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New periodic orbits in the planar equal-mass three-body problem
School of Mathematics and System Sciences, Beihang University, Beijing 100191, China |
It is known that there exist two sets of nontrivial periodic orbits in the planar equal-mass three-body problem: retrograde orbit and prograde orbit. By introducing topological constraints to a two-point free boundary value problem, we show that there exists a new set of periodic orbits for a small interval of rotation angle $ \mathit{\theta }$.
References:
[1] |
R. Broucke and D. Boggs,
Periodic orbits in the planar general three-body problem, Celestial Mech., 11 (1975), 13-38.
doi: 10.1007/BF01228732. |
[2] |
K. Chen,
Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses, Annals of Math., 167 (2008), 325-348.
doi: 10.4007/annals.2008.167.325. |
[3] |
K. Chen and Y. Lin,
On action-minimizing retrograde and prograde orbits of the three-body problem, Comm. Math. Phys., 291 (2009), 403-441.
doi: 10.1007/s00220-009-0769-5. |
[4] |
A. Chenciner and R. Montgomery,
A remarkable periodic solution of the three-body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901.
doi: 10.2307/2661357. |
[5] |
A. Chenciner, Action minimizing solutions in the Newtonian n-body problem: From homology to symmetry,
Proceedings of the International Congress of Mathematicians (Beijing, 2002), Higher Ed. Press, Beijing, (2002), 279-294. |
[6] |
W. B. Gordon,
A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971.
doi: 10.2307/2373993. |
[7] |
M. Hénon,
A family of periodic solutions of the planar three-body problem, and their stability, Celestial Mech., 13 (1976), 267-285.
doi: 10.1007/BF01228647. |
[8] |
W. Kuang, T. Ouyang, Z. Xie and D. Yan, The Broucke-Hénon orbit and the Schubart orbit in the planar three-body problem with equal masses, preprint, arXiv: 1607.00580. |
[9] |
C. Marchal,
How the method of minimization of action avoids singularities, Celest. Mech. Dyn. Astro., 83 (2002), 325-353.
doi: 10.1023/A:1020128408706. |
[10] |
T. Ouyang and Z. Xie,
Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem, Physica D, 307 (2015), 61-76.
doi: 10.1016/j.physd.2015.05.015. |
[11] |
B. Shi, R. Liu, D. Yan and T. Ouyang,
Multiple periodic orbits connecting a collinear configuration and a double isosceles configuration in the planar equal-mass four-body problem, Adv. Nonlinear Stud., 17 (2017), 819-835.
doi: 10.1515/ans-2017-6028. |
[12] |
G. Yu,
Simple choreography solutions of the Newtonian N-body problem, Arch. Rational Mech. Anal., 225 (2017), 901-935.
doi: 10.1007/s00205-017-1116-1. |
[13] |
G. Yu, Spatial double choreographies of the Newtonian $ 2n$-body problem, arXiv: 1608.07956. |
show all references
References:
[1] |
R. Broucke and D. Boggs,
Periodic orbits in the planar general three-body problem, Celestial Mech., 11 (1975), 13-38.
doi: 10.1007/BF01228732. |
[2] |
K. Chen,
Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses, Annals of Math., 167 (2008), 325-348.
doi: 10.4007/annals.2008.167.325. |
[3] |
K. Chen and Y. Lin,
On action-minimizing retrograde and prograde orbits of the three-body problem, Comm. Math. Phys., 291 (2009), 403-441.
doi: 10.1007/s00220-009-0769-5. |
[4] |
A. Chenciner and R. Montgomery,
A remarkable periodic solution of the three-body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901.
doi: 10.2307/2661357. |
[5] |
A. Chenciner, Action minimizing solutions in the Newtonian n-body problem: From homology to symmetry,
Proceedings of the International Congress of Mathematicians (Beijing, 2002), Higher Ed. Press, Beijing, (2002), 279-294. |
[6] |
W. B. Gordon,
A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971.
doi: 10.2307/2373993. |
[7] |
M. Hénon,
A family of periodic solutions of the planar three-body problem, and their stability, Celestial Mech., 13 (1976), 267-285.
doi: 10.1007/BF01228647. |
[8] |
W. Kuang, T. Ouyang, Z. Xie and D. Yan, The Broucke-Hénon orbit and the Schubart orbit in the planar three-body problem with equal masses, preprint, arXiv: 1607.00580. |
[9] |
C. Marchal,
How the method of minimization of action avoids singularities, Celest. Mech. Dyn. Astro., 83 (2002), 325-353.
doi: 10.1023/A:1020128408706. |
[10] |
T. Ouyang and Z. Xie,
Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem, Physica D, 307 (2015), 61-76.
doi: 10.1016/j.physd.2015.05.015. |
[11] |
B. Shi, R. Liu, D. Yan and T. Ouyang,
Multiple periodic orbits connecting a collinear configuration and a double isosceles configuration in the planar equal-mass four-body problem, Adv. Nonlinear Stud., 17 (2017), 819-835.
doi: 10.1515/ans-2017-6028. |
[12] |
G. Yu,
Simple choreography solutions of the Newtonian N-body problem, Arch. Rational Mech. Anal., 225 (2017), 901-935.
doi: 10.1007/s00205-017-1116-1. |
[13] |
G. Yu, Spatial double choreographies of the Newtonian $ 2n$-body problem, arXiv: 1608.07956. |



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