The time-dependent harmonic oscillator revisited
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866915106125774848 |
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| author | Fiore, Gaetano |
| author_facet | Fiore, Gaetano |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_01781 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The time-dependent harmonic oscillator revisited Fiore, Gaetano Mathematical Physics 37J65, 70H11, 34Cxx, 70J40, 70K28, 65L20 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. |
| title | The time-dependent harmonic oscillator revisited |
| topic | Mathematical Physics 37J65, 70H11, 34Cxx, 70J40, 70K28, 65L20 |
| url | https://arxiv.org/abs/2205.01781 |