Liouville's theorem on phase-space volume

Remarks

The vector field $X = (\partial H / \partial p, -\partial H / \partial q)$ has divergence $$\sum_{i=1}^n \left( \frac{\partial^2 H}{\partial q_i \partial p_i} - \frac{\partial^2 H}{\partial p_i \partial q_i} \right) = 0.$$ The derivative in $t$ of the volume of $\varphi_t(D)$ is the integral of this divergence over $\varphi_t(D)$, so the volume is constant. In symplectic language, the flow of a Hamiltonian vector field preserves the symplectic form $\omega$, hence also the volume form $\omega^n / n!$.

Example: for $H = \frac{1}{2}(q^2 + p^2)$ the flow rotates the plane, and rotations preserve area. For a damped oscillator $q' = p$, $p' = -q - p$ the divergence is $-1$, so areas shrink like $e^{-t}$; this system is not Hamiltonian.

An equivalent form concerns a density $\rho(t, q, p)$ carried by the flow: it satisfies $\frac{\partial \rho}{\partial t} + \{\rho, H\} = 0$, where $\{\cdot, \cdot\}$ is the Poisson bracket.

Going deeper

For a student

The theorem is the first sign that Hamiltonian flows preserve geometric structure: they preserve the symplectic form, and volume is its top power. It is the base of statistical mechanics, since it shows that the Liouville measure is invariant. With the Poincaré recurrence theorem it shows that if the motion stays in a region of finite volume, then almost every state returns arbitrarily close to where it started. In dimension $2n \ge 4$, preserving volume is much weaker than preserving $\omega$; Gromov's non-squeezing theorem shows the difference.