The masses in a symmetric centered solution of the -body problem

Author:
Florin Nicolae Diacu

Journal:
Proc. Amer. Math. Soc. **109** (1990), 1079-1085

MSC:
Primary 70F10

MathSciNet review:
1010798

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Abstract: We prove that if a planar solution of the -body problem has a symmetry axis, fixed with respect to the considered frame, and the center of mass of the particle system lies on this axis during the motion, then the symmetric masses must be equal. We also show that the set of initial conditions leading to symmetric solutions has *measure zero* and is *nowhere dense* relative to the set of all initial conditions that define solutions in the space.

**[1]**F. N. Diacu,*Improbability of collinear solutions in the 𝑛-body problem with generalized attraction law*, Astronom. Nachr.**309**(1988), no. 5, 341–344 (English, with German summary). MR**992725**, 10.1002/asna.2113090511**[2]**Florin N. Diacu,*Wintner’s collinear and flat solutions are nowhere dense*, Celestial Mech.**44**(1988/89), no. 3, 261–265. MR**993816**, 10.1007/BF01235539**[3]**-,*On the planar syzygy solutions of the**-body problem*, Celest. Mech. (to appear).**[4]**-,*The masses in a symmetric solution of the four body problem*, Celest. Mech. (to appear).**[5]**Florin Nicolae Diacu and Árpád Pál,*Improbability of some special solutions in the 𝑛-body problem with generalized attraction law*, Seminar of Celestial Mechanics and Space Research, Preprint, vol. 87, Univ. “Babeş-Bolyai”, Cluj-Napoca, 1987, pp. 19–24. MR**959377****[6]**Aurel Wintner,*The Analytical Foundations of Celestial Mechanics*, Princeton Mathematical Series, v. 5, Princeton University Press, Princeton, N. J., 1941. MR**0005824**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1990-1010798-3

Keywords:
-body problem,
celestial mechanics,
measure,
nowhere dense set

Article copyright:
© Copyright 1990
American Mathematical Society