blaszak,Multi-Hamiltonian Theory of Dynamical Systems, Springer, New York (1998).

    MATH  Google Scholar 

  18. N. G. Chebotarev,Introduction to the Theory of Algebraic Functions [in Russian], Gostekhizdat, Moscow (1949).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    A. M. Samoilenko & Ya A. Prikarpatskii

Authors
  1. A. M. Samoilenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ya A. Prikarpatskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1513–1528, November, 1999.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Samoilenko, A.M., Prikarpatskii, Y.A. Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable Hamiltonian systems. II. Ukr Math J 51, 1713–1728 (1999). https://doi.org/10.1007/BF02525274

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02525274

Keywords

Search

Navigation