Variational Structure of Collisions in the Three-Body Problem

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AbstractAward: DMS-0072336Principal Investigator: Richard W. MontgomeryThe principal investigator will search for new solutions to theNewtonian N-body problem using a combination of the direct methodof the calculus of variations, a detailed knowledge of ``shapespace'', and a careful investigation of the action functionalnear collisions of the bodies. By the ``shape space'' we meanthe space of either similarity classes or congruence classes ofN-gons. In recent joint work with Alain Chenciner, thisthree-pronged approach proved its utility by yielding a hithertounknown orbit for the three-body problem. In our new orbit allthree masses chase each other around the same figure eight shapedcurve in the plane. Our orbit turns out to be dynamically(actually KAM) stable. The chief technical difficulty to beovercome in successfully applying the method is that of avoidingcollisions between the masses. Unlike the action in problemswith strong-force potentials, the action with the Newtonianpotential admits finite-action solutions with collision. Oneknows very little about minimizers with collision. In particularthey need not be regularized in any of the various senses. Theproposer will focus on the collisions. If an action minimizingsequence tends to a curve with collisions, under whatcircumstance are those collisions Levi-Civita regularized? Arethere blow-up techniques which will enable us to betterunderstand such sequences tending toward collision? These aresome of the questions we will consider.The three-body problem is the problem of understanding the longterm behaviour of three masses (planets, stars, satellites)attracting each other according to Newton's laws of physics. Itis one of the oldest problems in mathematics,dating back toNewton. About 100 years ago the French mathematician Poincaremade fundamental progress. He showed that chaos exists in thethree-body problem, in contrast to the the two-body problem,where the motions are very regular (and well-approximated by thatof the earth around the sun). He also pointed out the centralimportance of periodic orbits to the problem. Periodic orbits aremotions of the masses which repeat the same pattern indefinitelylike a point going around a circle. We propose to find newperiodic solutions to the three-body problem and the N-body (N isfour, five, six,...) problem by using a combination ofmethods. The methods themselves are not new, but theircombination is. This approach has already proved successful inone instance -- by yielding a new solution in which three equalmasses chase each around a figure eight curve, never catchingeach other. Our work could lead to further significant advancesin the understanding of the N-body problem. The techniques mayprove to be useful in other dynamical situations. There is somepossibility that our orbits might be found to exist somewhere inthe universe, or used in space missions someday.