\`x^2+y_1+z_12^34\`
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Variational properties and linear stabilities of spatial isosceles orbits in the equal-mass three-body problem

  • * Corresponding author: Duokui Yan

    * Corresponding author: Duokui Yan

The second author is supported by NSFC No. 11432001

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

    Mathematics Subject Classification: Primary: 70F10, 70F15; Secondary: 70F07.

    Citation:

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

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

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

    Figure 7.  Broucke orbit

  •   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.
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