The pendulum that draws itself

March 22, 2026 · 4 min · calculus, teaching

A pendulum looks like it has one number: the angle. But an angle alone cannot tell you what happens next. A pendulum at the bottom of its swing might be about to fly leftward, fly rightward, or sit perfectly still — same angle, three futures. To know the future you need a second number, the speed, and the moment you admit that, the whole subject opens up. The pendulum toy shows the swing on the left and, on the right, those two numbers plotted against each other, drawing a shape as the bob moves.

Two numbers, one point

Call the angle θ and the angular speed ω. At any instant the pendulum’s entire state is the pair (θ, ω) — a single point on a flat map we call phase space. As the pendulum swings, that point moves, and the toy leaves its trail. This is a genuinely new way of looking, and I try not to rush past it: a thing moving in one dimension (a bob on an arc ) becomes a point moving in two. Motion in time turns into a curve in space, and the curve holds the whole story at a glance.

The shapes the motion makes

Start the pendulum with a gentle push and, in phase space, the point goes round and round a closed loop — a tidy oval. That loop is the pendulum minding its energy: at the ends of the swing it is high and slow (far left and right on the map, near ω = 0); at the bottom it is low and fast (top and bottom of the loop, biggest ω). Round and round, the same loop forever, if there is no friction. A closed curve in phase space is a promise of eternal repetition.

Push it harder and the oval swells and changes character, its top and bottom drawn out until, at the very edge of going right over the top, the loop stops closing altogether. That boundary curve — the swing that just reaches the vertical and stalls there for an eternity — is one of my favourite objects in all of school physics, because it is the exact border between swinging back and forth and spinning all the way round. The map shows you the border as a shape you can point at.

Where the energy goes

Now switch on the damping, and the promise breaks in the most beautiful way. Friction bleeds a little energy from every swing, so the pendulum never quite returns to where it started. On the map, the point no longer closes its loop — it falls slightly short each time and spirals inward, loop shrinking within loop, homing toward the single dead point at the centre where the pendulum hangs still.

That centre point is an attractor, and the spiral is the pendulum’s whole life story told in one curve: it will end at rest, and here is the path it takes to get there. Students who have watched the spiral wind in never again ask “but where does the energy go?” as an abstract question. They saw it go — each turn of the spiral a little smaller than the last, energy leaking out at the rim, the motion collapsing gracefully toward the middle.

The reason I built the phase portrait to draw itself, live, beside the swinging bob, is that the two pictures teach different halves of one idea. The swinging bob is what your eye already believes. The spiral is what your eye cannot see but the mathematics insists is true: that a pendulum is not really a thing on a string, it is a point on a map, obeying a rule that turns where it is and how fast into where it will be next. Every differential equation in the world works this way. The pendulum is just the first one honest enough to draw you its map.