# 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

## Issac Newton

Issac Newton is an authority who is credited to have "brought the laws of physics to the solar system."^{1} In regards to the multi-body problems of his system, Newton famously invoked divine intervention as his solution:

“ At the beginning of the 18th century, Newton famously wrote that the solar system needed occasional divine intervention (presumably a nudge here and there from the hand of God) in order to remain stable.^{11} This was interpreted to mean that Newton believed his mathematical model of the solar system—the *n* body problem—did not have stable solutions. Thus was the gauntlet laid down, and a proof of the stability of the *n* body problem became one of the great mathematical challenges of the age.^{11}Newton's remarks about divine intervention appear in Query 23 of the 1706 (Latin) edition of *Opticks*, which became Query 31 of the 1717 (2nd Edition) edition see Quote Q[New] in Appendix E). Similar 'theological' remarks are found in scholia of the 2nd and 3rd editions of *Principia*, and in at least one of Newton's letters. In a 1715 letter to Caroline, Princess of Wales, Leibniz observed sarcastically that Newton had not only cast the Creator as a clock-maker, and a faulty one, but now as a clock-repairman (see [Klo73], Part XXXIV, pp. 54-55). ”

^{1}The University of California San Diego credits Newton as providing the laws of physics for the Solar System:

“ Then came Isaac Newton (1642-1727) who brought the laws of physics to the solar system. Isaac Newton explained why the planets move the way they do, by applying his laws of motion, and the force of gravitation between any two bodies, letting the force decrease with the square of the distance between the two bodies. ”

## 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. ”