A new solution to the three body problem - and moreby Bill Casselman
3. The triangle construction
In order to explain what Chenciner and Montgomery have (and haven't) proven, I need to include a short digression about the topological nature of the system.
Richard Moeckel was apparently the first to suggestthata relatively simple but effective way to understand what is going on topologicallyis to track the triangle whose vertices are located atthe centers of the three bodies.With any legitimate choreography orbit, the bodiesfollowing the path will avoid collisions, which means thatat any moment all three of the sides of this triangle will have non-zero length.It turns out to be useful to focus attention onthe 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 parametrizinglabeled configurations of three points
Now some complex analyis.If
The equation shows that apositively oriented equilateral triangle corresponds to the point at infinity; this is because for such a trianglex=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 Riemannsphere, of
The principal result of Chenciner and Montgomery is thatin the homotopy class of the track of the pathexhibited above there does exist the track of a three-body system.
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