Liouville theorem (invariance of phase-space volume)

Hamiltonian time evolution preserves phase-space volume; equivalently, the Liouville measure dq dp is invariant under the Hamiltonian flow.

Liouville theorem (invariance of phase-space volume)

Statement

Let $\Gamma$ be the classical phase space with canonical coordinates $(q,p)$ and let $H(q,p)$ be a smooth Hamiltonian function generating Hamilton’s equations:

\dot q = \frac{\partial H}{\partial p},\qquad \dot p = -\frac{\partial H}{\partial q}.

Let $\Phi^t:\Gamma\to\Gamma$ be the associated flow map. Then Liouville’s theorem states that the standard phase-space volume measure (Liouville measure) is invariant:

\text{Vol}(\Phi^t(A))=\text{Vol}(A)

for every measurable set $A\subset\Gamma$ and all $t$ for which the flow exists. Equivalently, the Jacobian determinant satisfies

\det(D\Phi^t)=1,

and the phase-space divergence of the Hamiltonian vector field is zero.

Key hypotheses

Classical Hamiltonian dynamics on $\Gamma$ with sufficiently smooth $H$ so the flow exists and is differentiable.
Canonical (symplectic) coordinates on phase space.

Conclusions

Phase-space volume is conserved under time evolution: Hamiltonian dynamics is incompressible in phase space.
The induced invariant reference measure underlies equilibrium ensembles; for example, the microcanonical measure is stationary under Hamiltonian time evolution (assuming energy is conserved and appropriate regularity/ergodic assumptions are treated separately).

Proof idea / significance

Compute the divergence of the Hamiltonian vector field $(\dot q,\dot p)$ :

\nabla\cdot(\dot q,\dot p) ={} \sum_i \left(\frac{\partial \dot q_i}{\partial q_i} + \frac{\partial \dot p_i}{\partial p_i}\right) ={} \sum_i \left(\frac{\partial^2 H}{\partial q_i\partial p_i} - \frac{\partial^2 H}{\partial p_i\partial q_i}\right)=0.

Zero divergence implies the Jacobian determinant of the flow is constant in time, and at $t=0$ it equals $1$ , hence it is $1$ for all $t$ .
In statistical mechanics, this invariance justifies using uniform phase-space weighting (subject to constraints like fixed energy) and supports conservation of ensemble densities under the Liouville equation, connecting microscopic reversibility to macroscopic equilibrium descriptions.