Liouville's theorem

Liouville's theorem states that in a conservative physical system, such as a galaxy or a fluid made of particles, the density of points in phase space (a space combining positions and velocities) remains constant over time. Imagine particles moving around without any loss or gain of energy—no particles suddenly disappearing or appearing. If you look at a group of particles in phase space, their overall distribution doesn't thicken or thin out as they evolve, meaning the total "amount" of these points stays the same. This principle is fundamental in physics and helps us understand how systems evolve while conserving certain quantities.