Non-integrability criteria for Hamiltonians in the case of Lamé normal variational equations
type: Item
instance of: scholarly article
title: Non-integrability criteria for Hamiltonians in the case of Lamé normal variational equations
zbMATH Open document ID: 0866.58034
author: Juan J. Morales-Ruiz
author: Carles Simó
published in: Journal of Differential Equations
publication date: 1997-07-24T00:00:00Z
review text: Within the frame of the theorem of non-existence of the first integrals in a vicinity of a given solution obtained by S. L. Ziglin, it arises a question on solution of linear differential equations with time-dependent coefficients, called normal variational equations. In different applications to systems in two degrees of freedom, these equations are of Lamé type. In the paper under review the authors study a set of potentials of the classical Hamiltonian functions with an invariant plane such that an appropriate normal variational equation is of Lamé type. For these equations the conditions of non-integrability are obtained. The results are illustrated by examples from dynamics of Toda lattices of systems of two particles with cubic, quartic and rational potentials. The case of the homogeneous Henon-Heiles potential is studied in detail, including the dynamical meaning of the non-integrability.
Mathematics Subject Classification ID: 37J30
Mathematics Subject Classification ID: 37K05
Mathematics Subject Classification ID: 34M15
zbMATH DE Number: 933271
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: normal variational equations of Lamé type
zbMATH Keywords: nonintegrability differential Galois approach
reviewed by: G.V.Khmelevskaja-Plotnikova
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0113
OpenAlex ID: W2015267856
DOI: 10.1006/JDEQ.1996.0113
description: scientific article; zbMATH DE number 933271