 Solution

In phase-space coordinates $z=(x,v)$, the characteristic equations for a transport equation are the Hamilton equationsSmoothness of $H$ gives a unique local Hamiltonian flow ${\Phi }_t(\xi ,\eta )=(X(t),V(t))$. Along it the chain rule gives $d[f(X,V,t)]/dt=0$. ThusThe inverse is the backward autonomous flow, so this formula defines the unique local classical solution wherever the characteristic reaches the initial hypersurface. A smooth Hamiltonian on all of ${\mathbb{R}}^{2n}$ need not have a globally complete flow; the formula is understood on the interval and region where both flows exist. A common global time interval requires the corresponding completeness or growth assumption.