Everything so far has been a world of two bodies. Ellipses closed, positions came from a formula, and a thousand years ahead was the same calculation.
Add a third. Three equal masses, and nothing more. The paths never repeat themselves again.
No closed answer comes out. One extra body, and what could be written down cannot be.
That drop is the deepest thing in celestial mechanics. For two centuries after Newton wrote his formula, a great many people stopped here.
Why can it not be written? Counting shows why.
Following three bodies takes eighteen numbers of position and velocity. Conserved quantities are what reduce that count: six for the motion of the centre of mass, three for angular momentum, one for energy. Ten in all.
Eight short. With two bodies the conserved quantities exactly covered what was needed, and that is why it could be written as a formula.
Might there be others? Bruns in and Poincare in showed that there are not. They have not merely gone unfound: they are proved not to exist.
Unwritable is not incomputable. From the present positions and velocities, work out the forces and advance a short time. Repeat.
Each step introduces a small error, because the force is treated as constant across the step when in truth it is changing.
Finer steps make the error smaller without removing it, and there are then more steps for it to accumulate over. There is no escape by simply going finer and finer.
There are better ways of stepping. Schemes that keep the energy in check over long runs can carry a calculation a billion years. None of them replaces the work with a single formula.
It is not that no arrangement can be solved. Sufficiently orderly ones can be written down.
Euler found, in , a solution in which the three stay collinear as they turn. Lagrange found, in , one in which they hold an equilateral triangle. Neither changes shape, so both can be handled like a two-body problem.
In a figure-eight solution was found numerically: three bodies chasing one another round a single curve. Its existence was proved in .
More such solutions keep being found. All of them are confined to special arrangements. A formula for an arbitrary one still does not exist.
It is worth being exact about the word.
It does not mean incomputable. The positions of three bodies can be had to any precision wanted, and that is how spacecraft are navigated and planetary positions produced.
What it means is only that no closed formula exists. With two bodies, feeding a time into a formula gave a position. With three, the only route to a time is stepping all the way there.
And the difference is not merely one of labour. Having to step there is what leads into the rest of this chapter: that spacecraft are navigated all the same, that even among three bodies there are places that hold still, and that the errors open out.