Spectrum generating algebras and Lie groups in classical mechanics

Abstract

We give a general framework for a geometric foundation of time dependent classical mechanics. The theory is based on the concept of evolution space which is phase space extended by time. Lie algebras of constants of motion which may possess explicit time dependence are constructed, and general conditions for getting global Lie group actions from infinitesimal actions are derived. In a natural way these groups map solutions of the Hamiltonian equations of motion onto one another and act on the orbit space via symplectic transformations. The theory is applied to the nonrelativistic free particle, the harmonic and damped oscillator, nonstationary quadratic systems, and to the motion of a particle in constant electromagnetic fields.