论文标题
重新审视时间依赖时间的谐波振荡器
The time-dependent harmonic oscillator revisited
论文作者
论文摘要
我们指出了一种相当有效的方法，用于解决时间相关的谐波振荡器$ \ ddot q =-Ω^2 q $在各种规律性假设下。其中，$ω（t）$是$ c^1 $，将其简化为汉密尔顿方程的角度变量$ψ$ {\ it holy}（Action变量$ {\ cal I} $由Quadrature}获得\ IT}）。集成方程的固定点定理等效于$ψ（t）$的通用cauchy问题，得出序列$ \ {ψ^{（h）} \} \} _ {如果$ω$变化缓慢或较小，则已经$ψ^{（0）} $近似于$ψ$在相当长的时间里。 $ω$（如果有）的不连续性确定$ψ，{\ cal i} $的不连续性。 $ Q，\ dot Q $的零是通过Riccati方程研究的。我们的方法可以简化：解决方案上的上限和下限；琐碎的稳定性； $ω（t）$是周期性时的参数共振； $ {\ cal i} $的绝热不变性;缓慢的参数$ \ varepsilon $中的渐近扩展；时间依赖性驱动和阻尼的参数振荡器； ETC。
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.
