# American Institute of Mathematical Sciences

July  2017, 37(7): 3989-4018. doi: 10.3934/dcds.2017169

## Variational properties and linear stabilities of spatial isosceles orbits in the equal-mass three-body problem

 1 Department of Mathematics, Brigham Young University, Provo, UT 84602, USA 2 School of Mathematics and System Sciences, Beihang University, Beijing 100191, China

* Corresponding author: Duokui Yan

Received  September 2016 Revised  March 2017 Published  April 2017

Fund Project: The second author is supported by NSFC No. 11432001

We prove new variational properties of the spatial isosceles orbits in the equal-mass three-body problem and analyze their linear stabilities in both the full phase space $\mathbb{R}^{12}$ and a symmetric subspace Γ. We prove that each spatial isosceles orbit is an action minimizer of a two-point free boundary value problem with non-symmetric boundary settings. The spatial isosceles orbits form a one-parameter set with rotation angle θ as the parameter. This set of orbits always lies in a symmetric subspace Γ and we show that their linear stabilities in the full phase space $\mathbb{R}^{12}$ can be simplified to two separated sub-problems: linear stabilities in Γ and $(\mathbb{R}^{12} \setminus Γ) \cup \{0\}$. By applying Roberts' symmetry reduction method, we prove that the orbits are always unstable in the full phase space $\mathbb{R}^{12}$, but it is linearly stable in Γ when $θ ∈ [0.33π, 0.48 π] \cup [0.52 π, 0.78 π]$.

Citation: Tiancheng Ouyang, Duokui Yan. Variational properties and linear stabilities of spatial isosceles orbits in the equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3989-4018. doi: 10.3934/dcds.2017169
##### References:
 [1] R. Broucke, On the isosceles triangle configuration in the planar general three-body problem, Astron. Astrophys., 73 (1979), 303-313. [2] K. Chen, Existence and minimizing properties of retrograde orbits to the three-body problems with various choices of masses, Ann. of Math., 167 (2008), 325-348. doi: 10.4007/annals.2008.167.325. [3] K. Chen, Removing collision singularities from action minimizers for the N-body problem with free boundaries, Arch. Ration. Mech. Anal., 181 (2006), 311-331. doi: 10.1007/s00205-005-0413-2. [4] K. Chen, T. Ouyang and Z. Xia, Action-minimizing periodic and quasi-periodic solutions in the N-body problem, Math. Res. Lett., 19 (2012), 483-497. doi: 10.4310/MRL.2012.v19.n2.a19. [5] A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math., 152 (2000), 881-901. doi: 10.2307/2661357. [6] A. Chenciner, Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry, in Proceedings of the International Congress of Mathematiicans (Beijing, 2002), Higher Ed. Press, Beijing, 3 (2002), 279–294. [7] D. Ferrario and S. Terracini, On the existence of collisionless equivalent minimizers for the classical n-body problem, Inv. Math., 155 (2004), 305-362. doi: 10.1007/s00222-003-0322-7. [8] W. Gordon, A minimizing property of Keplerian orbits, American Journal of Math., 99 (1977), 961-971. doi: 10.2307/2373993. [9] 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, arXiv: 1607.00580. [10] Y. Long, Index Theory for Symplectic Paths with Applications Birkhäuser Verlag, Basel-Boston-Berlin, 2002. doi: 10.1007/978-3-0348-8175-3. [11] C. Marchal, How the method of minimization of action avoids singularities, Celest. Mech. Dyn. Astr., 83 (2002), 325-353. doi: 10.1023/A:1020128408706. [12] D. Offin and H. Cabral, Hyperbolicity for symmetric periodic orbits in the isosceles three body problem, Dis. Con. Dyn. Syst. Ser. S, 2 (2009), 379-392. doi: 10.3934/dcdss.2009.2.379. [13] G. Roberts, Linear Stability analysis of the figure-eight orbit in the three-body problem, Ergod. Th. & Dyn. Sys., 27 (2007), 1947-1963. doi: 10.1017/S0143385707000284. [14] M. Shibayama, Existence and stability of periodic solutions in the isosceles three-body problem, RIMS Kôkyûroku Bessatsu, B13 (2009), 141-155. [15] D. Yan, Existence of the Broucke periodic orbit and its linear stability, J. Math. Anal. Appl., 389 (2012), 656-664. doi: 10.1016/j.jmaa.2011.12.024. [16] D. Yan and T. Ouyang, New phenomena in the spatial isosceles three-body problem, Int. J. Bifurcation Chaos, 25 (2015), 1550116. doi: 10.1142/S0218127415501163. [17] D. Yan, R. Liu, X. Hu, W. Mao and T. Ouyang, New phenomena in the spatial isosceles three-body problem with unequal masses, Int. J. Bifurcation Chaos, 25 (2015), 1550169. doi: 10.1142/S0218127415501692. [18] Personal communications with chongchun zeng at Georgia institute of technology, 2008.

