Three Body Topology

Braids in Classical Dynamics

Christopher Moore

PRL 70, 3675--3679 (1993)

URL: http://link.aps.org/abstract/PRL/v70/p3675

This was a fascinating paper! I found it in the references of the paper on choreographic orbits. I never learned classical mechanics like this ...

In this paper, Moore analyzes orbits of N-body systems in 2+1 dimensions. He considers two-body potentials of the form

V(i,j) = A m(i) m(j) r(i,j)^X

as a function of the exponent X.

His approach to finding allowed orbits is novel. He suggests a topological classification. Basically, Moore starts out with a braid with N strands. (Every orbit in a plane defines a braid in 2+1 dimensions, and mathematicians know how to determine which braids are topologically equivalent.) Next, he defines an action for N-particle orbits whose minima are solutions of Newton's second law. He shows that starting with the braid and relaxing it toward a minimum of the action can result in only three things:

1) A particle flies off to infinity.
2) Two particles collide.
3) The braid relaxes to a topologically equivalent orbit that satisfies F = ma.

Armed with this tool, he examines the types of braids that yield solutions for a particular exponent X in the potential. First, he shows that escape is impossible if X < 2. That in itself is in interesting result. Next, he shows that for X <= -2, relaxation never leads to a collision. Thus, every braid is an allowed orbit for a potential that falls off at least as fast as 1/R^2.

The X=2 case is interesting because the system is integrable, and all particles orbit the center of mass with the same period. Moore shows that the only allowed braids are those in which any two particles have a winding number of +1 or -1 (as long as they don't collide at the origin). These are called harmonic braids.

After this, he reports numerical investigations of other braid types. He identifies solutions to solutions to the 3-body problem I didn't know about last week: the figure eight discussed in the choreographic orbits paper, and a braid I'll call the cross-circle. The table on page 3 is really interesting.

The application of topology to the 3-body problem is interesting and impressive. I'll have to get a copy of Arnol'd's book and see if I can learn a few of these methods.