Advertisement

Mathematical Notes

, Volume 58, Issue 3, pp 938–947 | Cite as

Integral invariants of the Hamilton equations

  • V. V. Kozlov
Article

Abstract

Conditions are found for the existence of integral invariants of Hamiltonian systems. For two-degrees-of-freedom systems these conditions are intimately related to the existence of nontrivial symmetry fields and multivalued integrals. Any integral invariant of a geodesic flow on an analytic surface of genus greater than 1 is shown to be a constant multiple of the Poincaré-Cartan invariant. Poincaré's conjecture that there are no additional integral invariants in the restricted three-body problem is proved.

Keywords

Hamiltonian System Analytic Surface Hamilton Equation Constant Multiple Geodesic Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Poincaré,Les Méthodes nouvelles de la mechanique céleste, Vol. 3, Gauthier-Villars, Paris (1892).Google Scholar
  2. 2.
    E. Cartan,Leçons sur les invariants intégraux, Hermann, Paris (1922).Google Scholar
  3. 3.
    C. Godbillon,Geometrie différentielle et mechanique analytique, Hermann, Paris (1969).Google Scholar
  4. 4.
    Hwa-Chung-Lee, “The universal integral invariants of Hamiltonian systems and application to the theory of canonical transformations,”Proc. Royal Soc. of Edinburgh. Sect. A.,LXII, No. 3, 237–246 (1946–48).Google Scholar
  5. 5.
    V. V. Kozlov, “The Liouville property of invariant measures in completely integrable systems and the Monge-Ampere equation,”Mat. Zametki [Math. Notes],53, No. 4, 45–52 (1993).zbMATHGoogle Scholar
  6. 6.
    V. V. Kozlov, “Dynamical systems determined by the Navier-Stokes equations,”Rus. J. Math. Phys.,1, No. 1, 57–69 (1993).zbMATHGoogle Scholar
  7. 7.
    N. N. Bogolyubov and Yu. L. Mitropol'skii,Asymptotic Methods in the Theory of Nonlinear Vibrations [in Russian], Nauka, Moscow (1974).Google Scholar
  8. 8.
    V. V. Kozlov,Qualitative Methods in Dynamics of Solids [in Russian], Izdat. Moskov. Univ., Moscow (1980).Google Scholar
  9. 9.
    C. L. Charlier,Die Mechanik des Himmels, W. de Gruyter, Berlin-Leipzig (1927).Google Scholar
  10. 10.
    D. V. Anosov and Ya. G. Sinai, “Some smooth ergodic systems,”Uspekhi Mat. Nauk [Russian Math. Surveys],22, No. 5, 107–172 (1967).MathSciNetGoogle Scholar
  11. 11.
    V. V. Kozlov, “Symmetry Groups of Dynamical Systems,”Prikl. Mat. Mekh. [J. Appl. Math. Mech.],52, No. 4, 531–541 (1988).zbMATHMathSciNetGoogle Scholar
  12. 12.
    V. V. Kozlov, “Symmetry groups of geodesic flows on closed surfaces,”Mat. Zametki [Math. Notes],48, No. 5, 62–67 (1990).zbMATHMathSciNetGoogle Scholar
  13. 13.
    V. M. Alekseev, “Final motions in the three-body problem and symbolic dynamics,”Uspekhi Mat. Nauk [Russian Math. Surveys],36, No. 4, 161–176 (1981).zbMATHMathSciNetGoogle Scholar
  14. 14.
    J. Llibre and C. Simo, “Oscillatory solutions in the planar restricted three-body problem,”Math. Ann. 248, No. 2, 153–184 (1980).MathSciNetGoogle Scholar
  15. 15.
    S. V. Bolotin and V. V. Kozlov, “Symmetry fields of geodesic flows,”Russ. J. Math. Phys. (to appear).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • V. V. Kozlov
    • 1
  1. 1.Moscow State UniversityUSSR

Personalised recommendations