Regular and Chaotic Dynamics

, Volume 21, Issue 6, pp 639–642 | Cite as

A generalization of Nekhoroshev’s theorem

On the 70th Birthday of Nikolai N. Nekhoroshev Special Memorial Issue. Part 1
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Abstract

Nekhoroshev discovered a beautiful theorem in Hamiltonian systems that includes as special cases not only the Poincaré theorem on periodic orbits but also the theorem of Liouville–Arnol’d on completely integrable systems [7]. Sadly, his early death precluded him publishing a full account of his proof. The aim of this paper is twofold: first, to provide a complete proof of his original theorem and second a generalization to the noncommuting case. Our generalization of Nekhoroshev’s theorem to the nonabelian case subsumes aspects of the theory of noncommutative complete integrability as found in Mishchenko and Fomenko [5] and is similar to what Nekhoroshev’s theorem does in the abelian case.

Keywords

periodic orbits Hamiltonian systems 

MSC2010 numbers

53D50 81S10 

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References

  1. 1.
    Abraham, R. and Marsden, J.E., Foundations of Mechanics, 2nd ed., Reading, Mass.: Benjamin/ Cummings, 1978.MATHGoogle Scholar
  2. 2.
    Bambusi, D. and Gaeta, G., On the Persistence of Invariant Tori and a Theorem of Nekhoroshev, Math. Phys. Electron. J., 2002, vol. 8, Paper No. 1, 13 pp.MathSciNetMATHGoogle Scholar
  3. 3.
    Cushman, R. H. and Bates, L.M., Global Aspects of Classical Integrable Systems, 2nd ed., Basel: Birkhäuser, 2015.CrossRefMATHGoogle Scholar
  4. 4.
    Duflo, M. and Vergne, M., Une propriété de la représentation coadjointe d’une algèbre de Lie, C. R. Acad. Sci. Paris, 1969, vol. 268, pp. 583–585.MathSciNetMATHGoogle Scholar
  5. 5.
    Mishchenko, A. S. and Fomenko, A. T., Generalized Liouville Method of Integration of Hamiltonian Systems, Func. Anal. Appl., 1978, vol. 12, no. 2, pp. 113–121; see also: Funktsional. Anal. i Prilozhen., 1978, vol. 12, no. 2, pp. 46–56.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Nekhoroshev, N. N., Action–Angle Variables and Their Generalization, Trans. Moscow Math. Soc., 1972, vol. 26, pp. 180–198; see also: Tr. Mosk. Mat. Obs., 1972, vol. 26, pp. 181–198.MATHGoogle Scholar
  7. 7.
    Nekhoroshev, N. N., The Poincaré–Lyapunov–Liouville–Arnold Theorem, Funct. Anal. Appl., 1994, vol. 28, no. 2, pp. 128–129; see also: Funktsional. Anal. i Prilozhen., 1994, vol. 28, no. 2, pp. 67–69.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

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