Automorphism groups of classical mechanical systems (Q1062774)

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.