Liouville's theorem
Under Hamiltonian flow, phase-space volume is exactly conserved. The foundation of classical statistical mechanics.
Definition
Liouville's theorem states that the flow generated by a Hamiltonian preserves phase-space volume: if you follow a region R(0) of initial conditions forward in time under Hamilton's equations, the volume of R(t) equals the volume of R(0) for all t. The proof is one-line: the divergence of the phase-space velocity field (q̇, ṗ) vanishes identically, because ∂q̇/∂q + ∂ṗ/∂p = ∂²H/∂q∂p − ∂²H/∂p∂q = 0.
The consequence is deep. Dissipative systems (friction, damping) shrink phase-space volume as they evolve; conservative Hamiltonian systems cannot. Classical statistical mechanics lives on exactly this geometric fact — the microcanonical ensemble is well-defined precisely because the phase-space measure is preserved by evolution.
History
Proved by Joseph Liouville in 1838.