Automorphism groups of classical mechanical systems
type: Item
instance of: scholarly article
title: Automorphism groups of classical mechanical systems
zbMATH Open document ID: 0573.70005
DOI: 10.1007/BF01339229
author: Ottmar Loos
published in: Monatshefte für Mathematik
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178254
review text: We study classical mechanical systems S consisting of a space-time manifold M of dimension \(n+1\) with a closed 1-form \(\theta\) (locally \(\theta =dt)\), a fibre metric g on the kernel of \(\theta\), and an equation of motion Z. The connection with \textit{J. M. Souriaus} description of mechanical systems [Structure des systèmes dynamiques. Maîtrises de mathematiques (1970; Zbl 0186.580)] in terms of a maximum rank 2-form on the \((2n+1)\)-dimensional subbundle \(\theta =1\) of T(M) is established. The automorphism group G of S is shown to be a Lie transformation group of dimension at most \((n+1)(n+2)\). In case G has this maximum dimension, a complete classification of the systems is given as follows. If M is simply connected, S is a damped harmonic oscillator; i.e., \(M={\mathbb{R}}^{n+1}\), and the equation of motion is ẍ\({}^ i=\lambda x^ i+\rho \dot x^ i\). If M is not simply connected, then it is a generalized Möbius strip with universal cover a free harmonic oscillator \((\rho =0\), \(\lambda <0)\). In this case, \(\theta\) is not exact.
Mathematics Subject Classification ID: 70G45
Mathematics Subject Classification ID: 58A15
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 22E70
zbMATH DE Number: 3915677
zbMATH Keywords: space-time manifold
zbMATH Keywords: closed 1-form
zbMATH Keywords: fibre metric
zbMATH Keywords: \textit{J. M. Souriaus} description of mechanical systems
zbMATH Keywords: maximum rank 2-form
zbMATH Keywords: automorphism group
zbMATH Keywords: Lie transformation group
zbMATH Keywords: classification
zbMATH Keywords: damped harmonic oscillator
zbMATH Keywords: generalized Möbius strip
zbMATH Keywords: free harmonic oscillator
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3915677