What you are looking at
A second pendulum hangs from the bob of the first. Each arm pulls on the other, and the result is motion
that swings, flips, and loops in a way that looks random — even though the rules are completely fixed and
the system has no randomness in it at all. The glowing trail is the path traced by the lower bob.
Deterministic, but chaotic
The double pendulum obeys exact equations derived from Newton's laws (via the Lagrangian). Given the arms'
lengths, the masses, and the starting angles, its future is
completely determined. Yet it
is the classic example of
chaos: it shows
sensitive dependence on initial
conditions. Two double pendulums released from
almost the same angle stay together for a
moment, then diverge and end up doing utterly different things — the "butterfly effect." Turn on the
chaos twin (a faint copy started just half a degree off) and watch the two separate.
tiny difference now → huge difference later
This is why the long-term motion is unpredictable in practice, even though it is perfectly deterministic in
principle. The single pendulum is regular; adding just one more joint is enough to unleash chaos.
Energy is still conserved
With no damping, the total mechanical energy — kinetic plus gravitational potential — stays
constant the whole time; the motion just shuffles it endlessly between the two arms. The
"Energy" readout barely budges, confirming the simulation is faithful (it uses a 4th-order Runge–Kutta
integrator). Add a little
damping and the energy bleeds away, the chaos calms, and the
pendulum eventually settles straight down.
Things to try
Release from a large angle for the wildest behaviour; from a small angle it stays comparatively tame. Make
the lower arm heavier or longer and the character of the motion changes completely. Above all, switch on the
chaos twin and release — the moment the two copies peel apart is the whole idea of chaos in one picture.