Integrable systems with linear periodic integral for the Lie algebra e(3)
type: Item
instance of: scholarly article
title: Integrable systems with linear periodic integral for the Lie algebra e(3)
zbMATH Open document ID: 1391.37046
DOI: 10.1134/S1995080217060063
author: A. A. Oshemkov
author: I. K. Kozlov
published in: Lobachevskii Journal of Mathematics
publication date: 2018-03-02T00:00:00Z
review text: The phase space topology of integrable systems is of primary interest in mechanics and has been studied by many authors over the years. The present paper provides a nice overview of the topological properties of integrable Hamiltonian systems with \(S^1\) symmetry and rigid body equations for the Euclidean Lie algebra \(e(3)\). Let us mention here that this setting covers the Lagrange and Kirchhoff integrable cases of rigid body dynamics, the dynamics of spin in the superfluid \(^3\mathrm{He}\) and the integrable case of the motion of a heavy ellipsoid on the horizontal plane. Starting with the definition of the Lie-Poisson brackets for the Euclidean Lie algebra \(e(3)\) the authors describe the above mentioned integrable systems and observe that all of them possess an additional integral which is linear function on the dual space \(e(3)^*\). The also provide the corresponding bifurcation analysis. The paper is clearly written and quite informative.
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37J15
Mathematics Subject Classification ID: 37J20
Mathematics Subject Classification ID: 53D20
Mathematics Subject Classification ID: 70E40
zbMATH DE Number: 6846086
zbMATH Keywords: integrable Hamiltonian system
zbMATH Keywords: periodic integral
zbMATH Keywords: bifurcation diagram
zbMATH Keywords: momentum mapping
zbMATH Keywords: Liouville tori
Wikidata QID: Q115247382
reviewed by: Ivaïlo Mladenov
MaRDI profile type: Publication
cites work: Q4821521
cites work: Dynamics of rigid body.
cites work: Q4011331
cites work: Topological classification of integrable Hamiltonian systems in a potential field on surfaces of revolution
cites work: Topological analysis of the motion of an ellipsoid on a smooth plane
description: scientific article; zbMATH DE number 6846086