The Geometric Phase in the Three-Body Problem

TL;DR

This paper derives a formula connecting the initial and later configurations of the three-body problem using geometric phases on a shape sphere, revealing insights into the system's rotational behavior.

Contribution

It introduces a simple integral formula for the overall rotation in the three-body problem based on geometric phase theory and shape space analysis.

Findings

Derived a universal integral formula for rotation

Connected shape space geometry with three-body dynamics

Extended geometric phase concepts to celestial mechanics

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.