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.