THE GEOMETRIC PHASE OF THE 3-BODY PROBLEM

Suppose that the initial triangle formed by the three moving masses of the three-body problem is similar to the triangle formed at some late r time. We derive a simple integral formula for the overall rotation r elating the two triangles. The formula is based on the fact that the s pace of similarity classes of triangles forms a two-sphere which we ca ll the shape sphere. The formula consists of a 'dynamic' and 'geometri c' term. The geometric term is the integral of a universal two-form on a 'reduced configuration space'. This space is a two-sphere bundle ov er the shape sphere. The fibring spheres are instantaneous versions of the angular momentum sphere appearing in rigid body motion. Our deriv ation of the formula is similar in spirit to our earlier reconstructio n formula for the rigid body motion.