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Syzygy sequences of the $N$ -center problem

  • KUO-CHANG CHEN (a1) and GUOWEI YU (a2)


The purpose of this paper is to consider the $N$ -center problem with collinear centers, to identify its syzygy sequences that can be realized by minimizers of the Lagrangian action functional and to count the number of such syzygy sequences. In particular, we show that the number of such realizable syzygy sequences of length $\ell$ greater than or equal to two for the 3-center problem is at least $F_{\ell +2}-2$ , where $\{F_{n}\}$ is the Fibonacci sequence. Moreover, with fixed length $\ell$ , the density of such realizable syzygy sequences of length $\ell$ for the $N$ -center problem approaches one as $N$ increases to infinity. Using reflection symmetry, the minimizers that we found can be extended to periodic solutions.



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Syzygy sequences of the $N$ -center problem

  • KUO-CHANG CHEN (a1) and GUOWEI YU (a2)


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