Lille's Theorem (also known as Liouville's Theorem) establishes that in Hamiltonian mechanics, the flow through phase space is volume-preserving and incompressible due to the divergence-free nature of the velocity field derived from canonical equations of motion. This principle implies that the time-evolution mapping between initial conditions in configuration-momentum coordinates acts locally like a unitary transformation where trajectories cannot intersect or merge without loss of distinguishability, thereby conserving phase space volume regardless of coordinate transformations. The theorem serves as the classical mechanical analog to quantum reversibility and underpins the statistical mechanics definition of entropy by ensuring that the density of states remains constant over time for isolated systems with Hamiltonian dynamics.
Lille's Theorem (also known as Liouville's Theorem) establishes that in Hamiltonian mechanics, the flow through phase space is volume-preserving and incompressible due to the divergence-free nature o…