October  2019, 39(10): 5785-5797. doi: 10.3934/dcds.2019254

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

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

Received  September 2018 Revised  February 2019 Published  July 2019

Fund Project: 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

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.

Citation: Yuika Kajihara, Misturu Shibayama. Variational proof of the existence of brake orbits in the planar 2-center problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5785-5797. doi: 10.3934/dcds.2019254
References:
[1]

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K.-C. Chen, Binary decompositions for planar N-body problems and symmetric periodic solutions, Arch. Ration. Mech. Anal., 170 (2003), 247-276. doi: 10.1007/s00205-003-0277-2. Google Scholar

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G. Yu, Periodic solutions of the planar N-center problem with topological constraints, Discrete Contin. Dyn. Syst., 36 (2016), 5131-5162. doi: 10.3934/dcds.2016023. Google Scholar

[11]

H. Urakawa, Calculus of Variations and Harmonic Maps, American Mathematical Society, 1993. doi: 1183532220. Google Scholar

show all references

References:
[1]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1. Google Scholar

[2]

K.-C. Chen, Binary decompositions for planar N-body problems and symmetric periodic solutions, Arch. Ration. Mech. Anal., 170 (2003), 247-276. doi: 10.1007/s00205-003-0277-2. Google Scholar

[3]

N.-C. Chen, Periodic brake orbits in the planar isosceles three-body problem, Nonlinearity, 26 (2013), 2875-2898. doi: 10.1088/0951-7715/26/10/2875. Google Scholar

[4]

W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971. doi: 10.2307/2373993. Google Scholar

[5]

R. MoeckelR. Montgomery and A. Venturelli, From brake to syzygy, Arch. Ration. Mech. Anal., 204 (2012), 1009-1060. doi: 10.1007/s00205-012-0502-y. Google Scholar

[6]

M. B. Sevryuk, Reversible Systems, Springer–Verlag, 1986. doi: 10.1007/BFb0075877. Google Scholar

[7] V. Szebehely, Theory of Orbits, the Restricted Problem of Three Bodies, Academic Press, 1967. Google Scholar
[8]

K. Tanaka, A prescribed-energy problem for a conservative singular Hamiltonian system., Arch. Ration. Mech. Anal., 128 (1994), 127-164. doi: 10.1007/BF00375024. Google Scholar

[9]

L. Tonelli, The calculus of variations, Bull. Amer. Math. Soc., 31 (1925), 163-172. doi: 10.1090/S0002-9904-1925-04002-1. Google Scholar

[10]

G. Yu, Periodic solutions of the planar N-center problem with topological constraints, Discrete Contin. Dyn. Syst., 36 (2016), 5131-5162. doi: 10.3934/dcds.2016023. Google Scholar

[11]

H. Urakawa, Calculus of Variations and Harmonic Maps, American Mathematical Society, 1993. doi: 1183532220. Google Scholar

Figure 1.  the domain $ D $
Figure 2.  $\mathit{\boldsymbol{q}}^\ast(t)\;(t \in [0,T])$
Figure 3.  a whole brake orbit
Figure 4.  minimizer with collosions
Figure 5.  the domain $D'$
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