What term describes the trajectory space splitting behavior when parametrizing a pendulum?

Question

So I was thinking about this post I made earlier: What is the second conserved Quantity of the Pendulum?

In which a pendulum appears two have significant properties. It's Kinetic Energy and its Phase.

I was thinking about pendulums and I wanted to characterize their trajectories (in a time forgetful way). One thought is to use the single parameter "Total Energy", by observing that $PE+KE$ is constant for the pendulum at all points in its trajectory, and therefore uniquely identifies the trajectory.

Or so I thought... I get into a problem when I consider pendulums whose $PE + KE$ exceeds a critical "roll around the top" constant $K(\text{mass}, \text{length})$. In this case it seems like the space of trajectories splits into a space that requires both $PE + KE$ as well as a binary $\leftarrow, \rightarrow$ orientation (since the Pendulum could be spinning to the left or to the right once its past this critical threshold).

Mathematically this is a very strange situation. What is the common mathematical terminology of this "space splitting" or "space transition" behavior? Are there other examples of physical systems or general ODES/PDES/Functional equations whose space of solutions have this property?

*Note the case of $PE+KE$ = Critical Constant, has an enormous plethora of solutions so the full picture is that below the constant a single real number suffices, at the constant the solution space becomes as big as function space itself, and above the constant the solution space is described by a single real number and a binary bit indicating left or right handedness.

Answer 1

You can describe the position of the pendulum with one degree of freedom, the angle $\theta$ which the pendulum mass makes with the vertical. Since $\theta$ is an angle, we identify $\theta$ and $\theta+2\pi$, meaning that (for example) $\theta=0$ and $\theta=2\pi$ are really the same physical point. Let's choose a convention where $\theta=0$ is the equilibrium position of the pendulum bob hanging straight down, and $\theta=\pm \pi$ is the top of the pendulum (the pendulum is pointing vertically up). I wrote $\theta=\pm \pi$ since $\theta=\pi$ and $\theta=-\pi$ differ by $2\pi$, and are therefore the same physical point. Then, it is convenient to imagine the space of possible positions of the pendulum as a line extending from $-\pi$ to $\pi$. Whenever $\theta$ crosses the boundary at $\pi$, it loops around to $-\pi$ like pacman crossing the edge of the screen.

For "small oscillations" of the pendulum (those which oscillate back and forth" around the vertical, rather than making a full orbit around the pendulum's pivot point), the absolute value of $\theta$ is always less than $\pi$, $|\theta|<\pi$. In this situation, $\theta$ never crosses the boundaries at $\pm\pi$. Then the pendulum undergoes oscillatory motion that repeats in time; if you walked into the pendulum long after someone started it moving, you could not infer what the initial conditions were (modulo the total energy that was given to the system originally, assuming there is no friction or air resistance).

For "large oscillations" (those which loop around the pivot point), $\theta$ crosses the boundary at $\pi$. Then (as you pointed out) there are two cases. $\theta$ can increase until it crosses the boundary at $\pi$ and move to $-\pi$, or $\theta$ can decrease until it crosses the boundary at $-\pi$ and move to $\pi$. This extra information is encoded in the state of motion, so if you walked into the room with the pendulum you could infer some extra information about the initial condition, namely whether it was set on a clockwise or counterclockwise orbit (although it is worth pointing out that if you viewed the pendulum from the other side of the plane of the pendulum, the sense of this rotation would be reversed).

The reason there are two different kinds of behavior is because the "phase space" has a non-trivial topology (or connectedness). The "small oscillation" orbits are "simply connected" (since they can be continuously shrunk to a point), while the orbits which wrap around the space are not simply connected.