Regular and Chaotic Dynamics

, Volume 17, Issue 1, pp 54–62 | Cite as

On the accuracy of conservation of adiabatic invariants in slow-fast Hamiltonian systems



Let the adiabatic invariant of action variable in a slow-fast Hamiltonian system with two degrees of freedom have limits along the trajectories as time tends to plus and minus infinity. The difference of these two limits is exponentially small in analytic systems. An isoenergetic reduction and canonical transformations are applied to transform the slow-fast system to form of a system depending on a slowly varying parameter in a complexified phase space. On the basis of this method an estimate for the accuracy of conservation of adiabatic invariant is given.


adiabatic invariant slow-fast Hamiltonian systems isoenergetic reduction 

MSC2010 numbers

70H11 70K70 34C29 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnold, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1989.Google Scholar
  2. 2.
    Arnold, V. I., Kozlov, V.V., and Neishtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006.MATHGoogle Scholar
  3. 3.
    Neishtadt, A. I., On the Change in the Adiabatic Invariant on Crossing a Separatrix in System with Two Degrees of Freedom, Prikl. Mat. Mekh., 1987, vol. 51, no. 5, pp. 750–757 [J. Appl. Math. Mech., 1987, vol. 51, no. 5, pp. 586–592].MathSciNetGoogle Scholar
  4. 4.
    Brown, J.W. and Churchill, R. V., Complex Variables and Applications, 6th ed., New York: McGraw-Hill, 1996.Google Scholar
  5. 5.
    Su, T., On the Accuracy of Conservation of Adiabatic Invariants in Slow-Fast Systems, arXiv:1103.1595v1 [math.DS] (2011).Google Scholar
  6. 6.
    Treschev, D.V., The Method of Continuous Averaging in the Problem of Fast and Slow Motions, Regul. Chaotic Dyn., 1997, vol. 12, nos. 3–4, pp. 9–20.Google Scholar
  7. 7.
    Benettin, G., Carati, A., and Gallavotti, G., A Rigorous Implementation of the Jeans-Landau-Teller Approximation for Adiabatic Invariants, Nonlinearity, 1997, vol. 10, no. 2, pp. 479–505.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Neishtadt, A. I., On the Accuracy of Persistence of Adiabatic Invariant in Single-Frequency Systems, Regul. Chaotic Dyn., 2000, vol. 5, no. 2, pp. 213–218.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Department of Mathematical SciencesLoughborough UniversityLoughboroughUK

Personalised recommendations