Hamiltonian evolution moves points through phase space without compressing or expanding phase-space volume. This is Liouville's theorem.

For canonical coordinates $(q_i,p_i)$, the phase-space velocity is $$\mathbf v_{\Gamma}=(\dot q_1,\ldots,\dot q_n,\dot p_1,\ldots,\dot p_n).$$ Using Hamilton's equations, $$\dot q_i=\frac{\partial H}{\partial p_i},\qquad \dot p_i=-\frac{\partial H}{\partial q_i},$$ the divergence is $$\nabla_{\Gamma}\cdot\mathbf v_{\Gamma} =\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^2H}{\partial q_i\partial p_i}

\frac{\partial^2H}{\partial p_i\partial q_i} \right)=0.$$ Thus an infinitesimal cloud of initial states can stretch and fold, but its total phase-space volume is preserved.

For a harmonic oscillator, an ensemble of nearby phase-space points rotates around the elliptical energy contours. The cloud changes orientation but does not steadily shrink onto the orbit or expand away from it.

If $\rho(q,p,t)$ is a probability density in phase space, conservation of probability leads to the Liouville equation $$\frac{\partial\rho}{\partial t}+{\rho,H}=0.$$ The probability density is transported by Hamiltonian flow rather than dissipatively compressed.

Liouville's theorem is the dynamical foundation of classical statistical mechanics. It explains why equilibrium ensemble densities can remain stationary under microscopic Hamiltonian evolution even while individual systems continue moving through phase space.