Geometric reduction and the three body problem

Abstract

This dissertation investigates a particular reduction of the three body problem, using a combination of Riemannian geometry and geometric invariant theory of three body motions in Euclidean space. Our point of departure is the reduction that is described in [HS07]. Here, we present this reduction from a new point of view. This viewpoint emphasizes the flexibility in the choice of geometric invariants of three body motions, within one particular class of systems of invariants. Many of our important calculations are based on the singular value decomposition of matrices, and we show that the flexibility of the geometric invariants is strongly related to the flexibility of the singular value decomposition. In addition, we go some steps further than [HS07]: In the context of the three dimensional three body problem, we calculate the reduced equations of motion in terms of our chosen system of invariants. The rotational part of this reduction is extended to the general case of many particle systems evolving in three dimensional space. We also include a large discussion on the conformal geometry of the shape invariants of the three body problem.