In Hamiltonian mechanics, the phase-space density is conserved along trajectories.
A blob of initial conditions evolves — it can stretch and contort, but its total
phase-space volume (area in 2D) never changes.
dρ/dt = 0, equivalently div(ẋ, ṗ) = 0.
Here, a colored blob of (q, p) points flows under H = p²/2 + ω²q²/2 + εq⁴/4.
Even for the anharmonic oscillator the area is preserved — but the blob shears dramatically.
The grid lines show the distortion of phase-space structure.