# Liouville's theorem (Hamiltonian)

In physics, **Liouville's theorem**, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that *the phase-space distribution function is constant along the trajectories of the system*—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability.[1]

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There are related mathematical results in symplectic topology and ergodic theory; systems obeying Liouville's theorem are examples of incompressible dynamical systems.

There are extensions of Liouville's theorem to stochastic systems.[2]