N Bodies

This will be familiar to anyone who reads John Baez’s This Week’s Finds in Mathematical Physics, but I can’t help but show these lovely exact solutions to the gravitational N-body problem. This one is beautiful in its simplicity: twenty-one point masses moving around in a figure-8.

The N-body problem is one of the most famous, and easily stated, problems in mathematical physics: find exact solutions to point masses moving under their mutual Newtonian gravitational forces (i.e. the inverse-square law). For N=2 the complete set of solutions is straightforward and has been known for a long time — each body moves in a conic section (circle, ellipse, parabola or hyperbola) around the center of mass. In fact, Kepler found the solution even before Newton came up with the problem!

But let N=3 and chaos breaks loose, quite literally. For a long time people recognized that the motion of three gravitating bodies would be a difficult problem, but there were hopes to at least characterize the kinds of solutions that might exist (even if we couldn’t write down the solutions explicitly). It became a celebrated goal for mathematical physicists, and the very amusing story behind how it was resolved is related in Peter Galison’s book Einstein’s Clocks and Poincare’s Maps. In 1885, a mathematical competition was announced in honor of the 60th birthday of King Oscar II of Sweden, and the three-body problem was one of the questions. (Feel free to muse about the likelihood of the birthday of any contemporary world leader being celebrated by mathematical competitions.) Henri Poincare was a favorite to win the prize, and he submitted an essay that demonstrated the stability of planetary motions in the three-body problem (actually the “restricted” problem, in which one test body moves in the gravitational field generated by two others). In other words, without knowing the exact solutions, we could at least be confident that the orbits wouldn’t go crazy; more technically, solutions starting with very similar initial conditions would give very similar orbits. Poincare’s work was hailed as brilliant, and he was awarded the prize.

But as his essay was being prepared for publication in Acta Mathematica, a couple of tiny problems were pointed out by Edvard Phragmen, a Swedish mathematician who was an assistant editor at the journal. Gosta Mittag-Leffler, chief editor, forwarded Phragmen’s questions to Poincare, asking him to fix up these nagging issues before the prize essay appeared in print. Poincare went to work, but discovered to his consternation that one of the tiny problems was in fact a profoundly devastating possibility that he hadn’t really taken seriously. What he ended up proving was the opposite of his original claim — three-body orbits were not stable at all. Not only were orbits not periodic, they didn’t even approach some sort of asymptotic fixed points. Now that we have computers to run simulations, this kind of behavior is less surprising (example here from Steve McMillan — note how the final “binary” is not made of the same “stars” as the original one), but at the time it came as an utter shock. In his attempt to prove the stability of planetary orbits, Poincare ended up inventing chaos theory.

But the story doesn’t quite end there. Mittag-Leffler, convinced that Poincare would be able to tie up the loose threads in his prize essay, went ahead and printed it. By the time he heard from Poincare that no such tying-up would be forthcoming, the journal had already been mailed to mathematicians throughout Europe. Mittag-Leffler swung into action, telegraphing Berlin and Paris in an attempt to have all copies of the journal destroyed. He basically succeeded, but not without creating a minor scandal in elite mathematical circles across the Continent. (The Wikipedia entry on Poincare tells a much less interesting, and less accurate, version of the story.)

However, just because the general solution to the three-body (and more-body) problem is chaotic, doesn’t mean we can’t find special exact solutions in highly-symmetric conditions, and that’s just what Cris Moore and Michael Nauenberg have recently been doing. The image at the top really is an exact solution to twenty-one equal-mass objects moving in a figure-eight under their mutual gravitational attraction. They’re moving in a plane, of course, but that’s not strictly necessary; here’s a close relative of the figure-8, perturbed outside the plane.

From there you can just go nuts; here’s an example with twelve objects orbiting with cubic symmetry — four distinct periodic paths with three particles each.

Knowledge of this exact solution, plus $3.50, will get you a grande latte at Starbucks. Mathematicians have all the fun.