Necessary conditions for central configurations of six-body problems (Q1879081)

This paper uses straightforward analytic techniques to study certain symmetric spatial central configurations the authors refer to as double pyramidal central configurations (d.p.c.c. for short). A d.p.c.c. is a central configuration of \(n+2\) bodies of which \(n\) bodies are lying in a plane \(P\) and the other two bodies are positioned symmetrically above and below \(P\). The authors restrict to the case \(n=4\) yielding spatial central configurations for the 6-body problem. A central configuration is a very special configuration of masses for which the force due to acceleration on each body points toward the center of mass. The forces are perfectly balanced so that the configuration will collapse homothetically (maintaining its initial shape) to total collision if given no initial velocity. The study of central configurations is an important subject in celestial mechanics and has a long and varied history. (See \textit{R. Moeckel} [Math. Z. 205, No. 4, 499--517 (1990; Zbl 0684.70005)] for a nice introduction to the subject.) Building on the work of \textit{N. Fayçal}'s paper [Proc. Am. Math. Soc. 124, No. 1, 249--258 (1996; Zbl 0844.70009)], the authors show that a d.p.c.c. with a rectangular base \(R\) and two masses positioned at equal distances from the center of \(R\) (one above, one below) must in fact have a square base containing equal masses. The masses of the non-coplanar bodies must be equal, although not necessarily equal to the masses on \(R\). The result is a generalization of Fayçal's earlier result on pyramidal central configurations (any pyramidal central configuration with a rectangular base must have a square base containing equal masses). The authors' result is proven by exploiting the symmetry of the setup and by taking various inner products with the equations for a central configuration. As a corollary to this result (and using an identity at the end of their proof), the regular octahedron must have equal masses in order to be a central configuration.