Variational proof of the existence of brake orbits in the planar 2-center problem

Yuika Kajihara; Misturu Shibayama

doi:10.3934/dcds.2019254

Article Contents

2019, Volume 39, Issue 10: 5785-5797. Doi: 10.3934/dcds.2019254

This issue Previous Article On a resonant and superlinear elliptic system Next Article Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations

Variational proof of the existence of brake orbits in the planar 2-center problem

Yuika Kajihara^, and
Misturu Shibayama

Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku Kyoto 606-8501, Japan

^* Corresponding author: cajihara@amp.i.kyoto-u.ac.jp
^* Corresponding author: cajihara@amp.i.kyoto-u.ac.jp

Received: August 31, 2018

Revised: January 31, 2019

Published: July 2019

The second 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

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Abstract

The restricted three-body problem is an important subject that deals with significant issues referring to scientific fields of celestial mechanics, such as analyzing asteroid movement behavior and orbit designing for space probes. The 2-center problem is its simplified model. The goal of this paper is to show the existence of brake orbits, which means orbits whose velocities are zero at some times, under some particular conditions in the 2-center problem by using variational methods.

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Mathematics Subject Classification: Primary: 37J45; Secondary: 70F15.

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Figure 1. the domain $ D $

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Figure 2. $\mathit{\boldsymbol{q}}^\ast(t)\;(t \in [0,T])$

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Figure 3. a whole brake orbit

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Figure 4. minimizer with collosions

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Figure 5. the domain $D'$

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