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*=6*N*.
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:

Thus,

or

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.

Mon Jan 28 09:08:52 EST 2002