On integrability in elementary functions of certain classes of nonconservative dynamical systems
type: Item
instance of: scholarly article
title: On integrability in elementary functions of certain classes of nonconservative dynamical systems
zbMATH Open document ID: 1188.37003
author: M. V. Shamolin
published in: Journal of Mathematical Sciences (New York)
publication date: 2010-02-19T00:00:00Z
review text: This survey paper integrates the results of the author on mathematics of such mechanical problems as the motion of a body in a resisting medium, being only partially embedded. This kind of problems results in systems with energy dissipation or anti-dissipation (pumping). The main mathematical model turns to be the pendulum second order equation in which there exists a linear dissipative force with variable coefficients which may change sign; in the mean the dissipation may be positive, negative (dispersive forces) or equal to zero - zero mean variable dissipation system (``almost conservative''). For such systems relative ``roughness'' in the sense of Andronov is introduced. The paper discusses zero mean variable dissipation systems with symmetries in both rough and non-rough cases. Systems on the sphere are considered and various phase portraits without limit cycles are presented. Higher order dynamics are considered on Lie algebra \(so(4)\) including symmetries. A comprehensive list of almost 400 references is given.
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 34-02
Mathematics Subject Classification ID: 37C99
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34M35
Mathematics Subject Classification ID: 37E99
Mathematics Subject Classification ID: 70E15
zbMATH DE Number: 5673498
zbMATH Keywords: integrability
zbMATH Keywords: dynamical system
zbMATH Keywords: dissipativeness
zbMATH Keywords: dispersiveness
reviewed by: Vladimir Răsvan
MaRDI profile type: Publication
full work available at URL: s10958-009-9597-6
OpenAlex ID: W2010545731
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On integrability in elementary functions of certain classes of nonconservative dynamical systems (DOI)
DOI: 10.1007/S10958-009-9597-6

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