 Liouville's theorem (Hamiltonian)

Liouville's theorem states that a smooth Hamiltonian flow preserves canonical phase-space volume. With coordinates $(q_i,p_i)$, the divergence of the phase-space velocity isprovided the mixed derivatives agree. This also holds for a smooth explicitly time-dependent Hamiltonian.

An evolving patch may stretch, fold or shear while retaining its volume, much as an incompressible fluid element can change shape. For a harmonic oscillator, an ellipse of initial positions and momenta rotates around the equilibrium without shrinking. Dissipative attraction toward a point therefore cannot be represented by a smooth finite-dimensional Hamiltonian flow of those same closed-system variables alone. In statistical mechanics, volume preservation supports the transport description of a phase-space distribution; a coarse observational description can still lose visible structure.

Liouville's theorem in the context of Hamiltonian mechanics is a fundamental result concerning the conservation of phase space volume in a dynamical system. The theorem states that the flow of a Hamiltonian system preserves the volume in phase space. More formally, consider a Hamiltonian system described by \( (q, p) \), where \( q \) represents the generalized coordinates and \( p \) represents the generalized momenta.