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

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 \begin{document}$\mathbb{R}^{12}$\end{document} 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 \begin{document}$\mathbb{R}^{12}$\end{document} can be simplified to two separated sub-problems: linear stabilities in Γ and \begin{document}$(\mathbb{R}^{12} \setminus Γ) \cup \{0\}$\end{document} . By applying Roberts' symmetry reduction method, we prove that the orbits are always unstable in the full phase space \begin{document}$\mathbb{R}^{12}$\end{document} , but it is linearly stable in Γ when \begin{document}$θ ∈ [0.33π, 0.48 π] \cup [0.52 π, 0.78 π]$\end{document} .