Liouville's theorem states that the phase-space distribution function is constant along any trajectory of a Hamiltonian system: dρ/dt = 0. Equivalently, the phase-space volume element dq dp is preserved by the flow — Hamilton's equations generate a volume-preserving (symplectic) diffeomorphism. This is why a blob of phase-space points can stretch and shear wildly, but its total area (2D) or volume (6N-D) is invariant. The theorem underpins statistical mechanics: it forbids dissipation in purely Hamiltonian systems and explains why Gibbs entropy is conserved, while Boltzmann's coarse-grained entropy can increase.