Picard-Vessiot theory and Ziglin's theorem
type: Item
instance of: scholarly article
title: Picard-Vessiot theory and Ziglin's theorem
zbMATH Open document ID: 0799.58035
published in: Journal of Differential Equations
publication date: 1994-11-17T00:00:00Z
review text: In 1983 S. L. Ziglin has given a criterium for non-integrability of Hamiltonian systems in terms of a monodromy group of normal and reduced variational equations along an integral curve. The authors of the paper apply the Picard-Vessiot differential Galois theory to linear differential equations of Ziglin's theorem which are linear Hamiltonian equations of Fuchsian type with coefficients in the field \(K\) of meromorphic functions on a compact Riemann surface. They consider the Picard-Vessiot extension \(P = K\langle y_ 1,\dots,y_{2n}\rangle\) of the field \(K\) by means of a fundamental system of solutions \(y_ 1,\dots,y_{2n}\) and the Galois group \(G\) of \(P\) over \(K\) made of all the automorphisms of \(P\) leaving invariant the functions of \(K\). The authors prove that under the hypothesis of Ziglin's theorem the identity component \(G_ 0\) of the Galois group \(G\) is a complete diagonal subgroup of \(\text{Sp}(n,\mathbb{C})\) and the index \([G:G_ 0]\) is finite. Using these results the authors give a full investigation of non- integrability for 2-dimensional Hamiltonian systems, in particular, Lamé and Hénon-Heiles equations.
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 522671
zbMATH Keywords: Picard-Vessiot theory
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: Ziglin's theorem
zbMATH Keywords: non- integrability
author: Juan J. Morales
author: Carles Simó
reviewed by: E. D. Belokolos
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1006
OpenAlex ID: W1976852307
DOI: 10.1006/JDEQ.1994.1006
description: scientific article; zbMATH DE number 522671