Three Body Problem
THE THREE BODY PROBLEM
For Tim, Cindy and Kathleen, Nov 2001
The “three body problem” was one of the great problems of 19th century science. The problem is simple to state: what is the motion of three bodies—two planets and a star, for example—due to the force of gravity? But it is much harder to solve.
Many scientists thought the answer would be straightforward because two centuries earlier, Isaac Newton had worked out how a single planet should move around a single star—a two-body problem. However, nobody could apply his ideas to three bodies in a way that gave meaningful solutions. Then, in 1889, scientists proved that there is no general solution to the three body problem, only rare solutions that apply in very special situations.
These equations show one of these 'restricted' situations. The largest body, designated by the subscript T for Tim, is a Sun-like mass, the second largest, with the subscript C for Cindy, is equivalent to a planet like Saturn, while the third, with the subscript K for Kathleen, is a tiny mass, like a particle of star dust. The bodies are all in the same plane. X and Y are the coordinates for each body at a time t, μ is a ratio between the two largest masses and R is the distance between the smallest mass and the two largest. The equations map out how these bodies move through space together influenced only by the attractive forces between them.
A discussion of the three body problem is at http://www.geom.uiuc.edu/~megraw/CR3BP_html/cr3bp_bg.html