Xia, Zhihong Central configurations with many small masses. (English) Zbl 0724.70015 J. Differ. Equations 91, No. 1, 168-179 (1991). Summary: By using the method of analytical continuation, we find the exact numbers of central configurations for some open sets of n positive masses for any choice of n. It turns out that the numbers increase dramatically as n increases; e.g., for some open set of 18 positive masses, some \(2.08766\times 10^{20}\) classes of distinctive central configurations are found. In the mean time, we obtained some results about the Hausdorff measure for the set of n positive masses where degenerate central configuration arises. Cited in 43 Documents MSC: 70G10 Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics 70F10 \(n\)-body problems 70-08 Computational methods for problems pertaining to mechanics of particles and systems Keywords:method of analytical continuation; central configurations; Hausdorff measure PDFBibTeX XMLCite \textit{Z. Xia}, J. Differ. Equations 91, No. 1, 168--179 (1991; Zbl 0724.70015) Full Text: DOI References: [3] Moeckel, R., Relative equilibria of the 4-body problem, Ergodic Theory Dynamical Systems. Ergodic Theory Dynamical Systems, Ergodic Theory Dynamical Systems, 5 (1985) · Zbl 0554.14004 [4] Palmore, J., Collinear relative equilibria of the planar \(n\)-body problem, Celestial Mech.. Celestial Mech., Celestial Mech., 28, 17-23 (1982) · Zbl 0511.70011 [5] Palmore, J., Measure of degenerate relative equilibria, I, Ann.of Math., 104, 421-429 (1976) · Zbl 0321.58014 [6] Easton, R., Some topology of \(n\)-body problems, J. Differential Equations, 19, 258-269 (1975) · Zbl 0317.58013 [7] Palmore, J., Classifying relative equilibria, I, Bull. Amer. Math. Soc.. Bull. Amer. Math. Soc., Bull. Amer. Math. Soc., 79, 904-907 (1973) · Zbl 0273.57016 [9] Saari, D., The manifold structure for collision and for hyperbolic-parabolic orbits in the \(n\)-body problem, J. Differential Equations. J. Differential Equations, J. Differential Equations, 55, 300-329 (1984) · Zbl 0571.70009 [10] Wintner, A., The Analytical Foundations of Celestial Mechanics (1941), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · JFM 67.0785.01 [11] Saari, D., On the role and properties of \(n\)-body central configurations, Celestial Mech., 21, 9-20 (1980) · Zbl 0422.70014 [12] Smale, S., Topology and mechanics, I, Invent. Math., 10, 305-331 (1970) · Zbl 0202.23201 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.