Concept of movement of the representative points in the phase space

Question

The derivation of Liouville's theorem (for example this one) makes use of the concept of "movement" of the points in the phase space.

Let it be a system of $N$ particles, with $q=\{q_i\}_{i=1}^{3N}$ the generalized coordinates and $p=\{p_i\}_{i=1}^{3N}$ the generalized momenta of the $N$ particles. If $(q,p)$ is a point of the phase space (also called representative point), a velocity vector $\vec{v} = (\dot{q}, \dot{p})$ is defined for $(q,p)$. This velocity, as time passes by, will give the direction of the trajectory of that point in the phase space.

This idea of the motion of representative points is something that confuses me, since in the three-dimensional space, for instance, the points $(x,y,z)$ of the space do not move anywhere (the point $(-3,9,1)$ will remain located at $(-3,9,1)$ in saecula saeculorum). In any case, what will move is the position vector $\vec{r}(t)=(x(t),y(t),z(t)$.

How should we understand the idea that, with time, representative points moves, entering and leaving a $\omega$ volume of phase space?

Answer 1

The best way to understand the idea is to take some simple example and then generalize it to prove the general result. Here I'll try to give the idea of representation point as you said:

This idea of the motion of representative points is something that confuses me.

The most simple example that every familiar with is of Simple pendulum but I'm involving some damping in it. The answer will make more sense if you go through the following visual.

Now consider an ensemble of pendulums with different initial position and momentum. When you set the motion On. The pendulums will start to move, and the system will evolve in time under a given Hamiltonian equation. The following is a nice picture of it:

Image Credited: 3B1B

Now for a second forget about all of them, just look at one of them, for say $\theta=30^o$ and $p_\theta= 2$ in suitable units. Now draw a point with this coordinate. Now let the system evolve in time and draw a point as they move (This can be done using Hamiltonian of the system). Now do the same for each state, remember each state is denoted by a point in phase space. What you get is a phase portrait of a system. That in this case looks like:

Image Credited: 3B1B

Each point in this space corresponds to some state of the system, that's what representation thing is.

Now what Liouville's theorem says (naively) is that If you suppose these state as fluid elements which are moving (or flowing) is phase space, then the volume of fluid element conserve as system evolve which is basically the incompressibility of fluid.

Image Credited: 3B1B