Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0661.70026
Finn, John M.
Lie transforms: A perspective. (English)
[A] 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.] \par 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. \par 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. \par 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 2000:: *70H15 Canonical transformations

Keywords: Lie transforms; generators; Lie series; infinite product Lie transforms; Campbell-Baker-Hausdorff formula; Normalization of Hamiltonian systems; zero frequencies of the linearized motion; Kolmogorov's superconvergent algorithm; superconvergence; Deprit Lie transforms

Citations: Zbl 0592.00030

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]