Fayçal, Nelly On the classification of pyramidal central configurations. (English) Zbl 0844.70009 Proc. Am. Math. Soc. 124, No. 1, 249-258 (1996). Summary: 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. Cited in 1 ReviewCited in 11 Documents MSC: 70F10 \(n\)-body problems Keywords:existence of central configurations; planes of symmetry; regular trapezoid; kite-shaped quadrilateral; rectangle PDFBibTeX XMLCite \textit{N. Fayçal}, Proc. Am. Math. Soc. 124, No. 1, 249--258 (1996; Zbl 0844.70009) Full Text: DOI References: [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. · Zbl 0758.68037 [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. · JFM 23.1222.01 [5] Kenneth R. Meyer and Glen R. Hall, Introduction to Hamiltonian dynamical systems and the \?-body problem, Applied Mathematical Sciences, vol. 90, Springer-Verlag, New York, 1992. · Zbl 0743.70006 [6] Richard Moeckel, On central configurations, Math. Z. 205 (1990), no. 4, 499 – 517. · Zbl 0684.70005 [7] P. Pizzetti, Casi particolari del problema dei tre corpi, Rendiconti 13 (1904), 17–26. · JFM 35.0724.03 [8] Dieter S. Schmidt, Central configurations in \?² and \?³, Hamiltonian dynamical systems (Boulder, CO, 1987) Contemp. Math., vol. 81, Amer. Math. Soc., Providence, RI, 1988, pp. 59 – 76. [9] A. Wintner, The analytical foundations of celestial mechanics, Princeton Univ. Press, Princeton, NJ, 1941. · JFM 67.0785.01 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.