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##### References:
 [1] R. Broucke, On the isosceles triangle configuration in the planar general three-body problem, Astron. Astrophys., 73 (1979), 303-313. [2] K. Chen, Existence and minimizing properties of retrograde orbits to the three-body problems with various choices of masses, Ann. of Math., 167 (2008), 325-348. doi: 10.4007/annals.2008.167.325. [3] K. Chen, Removing collision singularities from action minimizers for the N-body problem with free boundaries, Arch. Ration. Mech. Anal., 181 (2006), 311-331. doi: 10.1007/s00205-005-0413-2. [4] K. Chen, T. Ouyang and Z. Xia, Action-minimizing periodic and quasi-periodic solutions in the N-body problem, Math. Res. Lett., 19 (2012), 483-497. doi: 10.4310/MRL.2012.v19.n2.a19. [5] A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math., 152 (2000), 881-901. doi: 10.2307/2661357. [6] A. Chenciner, Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry, in Proceedings of the International Congress of Mathematiicans (Beijing, 2002), Higher Ed. Press, Beijing, 3 (2002), 279–294. [7] D. Ferrario and S. Terracini, On the existence of collisionless equivalent minimizers for the classical n-body problem, Inv. Math., 155 (2004), 305-362. doi: 10.1007/s00222-003-0322-7. [8] W. Gordon, A minimizing property of Keplerian orbits, American Journal of Math., 99 (1977), 961-971. doi: 10.2307/2373993. [9] 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, arXiv: 1607.00580. [10] Y. Long, Index Theory for Symplectic Paths with Applications Birkhäuser Verlag, Basel-Boston-Berlin, 2002. doi: 10.1007/978-3-0348-8175-3. [11] C. Marchal, How the method of minimization of action avoids singularities, Celest. Mech. Dyn. Astr., 83 (2002), 325-353. doi: 10.1023/A:1020128408706. [12] D. Offin and H. Cabral, Hyperbolicity for symmetric periodic orbits in the isosceles three body problem, Dis. Con. Dyn. Syst. Ser. S, 2 (2009), 379-392. doi: 10.3934/dcdss.2009.2.379. [13] G. Roberts, Linear Stability analysis of the figure-eight orbit in the three-body problem, Ergod. Th. & Dyn. Sys., 27 (2007), 1947-1963. doi: 10.1017/S0143385707000284. [14] M. Shibayama, Existence and stability of periodic solutions in the isosceles three-body problem, RIMS Kôkyûroku Bessatsu, B13 (2009), 141-155. [15] D. Yan, Existence of the Broucke periodic orbit and its linear stability, J. Math. Anal. Appl., 389 (2012), 656-664. doi: 10.1016/j.jmaa.2011.12.024. [16] D. Yan and T. Ouyang, New phenomena in the spatial isosceles three-body problem, Int. J. Bifurcation Chaos, 25 (2015), 1550116. doi: 10.1142/S0218127415501163. [17] D. Yan, R. Liu, X. Hu, W. Mao and T. Ouyang, New phenomena in the spatial isosceles three-body problem with unequal masses, Int. J. Bifurcation Chaos, 25 (2015), 1550169. doi: 10.1142/S0218127415501692. [18] Personal communications with chongchun zeng at Georgia institute of technology, 2008.
A demonstration of one piece of a spatial isosceles orbit with rotation angle $\theta$, from an Euler configuration ($t = 0$) to an isosceles configuration ($t = 1$). Body 2 reaches its lowest point on the z-axis at $t = 1$. The isosceles configuration at $t = 1$ lies in a plane which is an $\theta$ counterclockwise rotation of the xz plane.
Motion of a spatial isosceles orbit. The three dots represent the starting positions of the three bodies. The trajectory of each body is represented by a curve of its color. In every period, body 2 (the black dot) moves up and down on the z-axis and the other two bodies (red and blue dots) rotate about the z-axis symmetrically.
Linear stability of the spatial isosceles orbits in $\Gamma$ with respect to $\theta/\pi$. When $\theta/\pi \in [0.33, 0.48]$, the orbit is linearly stable in $\Gamma$; when $\theta/\pi \in [0.49, 0.51]$, it is unstable; when $\theta/\pi \in [0.52, 0.78]$, it becomes linearly stable again in $\Gamma$; when $\theta/\pi \in [0.79, 1)$, it is unstable.
Spatial isosceles orbit with $\theta = \pi/3$.
Spatial isosceles orbit with $\theta = \pi/2$.
Spatial isosceles orbit with $\theta=3\pi/4$.
Broucke orbit
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