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 $\mathcal{I}$ 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 $\psi ,\mathcal{I}$. 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 $\mathcal{I}$; asymptotic expansions in a slow time parameter $\epsilon$; time-dependent driven and damped parametric oscillators; etc.

31 pages. Latex file, 3 figures. Final version to appear in J. Phys. A

References in corpus (9)

• Carroll symmetry of plane gravitational waves
• Class of invariants for the 2D time-dependent Landau problem and harmonic oscillator in a magnetic field
• New BCH-like relations of the su(1, 1), su(2) and so(2, 1) Lie algebras
• Niederer's transformation, time-dependent oscillators and polarized gravitational waves
• Efficient algebraic solution for a time-dependent quantum harmonic oscillator
• On the impact of short laser pulses on cold diluted plasmas
• Lie transformation method on quantum state evolution of a general time-dependent driven and damped parametric oscillator
• Fuzzy hyperspheres via confining potentials and energy cutoffs
• Quantum-based solution of time-dependent complex Riccati equations

Cited by in corpus (1)

• The time-dependent quantum harmonic oscillator: a pedagogical approach via the Lewis-Riesenfeld dynamical invariant method