The time-dependent harmonic oscillator revisited

Abstract

We point out a rather effective approach for solving the time-dependent harmonic oscillator $\ddot q=-\omega^2 q$ under various regularity assumptions. Where $\omega(t )$ is $C^1$ this is reduced to Hamilton equation for the angle variable $\psi$ {\it alone} (the action variable ${\cal I}$ is obtained \it by quadrature}). The fixed point theorem for the integral equation equivalent to the generic Cauchy problem for $\psi(t )$ yields a sequence $\{\psi^{(h)}\}_{h\in\mathbb{N}_0}$ converging to $\psi$ rather fast; if $\omega$ varies slowly or little, already $\psi^{(0)}$ approximates $\psi$ well for rather long time lapses. The discontinuities of $\omega$, if any, determine those of $\psi,{\cal I}$. The zeros of $q,\dot q$ 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 $\omega(t )$ is periodic; the adiabatic invariance of ${\cal I}$; asymptotic expansions in a slow time parameter $\varepsilon$; time-dependent driven and damped parametric oscillators; etc.