American Institute of Mathematical Sciences

Advanced
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

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 & 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. Gordon, A minimizing property of Keplerian orbits, American Journal of Math., 99 (1977), 961-971.  doi: 10.2307/2373993.  Google Scholar

[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. Google Scholar

[10]

Y. Long, Index Theory for Symplectic Paths with Applications Birkhäuser Verlag, Basel-Boston-Berlin, 2002. doi: 10.1007/978-3-0348-8175-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. Shibayama, Existence and stability of periodic solutions in the isosceles three-body problem, RIMS Kôkyûroku Bessatsu, B13 (2009), 141-155.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

Personal communications with chongchun zeng at Georgia institute of technology, 2008. Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. Gordon, A minimizing property of Keplerian orbits, American Journal of Math., 99 (1977), 961-971.  doi: 10.2307/2373993.  Google Scholar

[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. Google Scholar

[10]

Y. Long, Index Theory for Symplectic Paths with Applications Birkhäuser Verlag, Basel-Boston-Berlin, 2002. doi: 10.1007/978-3-0348-8175-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. Shibayama, Existence and stability of periodic solutions in the isosceles three-body problem, RIMS Kôkyûroku Bessatsu, B13 (2009), 141-155.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

Personal communications with chongchun zeng at Georgia institute of technology, 2008. Google Scholar

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.
Figure Options

Download full-size image

Download as PowerPoint slide

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.
Figure Options

Download full-size image

Download as PowerPoint slide

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.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Spatial isosceles orbit with $\theta = \pi/3$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Spatial isosceles orbit with $\theta = \pi/2$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Spatial isosceles orbit with $\theta=3\pi/4$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Broucke orbit
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Xiaojun Chang, Tiancheng Ouyang, Duokui Yan. Linear stability of the criss-cross orbit in the equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5971-5991. doi: 10.3934/dcds.2016062
[2]

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
[3]

Kuo-Chang Chen. On Chenciner-Montgomery's orbit in the three-body problem. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 85-90. doi: 10.3934/dcds.2001.7.85
[4]

Qinglong Zhou, Yongchao Zhang. Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1763-1787. doi: 10.3934/dcds.2017074
[5]

Samuel R. Kaplan, Mark Levi, Richard Montgomery. Making the moon reverse its orbit, or, stuttering in the planar three-body problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 569-595. doi: 10.3934/dcdsb.2008.10.569
[6]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463
[7]

Edward Belbruno. Random walk in the three-body problem and applications. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 519-540. doi: 10.3934/dcdss.2008.1.519
[8]

Daniel Offin, Hildeberto Cabral. Hyperbolicity for symmetric periodic orbits in the isosceles three body problem. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 379-392. doi: 10.3934/dcdss.2009.2.379
[9]

Regina Martínez, Carles Simó. On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb{S}^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1157-1175. doi: 10.3934/dcds.2013.33.1157
[10]

Hadia H. Selim, Juan L. G. Guirao, Elbaz I. Abouelmagd. Libration points in the restricted three-body problem: Euler angles, existence and stability. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 703-710. doi: 10.3934/dcdss.2019044
[11]

Richard Moeckel. A topological existence proof for the Schubart orbits in the collinear three-body problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 609-620. doi: 10.3934/dcdsb.2008.10.609
[12]

Mitsuru Shibayama. Non-integrability of the collinear three-body problem. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 299-312. doi: 10.3934/dcds.2011.30.299
[13]

Richard Moeckel. A proof of Saari's conjecture for the three-body problem in $\R^d$. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 631-646. doi: 10.3934/dcdss.2008.1.631
[14]

Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229
[15]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090
[16]

Abimael Bengochea, Manuel Falconi, Ernesto Pérez-Chavela. Horseshoe periodic orbits with one symmetry in the general planar three-body problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 987-1008. doi: 10.3934/dcds.2013.33.987
[17]

Niraj Pathak, V. O. Thomas, Elbaz I. Abouelmagd. The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 849-875. doi: 10.3934/dcdss.2019057
[18]

Martha Alvarez-Ramírez, Joaquín Delgado. Blow up of the isosceles 3--body problem with an infinitesimal mass. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1149-1173. doi: 10.3934/dcds.2003.9.1149
[19]

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
[20]

Frederic Gabern, Àngel Jorba, Philippe Robutel. On the accuracy of restricted three-body models for the Trojan motion. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 843-854. doi: 10.3934/dcds.2004.11.843

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (18)
  • HTML views (6)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences