The pendulum, the cat's eye, and a little friction

The pendulum is what you get when you stop lying about the spring. Everybody’s first pendulum in a physics class is secretly a spring in disguise, because the very first thing the textbook does is wave its hands and replace $\sin\theta$ with $\theta$. That’s fine for a wristwatch, where the swing never gets big, but it throws away everything interesting. I wanted to keep the sine, so the widget is drawing the honest equation this time.

Here’s where it comes from. Hang a mass on a rigid rod of length $L$ and let $\theta$ be the angle off straight down. Gravity pulls the mass straight down with force $mg$, but the rod only lets it move along its arc, and the part of gravity that actually pushes it along that arc is $mg\sin\theta$. It points back toward the bottom, so once again there’s a minus sign. Balancing that against the mass’s acceleration along the arc and canceling a length gives

$\ddot{\theta} = -\tfrac{g}{L}\sin\theta.$

Same shape as the spring, restoring force pulling back toward the middle, except the strength of the pull is $\sin\theta$ instead of $\theta$. For small swings the two are nearly equal and you’re back to the spring. For big ones they part ways, and near the very top $\sin\theta$ rolls back to zero, which is the pendulum telling you that balanced perfectly upside down it feels no push at all. Split it into a state pair the same way, angle and angular velocity $(\theta, \omega)$:

$\dot{\theta} = \omega, \qquad \dot{\omega} = -\tfrac{g}{L}\sin\theta.$

interactive · the pendulum in phase spaceopen ↗

Leave the damping at zero. Down the middle you get the same nested loops the spring had, because a pendulum making small swings is a spring. But follow them outward and they stop being ellipses. They stretch, flatten along the top, and eventually a special curve comes through and pinches everything into the shape that made me want to build this in the first place, the row of cat’s eyes. The green dots at the bottom of each eye are the stable resting points, the pendulum hanging straight down, and they repeat across the picture because turning the pendulum all the way around, $2\pi$ in angle, brings it right back to where it started. The red dots sitting between the eyes are the pendulum balanced upside down. Those are fixed points too, but the unstable kind, the saddle. Land exactly on one and you’d hang there forever, but the faintest nudge and you’re gone.

The red curve threading through the saddles is the separatrix, and it’s the most important line on the plot. Inside an eye the pendulum is swinging back and forth, over and over, never making it to the top. That’s libration. Outside the eyes, the flow stops looping and streams across the picture in one direction, and that’s the pendulum with enough energy to go over the top and just keep going around and around like a propeller. The separatrix is the exact border between those two lives, the razor’s edge where the pendulum crawls up to the top and arrives with precisely nothing left. Everything the pendulum can do is sorted by which side of that curve you drop it on.

Now bring the damping slider up. Friction bleeds energy the same way it did for the spring, so the closed eyes unwind into spirals and everything drifts down toward the bottom of the nearest well. A pendulum that started out flying over the top loses a little each pass until one time it doesn’t quite clear it, drops into an eye, and rings itself down to rest hanging straight down. On the plot you can watch a streak cross the old separatrix, get captured, and spiral in. The upside-down saddles stay exactly where they were, unwelcoming as ever, because friction doesn’t make balancing on top any easier, it just guarantees you won’t be circling up there for long.

This is also the doorway to the messy stuff. Give the pendulum a push on a timer instead of letting it wind down and that clean sorted picture starts to fold over on itself, and you’re standing at the edge of chaos. I’ve poked at that end of it before. But even the plain damped pendulum, sitting here doing nothing but swinging and stopping, has more geometry packed into it than the spring ever did, and all of it came from refusing to erase one sine.