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Central configurations with many small masses. (English) Zbl 0724.70015

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.

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
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[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
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