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Nilpotent flows of S1-invariant Hamiltonian systems on 4-dimensional symplectic manifolds

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Abstract

We investigate S1-invariant Hamiltonian systems on compact 4-dimensional symplectic manifolds with free symplectic action of a circle. We show that, in a rather general case, such systems generate ergodic flows of types (quasiperiodic and nilpotent) on their isoenergetic surfaces. We solve the problem of straightening of these flows.

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References

  1. N. M. Krylov, and N. N. Bogolyubov, Application of the Methods of Nonlinear Mechanics to the Theory of Stationary Oscillations [in Russian], Ukrainian Academy of Science, Kiev (1934).

    Google Scholar 

  2. A. N. Kolmogorov, “On dynamical systems with integral invariant on a torus,” Dokl. AN SSSR, 93, No. 4, 763–766 (1953).

    MATH  MathSciNet  Google Scholar 

  3. A. N. Kolmogorov, “On a preservation of conditionally-periodic motions under a small change of the Hamilton function”, Dokl. AN SSSR, 98, No. 4, 527–530 (1954).

    MATH  MathSciNet  Google Scholar 

  4. V. I. Amold, “Small denominators and the problems of stability of motion in classical and celestial mechanics,” Usp. Mat. Nauk,, 18, No. 6, 91–192 (1963).

    Google Scholar 

  5. 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 

  6. J. Moser, “Convergent senes expansions for quasi-periodic motions,” Math. Ann., 169, 136–176 (1967).

    Article  MATH  MathSciNet  Google Scholar 

  7. A. M. Samoilenko, Elements of the Mathematical Theory of Multi-Frequency Oscillations, Kluwer, Dordrecht-Boston-London (1993).

    Google Scholar 

  8. Yu. A. Mitropol’skii, “On the construction of a general solution of nonlinear differential equations by means of the method providing “accelerated” convergence,” Ukr. Mat. Zh., 16, No. 4, 475–501 (1964).

    Article  Google Scholar 

  9. Yu. A. Mitropol’skii, “Method of accelerated convergence in the problems of nonlinear mechanics,” Funkc. Ekvac., 9, No. 1-3, 27–42 (1966).

    Google Scholar 

  10. Yu. A. Mitropol’skii, “On the construction of solutions and reducibility of differential equations with quasiperiodic coefficients,” in: Third All-Union Congress on Theoretical and Applied Mechanics [in Russian]. Nauka, Moscow (1968)), pp. 212–213.

    Google Scholar 

  11. L. Auslender, L. Green, and F. Hahn, Flows on Homogeneous Spaces, Princeton University Press, Princeton (1963).

    Google Scholar 

  12. D. McDuff, “The moment map for circle action on symplectic manifold,” J. Geometr. Phys., 5, 149–161 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  13. M. Audin, “Hamiltoniens periodique sur les varietes symplectique compactes de dimension 4,” Lect. Notes Math., 1416, 1–25 (1990).

    Article  MathSciNet  Google Scholar 

  14. K. Ahara and A. Hattori, “4-Dimensional sympletics S 1-manifolds admitting moment map,” J. Fac. Sci. Univ. Tokyo, 38, No. 2, 251–298 (1991).

    MATH  MathSciNet  Google Scholar 

  15. M. Fernandes, A. Gray, and J. W. Morgan, “Compact symplectic manifolds with free circle action, and Massey products,” Mich. Math. J., 38, No. 2, 271–283 (1991).

    Article  Google Scholar 

  16. I. O. Parasyuk, “On isoenergetic surfaces of S 1-invariant Hamiltonian systems on 4-dimensional compact symplectic products,” Dokl. Akad. Nauk Ukr., Ser. A, No. 11, 13–16 (1993).

    MathSciNet  Google Scholar 

  17. S. P. Novikov, “Hamiltonian formalism and many-valued analog of Morse theory,“ Usp. Mat. Nauk, 37, No. 5, 3–49 (1982).

    Google Scholar 

  18. S. Lang, Introduction to Diophantine Approximations, Addison-Wesley, Reading (1966).

  19. V. I. Amold, Mathematical Methods of Classical Mechanicsc [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  20. B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry. Methods of the Homology Theory [in Russian], Nauka, Moscow (1984).

    Google Scholar 

  21. A. T. Fomenko and D. B. Fuks, A Course in Homotopic Topology [in Russian], Nauka, Moscow (1989).

    MATH  Google Scholar 

  22. L. A. Kaluzhnin, V. A. Vyshenskii, and Ts. O. Shub, Linear Spaces [in Russian], Vyshcha Shkola, Kiev (1971).

    Google Scholar 

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Authors and Affiliations

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

    A. M. Samoilenko

  2. National University, Kiev

    I. O. Parasyuk

Authors
  1. A. M. Samoilenko
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  2. I. O. Parasyuk
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 1, pp. 122–140, January, 1997.

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Samoilenko, A.M., Parasyuk, I.O. Nilpotent flows of S1-invariant Hamiltonian systems on 4-dimensional symplectic manifolds. Ukr Math J 49, 135–155 (1997). https://doi.org/10.1007/BF02486622

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  • DOI: https://doi.org/10.1007/BF02486622

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