Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)



Types of integrability on a submanifold and generalizations of Gordon's theorem

Author: N. N. Nekhoroshev
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 66 (2005).
Journal: Trans. Moscow Math. Soc. 2005, 169-241
MSC (2000): Primary 37J05, 70H12; Secondary 37J15, 37J35, 37J45
Published electronically: November 9, 2005
MathSciNet review: 2193433
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: At the beginning of the paper the concept of Liouville integrability is analysed for systems of general form, that is, ones that are not necessarily Hamiltonian. On this simple basis Hamiltonian systems are studied that are integrable only on submanifolds $ N$ of the phase space, which is the main subject of the paper. The study is carried out in terms of $ k$-dimensional foliations and fibrations defined on $ N$ by the Hamiltonian vector fields corresponding to $ k$ integrals in involution. These integrals are said to be central and may include the Hamiltonian function of the system. The parallel language of sets of functions is also used, in particular, sets of functions whose common level surfaces are the fibres of fibrations.

Relations between different types of integrability on submanifolds of the phase space are established. The main result of the paper is a generalization of Gordon's theorem stating that in a Hamiltonian system all of whose trajectories are closed the period of the solutions depends only on the value of the Hamiltonian. Our generalization asserts that in the case of the strongest ``Hamiltonian'' integrability the frequencies of a conditionally periodic motion on the invariant isotropic tori that form a fibration of an integrability submanifold depend only on the values of the central integrals. Under essentially weaker assumptions on the fibration of the submanifold into such tori it is proved that the circular action functions also have the same property. In addition, certain general recipes for finding the integrability submanifolds are given.

References [Enhancements On Off] (What's this?)

