LIOUVILLE'S THEOREM a cloud of states flows through phase space without ever changing its volume
In phase space — positions and momenta together — a system's state is a point, and a whole ensemble of possible states is a cloud. Liouville's theorem says that as Hamiltonian dynamics carries the cloud forward in time, its volume is exactly preserved: it may stretch, shear, and fold into fantastically thin filaments, but its phase-space volume never grows or shrinks. The flow is incompressible. This is the invariant beneath statistical mechanics (it justifies the microcanonical ensemble) and the reason a symplectic integrator — like the Verlet method (dart 082) — is trusted for long simulations: it respects the volume the true dynamics conserve. The transformation is time itself; the invariant is volume.
THE TECHNIQUE Hamiltonian flow preserves phase-space volume (Jacobian = 1)
A shear flow (x,p) → (x+p, p) — a step of Hamiltonian evolution. The demo computes its Jacobian and shows the area of a state-cloud is unchanged: live demo
HISTORY & CREDIT Liouville 1838 · Jacobi 1842
“A spreading cloud of states grows in volume.” — under Hamiltonian flow it cannot: it stretches thin in one direction exactly as much as it narrows in another. The volume is conserved even as the shape is destroyed. cited
1838 · Joseph Liouville — the underlying theorem, proved for a class of ODEs (not yet mechanics); Jacobi (1842) cast it as phase-space-volume conservation for Hamiltonian flow, and Gibbs (1902) gave the statistical-mechanics reading. structure · the deep reason is the symplectic form: Hamiltonian flows are area-preserving maps. now · statistical mechanics, symplectic integrators (Verlet, dart 082), accelerator and plasma physics.
Time stretches and folds the cloud without ever changing its volume — the flow is incompressible because it is symplectic. The shape is free; the volume is fixed. Liouville 1838
RECOMMEND FOR I-13 phase-space volume under a Hamiltonian step, computed
On the canonical compiler, the shear step (x,p)→(x+p,p) has Jacobian determinant 1, so a state-cloud of area 6 stays area 6:
$ i13 run inv_liouville.i13 # Jacobian of a Hamiltonian (shear) step
jacobian = 1
area_before = 6
area_after = 6 -- phase-space volume preserved
Recommend: Liouville is the determinant invariant (dart 279) promoted to a law of motion, and it ties the batch back to a keeper. A Hamiltonian step is a shear, whose Jacobian i13 computes as 1, so the phase-space area is preserved exactly — the same volume-invariance the determinant measures, now enacted as time evolution. This is precisely why Verlet integration (the batch-16 keeper) is trustworthy: it is symplectic, so it respects Liouville's volume where a naive integrator would leak it. i13 verifies the invariant with a single determinant; the physics is that time cannot compress phase space.