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A new solution to the three body problem - and more
by Bill Casselman
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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.
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Are the loops convex?
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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?
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Are there any other periodic three body choreographies?
Any other periodic three body systems at all? Can they
be produced by action minimization?
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What other techniques can produce numerically
convincing candidate orbits? Can one
construct saddle-points 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
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Ernest W. Brown, An introductory treatise on the lunar theory,
Cambridge University Press, 1896.
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Sections II.18-24 and Chapter XI
deal with Hill's solution.
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Alain Chenciner, Joseph Gerver,
Richard Montgomery,
and Carles Simó,
Simple choreographic motions of N bodies:
A preliminary study, preprint.
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Alain Chenciner and Richard Montgomery,
A remarkable solution of the three-body problem
in the case of equal masses, to appear in the Annals of Mathematics.
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Marshall Hampton,
http://www.math.washington.edu/~hampton/Lagrange.html
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An animation of a Lagrangean system with near collisions..
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Richard Moeckel,
Some qualitative features of the three-body problem, pp. 1-21 in
Hamiltonian Dynamical Systems, Proceedings of a
summer research conference held June 21-27, 1981.
Edited by Kenneth R. Mayer and Donald G. Saari.
Contemporary Mathematics 81.
American Mathematical Society, 1988.
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Richard Montgomery, A new solution to the three-body problem,
to appear in the Notices of the A. M. S., May 2001.
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A forthcoming survey article.
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Christopher Moore, Braids in classical gravity,
Physical Review Letters 70 (1993), pp. 3675-3679.
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Includes a general discussion of how to classify periodic
orbits by brids, and includes sketches of several periodic three-body
orbits found numerically, including the figure eight.
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Henri Poincaré, Sur les solutions périodiques et le principe de
moindre action, C. R. A. S. 123 (1896), pp. 815-918.
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Poincaré shows, essentially, that every homology class of
non-colliding orbit triples contains at least one periodic solution
of the three-body problem.
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Carles Simó,
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New families of solutions in N-body problems,
preprint. To appear in the Proceedings of the ECM 2000,
which was held in Barcelona in July of 2000.
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Periodic orbits of the planar N-body problem
with equal masses and all bodies on the same path,
preprint.
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