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Lie transforms: A perspective. (English) Zbl 0661.70026
Local and global methods of nonlinear dynamics, Proc. Workshop, Silver Spring/Md. 1984, Lect. Notes Phys. 252, 63-86 (1986).

[For the entire collection see Zbl 0592.00030.]

Some of the properties of both types of Lie transforms are discussed and some manipulations that are useful in practice are shown. The two methods are compared and a method of obtaining the generators of each type of transform from the generators of the other is presented. At the least, this last exercise shows that the two formalisms are nontrivially related. This material, which follows on section 2 on the basics of Lie series, is contained in Sections 3 and 4.

Section 5 contains some examples of direct applications of Lie transforms, i.e., applications where the actual time evolution of the system is described in terms of infinite product Lie transforms. Included is a comact rederivation of the Campbell-Baker-Hausdorff formula. Normalization of Hamiltonian systems is the most widely used application of Lie tranform techniques and the subject of the last section. This material contains dicussions of nonresonant and resonant cases, time dependent cases with either small amplitude or adiabatic evolution, cases with zero frequencies of the linearized motion, and Kolmogorov’s superconvergent algorithm.

All of these applications are discussed in terms of infinite product Lie transforms, although the basic framework for normalization (including superconvergence) is similar to that using Deprit Lie transforms.

MSC:
70H15Canonical and symplectic transformations in particle mechanics