Rough Classification of Integrable Hamiltonians on Four-Dimensional Symplectic Manifolds

  • A. T. Fomenko

Abstract

This work develops, in particular, important ideas from the famous paper of S. Smale “Topology and mechanics” [1]. The theory which is the subject of the present paper gives, in particular, the answer to some questions formulated by Marsden, Ratiu, Novikov, Arnold, Weinstein, Adler, van Moerbeke, Scovel, Flaschka, Kozlov, Gray, McLaughlin, and Haine. These scientific discussions were important and the author expresses his thanks to all these specialists.

Keywords

Manifold Weinstein 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Smale S. Topology and mechanics: I, Invent. Math. 10 (1970).Google Scholar
  2. 2.
    Abraham R. and Marsden J.E., Foundations of Mechanics, 2nd ed., Benjamin/ Cummings, New York, 1978.MATHGoogle Scholar
  3. 3.
    Lerman L. M. and Umansky Ya.L., Integrable Hamiltonian systems and Poisson action, Methods of Qualitative Theory of Differential Equations, Gorjky University Press, Gorjky, 1984, pp. 126–139.Google Scholar
  4. 4.
    Marsden J., Ratiu T., Weinstein A., Semidirect products and reduction in mechanics, Trans. Amer. Math. Soc. 281 (1984), 147–178.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Fomenko A. T., The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability, Math. USSR Izvestiya 29 (1987), 629–658 (English).MATHCrossRefGoogle Scholar
  6. 6.
    Fomenko A. T., Topological invariants of Hamiltonian systems integrable in Liouville sense, Funct. Anal. Appl. 22 (4) (1988), 38–51. (See the corresponding English translation.)MathSciNetGoogle Scholar
  7. 7.
    Fomenko A. T., Symplectic topology of completely integrable Hamiltonian systems, Uspechi Matem. Nauk. 44 (1) (1989), 145–173. (See the corresponding English translation.)MathSciNetMATHGoogle Scholar
  8. 8.
    Fomenko A. T. and Zieschang H., Criterion of topological equivalence of integrable Hamiltonian systems with two degree of freedom, Izvestiya Akad. Nauk. SSSR (1990). (See the corresponding English translation.) [See also the English paper: Instituí Hautes Etudes Sci. 35 (1988), 1–45.]Google Scholar
  9. 9.
    Bolsinov A. V., Fomenko A. T., Matveev S.V., Topological classification and arrangement with respect to complexity of integrable Hamiltonian systems of differential equations with two degrees of freedom. The list of all integrable systems of low complexity. Uspechi Matem. Nauk (1990).Google Scholar
  10. 10.
    Fomenko A. T. and Zieschang H., On typical topological properties of integrable Hamiltonian systems, USSR Izv. 31 (2) (1989), 385–412. (English).MathSciNetCrossRefGoogle Scholar
  11. 11.
    Matveev S. V. and Fomenko A. T., Constant energy surfaces of Hamiltonian systems, enumeration of three-dimensional manifolds in increasing order of complexity, and computation of volumes of closed hyperbolic manifolds, Russian Math. Surv. 43(1) (1988), 3–24.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Fomenko A. T., Topological classification of all integrable Hamiltonian differential equations of general type with two degrees of freedom. The Geometry of Hamiltonian Systems, T. Ratiu, ed., Math. Sci. Res. Inst. Pub., v. 22, Springer, N.Y., 1989Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

  • A. T. Fomenko

There are no affiliations available

Personalised recommendations