The time-dependent harmonic oscillator revisited

TL;DR

This work presents a novel reduction of the time-dependent harmonic oscillator to action-angle variables, where the angle $\psi$ obeys a decoupled first-order equation and the action $\mathcal I$ is obtained by quadrature. By recasting the $\psi$-dynamics as a Volterra integral equation, the authors prove rapid convergence of a Picard-iteration sequence $\{\psi^{(h)}\}$ on compact intervals, with explicit error bounds tied to the cumulative frequency variation $g(t)=\int |\dot\omega/\omega|$. They further develop a Riccati-based reduction to study zeros of $q$ and $\dot q$, provide adiabatic invariance and slow-time asymptotics, and derive upper/lower bounds for solutions, including parametric resonance analysis for periodic $\omega(t)$ via Floquet theory. The framework accommodates frequency discontinuities through matching conditions and yields practical insights into stability, damping, and beating, with potential applications to Hill-type systems and quantum settings. Overall, the approach offers a unified, efficient route to long-time behavior, rigorous bounds, and resonance phenomena for non-autonomous linear oscillators.

Abstract

We point out a rather effective approach for solving the time-dependent harmonic oscillator $\ddot q=-ω^2 q$ under various regularity assumptions. Where $ω(t )$ is $C^1$ this is reduced to Hamilton equation for the angle variable $ψ$ {\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 $ψ(t )$ yields a sequence $\{ψ^{(h)}\}_{h\in\mathbb{N}_0}$ 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 $ψ,{\cal I}$. 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 $ω(t )$ is periodic; the adiabatic invariance of ${\cal I}$; asymptotic expansions in a slow time parameter $\varepsilon$; time-dependent driven and damped parametric oscillators; etc.

Figures (3)

• Figure 1: The angle $\psi$. We order the quadrants clockwise, i.e. in the direction of the motion.
• Figure 2: Mathieu equation with $\omega(t)=\bar{\omega}\sqrt{1+ 0.5\, \sin(0.2 t)}$ (this is a non-resonant case). Up: graph of $\omega$; center: graphs of the two solutions $v_a$ ($a=1,2$) of (\ref{['eq1']}) fulfilling $v_1(0)=\dot v_2(0)=1$, $v_2(0)=\dot v_1(0)=0$ (see section \ref{['preli']}) and of their approximations $\hat{v}_a,\tilde{v}_a$, with $a=1$ on the left, $a=2$ on the right; down: graphs of the corresponding action variable ${\cal I}_a$ and of their approximations $\hat{{\cal I}}_a,\tilde{{\cal I}}_a$. As we can see, $\hat{v}_a\simeq v_a$, $\hat{{\cal I}}_a\simeq {\cal I}_a$, namely the approximation (\ref{['def-hat-q']}) is rather good, and much better than the one (\ref{['tilde-approx']}).
• Figure 3: Mathieu equation with $\omega(t)=\bar{\omega}\sqrt{1+ 0.2\, \sin(2 t)}$ (this is a resonant case). Up: graph of $\omega$; center: graphs of the two solutions $v_a$ ($a=1,2$) of (\ref{['eq1']}) fulfilling $v_1(0)=\dot v_2(0)=1$, $v_2(0)=\dot v_1(0)=0$ (see section \ref{['preli']}) and of their approximations $\hat{v}_a,\tilde{v}_a$, with $a=1$ on the left, $a=2$ on the right; down: graphs of the corresponding action variable ${\cal I}_a$ and of their approximations $\hat{{\cal I}}_a,\tilde{{\cal I}}_a$. Again, as we can see, $\hat{v}_a\simeq v_a$, $\hat{{\cal I}}_a\simeq {\cal I}_a$, namely the approximation (\ref{['def-hat-q']}) is rather good, and much better than the one (\ref{['tilde-approx']}).

Theorems & Definitions (5)

• Proposition 1
• Proposition 2
• Proposition 3
• Corollary 1
• Lemma 1