Theoretical and Mathematical Physics

, Volume 136, Issue 3, pp 1325–1335 | Cite as

Evolution of Measures in the Phase Space of Nonlinear Hamiltonian Systems

  • V. V. Kozlov
  • D. V. Treshchev


We establish the existence of weak limits of solutions (in the class Lp, p%thinsp;≥ 1) of the Liouville equation for nondegenerate quasihomogeneous Hamilton equations. We find the limit probability distributions in the configuration space. We give conditions for a uniform distribution of Gibbs ensembles for geodesic flows on compact manifolds.

quasihomogeneous Hamiltonian system geodesic flow weak limit Gibbs ensemble uniform distribution 


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  1. 1.
    M. A. Ackoglu, Canad. J. Math., 27, 1075–1082 (1975).Google Scholar
  2. 2.
    A. Ionescu-Tulcea, Bull. Amer. Math. Soc., 70, 366–371 (1964).Google Scholar
  3. 3.
    V. V. Kozlov, Dokl. Rossiiskoi Akad. Nauk, 382, 602–605 (2002).Google Scholar
  4. 4.
    V. V. Kozlov and D. V. Treshchev, Theor. Math. Phys., 134, 339–350 (2003).Google Scholar
  5. 5.
    H. Poincaré, J. Phys. Théor. Appl. (4th Ser.), 5, 369–403 (1906).Google Scholar
  6. 6.
    V. V. Kozlov, Regul. Chaotic Dyn., 6, 235–251 (2001).Google Scholar
  7. 7.
    V. V. Kozlov and D. V. Treshchev, Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts [in Russian], MSU Publ., Moscow (1991); English transl.: (Transl. Math. Monographs, Vol. 89), Am. Math. Soc., Providence, R. I. (1991).Google Scholar
  8. 8.
    Ya. G. Sinai, Russ. Math. Surv., 25, 137–189 (1970).Google Scholar
  9. 9.
    D. Szász, Sci. Math. Hungarica, 31, No. 1-3, 299–322 (1996).Google Scholar
  10. 10.
    U. Krengel, Ergodic Theorems, Gruyter, Berlin (1985).Google Scholar
  11. 11.
    V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], Gostekhizdat, Moscow (1949); English transl.: (Princeton Math. Series), Princeton Univ. Press, Princeton, N. J. (1960).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • V. V. Kozlov
    • 1
  • D. V. Treshchev
    • 1
  1. 1.Moscow State UniversityMoscowRussia

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