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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

Tiancheng Ouyang 1, and  Duokui Yan 2,,

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 $θ ∈ [0.33π, 0.48 π] \cup [0.52 π, 0.78 π]$.

Keywords: Three-body problem, spatial isosceles orbit, variational method, linear stability.
Mathematics Subject Classification: Primary: 70F10, 70F15; Secondary: 70F07.
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:
[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.

show all references

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.

Figure 1.  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.
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Figure 2.  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.
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Figure 3.  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.
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Figure 4.  Spatial isosceles orbit with $\theta = \pi/3$.
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Figure 5.  Spatial isosceles orbit with $\theta = \pi/2$.
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Figure 6.  Spatial isosceles orbit with $\theta=3\pi/4$.
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Figure 7.  Broucke orbit
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