    August  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 and Continuous Dynamical Systems, 2017, 37 (7) : 3989-4018. doi: 10.3934/dcds.2017169
##### References:
  R. Broucke, On the isosceles triangle configuration in the planar general three-body problem, Astron. Astrophys., 73 (1979), 303-313. 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.   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.   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.   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.   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.  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.   W. Gordon, A minimizing property of Keplerian orbits, American Journal of Math., 99 (1977), 961-971.  doi: 10.2307/2373993.   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. Y. Long, Index Theory for Symplectic Paths with Applications Birkhäuser Verlag, Basel-Boston-Berlin, 2002. doi: 10.1007/978-3-0348-8175-3.   C. Marchal, How the method of minimization of action avoids singularities, Celest. Mech. Dyn. Astr., 83 (2002), 325-353.  doi: 10.1023/A:1020128408706.   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.   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.   M. Shibayama, Existence and stability of periodic solutions in the isosceles three-body problem, RIMS Kôkyûroku Bessatsu, B13 (2009), 141-155.  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.   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.   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.   Personal communications with chongchun zeng at Georgia institute of technology, 2008. show all references

##### References:
  R. Broucke, On the isosceles triangle configuration in the planar general three-body problem, Astron. Astrophys., 73 (1979), 303-313. 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.   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.   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.   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.   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.  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.   W. Gordon, A minimizing property of Keplerian orbits, American Journal of Math., 99 (1977), 961-971.  doi: 10.2307/2373993.   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. Y. Long, Index Theory for Symplectic Paths with Applications Birkhäuser Verlag, Basel-Boston-Berlin, 2002. doi: 10.1007/978-3-0348-8175-3.   C. Marchal, How the method of minimization of action avoids singularities, Celest. Mech. Dyn. Astr., 83 (2002), 325-353.  doi: 10.1023/A:1020128408706.   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.   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.   M. Shibayama, Existence and stability of periodic solutions in the isosceles three-body problem, RIMS Kôkyûroku Bessatsu, B13 (2009), 141-155.  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.   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.   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.   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.
  Xiaojun Chang, Tiancheng Ouyang, Duokui Yan. Linear stability of the criss-cross orbit in the equal-mass three-body problem. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5971-5991. doi: 10.3934/dcds.2016062  Hiroshi Ozaki, Hiroshi Fukuda, Toshiaki Fujiwara. Determination of motion from orbit in the three-body problem. Conference Publications, 2011, 2011 (Special) : 1158-1166. doi: 10.3934/proc.2011.2011.1158  Kuo-Chang Chen. On Chenciner-Montgomery's orbit in the three-body problem. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 85-90. doi: 10.3934/dcds.2001.7.85  Qinglong Zhou, Yongchao Zhang. Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1763-1787. doi: 10.3934/dcds.2017074  Samuel R. Kaplan, Mark Levi, Richard Montgomery. Making the moon reverse its orbit, or, stuttering in the planar three-body problem. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 569-595. doi: 10.3934/dcdsb.2008.10.569  Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463  Edward Belbruno. Random walk in the three-body problem and applications. Discrete and Continuous Dynamical Systems - S, 2008, 1 (4) : 519-540. doi: 10.3934/dcdss.2008.1.519  Regina Martínez, Carles Simó. On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb{S}^2$. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1157-1175. doi: 10.3934/dcds.2013.33.1157  Hadia H. Selim, Juan L. G. Guirao, Elbaz I. Abouelmagd. Libration points in the restricted three-body problem: Euler angles, existence and stability. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 703-710. doi: 10.3934/dcdss.2019044  Daniel Offin, Hildeberto Cabral. Hyperbolicity for symmetric periodic orbits in the isosceles three body problem. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 379-392. doi: 10.3934/dcdss.2009.2.379  Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3759-3779. doi: 10.3934/dcds.2021015  Richard Moeckel. A topological existence proof for the Schubart orbits in the collinear three-body problem. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 609-620. doi: 10.3934/dcdsb.2008.10.609  Mitsuru Shibayama. Non-integrability of the collinear three-body problem. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 299-312. doi: 10.3934/dcds.2011.30.299  Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229  Richard Moeckel. A proof of Saari's conjecture for the three-body problem in $\R^d$. Discrete and Continuous Dynamical Systems - S, 2008, 1 (4) : 631-646. doi: 10.3934/dcdss.2008.1.631  Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090  Niraj Pathak, V. O. Thomas, Elbaz I. Abouelmagd. The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 849-875. doi: 10.3934/dcdss.2019057  Abimael Bengochea, Manuel Falconi, Ernesto Pérez-Chavela. Horseshoe periodic orbits with one symmetry in the general planar three-body problem. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 987-1008. doi: 10.3934/dcds.2013.33.987  Martha Alvarez-Ramírez, Joaquín Delgado. Blow up of the isosceles 3--body problem with an infinitesimal mass. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1149-1173. doi: 10.3934/dcds.2003.9.1149  Jean-Baptiste Caillau, Bilel Daoud, Joseph Gergaud. Discrete and differential homotopy in circular restricted three-body control. Conference Publications, 2011, 2011 (Special) : 229-239. doi: 10.3934/proc.2011.2011.229

2021 Impact Factor: 1.588