Abstract
We prove that there is an unique convex noncollinear central configuration of the planar Newtonian four-body problem when two equal masses are located at opposite vertices of a quadrilateral and, at most, only one of the remaining masses is larger than the equal masses. Such a central configuration possesses a symmetry line and it is a kite-shaped quadrilateral. We also show that there is exactly one convex noncollinear central configuration when the opposite masses are equal. Such a central configuration also possesses a symmetry line and it is a rhombus.
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References
Albouy A. (1995). Symétrie des configurations centrales de quatre corps. C.R. Acad. Sci Paris. 320: 217–220
Albouy A. (1996). The Symmetric Central Configurations of Four Equal Masses. Contemp Math. V 198: 131–135
Albouy A. (2003). On a Paper of Moeckel on Central Configurations. Reg. Chaotic Dynamics. 8: 133–142
Albouy, A.: Mutual Distances in Celestial Mechanics. preprint (2004)
Albouy A. and Chenciner A. (1988). Le probléme des n corps et les distances mutuells. Invent. Mat. 131: 151–184
Albouy, A., Fu, Y.:Euler Configurations and Quasi-Polynomial Systems. preprint (2006)
Chazy J. (1918). Sur Certaines trajectoires du probléme des n corps. Bull. Astron. 35: 321–389
Hampton M. and Moeckel R. (2006). Finiteness of Relative Equilibria of the Four-Body Problem. Invent.math. 163: 289–312
Leandro E.S.G. (2003). Finitness and Bifurcations of some Symmetrical Classes of Central Configurations. Arch. Rational Mech Anal. 167: 147–177
Long Y. and Sun S. (2002). Four-Body Central Configurations with some Equal Masses. Arch. Rational Mech Anal. 162: 24–44
Moulton F.R. (1910). The Straight Line Solutions of the Problem of n-bodies. Ann. Math. 12: 1–17
Smale S. (1970). Topology and Mechanics II. Inv. Math. 11: 45–64
Smale S. (1998). Mathematical Problems for the Next Century. Math. Intell. 20: 7–15
Tien, F.: Recursion Formulas of Central Configurations, Thesis, University of Minnesota 1993
Uspensky J.V. (1948). Theory of Equations. McGraw Hill, New York
Wintner, A.:The Analytical Foundations of Celestial Mechanics. Princeton Math. Series 5. Princeton University Press, Princeton NJ, 1941
Xia Z. (1991). Central Configurations with Many Small Masses. J. Diff. Equations. 91: 168–179
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Communicated by P. Rabinowitz
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Perez-Chavela, E., Santoprete, M. Convex Four-Body Central Configurations with Some Equal Masses. Arch Rational Mech Anal 185, 481–494 (2007). https://doi.org/10.1007/s00205-006-0047-z
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DOI: https://doi.org/10.1007/s00205-006-0047-z