Variational Structure of Collisions in the Three-Body Problem

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

Abstract Award: DMS-0072336 Principal Investigator: Richard W. Montgomery The principal investigator will search for new solutions to the Newtonian N-body problem using a combination of the direct method of the calculus of variations, a detailed knowledge of ``shape space'', and a careful investigation of the action functional near collisions of the bodies. By the ``shape space'' we mean the space of either similarity classes or congruence classes of N-gons. In recent joint work with Alain Chenciner, this three-pronged approach proved its utility by yielding a hitherto unknown orbit for the three-body problem. In our new orbit all three masses chase each other around the same figure eight shaped curve in the plane. Our orbit turns out to be dynamically (actually KAM) stable. The chief technical difficulty to be overcome in successfully applying the method is that of avoiding collisions between the masses. Unlike the action in problems with strong-force potentials, the action with the Newtonian potential admits finite-action solutions with collision. One knows very little about minimizers with collision. In particular they need not be regularized in any of the various senses. The proposer will focus on the collisions. If an action minimizing sequence tends to a curve with collisions, under what circumstance are those collisions Levi-Civita regularized? Are there blow-up techniques which will enable us to better understand such sequences tending toward collision? These are some of the questions we will consider. The three-body problem is the problem of understanding the long term behaviour of three masses (planets, stars, satellites) attracting each other according to Newton's laws of physics. It is one of the oldest problems in mathematics,dating back to Newton. About 100 years ago the French mathematician Poincare made fundamental progress. He showed that chaos exists in the three-body problem, in contrast to the the two-body problem, where the motions are very regular (and well-approximated by that of the earth around the sun). He also pointed out the central importance of periodic orbits to the problem. Periodic orbits are motions of the masses which repeat the same pattern indefinitely like a point going around a circle. We propose to find new periodic solutions to the three-body problem and the N-body (N is four, five, six,...) problem by using a combination of methods. The methods themselves are not new, but their combination is. This approach has already proved successful in one instance -- by yielding a new solution in which three equal masses chase each around a figure eight curve, never catching each other. Our work could lead to further significant advances in the understanding of the N-body problem. The techniques may prove to be useful in other dynamical situations. There is some possibility that our orbits might be found to exist somewhere in the universe, or used in space missions someday.