Liouville's theorem

Liouville's Theorem

Liouville’s theorem in Hamiltonian mechanics expresses the conservation of phase space volume under Hamiltonian evolution. It can be formulated in three equivalent ways:

1. Geometric formulation (symplectic structure)

Let $ω = \sum_{i} d q_{i} \land d p_{i}$ be the canonical symplectic form, and let $X_{H}$ be the Hamiltonian vector field generated by the Hamiltonian $H (q, p)$ . Then:

L_{X_{H}} ω = 0.

This says that the flow of $X_{H}$ preserves the symplectic form. Since the volume form is $ω^{n}$ , this implies that phase space volume is preserved:

L_{X_{H}} (ω^{n}) = 0.

2. Analytic formulation (divergence-free flow)

The Hamiltonian flow is divergence-free with respect to the standard volume measure on phase space. That is:

div X_{H} = \sum_{i} (\frac{\partial {\dot{q}}_{i}}{\partial q_{i}} + \frac{\partial {\dot{p}}_{i}}{\partial p_{i}}) = 0.

Using the Hamiltonian equations:

{\dot{q}}_{i} = \frac{\partial H}{\partial p_{i}}, {\dot{p}}_{i} = - \frac{\partial H}{\partial q_{i}},

we get:

div X_{H} = \sum_{i} (\frac{\partial^{2} H}{\partial q_{i} \partial p_{i}} - \frac{\partial^{2} H}{\partial p_{i} \partial q_{i}}) = 0.

3. Statistical formulation (Liouville equation)

Let $ρ (q, p, t)$ be the density of an ensemble of systems in phase space. Then along the flow of the system, the total derivative of $ρ$ vanishes:

\frac{d ρ}{d t} = \frac{\partial ρ}{\partial t} + \sum_{i} (\frac{\partial ρ}{\partial q_{i}} {\dot{q}}_{i} + \frac{\partial ρ}{\partial p_{i}} {\dot{p}}_{i}) = 0.

This is the Liouville equation in classical statistical mechanics. It ensures the conservation of probability in Classical Statistical Mechanics.

Summary

All three formulations express that Hamiltonian evolution preserves phase space volume:

$L_{X_{H}} ω = 0$ : symplectic geometry.
$div X_{H} = 0$ : analytic, local coordinate expression.
$\frac{d ρ}{d t} = 0$ : statistical interpretation for ensembles.