A new solution to the three body problem - and moreby Bill Casselman
6. Some further questions; References
Some further questions
The principal result of Chenciner and Montgomery is that in the homotopy class of the track of the path above there does exist the track of a three-body system. It is known that the loops are star-shaped, in the technical sense that a radius drawn from a center hits the loop once. But much more is not known, unless I am mistaken. We start with a simple question and get tougher.
These questions are presumably much easier for striong potentials, in which case Poincaré already knew that there were lots of periodic orbits. But how many? How to construct them numerically?
Richard Montgomery's survey article (see below) includes a detailed discussion of the relations between systems and their paths on the Riemann sphere.
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