On a Generalization of the Lewis Invariant for the Time-Dependent Harmonic Oscillator

An exact invariant is found for the one dimensional oscillator with equation of motion q + f( t ) q + ω ² ( t )q = 0. The method used is that of linear canonical transformations with time-dependent coefficients. These transformations are members of the symplectic group. The Lewis invariant, I = 1/2[ ρ - 2 q² + ( ρ p - ρ q )² ] where ρ + ω ² ( t )ρ = ρ - 3, is a particular case of this invariant. The significance of extension to higher dimensions of these results is indicated, in particular for the existence of noninvariance dynamical symmetry groups.