Xijun Hu
Jinan, Shandong 250100, The People’s Republic of China
Department of Mathematics, Shandong University
Publications:
Hu X., Ou Y., Tang X.
Linear Stability of an Elliptic Relative Equilibrium in the Spatial $n$Body Problem via Index Theory
2023, vol. 28, nos. 45, pp. 731755
Abstract
It is well known that a planar central configuration of the $n$body problem gives rise to a solution where each particle moves in a Keplerian orbit with a common eccentricity $\mathfrak{e}\in[0,1)$. We call this solution an elliptic relative equilibrium (ERE for short). Since each particle of the ERE is always in the same plane, it is natural to regard it as a planar $n$body problem. But in practical applications, it is more meaningful to consider the ERE as a spatial $n$body problem (i.\,e., each particle belongs to $\mathbb{R}^3$). In this paper, as a spatial $n$body problem, we first decompose the linear system of ERE into two parts, the planar and the spatial part. Following the Meyer\,\,Schmidt coordinate~\cite{Meyer}, we give an expression for the spatial part and further obtain a rigorous analytical method to study the linear stability of the spatial part by the Maslovtype index theory. As an application, we obtain stability results for some classical ERE, including the elliptic Lagrangian solution, the Euler solution and the $1+n$gon solution.

Hu X., Ou Y.
An Estimation for the Hyperbolic Region of Elliptic Lagrangian Solutions in the Planar Threebody Problem
2013, vol. 18, no. 6, pp. 732741
Abstract
It is well known that the linear stability of elliptic Lagrangian solutions depends on the mass parameter $\beta = 27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2 \in [0,9]$ and the eccentricity $e \in [0,1)$. Based on new techniques for evaluating the hyperbolicity and the recently developed trace formula for Hamiltonian systems [9], we identify regions for $(\beta,e)$ such that elliptic Lagrangian solutions are hyperbolic. Consequently, we have proven that the elliptic relative equilibrium of square central configurations is hyperbolic with any eccentricity.
