 Solution

For the isotropic harmonic oscillator flow, Hamilton's equations are $\dot{X}=V$, $\dot{V}=-{\omega }^{2}X$. In dimension three these are three identical uncoupled pairs. For $\omega \ne 0$, put $c_t=\cos (\omega t)$, $s_t=\sin (\omega t)$. The solution from $(x,v)$ at time zero isThe inverse flow is obtained by replacing $t$ by $-t$:Integrating the source along the backward characteristic gives the Duhamel formula for Hamiltonian transport:Indeed $f(t,S_{t}z)=f_0(z)+{\int }_0^{t}h(s,S_{s}z)ds$, and setting $z=S_{-t}(x,v)$ gives the formula. It has the prescribed initial value and differentiation along the characteristic gives the source.

The zero-frequency limit has $s_t/\omega \to t$, $\omega s_t\to 0$, so $S_t(x,v)=(x+tv,v)$ and