zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

70H15Canonical and symplectic transformations in particle mechanics