Liouville's Theorem — Phase Space Incompressibility
Hamiltonian flow preserves phase-space volume · dρ/dt = 0
Area: —
Liouville's theorem: for any Hamiltonian system, the phase-space density ρ(q,p,t) is conserved along trajectories: dρ/dt = ∂ρ/∂t + {ρ,H} = 0. The flow is incompressible — volumes are preserved (Poincaré). Here: H = p²/2 + ω²q²/2 + εq⁴/4. Left: position space. Right: phase space (q,p). Watch the blob deform but conserve area.