How do i read a pendulum phase diagram? I'm trying to understand how the intermediate axis theorem works. And in one of the works that I found, they used a pendulum phase diagram, but idk how to read it. Can anybody help please?
The work that I'm looking at is https://arxiv.org/abs/1606.08237 and the diagram I'm looking at is on page 8.
But it looks similar to this:

 A: This is a diagram in two dimensional phase space. The horizontal axis $q$ is the angle that the pendulum makes with the vertical. The vertical axis $p$ is the associated momentum i.e. the angular momentum of the pendulum.
In each of the orbits inside the red line the pendulum swings to and fro between a position $A$ with angle $-\theta$ and a position $B$ with angle $\theta$. At $A$ and $B$ the pendulum’s angular momentum is zero. As it swings from $A$ to $B$ its angular momentum is positive, reaching a maximum when $q=0$. As it swings from $B$ back to $A$ its angular momentum is  negative, reaching a minimum when $q=0$.
The orbits outside the red line are when the pendulum has sufficient momentum to swing “over the top” and continue rotating for ever. The orbits above the red line are rotations with positive momentum. Those below the red line are rotations in the opposite direction, with negative momentum.
The phase space diagram is laid out on a plane, but if you imagine it wrapped around so that the line $q=\pi$ meets the line $q=-\pi$ then you can see that the phase space of a simple pendulum is actually a cylinder.
