Extensions to Chen’s minimizing equal mass parallelogram solutions.(English)Zbl 1444.70006

Summary: In this paper, we study the extension of the minimizing equal mass parallelogram solutions which was derived by K.-C. Chen [Arch. Ration. Mech. Anal. 158, No. 4, 293–318 (2001; Zbl 1028.70009)]. Chen’s solution was minimizing for one quarter of the period $$[0,T]$$, where numerical integration had been used in his proof. In this paper we extend Chen’s solution in the reduced space to $$[0,4T]$$ and we show that this extension is also minimizing over the intervals $$[0,2T]$$ and $$[0,4T]$$. The minimizing property of the extension is proved without using numerical integration.

MSC:

 70F10 $$n$$-body problems 34C25 Periodic solutions to ordinary differential equations 70H12 Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics

Zbl 1028.70009
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References:

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