A geometric study of many body systems

Abstract

A n n-body system is a labelled collection of n point masses in Euclidean space, and their congruence and internal symmetry properties involve a rich mathematical structure which is investigated in the framework of equivariant Riemannian geometry. Some basic concepts are n-configuration, configuration space, internal space, shape space, Jacobi transformations and weighted root system. The latter is a generalization of the root system of SU(n), which provides a bookkeeping for expressing the mutual distances of the point masses in terms of the Jacobi vectors. Moreover, its application to the study of collinear central n-configurations yields a simple proof of Moulton's enumeration formula. A major topic is the general study of matrix spaces representing the shape space of many body systems in Euclidean k-space, the structure of the m-universal shape space and its O(m)-equivariant linear model.This also leads to those orbital fibrations where SO(m) or O(m) act on a sphere with a sphere as orbit space. Some examples of this kind are encountered in the literature, e.g. the special case of the 5-sphere mod O(2), which equals the 4-sphere, was analyzed independently by Arnold, Kuiper and Massey in the 1970's.