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Multiple symmetric periodic solutions to the \(2n\)-body problem with equal masses. (English) Zbl 1260.70006

Summary: Using the variational method, A. Chenciner and R. Montgomery [Ann. Math. (2) 152, No. 3, 881–901 (2000; Zbl 0987.70009)] proved the existence of an eight-shaped orbit of the planar three-body problem with equal masses. Since then a number of solutions to the N-body problem have been discovered. In particular, D. L. Ferrario and S. Terracini [Invent. Math. 155, No. 2, 305–362 (2004; Zbl 1068.70013)] proved the existence of symmetric periodic solutions under a quite general setting. In this paper we consider the 2n-body problem with a certain symmetry and use the results of Ferrario and Terracini to prove the existence of multiple solutions for each n. Some of the solutions we find were already obtained by K.-C. Chen [Arch. Ration. Mech. Anal. 170, No. 3, 247–276 (2003; Zbl 1036.70007)] and Ferrario and Terracini (op. cit.), but their argument does not allow us to distinguish the solutions obtained in this paper. By reducing the problem to the quotient space of the configuration space under the action of the group of symmetries and by observing boundary conditions of the solutions in the quotient space, we are able to distinguish these solutions and conclude that they are indeed distinct solutions. As a by-product, we can also determine the sign of the angular momentum of masses.

MSC:

70F10 \(n\)-body problems
37C80 Symmetries, equivariant dynamical systems (MSC2010)
70G75 Variational methods for problems in mechanics
70K42 Equilibria and periodic trajectories for nonlinear problems in mechanics
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