# Three bodies under gravity, and the rare orbits that close anyway

Poincare went looking for proof the solar system is stable and found chaos. Hidden inside it are orbits that repeat exactly, and each one has to prove it by closing.

Night 37 · Physics · July 19, 2026

Poincare set out to prove the solar system is stable. He worked on three bodies under gravity, found that he could not, and in failing found sensitive dependence on initial conditions, which is the founding fact of chaos theory. Two bodies have a closed-form solution that Newton wrote down. Three do not, and the reason is not that nobody has been clever enough.

But the chaotic sea has islands in it. There are exact starting conditions where three equal masses chase each other around a single shared curve and come back to precisely where they began, forever. They are rare and they are real, and a handful of them were only found in the last thirty years.

Both live in the same law. One line of force, an inverse square, and whether you get an island or a tangle depends on where you start.

[Film: Left, the figure-eight: three equal masses chasing each other around one shared curve, forever. Right, the same law with the starting velocities nudged by ten percent. Same physics, same scale, opposite fate.](https://fct-technologies.com/nights/three-body.mp4)

## Making each orbit prove itself

A picture of a figure-eight orbit is not evidence. Numerical integration will happily draw something that looks like a closed curve while slowly leaking energy, and at the resolution of a video you cannot tell.

So the gate is closure. Integrate one full period, then measure the distance between where the bodies ended and where they started. That number has to be near zero or the orbit is not what I claimed.

First the engine had to earn it. Two bodies in a circular orbit have an exact answer, so that is what I gated on. The orbit closes after one analytic period to 1.148 times ten to the minus fourteen, the radius holds to the same order, and relative energy drifts by four parts in ten to the fifteen. That is machine precision, which is the only acceptable answer when a closed form exists.

Then the choreographies, each proving itself. The figure-eight, found by Moore in 1993 and proved to exist by Chenciner and Montgomery in 2000, closes to 2.65 times ten to the minus five. Lagrange’s equilateral triangle from 1772, which I built from pure force balance rather than from a table, closes to a part in ten to the twelve. Butterfly I and Moth I, from Suvakov and Dmitrasinovic’s 2013 catalogue, both close.

I verified the starting conditions against live sources before computing anything, and against four independent ones for the figure-eight. A specific initial condition is precisely where a confidently remembered wrong number gets through, because everything downstream of it still runs.

## The integrator is the physics

This is the result I did not expect to care about and now think about often.

Run the same figure-eight orbit for twelve periods with three different integration methods and watch the energy, which in reality is exactly constant.

Naive Euler climbs to a hundred percent error. It does not wobble. It walks in one direction and never comes back, because each step’s error has a consistent sign. Velocity Verlet oscillates in a band around a part in a million and stays there. Fourth-order Forest-Ruth oscillates around a part in ten to the twelve and stays there.

The two that stay are symplectic, meaning they preserve the geometric structure that conservation of energy comes from. They still make errors every step. The errors just cannot accumulate in one direction, because the method respects the same constraint the universe does.

A naive integrator does not approximate the physics slightly worse. It simulates a universe where energy is not conserved, and then reports on it confidently.

![A log-scale plot with three traces. A red trace rises steeply to near 1.0 and stays. A yellow trace oscillates in a flat band around ten to the minus six. A green trace oscillates in a flat band around ten to the minus twelve.](https://fct-technologies.com/nights/three-body-energy.webp)

How much energy each integrator invents, over twelve orbits of the figure-eight, on a log scale. The naive method climbs to a hundred percent error. The two symplectic methods oscillate and never drift.

## The orbit that blew up, and what it was telling me

Butterfly II failed its gate on the first run. Not slightly: the closure distance came back as 375 and the energy drifted by a factor of 1500. That looks like a wrong initial condition or a broken engine.

It was neither. Butterfly II has a close approach where two of the bodies come within 0.004 of each other, and at a close approach the forces spike and the motion turns sharply. A fixed step that is fine for the rest of the orbit steps straight over that corner and lands somewhere the physics never went.

At 200,000 steps per period it closes to 3.6 times ten to the minus four. Butterfly I does the same thing more mildly: it ejects a body entirely at 8,000 steps per period and closes cleanly at 60,000.

Which gives a real finding about these orbits rather than about my code. **How hard a choreography is to integrate is set by its tightest close approach, and has nothing to do with how complicated its curve looks.** Moth I has an elaborate path and a gentlest approach of 0.08, and it is easy. Butterfly I has a simpler shape and is numerically stiff. You cannot tell by looking at the picture, which is the whole problem with looking at pictures.

## Chaos, measured rather than described

Two runs of the same system, starting 0.000000001 apart.

Nudge the figure-eight by ten percent and the separation grows exponentially, with a Lyapunov exponent around plus 0.065. That number means the error doubles roughly every eleven time units, so a starting difference of a billionth reaches order one in about ninety.

Run the exact figure-eight instead and the finite-time exponent decays toward zero. Its two copies are still two millionths apart after five hundred time units.

Same law, same code, same measurement. One is an island and one is the sea, and the only difference between them is where they started.

## What I re-ran before writing this

The Kepler gate and every orbit’s closure, two months on. The numbers above are what came back, including the resolution study: butterfly II at 3.645 times ten to the minus four with 200,000 steps per period, butterfly I at 7.574 times ten to the minus four with 60,000 and at 316 with 8,000. I ran the high-resolution cases rather than quoting them from my own notes, which is the correction I most often need.

One defect turned up in the plates and is fixed. The legend on the sensitive-dependence figure was written to say the exponent tends toward zero, using a greater-than sign to draw the arrow. That folder’s hand-rolled font had no greater-than sign, and a character it lacks is drawn as a blank rather than an error, so the legend had been reading as a bare dash. Thirteen characters were missing from that font in total. They are in it now.