# Double Pendulum

Bodies · mechanical systems

```
pendulum
l=1  m=1  a=2.4
l=1  m=1  a=2.4
```

[Open in the app](https://www.wavelace.com/app#p=86) · [This page](https://www.wavelace.com/presets/double-pendulum)

### The setup

Two rods hang in a chain, both of length 1 and mass 1, both released from rest at `a = 2.4` radians. That last column is the starting angle, measured from straight down, so `2.4` radians is `137.5°`. Both joints therefore begin above the pivot, which is why the first swing is a drop rather than a swing. The chain hangs from a pivot at the height of its total length, so stretched out it just touches the floor.

### Why it is unpredictable

Two rods need two angles, and the equation that governs them, `M(θ) θ̈ + C(θ, θ̇) + G(θ) = 0`, is nonlinear in both. With this much energy the lower joint can swing over the top, and it does. The upper rod alone turns twelve full circles in the first minute. Nothing repeats.

The sharper statement is how fast a difference grows. Start a second run a thousandth of a radian away, and the two are a whole radian apart by `t ≈ 4.5`. Start it a hundred-thousandth away, a hundredfold better, and that only postpones the parting to `t ≈ 6.3`. Every extra digit of the starting angle buys the same small fixed stretch of agreement, which is what a positive Lyapunov exponent means in practice.

### Small swings

Set both angles small and the same system turns orderly. Its two normal modes are the rods swinging in step, with a period of `2π/√((2 − √2) g) ≈ 2.62` seconds, and the rods swinging opposed, at `≈ 1.09` seconds. Any small motion is a mixture of the two, and it beats between them forever.

### Try

- `Front` in the deck: the swing plane seen face on, the way a pendulum is drawn.
- Tame it with `a=0.2` on both lines. A thousandth of a radian of difference stays a thousandth, and the two-mode beat is plain.
- Lengthen the lower rod, `l=1.6` on the second line: the same wildness, a different shape of path.
- Lighten the lower rod instead, `m=0.05` on the second line, and the upper rod swings almost as a simple pendulum.
- `Rewind` runs the identical path again. The motion is unpredictable, not random.

### Read more

- [Double pendulum](https://en.wikipedia.org/wiki/Double_pendulum)
- [Chaos theory](https://en.wikipedia.org/wiki/Chaos_theory)
- [Lyapunov exponent](https://en.wikipedia.org/wiki/Lyapunov_exponent)
- [Normal mode](https://en.wikipedia.org/wiki/Normal_mode)
- [Lagrangian mechanics](https://en.wikipedia.org/wiki/Lagrangian_mechanics)