Non-integrability in Hamiltonian mechanics
type: Item
instance of: scholarly article
title: Non-integrability in Hamiltonian mechanics
zbMATH Open document ID: 0884.58045
DOI: 10.1007/BF00048436
published in: Celestial Mechanics and Dynamical Astronomy
publication date: 1998-04-01T00:00:00Z
review text: This paper is a survey on theorems for proving non-integrability of Hamiltonian dynamical systems. In this review, the following two points of view are presented: 1. Properties of the variational equations around periodic solutions, in the real or complex time domain. 2. Destruction of the resonant tori of an integrable system under a perturbation. Let \(\dot x=J \nabla H\) be a Hamiltonian system, where \(x\) is a set of canonical coordinates and \(J\) is the standard symplectic structure. Further, let \(\dot z=A(t)z\) be the corresponding system of variational equations, where \(A=J \nabla^2H\) is a periodic matrix, evaluated on a periodic solution. The first reviewed theorem belongs to Poincaré (1892): If \(H\) possesses \(n\) integrals in involution, independent on a periodic solution, then it is non-isolated, i.e., all its characteristic exponents are zero. The last reviewed theorem (Ichtiaroglou and Meletlidou 1996) gives the conditions, under which the perturbed Hamiltonian \(H= H_0(I_j)+ \varepsilon H_1 (I_j,w_j)\) (where \((I_j,w_j)\) are action-angle variables of the integrable part \(H_0)\) does not possess a complete set of analytic integrals of motion.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 70H05
zbMATH DE Number: 1051498
zbMATH Keywords: variational equations
zbMATH Keywords: resonant tori
zbMATH Keywords: monodromy matrix
zbMATH Keywords: integrability coefficient
zbMATH Keywords: non-integrability
zbMATH Keywords: Hamiltonian dynamical systems
author: Simos Ichtiaroglou
reviewed by: A. Klíč
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00048436
OpenAlex ID: W2021989321
description: scientific article; zbMATH DE number 1051498