Mathematical Notes

, Volume 61, Issue 2, pp 146–163 | Cite as

Exact smooth classification of hamiltonian vector fields on two-dimensional manifolds

  • B. S. Kruglikov
Article

Abstract

A complete exact classification of Hamiltonian systems with Morse Hamiltonians on two-dimensional manifolds is given, i.e., the systems are classified up to diffeomorphisms mapping vector fields into vector fields. The classification imposes no restrictions on Morse functions.

Key words

Hamiltonian vector field two-dimensional manifold germ Morse function letter-atom deformation retract 

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References

  1. 1.
    A. V. Bolsinov and A. T. Fomenko, “Trajectory equivalence of integrable Hamiltonian systems with two degrees of freedom: classification theorem. I”,Mat. Sb. [Russian Acad. Sci. Sb. Math.],185, No. 5, 27–78 (1994).Google Scholar
  2. 2.
    A. V. Bolsinov, “Smooth trajectory classification of integrable Hamiltonian systems with two degrees of freedom”,Mat. Sb. [Russian Acad. Sci. Sb. Math.],186, No. 1, 3–28 (1995).MATHMathSciNetGoogle Scholar
  3. 3.
    H. Ito, “Action-angle coordinates at singularities for analytic integrable systems”,Math. Z.,206, 363–407 (1991).MATHMathSciNetGoogle Scholar
  4. 4.
    L. H. Elliason, “Normal forms for Hamiltonian systems with Poisson commuting integrals. Elliptic case”,Comm. Math. Helv.,65, No. 1, 4–35 (1990).Google Scholar
  5. 5.
    J. Colin De Verdiére and J. Vey, “Le lemme de Morse isochore”,Topology,18, 283–293 (1979).MathSciNetGoogle Scholar
  6. 6.
    J. Moser, “On the volume elements on a manifold”,Trans. Amer. Math. Soc.,120, No. 2, 286–294 (1965).MATHMathSciNetGoogle Scholar
  7. 7.
    A. T. Fomenko,Symplectic Geometry: Methods and Applications [in Russian], Izd. Moskov. Univ., Moscow (1966).Google Scholar
  8. 8.
    A. V. Bolsinov, S. V. Matveev, and A. T. Fomenko, “Topological classification of integrable Hamiltonian systems with two degrees of freedom: a list of systems of small complexity”,Uspekhi Mat. Nauk [Russian Math. Surveys],45, No. 2(272), 59–77 (1990).MathSciNetGoogle Scholar
  9. 9.
    J.-P. Dufour, P. Molino, and A. Toulet, “Classification des systèmes intégrables en dimension 2 et invariants des modèles de Fomenko”,C. R. Acad. Sci. Paris. Sér. A,318, No. 10, 949–952 (1994).MathSciNetGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • B. S. Kruglikov
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityMoscowUSSR

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