Kinematic geometry of triangles and the study of the three-body problem

Abstract

This monograph describes a Riemannian geometric reduction approach to the three-body problem. The fundamental theorems are presented in the introductory part, whereas their proofs are provided in later chapters where specific topics are analyzed in more detail. The basic idea is to reduce the kinematic and dynamics of mass triangles to the congruence moduli space and the shape space, representing SO(3)-congruence and similarity classes respectively.In particular, the Euler equations for mass triangle motions are worked out, generalizing the classical Euler equations for rigid body motions. The last chapters are devoted to the study of Newtonian triple collision motions, in the above geometric setting, and their shape curves are analyzed in special cases.