Variational and Topological Approaches to the Three-body Problem

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Abstract. VARIATIONAL AND TOPOLOGICAL APPROACHES TO THE THREE-BODYPROBLEM.The proposer will investigate several aspectsof the Newtonian three-body problems usingtechniques from geometric analysis and classical ODEs. The main new direction proposed is to understand scattering in theNewtonian three-body problem using the technique of ``gluing'' borrowedfrom geometric PDE. An unbounded solution with negative energy consist of two ``bound masses'' orbiting each otherin a nearly Keplerian orbit while the third masssails away, asymptoting to a Keplerianhypebolic orbit. Each of the two Jacobi vectors asymptotes to a solution to a Keplertwo-body problems -- an elliptic one for the bound pair,and a hyperbolic one for the vector joining the receding mass to the center of mass of the bound pair.If the solution is unbounded in both the past and future,there can, and typically will, be different Keplerian orbits in the pastandthe future -- indeed the pair which is bound in the distant pastmight be different from the pair bound in the future. Which Kepler parameters can be connected to which by an unbounded three-body orbit? This scattering problem is one of the central questions we will investigate. Other problems to be investigatedfollow past lines of investigation initiated by the proposer.We give two examples. Planar three body solutions can be partially encodedbytheir syzygy sequence. Is every syzygy sequence possible?Initial computer investigations suggest not. Is it true that every zero-angular momentum negative energysolution which tends to the Lagrange homothety orbiteither is the Lagrange homothety orbit, or suffers a syzygy? The Newtonian three-body problem concerns the dynamics ofthree bodies, modelled as point masses, attracting each other by Newton's$1/r^2$ gravitational force. Think of the bodiesas the earth, moon, and sun, or as three stars. The ``problem'' is not a single problem,but rather a large collectionof problems concering the long-term qualitative behavior of such a system of masses. A solution is called ``bounded'' if the three interbody distancesremain finite and bounded by some fixed distance for all time. A central open question is: howbig is the set of bounded solutions? Is it true that arbitrarily close to a bound solution there is anunbounded one? The traditional approach to this question has involved ``Arnol'ddiffusion''-- a mechanism discovered by V.I. Arnol'd through which seemingly stableregions and orbits become unstable and perhaps eventually unbounded.The Arnol'd diffusion method or mechanism is in essence perturbative: onetries to perturb away from an apparently stable situation.As an alternative, we propose to start at infinity and sweep in frominfinity to see what parts of phase space are swept out. ``Starting at infinity'' means thinking of the scattering problem --the situation of one body infinitely distant from the other two.The most interesting case is that in which two of the masses remain bounded, orbiting each other in a nearly Keplerian orbit as the third massrecedes.Imagine beginning in such a manner in the infinite past, and ending up insuch a manner in the infinite future, but with the past and future bound pairs perhaps different,or their Keplerian ellipse having different eccentricities, energies, etc. In this way we get a kind of ``scattering map'' fromKepler orbits(or parameters) to Kepler orbits. What does this scattering map look like? This ``scattering problem''is one of the main new directions we propose to investigate.In addition, we propose several questions more in line withour past investigations. Here is one example. A ``syzygy'' is an instantat which all three masses lie on a line. At such an instant one mass ``eclipses'' (lies between) the other two.Label the syzygies by the mass doing the eclipsing.List the labelled syzygies in order of appearance, thus associating toeach solutiona syzygy sequence. In the planar three-body problem, are all syzygy sequences possible?This is an open problem. The proposer has made partial progress. Initialcomputer investigations withthe help of an undergraduate researcher suggest that the answer might be``no''.