Invariants for forced time-dependent oscillators and generalizations

Abstract

Several methods for deriving invariants for time-dependent systems have been developed. Four of these methods are as follows: (1) dynamical algebra, (2) Noether's theorem, (3) transformation group, and (4) Ermakov's method. These four methods are related by their use in deriving invariants for the forced, time-dependent oscillator. For this system the methods give the same results. We also discuss forced, nonlinear oscillators possessing analogous invariants. The nonlinear superposition laws for the forced oscillator are derived and discussed.