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Syzygy sequences of the $N$-center problem

Published online by Cambridge University Press:  22 September 2016

KUO-CHANG CHEN  and
GUOWEI YU
KUO-CHANG CHEN
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan email kchen@math.nthu.edu.tw
GUOWEI YU
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada email yu@math.utoronto.ca
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Abstract

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.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 2 , April 2018 , pp. 566 - 582
Copyright
© Cambridge University Press, 2016 

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