# Lagrange Triangle

Bodies · mechanical systems

```
gravity
m=1  x=1     y=0        vx=0        vy=0.7598
m=1  x=-0.5  y=0.8660   vx=-0.6580  vy=-0.3799
m=1  x=-0.5  y=-0.8660  vx=0.6580   vy=-0.3799
```

[Open in the app](https://www.wavelace.com/app#p=83) · [This page](https://www.wavelace.com/presets/lagrange-triangle)

### The setup

Three equal masses sit at the corners of an equilateral triangle inscribed in the unit circle, so every side is `√3 ≈ 1.732`. Each velocity has the same length, `0.7598`, turned a quarter turn from its own radius. That sets the whole figure spinning about the centre without changing its shape.

### Why it turns

Take one corner. Its two neighbours are each at distance `√3`, pulling with `1/3` apiece, and the two pulls are `60°` apart. They add to `2 · (1/3) · cos 30° = 1/√3`, aimed straight at the centre. A circle of radius 1 needs `v²` for that, which fixes `v = √(1/√3) ≈ 0.7598`. One turn then takes `2π/v ≈ 8.27` seconds. The same sum works for a triangle of any size.

### History

Lagrange found the equilateral configuration in 1772, one of the few exact solutions of the three-body problem. The Trojan asteroids, sixty degrees ahead of and behind Jupiter along its orbit, are the case nature runs.

### Why it does not last

Exact is not the same as stable. The equilateral solution survives a nudge only when one mass dominates the other two, and equal masses fail that test badly. Here the sides hold to within a percent for some thirteen seconds, a turn and a half. By twenty-one seconds the triangle is unrecognisable. Two bodies then pair off into a tight binary and the third is flung away.

### Try

- `Top` in the deck: the triangle flat, turning as one piece for the first two turns.
- Carry more digits, `y=0.8660254` and speeds from `vy=0.7598357` on down. The figure lasts three and a half turns instead of one and a half. Digits buy turns, never stability.
- Delete every `vx` and `vy`. With nothing said the speeds start at zero, so the three fall straight through the middle and are slung back out.
- Raise `Span` once the triangle has broken, to follow the body that leaves.

### Read more

- [Lagrange point](https://en.wikipedia.org/wiki/Lagrange_point)
- [Trojan (celestial body)](https://en.wikipedia.org/wiki/Trojan_(celestial_body))
- [Central configuration](https://en.wikipedia.org/wiki/Central_configuration)
- [Three-body problem](https://en.wikipedia.org/wiki/Three-body_problem)