Past exam of the classical mechanics course of Part B of the physics course of the University of Oxford / 2016 / 3 / b / Solution

For $n$ degrees of freedom, phase space is the $2n$-dimensional space of canonical coordinates $(q_1,\dots ,q_n,p_1,\dots ,p_n)$. Initial data at one point determine a trajectory under the Hamilton equations.

Liouville's theorem in Hamiltonian mechanics states that this flow preserves the canonical volume element $d\Gamma ={\prod }_{i}d{q}_{i}d{p}_i$. Its divergence isIf $J(t)$ is the Jacobian determinant of the flow from initial data, the determinant evolution formula gives $\dot{J}=J\nabla \cdot (\dot{q},\dot{p})=0$. Since $J(0)=1$, it remains one, proving preservation of every advected phase volume. The proof also applies to smooth explicitly time-dependent Hamiltonians.

For an ensemble density $\rho (q,p,t)$, probability conservation and the same vanishing divergence giveThus the fine-grained density is constant along the Hamiltonian trajectories, although the ensemble may shear, stretch and develop fine structure. This is not a claim that coordinate-space volume or every lower-dimensional projection is conserved.