Theory of Orbits pp 125-178 | Cite as

# Lie Transform Perturbation Theory

## Abstract

Though the concept of the Lie transform dates back to more than a century ago, it is only in about the last thirty years that this concept has been introduced into perturbative theories and then applied on a vast scale in various fields of physics. As we recall in the course of this chapter, the field where the concept of the Lie transform was introduced for the first time is celestial mechanics and, incredibly, this concept is the only development of perturbation theory which cannot in some way be made to date back to Poincaré. Equally surprising is that the “old” canonical perturbation theory, in spite of the awkwardness involved by the use of a generating function with “mixed” variables, has ruled up until now, never falling into discredit, not even as a consequence of exaggerations like that of Delaunay, who had to calculate no less than 505 successive canonical transformations. We think that the record for absurdity has been set in plasma physics, where the use was established by somebody of quantizing classical systems, applying quantum perturbation theory (which provides more practical rules) and then letting the Planck constant *h* → 0 in the result. That was the situation until three decades ago. For these reasons, we thought it right to follow, in our exposition, wherever it has been possible, the chronological order in which the various contributions have appeared, at the end showing how the Lie transform method is substantially the *right* method for implementing KAM techniques. In this chapter, as in the preceding ones, we have tried to isolate what appeared to us to be the fundamental concepts and to insist on them, instead of dwelling upon the exposition of complicated examples or involved formulae for calculations. For the latter, the reader will find all the necessary information in the bibliographical notes.

## Keywords

Hamiltonian System Null Space Canonical Transformation Celestial Mechanic Homological Equation## Preview

Unable to display preview. Download preview PDF.

## References

- 1.G. Hori: Theory of general perturbations with unspecified canonical variables,
*Publ. Astron. Soc. Japan***18**, 287–296 (1966)ADSGoogle Scholar - A. Deprit: Canonical transformations depending on a small parameter,
*Celestial Mechanics***1**, 12–30 (1969).MathSciNetADSzbMATHCrossRefGoogle Scholar - 2.See, for instance, M. Hausner, J. T. Schwarz:
*Lie Groups, Lie Algebras*( Gordon and Breach, New York, 1968 )zbMATHGoogle Scholar - B. G. Wybourne:
*Classical Groups for Physicists*( Wiley, New York, 1974 ).zbMATHGoogle Scholar - 4.For a demonstration of this and the subsequent propositions regarding Lie series, see W. Gröbner:
*Serie di Lie e loro applicazioni*(Cremonese, Roma, 1973)Google Scholar - W. Gröbner:
*Die Lie-Reihen und ihre Anwendungen*( VEB Deutscher Verlag der Wissenschaften, Berlin, 1960 ).zbMATHGoogle Scholar - 6.H. Shniad: The equivalence of Von Zeipel mappings and the Lie transform,
*Celestial Mechanics***2**, 114–120 (1970).MathSciNetADSzbMATHCrossRefGoogle Scholar - 7.A. A. Kamel: Expansion formulae in canonical transformations depending on a small parameter,
*Celestial Mechanics***1**, 190–199 (1969);MathSciNetADSzbMATHCrossRefGoogle Scholar - A. A. Kamel: Perturbation method in the theory of non-linear oscillations,
*Celestial Mechanics***3**, 90–106 (1970).MathSciNetADSzbMATHCrossRefGoogle Scholar - 8.See, for instance, J. K. Hale:
*Oscillations in Nonlinear Systems*(Dover, 1992 ), pp. 57–59.Google Scholar - 9.The reader can find the (positive) result of this comparison in A. H. Nayfeh:
*Perturbation Methods*(Wiley, 1973 ) p. 211.Google Scholar - 12.See the (already quoted) paper by O. E. Gerhard, P. Saha: Recovering galactic orbits by perturbation theory,
*Mon. Not. R. Astr. Soc.*,**251**, 449–467 (1991).ADSzbMATHGoogle Scholar - 13.A. J. Dragt, J. M. Finn: Lie series and invariant functions for analytic symplectic maps,
*Journal of Math. Phys.***17**, 2215–2227 (1976).MathSciNetADSzbMATHCrossRefGoogle Scholar - 14.For the definition of p(n), see, for instance, M. Abramowitz, I. A. Stegun, eds.,
*Handbook of Mathematical Functions*(Dover, 1968), Sect. 24.Google Scholar - 15.For a detailed comparison of the Deprit and Dragt-Finn transforms, see P. V. Koseleff: Comparison between Deprit and Dragt-Finn perturbation methods,
*Celestial Mechanics and Dynamical Astronomy***58**, 17–36 (1994).MathSciNetADSCrossRefGoogle Scholar - 16.A. Milani: Secular perturbations of planetary orbits and their representation as series, in
*Long-Term Dynamical Behaviour of Natural and Artificial N-Body Systems*, ed. by A. E. Roy (Kluwer, 1988 ) pp. 73–108.Google Scholar - 17.The example is taken from F. Verhulst:
*Nonlinear Differential Equations and Dynamical Systems*(Springer, 1990), Sect. 13. 2.Google Scholar - 18.V. I. Arnold:
*Geometrical Methods in the Theory of Ordinary Differential Equations*(Springer, 1983), chap. 5. J. Guckenheimer, P. Holmes:*Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vectors Fields*(Springer, 1983), Sect. 3. 3.Google Scholar - 19.For a deeper understanding of the subject, see F. G. Gustayson: On constructing formal integrals of a Hamiltonian system near an equilibrium point,
*The Astronomical Journal***71**, 670–686 (1966)ADSCrossRefGoogle Scholar - J. M. Finn: Lie transform: a perspective, in
*Local and Global Methods of Nonlinear Dynamics*(Lecture Notes in Physics, 252) ed. by A. W. Sâenz, W. W. Zachary, R. Cawley (Springer, 1985 ).Google Scholar - 20.For a demonstration, see, for example, V. I. Arnold:
*Mathematical Methods of Classical Mechanics*, 2nd edn ( Springer, 1989 ), Appendix 7.CrossRefGoogle Scholar