Liouville's theorem can be thought of as conservation of information in classical mechanics.

Suppose you have a bowl, perhaps of some slightly-wonky shape, and a marble that can roll around in the bowl. You put the marble down somewhere and give it a push. (We'll call the this the initial state.) The marble does its thing. Using physics, you can predict where it will be and how fast it will be going 10 seconds from now. (We'll call this the final state.)

However, you might not know the initial state perfectly. Instead there is some range of possible initial states. The size of the range of initial states represents your uncertainty.

Due to the uncertainty, you can't calculate the exact final state. Instead, there is uncertainty about the final state. Liouville's theorem says that you have the same amount of uncertainty about the initial and final states.

The size of the uncertainty is a measure of how much information you have, so Liouville's theorem says that you neither gain nor lose information, i.e. information is conserved. (Specifically, the information you have is the negative of the logarithm of the uncertainty.)

This is best-visualized in phase space. The marble is now a point that moves through some trajectory or other. Your uncertainty is a region made of many points all following their own trajectories. Imagine a very small rectangular initial uncertainty propagating a small timestep forward.

Your initial uncertainty is the small rectangle with four different-colored dots at the corners. This turns into the parallelogram. Each of the corners has some [math](\mathrm{d}p,\mathrm{d}q)[/math] vectors that describes how it moves, but I've drawn that vector for only the bottom-left corner.

The change in volume of the rectangle is the divergence

[math]\mathrm{d}V = \frac{\partial \mathrm{d}q}{\partial q} + \frac{\partial \mathrm{d}p}{\partial p}[/math]

Hamilton's equations tell us

[math]\mathrm{d}q = \frac{\partial H}{\partial p} \mathrm{d}t[/math]

[math]\mathrm{d}p = -\frac{\partial H}{\partial q} \mathrm{d}t[/math]

for some function [math]H[/math], called the Hamiltonian. This makes it clear that [math]\mathrm{d}V=0[/math] for our small rectangle.

Because a large initial uncertainty is just a bunch of small rectangular uncertainties added up and a long time is just a bunch of short times added up, the volume of our uncertainty in phase space doesn't change for any Hamiltonian system. This is Liouville's theorem.

In chaotic systems, the volume and topology of the uncertainty region don't change, but it can get stretched and strung out in very complicated ways, so that in practice we lose information, even though in theory we do not.

The analog to Liouville's theorem in quantum mechanics is Unitarity (physics).