A new solution to the three body problem - and more
by Bill Casselman
Richard Moeckel was apparently the first to suggest that a relatively simple but effective way to understand what is going on topologically is to track the triangle whose vertices are located at the centers of the three bodies. With any legitimate choreography orbit, the bodies following the path will avoid collisions, which means that at any moment all three of the sides of this triangle will have non-zero length. It turns out to be useful to focus attention on the shapes of the triangles, that is to say, to consider similar triangles as being equivalent. Furthermore, it is useful to keep track of labels for the vertices. There is a very pretty way to parametrize such configurations.
We are interested in parametrizing
labeled configurations of three points
Now some complex analyis.
The equation shows that a
positively oriented equilateral triangle corresponds to
the point at infinity; this is because for such a triangle
x=1; similarly a triangle with
If we are given any periodic collisionless system of three bodies,
as they move the point corresponding to the triangle they form
will trace out a path in the complement, on the Riemann
|The Möbius transformation maps a labeled triangle ABC to a point on the Riemann sphere (the complex plane together with a point at infinity), where we have singled out the three cube roots of 1: a=1, b, c, in counterclockwise order. Triangles with A=C map to the point b, etc., so the triangles formed by a collisionless choreography will always lie in the complement of a=1, b, and c. Positively oriented triangles map outside the unit circle, negatively oriented ones inside, triangles with the three vertices collinear map to the circle itself. Isosceles triangles map to one of three lines running through the origin and a=1, b, or c. Similar (and similarly labeled) triangles map to the same point on the Riemann sphere.|
The principal result of Chenciner and Montgomery is that in the homotopy class of the track of the path exhibited above there does exist the track of a three-body system.
) Copyright 2001, American Mathematical Society