Consider a phase space volume element at t=0, containing a small collection of initial conditions on a set of trajectories. The trajectories evolve in time according to Hamilton's equations of motion, and at a time t later will be located in a new volume element as shown in the figure below:
How is related to ? To answer this, consider a trajectory starting from a phase space vector in and having a phase space vector at time t in . Since the solution of Hamilton's equations depends on the choice of initial conditions, depends on :
Thus, the phase space vector components can be viewed as a coordinate transformation on the phase space from t=0 to time t. The phase space volume element then transforms according to
where is the Jacobian of the transformation:
where n=6N. The precise form of the Jacobian can be determined as will be demonstrated below.
The Jacobian is the determinant of a matrix M,
whose matrix elements are
Taking the time derivative of the Jacobian, we therefore have
The matrices M and can be seen to be given by
Substituting into the expression for dJ/dt gives
where the chain rule has been introduced for the derivative . The sum over i can now be performed:
The initial condition on this differential equation is . Moreover, for a Hamiltonian system . This says that dJ/dt=0 and J(0)=1. Thus, . If this is true, then the phase space volume element transforms according to
which is another conservation law. This conservation law states that the phase space volume occupied by a collection of systems evolving according to Hamilton's equations of motion will be preserved in time. This is one statement of Liouville's theorem.
Combining this with the fact that df/dt=0, we have a conservation law for the phase space probability:
which is an equivalent statement of Liouville's theorem.