# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2019, 39 (10) : 5785-5797. doi: 10.3934/dcds.2019254
##### References:
 [1] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1. [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. [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. [4] W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971.  doi: 10.2307/2373993. [5] R. Moeckel, R. Montgomery and A. Venturelli, From brake to syzygy, Arch. Ration. Mech. Anal., 204 (2012), 1009-1060.  doi: 10.1007/s00205-012-0502-y. [6] M. B. Sevryuk, Reversible Systems, Springer–Verlag, 1986. doi: 10.1007/BFb0075877. [7] V. Szebehely, Theory of Orbits, the Restricted Problem of Three Bodies, Academic Press, 1967. [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. [9] L. Tonelli, The calculus of variations, Bull. Amer. Math. Soc., 31 (1925), 163-172.  doi: 10.1090/S0002-9904-1925-04002-1. [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. [11] H. Urakawa, Calculus of Variations and Harmonic Maps, American Mathematical Society, 1993. doi: 1183532220.

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. [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. [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. [4] W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971.  doi: 10.2307/2373993. [5] R. Moeckel, R. Montgomery and A. Venturelli, From brake to syzygy, Arch. Ration. Mech. Anal., 204 (2012), 1009-1060.  doi: 10.1007/s00205-012-0502-y. [6] M. B. Sevryuk, Reversible Systems, Springer–Verlag, 1986. doi: 10.1007/BFb0075877. [7] V. Szebehely, Theory of Orbits, the Restricted Problem of Three Bodies, Academic Press, 1967. [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. [9] L. Tonelli, The calculus of variations, Bull. Amer. Math. Soc., 31 (1925), 163-172.  doi: 10.1090/S0002-9904-1925-04002-1. [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. [11] H. Urakawa, Calculus of Variations and Harmonic Maps, American Mathematical Society, 1993. doi: 1183532220.
the domain $D$
$\mathit{\boldsymbol{q}}^\ast(t)\;(t \in [0,T])$
a whole brake orbit
minimizer with collosions
the domain $D'$
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