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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the classification
of pyramidal central configurations


Author: Nelly Fayçal
Journal: Proc. Amer. Math. Soc. 124 (1996), 249-258
MSC (1991): Primary 70F15, 70F10
MathSciNet review: 1301024
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Abstract: We present some results associated with the existence of central configurations (c.c.'s) in the classical gravitational $N$-body problem of Newton. We call a central configuration of five bodies, four of which are coplanar, a $pyramidal$ central configuration (p.c.c). It can be shown that there are only three types of p.c.c.'s, admitting one or more planes of symmetry, viz. (i) the case where the planar bodies lie at the vertices of a regular trapezoid, (ii) the case where the bodies lie at the vertices of a kite-shaped quadrilateral, and (iii) the case where the bodies lie at the vertices of a rectangle. In this paper we classify all p.c.c.'s with a rectangular base and, in fact, prove that there is only one such c.c., namely, the square-based pyramid with equal masses at the corners of the square. The classification of all p.c.c.'s satisfying either (i) or (ii) will be discussed in subsequent papers.


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Additional Information

Nelly Fayçal
Affiliation: Carleton University, Department of Mathematics and Statistics, Ottawa, Ontario, Canada K1S 5B6

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03135-8
PII: S 0002-9939(96)03135-8
Received by editor(s): July 25, 1994
Additional Notes: This research was partially supported by an NSERC, PGS3 Scholarship
Communicated by: Hal L. Smith
Article copyright: © Copyright 1996 American Mathematical Society