On the classification

of pyramidal central configurations

Author:
Nelly Fayçal

Journal:
Proc. Amer. Math. Soc. **124** (1996), 249-258

MSC (1991):
Primary 70F15, 70F10

DOI:
https://doi.org/10.1090/S0002-9939-96-03135-8

MathSciNet review:
1301024

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Abstract | References | Similar Articles | Additional Information

Abstract: We present some results associated with the existence of central configurations (c.c.'s) in the classical gravitational -body problem of Newton. We call a central configuration of five bodies, four of which are coplanar, a 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.

**1**B.W. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monagan, and S.M. Watt,*First leaves: A tutorial introduction to Maple V*, Springer-Verlag, New York, 1992.**2**L. Euler,*De mot rectilineo trium corporum se mutuo attahentium*, Novi Comm. Acad. Sci. Imp. Petrop.**11**(1767), 144--151.**3**J.L. Lagrange,*Oeuvres*, vol. 6, Gauthier-Villars, Paris, 1873, 272--292.**4**Lehmann-Filhés,*Ueber zwei Fälle des Vielkörperproblems*, Astr. Nachr.**127**(1891), 137--144.**5**K.R. Meyer and G.R. Hall,*Introduction to Hamiltonian dynamical systems and the N-body problem*, Appl. Math. Sci., vol. 90, Springer-Verlag, New York and Berlin, 1992. MR**93b:70002****6**R. Moeckel,*On central configurations*, Math. Z.**205**(1990), 499--517. MR**92b:70012****7**P. Pizzetti,*Casi particolari del problema dei tre corpi*, Rendiconti**13**(1904), 17--26.**8**D. S. Schmidt,*Central configurations in and*, Contemp. Math., vol. 81, Amer. Math. Soc., Providence, RI, 1988, pp. 59--76 MR**90d:70028**.**9**A. Wintner,*The analytical foundations of celestial mechanics*, Princeton Univ. Press, Princeton, NJ, 1941. MR**3:215b**

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Additional Information

**Nelly Fayçal**

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

DOI:
https://doi.org/10.1090/S0002-9939-96-03135-8

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