Automorphism groups of classical mechanical systems

The automorphism group of a classical mechanical system withn degrees of freedom is in a natural way a Lie group of dimension at most\(\frac{1}{2}\) (n+1) (n+2). Systems whose automorphism group has this maximal dimension are classified as follows. If the system is simply connected, it is a damped harmonic oscillator with equation of motion\(\ddot x^i = \lambda x^i + \varrho \dot x^i \). If not, it is obtained from such an oscillator with γ=0 and λ<0 by passing to the quotient with respect to the infinite cyclic group generated by for some positive integerl.