Convergence of Birkhoff normal forms for integrable systems
type: Item
instance of: scholarly article
title: Convergence of Birkhoff normal forms for integrable systems
zbMATH Open document ID: 0686.58021
DOI: 10.1007/BF02564686
author: Hidekazu Ito
published in: Commentarii Mathematici Helvetici
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140164
review text: The object of this article is to give a contribution to the study of normal forms of Hamiltonian systems near an equilibrium point of a periodic motion. It is considered analytic (or real analytic) Hamiltonian systems near a non-resonant equilibrium point (there is a canonical transformation which takes the Hamiltonian into the form \(H=\sum_{i}\lambda_ ix_ iy_ i-...\), \(i=1,...,n\) where (1) \((x,y)=z\) are coordinates in a neighborhood of the origin which is supposed to be in equilibrium and (2) the term not written out explicitly in H denotes a power series containing terms of order \(\geq 3)\). The higher order terms of H can be normalized by a canonical transformation f, called Birkhoff. In general f is divergent. A Hamiltonian system is called integrable if the Birkhoff transformation is convergent. Inspired by a result of \textit{J. Vey} [Am. J. Math. 100, 591-614 (1978; Zbl 0384.58012)] the author shows that if the system is integrable near a non- resonant equilibrium point then there is an analytic (convergent) Birkhoff transformation. A similar result is obtained for the real analytic case.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 4124653
zbMATH Keywords: Birkhoff normal forms
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: equilibrium point
zbMATH Keywords: periodic motion
reviewed by: Paulo R. Rodrigues
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4124653