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Variables of the action-angle type on symplectic manifolds stratified by coisotropic tori

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Abstract

A symplectic manifold is considered under the assumption that a smooth symplectic action of a commutative Lie group with compact coisotropic orbits is defined on it. The problem of existence of variables of the action-angle type is investigated with a view to giving a detailed description of flows in Hamiltonian systems with invariant Hamiltonians. We introduce the notion of a nonresonance symplectic structure for which the problem of recognition of resonance and nonresonance tori is solved.

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References

  1. V. I. Arnol'd and A. B. Givental', “Symplectic geometry“, in:VINITI Series on Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 4, VINITI, Moscow (1985), pp. 4–139.

    Google Scholar 

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

    Google Scholar 

  3. J. J. Duistermaat, “On global action-angle coordinates,”Commun. Pure Appl. Math.,33, 687–706 (1980).

    Google Scholar 

  4. I. O. Parasyuk, “Deformation of a symplectic structure and coisotropic invarianttori of Hamiltonian systems,”Mat. Fiz. Nelinein. Mekh., Issue 12, 35–37 (1989).

    Google Scholar 

  5. I. O. Parasyuk, “Coisotropic invariant tori of Hamiltonian systems in quasiclassical theory of the motion of conductivity electrons,”Ukr. Mat. Zh.,42, No. 3, 346–351 (1990).

    Google Scholar 

  6. M. V. Karasev and V. P. Maslov,Nonlinear Poisson Brackets. Geometry and Quantization [in Russian], Nauka, Moscow (1991).

    Google Scholar 

  7. Sh. Koboyasi and K. Nomidzu,Foundations of Differential Geometry [Russian translation], Vol. 1, Nauka, Moscow (1981).

    Google Scholar 

  8. V. Guillemin and S. Sternberg,Geometric Asymptotics [Russian translation], Mir, Moscow (1981).

    Google Scholar 

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Authors and Affiliations

  1. Kiev University, Kiev

    I. O. Parasyuk

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  1. I. O. Parasyuk
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Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 77–85, January, 1993.

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Parasyuk, I.O. Variables of the action-angle type on symplectic manifolds stratified by coisotropic tori. Ukr Math J 45, 85–93 (1993). https://doi.org/10.1007/BF01062041

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  • DOI: https://doi.org/10.1007/BF01062041

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