orbits6 A new solution to the three body problem  and more by 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 threebody system. It is known that the loops are starshaped, 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.  Are the loops convex?
 Does the algorithm which produces the picture of the figure eight orbit (action minimization from a simple Lissajous figure) converge to the path certified by Chenciner and Montgomery?
 Are there any other periodic three body choreographies? Any other periodic three body systems at all? Can they be produced by action minimization?
 What other techniques can produce numerically convincing candidate orbits? Can one construct saddlepoints of the action easily?
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. References  Ernest W. Brown, An introductory treatise on the lunar theory, Cambridge University Press, 1896.
Sections II.1824 and Chapter XI deal with Hill's solution.   Alain Chenciner, Joseph Gerver, Richard Montgomery, and Carles Simó, Simple choreographic motions of N bodies: A preliminary study, preprint.
 Alain Chenciner and Richard Montgomery, A remarkable solution of the threebody problem in the case of equal masses, to appear in the Annals of Mathematics.
 Marshall Hampton, http://www.math.washington.edu/~hampton/Lagrange.html
An animation of a Lagrangean system with near collisions..   Richard Moeckel, Some qualitative features of the threebody problem, pp. 121 in Hamiltonian Dynamical Systems, Proceedings of a summer research conference held June 2127, 1981. Edited by Kenneth R. Mayer and Donald G. Saari. Contemporary Mathematics 81. American Mathematical Society, 1988.
 Richard Montgomery, A new solution to the threebody problem, to appear in the Notices of the A. M. S., May 2001.
A forthcoming survey article.   Christopher Moore, Braids in classical gravity, Physical Review Letters 70 (1993), pp. 36753679.
Includes a general discussion of how to classify periodic orbits by brids, and includes sketches of several periodic threebody orbits found numerically, including the figure eight.   Henri Poincaré, Sur les solutions périodiques et le principe de moindre action, C. R. A. S. 123 (1896), pp. 815918.
Poincaré shows, essentially, that every homology class of noncolliding orbit triples contains at least one periodic solution of the threebody problem.   Carles Simó,
 New families of solutions in Nbody problems, preprint. To appear in the Proceedings of the ECM 2000, which was held in Barcelona in July of 2000.
 Periodic orbits of the planar Nbody problem with equal masses and all bodies on the same path, preprint.

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