Abstract. VARIATIONAL AND TOPOLOGICAL APPROACHES TO THE THREE-BODY PROBLEM.
The proposer will investigate several aspects of the Newtonian three-body problems using techniques from geometric analysis and classical ODEs. The main new direction proposed is to understand scattering in the Newtonian three-body problem using the technique of ``gluing'' borrowed from geometric PDE. An unbounded solution with negative energy consist of two ``bound masses'' orbiting each other in a nearly Keplerian orbit while the third mass sails away, asymptoting to a Keplerian hypebolic orbit. Each of the two Jacobi vectors asymptotes to a solution to a Kepler two-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 past and the future -- indeed the pair which is bound in the distant past might 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 investigated follow past lines of investigation initiated by the proposer. We give two examples. Planar three body solutions can be partially encoded by their syzygy sequence. Is every syzygy sequence possible? Initial computer investigations suggest not. Is it true that every zero-angular momentum negative energy solution which tends to the Lagrange homothety orbit either is the Lagrange homothety orbit, or suffers a syzygy?
The Newtonian three-body problem concerns the dynamics of three bodies, modelled as point masses, attracting each other by Newton's $1/r^2$ gravitational force. Think of the bodies as the earth, moon, and sun, or as three stars. The ``problem'' is not a single problem, but rather a large collection of problems concering the long-term qualitative behavior of such a system of masses. A solution is called ``bounded'' if the three interbody distances remain finite and bounded by some fixed distance for all time. A central open question is: how big is the set of bounded solutions? Is it true that arbitrarily close to a bound solution there is an unbounded one? The traditional approach to this question has involved ``Arnol'd diffusion'' -- a mechanism discovered by V.I. Arnol'd through which seemingly stable regions and orbits become unstable and perhaps eventually unbounded. The Arnol'd diffusion method or mechanism is in essence perturbative: one tries to perturb away from an apparently stable situation. As an alternative, we propose to start at infinity and sweep in from infinity 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 mass recedes. Imagine beginning in such a manner in the infinite past, and ending up in such 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'' from Kepler 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 with our past investigations. Here is one example. A ``syzygy'' is an instant at 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 to each solution a syzygy sequence. In the planar three-body problem, are all syzygy sequences possible? This is an open problem. The proposer has made partial progress. Initial computer investigations with the help of an undergraduate researcher suggest that the answer might be ``no''.
