Completely integrable Hamiltonian systems
type: Item
instance of: scholarly article
title: Completely integrable Hamiltonian systems
zbMATH Open document ID: 1243.37050
published in: Aequationes Mathematicae
publication date: 2011-10-11T00:00:00Z
review text: Integrable Hamiltonian systems are nonlinear ordinary differential equations described by a Hamiltonian function and possessing sufficiently many independent constants of motion in involution. The regular compact level manifolds defined by the intersection of the constants of motion are diffeomorphic to a real torus on which the motion is quasi-periodic as a consequence of the following purely differential-geometric fact: a compact and connected \(n\)-dimensional manifold on which there exist \(n\) vector fields which commute and are independent at every point is diffeomorphic to an \(n\)-dimensional real torus and each vector field will define a linear flow there. The connection with the concept of completely integrable systems is studied and applied to solve several classical problems. This work consists of the following parts: introduction; the Arnold-Liouville theorem; rigid body rotation around a fixed point, the Euler case, the Lagrange case, the Kovalevskaya case; the motion of a rigid body in ideal fluid, the Clebsch case, the Lyapunov-Steklov case; geodesic flow on the group \(\mathrm{SO}(4)\); hierarchy of integrable systems.
reviewed by: Vladislav Nikolaevich Dumachev
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 70H06
Mathematics Subject Classification ID: 14H70
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 70E17
zbMATH DE Number: 5956222
zbMATH Keywords: completely integrable systems
zbMATH Keywords: topological structure of phase space
zbMATH Keywords: methods of integration
author: Ahmed Lesfari
MaRDI profile type: Publication
full work available at URL: s00010-011-0078-x
OpenAlex ID: W2086160442
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DOI: 10.1007/S00010-011-0078-X
description: scientific article; zbMATH DE number 5956222