# Three Body Problem

The **Three Body Problem** is a four hundred year old problem of mathematics which has its roots in the attempts to simulate a heliocentric Sun-Earth-Moon system.

# History

## Henri Poincaré

http://n.ethz.ch/~stiegerc/HS09/Mechanik/Unterlagen/Lecture13.pdf

“ 9.2.1 History

In 1885, Poincaré entered a contest formulated by the King Oscar II of Sweden in honor of his 60th birthday. One of the questions was to show the solar system, as modeled by Newton’s equations, is dynamically stable. The question was nothing more than a generalization of the famous three body problem, which was considered one of the most difficult problems in mathematical physics. In essence, the three body problem consists of nine simultaneous differential equations. The difficulty was in showing that a solution in terms of invariants converges.(This isn’t likely to happen today - that the birthday of any contemporary world leader is celebrated by a mathematical competition!) 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. Poincaré'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 Phragmn, a Swedish mathematician who was an assistant editor at the journal. Gsta Mittag-Leffler, chief editor, forwarded Phragmn's questions to Poincaré, asking him to fix up these nagging issues before the prize essay appeared in print. Poincaré 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 the orbits none 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, but at the time it came as an utter shock. In his attempt to prove the stability of planetary orbits, Poincaré ended up inventing chaos theory. ”