The Liouville Theorem is in fact quite easy to understand ... though a formal proof might be more complicated.
Anyway. Physicists like to look at problems in phase space
, that is, a coordinate system of spatial and momentum coordinates. In that space an object therefore usually has 6 coordinates - 3 for space (x1,x2,x3) and 3 for momentum (p1,p2,p3). These values determine the future of the object - using the Hamilton formalism
one can calculate its trajectory
in phase space.
Now suppose you have a number of points in phase space like this:
p /\ .
If now one of the inner points' trajectories were to touch one of the outer points', from then on it would evolve
right along with that other point. That means it can not cross!
And in fact, two trajectories cannot even touch because there are always other trajectories in between (the coordinates are real number
s) which would have to be crossed first!
Therefore an inner point will always remain an inner point ... the group of points may change its shape, but the area (or volume in 6d) has to stay constant because there's no place to go for the inner points. A phase space volume
behaves like an incompressible liquid
and the phase space density is constant!
PS: It seems that the theorem ariels
refers to is a different one, also called Liouville's Boundedness Theorem