Liouville's Theorem
Phase space volume conservation under Hamiltonian flow
System
Pendulum
HO
Duffing
N particles:
300
Blob size:
0.20
Energy E:
1.0
Speed:
1.0
⏸ Pause
Reset blob
Phase area (init)
—
Phase area (curr)
—
Area ratio
—
Liouville's theorem:
dρ/dt = 0 — phase space density is conserved along trajectories (∇·v = 0 for Hamiltonian flows).
The
shape
of a phase-space blob distorts wildly, but its
volume
(area in 2D) is exactly conserved.