Math · chapter 3 of 13
Add one rock
The equations for three bodies fit on one line and hide nothing. What breaks is not the writing down. It is the solving.
Three bodies. Each one feels the pull of the other two, added as arrows. That is the entire physics, and here it is, written out.
Nothing is missing from that line. No approximation, no fudge, no term dropped for being small. If you know where the three are and how fast each is going, the line tells you what happens next, to as many decimal places as you care to carry.
Count what you have to keep track of. Three bodies, three coordinates each, is nine numbers for position and nine for velocity: eighteen numbers, and the line above says how each of the eighteen changes. In a flat plane it is twelve. That is a small bookkeeping job.
And yet. There is no formula that takes the eighteen numbers and the time and hands back where everything is. Not a hard one, not a long one — there is none, and the next two chapters are about how that got proved rather than assumed.
Newton knew. Book I of the Principia, Proposition 66, is him going at the Sun–Earth–Moon case by hand, with twenty-two corollaries and no clean answer. Three and a half centuries later the plain statement has not changed much: you can say what the rules are, and to find out what they do you have to watch.
Reading it: For each body: add up, over the other two, the other one's weight times the direction toward it, divided by the distance cubed. That sum is how fast this body's velocity is changing.
The cube on the bottom is the inverse square plus one more power to turn the arrow on top into a pure direction. Same law as chapter one, written for a crowd.
Reading it: Enough equations to fix the answer, and no way to write the answer as a formula.
Existence and uniqueness are not in doubt: the path is there and it is the only one. Writing it in closed form is the part that fails.
This one runs in the browser, with JavaScript on. The words above it stand on their own.