Abstract:
Using the variational method, Chenciner and Montgomery (2000 Ann. Math. 152 881-901) 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. On the other hand, symbolic dynamics is one of the most useful methods for understanding chaotic dynamics. The Sitnikov problem is a special case of the three-body problem. The system is known to be chaotic and was studied by using symbolic dynamics (J.Moser, Stable and random motions in dynamical systems, Princeton University Press, 1973).
In this paper, we study the limiting case of the Sitnikov problem. By using the variational method, we show the existence of various kinds of solutions in the planar Sitnikov problem. For a given symbolic sequence, we show the existence of orbits realizing it. We also prove the existence of periodic orbits.
The author is supported by the Japan Society for the Promotion of Science (JSPS), Grant-in-Aid for Young Scientists (B) No. 26800059 and Scientific Research (C) No. 18K03366.
\Bibitem{Shi19}
\by Mitsuru Shibayama
\paper Variational Construction of Orbits Realizing Symbolic Sequences in the Planar Sitnikov Problem
\jour Regul. Chaotic Dyn.
\yr 2019
\vol 24
\issue 2
\pages 202--211
\mathnet{http://mi.mathnet.ru/rcd454}
\crossref{https://doi.org/10.1134/S1560354719020060}
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This publication is cited in the following 3 articles:
Shota Iguchi, Yuika Kajihara, Mitsuru Shibayama, “Variational proof of the existence of periodic orbits in the spatial Hill problem and its constrained problems”, Japan J. Indust. Appl. Math., 40:1 (2023), 513
宝婷 刘, “A Study on the Periodic Solutions of a Class of Newton Equations with Special Symmetry”, AAM, 12:05 (2023), 2200
Yuika Kajihara, Mitsuru Shibayama, “Variational existence proof for multiple periodic orbits in the planar circular restricted three-body problem”, Nonlinearity, 35:3 (2022), 1431
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