Poincaré's méthode nouvelle by skew composition
type: Item
instance of: scholarly article
title: Poincaré's méthode nouvelle by skew composition
zbMATH Open document ID: 0958.70015
DOI: 10.1023/A:1008343529347
author: Etienne Deprit
author: André Deprit
published in: Celestial Mechanics and Dynamical Astronomy
publication date: 2001-04-01T00:00:00Z
review text: Poincaré's méthode nouvelle treats conservative Hamiltonians in the form of power series in a small parameter \(\varepsilon \), and constructs the transformed Hamiltonian and generating function also as power series in \(\varepsilon \). The lack of a systematic symbolic algorithm and some other fundamental deficiencies compelled dynamicists to abandon this method and to decide for techniques based on Lie transformations. Here the authors propose several improvements in order to give to Poincaré's method the flexibility of perturbation procedures based on Lie transformations. The improvements are mainly based on two new operations on power series (a skew composition to expand series whose coefficients are also series, and a skew inversion to solve implicit vector equations involving power series), include a procedure to express the Hamiltonian in terms of partial derivatives of generator, and use an algorithm to make explicit the direct and inverse transformations defined implicitly by the generating function. Some examples show in detail how to perform these programs.
reviewed by: Nicolae Cotfas
Mathematics Subject Classification ID: 70H09
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 37N05
zbMATH DE Number: 1442322
zbMATH Keywords: Hamiltonian dynamics
zbMATH Keywords: perturbation theory
zbMATH Keywords: implicit equations
zbMATH Keywords: Lagrange's formula
zbMATH Keywords: symbolic algebra
zbMATH Keywords: Poincaré méthode nouvelle
zbMATH Keywords: conservative Hamiltonians
zbMATH Keywords: power series
zbMATH Keywords: small parameter
zbMATH Keywords: generating function
zbMATH Keywords: skew composition
zbMATH Keywords: skew inversion
MaRDI profile type: Publication
full work available at URL: a:1008343529347
OpenAlex ID: W1622082387
description: scientific article; zbMATH DE number 1442322