Canonical Perturbation Theory

  • Dino Boccaletti
  • Giuseppe Pucacco
Part of the Astronomy and Astrophysics Library book series (AAL)


To continue with the metaphor used when introducing Chapter 6, we can say that the present chapter deals with the maturity (which is extraordinarily long indeed) of perturbation theory, that is, with canonical perturbation theory. The introduction of the Hamiltonian formalism into perturbation theory occurred through two fundamental steps. The first, the introduction of a near-identity canonical transformation, has enabled one to write the equations of the perturbed motion at any order as canonical equations. The second, the introduction of the action-angle variables for the unperturbed system (which is always assumed integrable), has enabled one to identify in an n-dimensional torus the manifold on which the motion in the phase space of the representative point of a system with n-degrees of freedom is confined. It is from this that the geometrical interpretation of the perturbation has resulted. The perturbation then consists of a deformation, or even destruction, of the tori which stayed invariant when the system was not perturbed. The classical canonical theory, that is, before the KAM theory, leaves the problem of small denominators unsolved; the KAM theory, on the other hand, establishes what conditions (very restrictive ones) are required to be fulfilled in order that the small denominators may be neutralized and the convergence of the series guaranteed. The last section of this chapter finds its continuation in Chapter 11, where chaos in Hamiltonian systems is dealt with.


Hamiltonian System Rotation Number Canonical Transformation Invariant Torus Unperturbed System 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    C. G. J. Jacobi: Vorlesungen Über Dynamik, 2nd rev. edn. ( Reiner, Berlin, 1884 )Google Scholar
  2. 2.
    F. Tisserand: Traité de Mécanique Céleste, Vol. I (Paris, Gauthier-Villars, 1889 ), p. 163.Google Scholar
  3. 4.
    H. Von Zeipel: Recherches sur le mouvement des petites planètes, Ark. Mat. Astron. Fysik, Stockholm 11, no. 1, 1–58; no. 7, 1–62 (1916). Ibid. 12, no. 9, 1–89; 13, no. 3, 1–93 (1917).Google Scholar
  4. 9.
    The reader can find a “modern” demonstration of Poincaré’s theorem in: Dynamical Systems,III, ed. by V. I. Arnold, (Springer, 1988), Chap. 6. A very readable proof of a reduced form of Poincaré’s theorem is contained in the paper “Poincaré’s non-existence theorem and classical perturbation theory for nearly integrable Hamiltonian systems” by G. Benettin, L. Galgani, A. Giorgilli in Advances in Nonlinear Dynamics and Stochastic Processes ed. by R. Livi, A. Politi (World Scientific, 1985).Google Scholar
  5. 11.
    G. D. Birkhoff: Dynamical System (Am. Math. Soc., New York, 1927 ).Google Scholar
  6. 12.
    N. N. Nekhoroshev: An exponential estimate of the time stability of nearly-integrable Hamiltonian systems Russian Math. Surveys, 32, 6, 1–65 (1977).ADSzbMATHCrossRefGoogle Scholar
  7. 13.
    J. E. Littlewood: On the equilateral configuration in the restricted problem of three bodies, Proc. London Math. Soc. (3) 9, 343–372 (1959).MathSciNetCrossRefGoogle Scholar
  8. 14.
    A. N. Kolmogorov: Doklady Akad. Nauk. 98, 527 (1954). English translation in: G. Casati. J. Ford (eds): Stochastic Behavior in Classical and Quantum Hamiltonian Systems (Springer, 1979 ), pp. 51–56.Google Scholar
  9. 15.
    This version is from V. I. Arnold (ed): Dynamical Systems III (Springer, 1988 ) p. 183.Google Scholar
  10. 16.
    V. I. Arnold: Proof of A. N. Kolmogorov’s theorem on the preservation of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surv. 18, 9–36 (1963).CrossRefGoogle Scholar
  11. 17.
    J. Moser: On invariant curves of area-preserving mappings on an annulus, Nachr. Akad. Wiss. Göttingen Math. Phys. K 1, 1–20 (1962).Google Scholar
  12. 19.
    M. V. Berry: Regular and Irregular Motion, in American Institute of Physics Conference Proceedings vol. 46, ed. by S. Jorna (1978), pp. 38–42.Google Scholar
  13. 20.
    J. Moser: Near integrable and integrable systems, in American Institute of Physics Conference Proceedings,vol. 46, ed. by S. Jorna (1978), pp. 1–15Google Scholar
  14. 22.
    See: A. Ya. Khinchin: Continued Fractions (University of Chicago Press, 1964), or: C. D. Olds: Continued Fractions (Random House, 1963 ).Google Scholar
  15. 24.
    See: J. Moser: Lectures on Hamiltonian systems, Mem. Ann. Math. Soc. 81, 1–60 (1968), pp. 40 and ff.Google Scholar
  16. 26.
    J. Wisdom: The origin of the Kirkwood gaps: a mapping for asteroidal motion near the 3/1 commensurability, Astron. J. 87, 577–593 (1982);MathSciNetADSCrossRefGoogle Scholar
  17. J. Wisdom: A per-turbative treatment of motion near the 3/1 commensurability, Icarus 63, 272–289 (1985);ADSCrossRefGoogle Scholar
  18. J. Koiller, J. M. Balthazar, T. Yokoyama: Relaxation-chaos phenomena in celestial mechanics–Physica 26 D, 85–122 (1987);Google Scholar
  19. A. J. Neishtadt: Jumps in the adiabatic invariant on crossing the separatrix and the origin of the 3/1 Kirkwood gap–Soy. Phys. Dokl. 32, 571–573 (1987).ADSGoogle Scholar
  20. 29.
    For this subject, we refer the reader to A. J. Lichtenberg, M. A. Lieberman: Regular and Stochastic Motion (Springer, 1983), chap. 3.Google Scholar
  21. 30.
    H. Poincaré: Sur un théorème de géometrie, Rend. Circ. Mat. Palermo 33, 375–407 (1912);CrossRefGoogle Scholar
  22. G. D. Birkhoff: Proof of Poincaré’s geometric theorem, Trans. Am. Math. Soc. 14, 14–22 (1913).MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Dino Boccaletti
    • 1
  • Giuseppe Pucacco
    • 2
  1. 1.Dipartimento di Matematica “Guido Castelnuovo”Università degli Studi di Roma “La Sapienza”RomaItaly
  2. 2.Dipartimento di FisicaUniversità degli Studi di Roma “Tor Vergata”RomaItaly

Personalised recommendations