Visualizing Louisville Theorem: Explanation

[Winzer]

I am trying visualize(graphically) what Louisville theorem is saying. So if we have some system that can be described in space, imagine the line phase space lines. It says that the density is some volume element is constant? Please give a thorough explanation.

 

[confinement]

It is best to start understanding the theorem for a particle in one-dimension, so that the phase space is 2-dimensional and Louiville's THM says that Hamiltonian evolution preserves areas in phase space. Since an initial condition is just a point, an area in phase space corresponds to a set of initial conditions. Each of these initial conditions becomes a trajectory, and at some later timethe points will all have moved somewhere else but they will still occupy the same total volume in phase space.

For contrast, take a damped pendulum, which is typically not a Hamiltonian system. Since the pendulum bob eventually comes to rest at a single point in phase space, no matter how large your initial area is it will shrink until it is only the size of one point.