Poisson Bracket - Constants of Motion

Constants of Motion

An integrable dynamical 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 the Hamilton's equations of motion, then one has that along that trajectory. Then one has

where, as above, the intermediate step follows by applying the equations of motion. This equation is known as the Liouville equation. The content of Liouville's theorem is that the time evolution of a measure (or "distribution function" on the phase space) is given by the above.

In order for a Hamiltonian system to be completely integrable, all of the constants of motion must be in mutual involution.