Some physical applications of systems of differential equations admitting a superposition rule
type: Item
instance of: scholarly article
title: Some physical applications of systems of differential equations admitting a superposition rule
zbMATH Open document ID: 1020.34009
DOI: 10.1016/S0034-4877(01)80063-X
author: J. F. Cariñena
author: J. Grabowski
author: Giuseppe Marmo
published in: Reports on Mathematical Physics
publication date: 2002-09-12T00:00:00Z
review text: The authors refer to old results of S. Lie and G. Scheffers concerning systems of ordinary differential equations admitting a superposition rule. Such systems are associated with the action of a Lie group \(G\) on a manifold \(M\) being in general a homogeneous space \(G/H\) and are called Lie-Scheffers systems. The basic point is (following \textit{J. Wei} and \textit{E. Norman} [J. Math. Phys. 4, 575-581 (1963; Zbl 0133.34202)]) to regard equations of the type \[ \dot g(t)\cdot g(t)^{-1}= a(t)= \sum^r_{\alpha=1} A_\alpha(t)\cdot a_\alpha\in T_e(G), \] in a Lie group \(G\), where the r.-h.s. is formed in a certain subalgebra of the Lie algebra of \(G\) and \(g(t)\) is locally written as the product \(g(t)= \prod^r_{\alpha=1} \exp(u_\alpha(t)\cdot a_\alpha)\). Several examples, also with physical relevance, are given and demonstrate the general character of the construction.
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34A26
Mathematics Subject Classification ID: 34C14
zbMATH DE Number: 1799969
zbMATH Keywords: superposition rule
zbMATH Keywords: Riccati equation
zbMATH Keywords: Lie-Scheffers systems
reviewed by: Günter Czichowski
MaRDI profile type: Publication
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full work available at URL: s0034-4877(01)80063-x
OpenAlex ID: W2019621538
description: scientific article; zbMATH DE number 1799969