  • 1. V. I. Arnol'd, Mathematical methods of classical mechanics, 3rd ed., Nauka, Moscow, 1989; English transl. of 2nd ed., Springer-Verlag, New York, 1989. MR 1037020 (93c:70001)
  • 2. V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt,Mathematical aspects of classical and celestial mechanics, URSS, Moscow, 2002; English transl. of the 1985 Russian original, Springer-Verlag, Berlin, 1997. MR 1656199 (2000b:37054)
  • 3. V. I. Arnol'd and A. B. Givental', Symplectic geometry, VINITI, Moscow, 1985; English transl., Springer-Verlag, Berlin, 2001. MR 0842908 (88b:58044)
  • 4. N. N. Nekhoroshev, Two theorems on action-angle variables, Uspekhi Mat. Nauk 24 (1969), no. 5, 237-238. (Russian) MR 0261824 (41:6435)
  • 5. N. N. Nekhoroshev, Action-angle variables and their generalizations, Trudy Moscow Math. Soc. 26 (1972), 181-198. (Russian) MR 0365629 (51:1881)
  • 6. N. N. Nekhoroshev, Families of invariant isotropic tori of Hamiltonian systems, Uspekhi Mat. Nauk 40 (1985), no. 5, 217. (Russian)
  • 7. N. N. Nekhoroshev, The Poincaré-Lyapunov-Liouville-Arnol'd theorem, Funktsional. Anal. i Prilozhen. 28 (1994), no. 2, 67-69; English transl. in Funct. Anal. Appl. 28 (1994), 128-129. MR 1283258 (95e:58072)
  • 8. N. N. Nekhoroshev, Generalizations of Gordon's theorem, Proc. Symmetry and perturbation theory (Cala Gonone, Sardinia, Italy, 2001), World Sci. Publ., River Edge, NJ, 2001, pp.137-142. MR 1875487 (2002g:37002)
  • 9. N. N. Nekhoroshev, D. A. Sadovskii, and B. I. Zhilinskii, Fractional monodromy of resonant classical and quantum oscillators, C.R.Math. Acad. Sci. Paris 335 (2002), 985-988. MR 2030223 (2004j:37143)
  • 10. N. N. Nekhoroshev, Generalizations of Gordon theorem, Regul. Chaotic Dyn. 7 (2002), 239-247. MR 1931395 (2003j:37082)
  • 11. V. Guillemin and S. Sternberg, Geometric asymptotics, Amer. Math. Soc., Providence, RI, 1977. MR 0523209 (58:25636)
  • 12. W. Gordon, On the relation between period and energy in periodic dynamical systems, J. Math. Mech. 19 (1969/1970), 111-114. MR 0245930 (39:7236)
  • 13. V.V. Kozlov, Symmetries, topology and resonances in Hamiltonian mechanics, Udmurt Univ., Izhevsk, 1995; English transl., Springer-Verlag, Berlin, 1996. MR 1443434 (97j:70010b)
  • 14. M.V. Karasëv and V.P. Maslov, Nonlinear Poisson brackets. Geometry and quantization, Nauka, Moscow, 1991; English transl., Amer. Math. Soc., Providence, RI, 1993. MR 1214142 (94a:58072)
  • 15. A.S. Mishchenko and A.T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funkcional. Anal. i Prilozhen. 12 (1978), no. 2, 46-56; English transl. in Funct. Anal. Appl. 12 (1978), 113-121. MR 0516342 (58:24357)
  • 16. A.T. Fomenko, Symplectic geometry, Moscow Univ., Moscow, 1988; English transl., Gordon and Breach, Luxembourg, 1995. MR 1673400 (99k:58068)
  • 17. A.D. Bruno, The restricted $ 3$-body problem: plane periodic orbits, Nauka, Moscow, 1990; English transl., Walter de Gruyter, Berlin, 1994. MR 1301328 (95g:70007)
  • 18. G.N. Duboshin, Celestial mechanics. Analytic and qualitative methods, Nauka, Moscow, 1978. (Russian)
  • 19. R.L. Bishop and R.J. Crittenden, Geometry of manifolds, AMS Chelsea Publishing, Providence, RI, 2001. MR 1852066 (2002d:53001)
  • 20. A.V. Borisov and I.S. Mamaev, Poisson structures and Lie algebras in Hamiltonian mechanics, Udmurt. Univ., Izhevsk, 1999. (Russian) MR 1707317 (2000k:37073)
  • 21. J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Commun. Pure Appl. Math. 23 (1970), 609-636. MR 0269931 (42:4824)
  • 22. E.A. Kudryavtseva, Generalization of geometric Poincaré theorem for small perturbations, Regul. Chaotic Dyn. 3 (1998), no. 2, 46-66. MR 1693474 (2000d:37072)
  • 23. J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), no. 1, 121-130. MR 0402819 (53:6633)
  • 24. G.V. Gorr, L.V. Kudryashova, and L.A. Stepanova, Classical problems in the theory of solid bodies. Their development and current state, Naukova Dumka, Kiev, 1978. (Russian) MR 0519067 (80m:70001)
  • 25. F. Pacella, Central configurations of the $ N$-body problem via equivariant Morse theory, Arch. Rational Mech. Anal. 97 (1987), no. 1, 59-74. MR 0856309 (87k:70014)
  • 26. A. Chenciner and A. Venturelli, Minima de l'intégrale d'action du problème newtonien de $ 4$ corps de masses égales dans $ {\bf R}^3$: orbites ``hip-hop'', Celestial Mech. Dynam. Astronom. 77 (2000), no. 2, 139-152. MR 1820355 (2001k:70012)
  • 27. A. Albouy and A. Chenciner, Le problème des $ n$ corps et les distances mutuelles, Invent. Math. 131 (1998), 151-184. MR 1489897 (98m:70017)
  • 28. D. Bambusi and G. Gaeta, On persistence of invariant tori and a theorem by Nekhoroshev, Math. Phys. Electron. J. 8 (2002), Paper 1, 13 pp. (electronic). MR 1922423 (2003f:37097)
  • 29. D. Bambusi and D. Vella, Quasi periodic breathers in Hamiltonian lattices with symmetries, Discrete Contin. Dyn. Syst. Ser. B 2 (2002), 389-399. MR 1898321 (2003b:37117)
  • 30. G. Gaeta, The Poincaré-Lyapounov-Nekhoroshev theorem, Ann. Physics 297 (2002), no. 1, 157-173. MR 1900063 (2003b:37085)
  • 31. G. Gaeta, The Poincare-Nekhoroshev map, J. Nonlinear Math. Phys. 10 (2003), no. 1, 51-64. MR 1943943 (2003j:37107)

Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2000): 37J05, 70H12, 37J15, 37J35, 37J45

Retrieve articles in all journals with MSC (2000): 37J05, 70H12, 37J15, 37J35, 37J45

Additional Information

N. N. Nekhoroshev
Affiliation: Lomonosov Moscow State University, Leninskie Gory, Moscow, GSP-2, 119992, Russia

Keywords: Integrable, Hamiltonian, Gordon's theorem, integrability submanifold, conditionally periodic motion, invariant tori, vector field, integrals in involution, symplectic structure, circular action functions, frequency, trajectory, isotropic tori
Published electronically: November 9, 2005
Additional Notes: This paper was written with partial support of the INTAS grant no. 00-221 and the research was partially carried out during the author’s stay at the Littoral University, Laboratory UMR 8101 of CNRS, Dunkerque, France, and at the Milan University, Italy.
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society