A geometric study of many body systems

E. Straume

Lobachevskii journal of mathematics, Tome 24 (2006), pp. 73-134

Résumé

An $n$-body system is a labelled collection of $n$ point masses in a 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 transformation 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 study of matrix spaces representing the shape space of $n$-body configurations 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. A few of these examples are encountered in the literature, e.g. the special case $S^5/O(2)\approx S^4$ was analyzed independently by Arnold, Kuiper and Massey in the 1970's.