Past exam of the theoretical physics I course of the University of Cambridge / 2015 / Paper 1 / 3 / d / Solution

The perturbed Hamiltonian is $H^{\prime }=(Q^2+P^2)/2-g(t)Q$. Hamilton's equations give $\dot{Q}=P$, $\dot{P}=-Q+g(t)$, henceIn the paper's time units the natural frequency is one. The retarded Green function is $G_R(t)=\Theta (t)\sin t$: it vanishes before the source, is continuous at zero and has a unit derivative jump there. With Fourier convention $e^{-i\omega t}$,Both real-frequency poles are displaced below the real axis. Causality therefore selects the convolutionFor $t<0$ the exponential rises as $e^{{\omega }_0{t}^{\prime }}$ throughout the integral. For $t\ge 0$, split the integral at $t^{\prime }=0$. Evaluation yieldsThe response is nonzero before $t=0$ because the prescribed force is already nonzero then; this does not violate causality. Both $Q$ and $\dot{Q}$ are continuous at zero. The force eventually decays, leaving the free oscillation ${\omega }_0^2\sin t/(1+{\omega }_0^2)$. A general solution adds $A\cos t+B\sin t$, but the retarded condition of no incoming free oscillation as $t\to -\infty$ fixes $A=B=0$.

The oscillator equation is well-defined on the full $(Q,P)$ plane. Interpreted through the positive-$p$ chart of part (c), it describes the original real variables only while $P>|Q|$, with $P=\dot{Q}$. The requested retarded oscillator solution is a continuation in the new variables; it is not a claim that this canonical chart remains valid at all times for every ${\omega }_0$.