z = (x, v)

, the characteristic equations for a transport equation are the Hamilton equations

\dot{X} = \nabla_{v} H (X, V), \dot{V} = - \nabla_{x} H (X, V), (X (0), V (0)) = (ξ, η) .

Smoothness of

H

gives a unique local Hamiltonian flow

Φ_{t} (ξ, η) = (X (t), V (t))

. Along it the chain rule gives

d [f (X, V, t)] / d t = 0

. Thus

f (x, v, t) = f_{I} (Φ_{- t} (x, v)) .

(2)

The 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

R^{2 n}

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.