The time-dependent harmonic oscillator revisited

We point out a rather effective approach for solving the time-dependent harmonic oscillator q ¨ = − ω 2 q under various regularity assumptions. Where ω ( t ) is C1 this is reduced to Hamilton equation for the angle variable ψ alone (the action variable I is obtained by quadrature). The fixed point theorem for the integral equation equivalent to the generic Cauchy problem for ψ ( t ) yields a sequence { ψ ( h ) } h ∈ N 0 converging to ψ rather fast; if ω varies slowly or little, already ψ ( 0 ) approximates ψ well for rather long time lapses. The discontinuities of ω, if any, determine those of ψ , I . The zeros of q , q ˙ are investigated via Riccati equations. Our approach may simplify the study of: upper and lower bounds on the solutions; the stability of the trivial one; parametric resonance when ω ( t ) is periodic; the adiabatic invariance of I ; asymptotic expansions in a slow time parameter ɛ; time-dependent driven and damped parametric oscillators; etc.