The Geometric Phase in the Three-Body Problem

Abstract

Suppose that the initial triangle formed by the three moving masses of the three-body problem is similar to the triangle formed at some later time. We derive a simple integral formula for the overall rotation relating the two triangles. The formula is based on the fact that the space of similarity classes of triangles forms a two-sphere which we call the shape sphere. The formula consists of a ``dynamic'' and ``geometric'' term. The geometric term is the integral of a universal two-form on a``reduced configuration space''. This space is a two-sphere bundle over the shape sphere. The fibering spheres are instantaneous versions of the angular momentum sphere appearing in rigid body motion. Our derivation of the formula is similar in spirit to our earlier reconstruction formula for the rigid body motion.