An integrable system will have constants of motion in addition to the energy. Such constants of motion will commute with the Hamiltonian under the Poisson bracket. Suppose some function f ( p , q ) is a constant of motion. This implies that if p ( t ) , q ( t ) is a trajectory or solution to Hamilton's equations of motion, then 0 = d f d t along that trajectory. Then 0 = d d t f ( p , q ) = { f , H } where, as above, the intermediate step follows by applying the equations of motion and we assume that f does not explicitly depend on time. This equation is known as the Liouville equation. The content of Liouville's theorem is that the time evolution of a measure given by a distribution function f is given by the above equation.