DRAFT. April

Fitting Hyperbolic pants to a three-body problem. Abstract. Consider the three-body problem with an attractive 1/r 2 potential. Modulo symmetries, the dynamics of the bounded zero-angular momentum solutions is equivalent to a geodesic flow on the thrice-punctured sphere, or " pair of pants ". The sphere is the shape sphere. The punctures are the binary collisions. The metric generating the geodesics is the Jacobi-Maupertuis metric. The metric is complete, has infinite area, and its ends, the neighborhoods of the punctures, are asymptotically cylindrical. Our main result is that when the three masses are equal then the metric has negative curvature everywhere except at two points (the Lagrange points). A corollary of this negativity is the uniqueness of the 1/r 2 figure eight, a complete symbolic dynamics for encoding the collision-free solutions, and the fact that collision solutions are dense within the bound solutions. 1. Introduction and Results. We study the planar three-body problem with an attractive 1/r 2 potential. According to the Lagrange-Jacobi identity (eq. (3.7) below) every bounded solution must have zero energy and constant moment of interia I , and conversely, if an initial condition has zero energy and ˙ I(0) = 0 then that solution is bounded. Setting the moment of inertia I equal to a constant defines a three-sphere in configuration space. Rotations act on this sphere according to the Hopf flow so that the quotient of the three-sphere by rotations is the two-sphere or shape sphere. See figure 1a. Points of this shape sphere represent oriented similarity classes of triangles. Newton's equations, for solutions with H = 0, ˙ I = 0, push down to the shape sphere to yield a a family of second-order ODEs parameterized by the angular momentum. These ODEs have singularities at the three points representing the three types of binary collisions. Upon deleting the collision points we arrive at dynamics on the pair of pants, – the two-sphere minus three points. When the angular momentum is zero the resulting dynamical system is, after a time reparameterization, the geodesic flow for a certain Riemannian metric on the pair of pants. This metric is the (reduced) Jacobi-Maupertuis metric for energy 0.