Xijun Hu

Jinan, Shandong 250100, The People’s Republic of China
Department of Mathematics, Shandong University


Hu X., Ou Y., Tang X.
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 Maslov-type 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.
Keywords: linear stability, elliptic relative equilibrium, Maslov-type index, spatial $n$-body problem
Citation: Hu X., Ou Y., Tang X.,  Linear Stability of an Elliptic Relative Equilibrium in the Spatial $n$-Body Problem via Index Theory, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 731-755
Hu X., Ou Y.
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.
Keywords: central configurations, elliptic relative equilibrium, linear stability, hyperbolicity, $n$-body problem
Citation: Hu X., Ou Y.,  An Estimation for the Hyperbolic Region of Elliptic Lagrangian Solutions in the Planar Three-body Problem, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 732-741

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