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The periodic solutions for planar \(2N\)-body problems. (English) Zbl 1331.70029

Summary: Based on the works of L. M. Perko and E. L. Walter [Proc. Am. Math. Soc. 94, 301–309 (1985; Zbl 0583.70008)], R. Moeckel and C. Simó [SIAM J. Math. Anal. 26, No. 4, 978–998 (1995; Zbl 0830.34032)], and S. Zhang and Q. Zhou [Proc. Am. Math. Soc. 131, No. 7, 2161–2170 (2003; Zbl 1028.34039)], we study the necessary conditions and sufficient conditions for the uniformly rotating planar nested regular polygonal periodic solutions for the \(2N\)-body problems.

MSC:

70F10 \(n\)-body problems
34C25 Periodic solutions to ordinary differential equations
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References:

[1] Perko, L. M.; Walter, E. L., Regular polygon solutions of the \(N\)-body problem, Proceedings of the American Mathematical Society, 94, 2, 301-309 (1985) · Zbl 0583.70008
[2] Moeckel, R.; Simó, C., Bifurcation of spatial central configurations from planar ones, SIAM Journal on Mathematical Analysis, 26, 4, 978-998 (1995) · Zbl 0830.34032
[3] Zhang, S.; Zhou, Q., Periodic solutions for planar \(N\)-body problems, Proceedings of the American Mathematical Society, 131, 7, 2161-2170 (2003) · Zbl 1028.34039
[4] Marcus, M.; Minc, H., A Survey of Matrix Theory and Matrix Inequalities, xvi+180 (1964), Boston, Mass, USA: Allyn and Bacon, Boston, Mass, USA · Zbl 0126.02404
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