🎢 The double pendulum

What you’re seeing

Two double pendulums hang from the same pin: a rod with a weight, and a second rod with a weight hanging from the first. Pendulum B starts with its upper rod turned a tiny bit more than pendulum A, by default one millionth of a radian. That is far less than the width of a hair at this size.

At first they move as one, and you only see one. Then, after a few seconds, they split and move in completely different ways. The red line below shows the gap between them on a log scale, where each grid line is 100,000 times the one below it. The line climbs at a fairly steady slant, then levels off once the gap is as big as it can get.

Why it works

The equations are exact. There is no randomness here. Newton’s laws give two equations for the two angles $\theta_1$ and $\theta_2$. With equal rods of length 1 m, equal weights and gravity $g$, the first one is

$$\theta_1'' = \frac{-3g\sin\theta_1 - g\sin(\theta_1 - 2\theta_2) - 2\sin(\theta_1-\theta_2)\left(\omega_2^2 + \omega_1^2\cos(\theta_1-\theta_2)\right)}{3 - \cos(2\theta_1 - 2\theta_2)}$$

where $\omega_1 = \theta_1'$ and $\omega_2 = \theta_2'$ are the spinning speeds. The second is similar. Same start, same motion, every time.

But small gaps grow. For a chaotic system, a small gap $\delta_0$ tends to grow roughly like

$$\delta(t) \approx \delta_0\, e^{\lambda t}$$

for some rate $\lambda > 0$. Growth like this is a straight slanted line on a log scale, which is what the chart shows. Doubling your precision buys only a fixed amount of extra time. If the gap is $10^{-6}$ and it grows by a factor of 10 every second, you have about 6 seconds before the two disagree completely. Start at $10^{-9}$, a thousand times more precise, and you get only about 3 more seconds.

Gentle swings are tame. Start with small angles, and the pendulum behaves almost like two linked springs: regular, repeating, and predictable. Chaos needs enough energy for the rods to whip around. Try a start angle of 30° and then 150°.

How the computer does it. We can’t solve these equations with a formula, so the page steps them forward 600 times a second of motion with the Runge–Kutta method (RK4). The method has its own tiny errors, and in a chaotic system those grow too. So the exact path shown after a minute is not the true path; what is reliable is the behavior: the split, its timing, and the kind of motion.

Chaos in one sentence. A system is called chaotic when it follows fixed rules but tiny differences in where it starts grow fast, so long-range prediction is impossible in practice. Weather is the famous example; see the Lorenz attractor in this gallery.

Try this

• Set the start angle to 30°. Do the pendulums ever split? Now try 150°.
• Change the difference from $10^{-6}$ to $10^{-9}$. How much longer do they stay together?
• Change it to $10^{-2}$ (about half a degree). They split almost at once.
• Use the Step button to walk through the moment they part.