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

Tiancheng Ouyang; Duokui Yan

doi:10.3934/dcds.2017169

Article Contents

2017, Volume 37, Issue 7: 3989-4018. Doi: 10.3934/dcds.2017169

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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
* Corresponding author: Duokui Yan

Received: August 31, 2016

Revised: February 28, 2017

Published: August 2017

The second author is supported by NSFC No. 11432001

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Abstract

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

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

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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.

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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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