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ЖУРНАЛЫ // Regular and Chaotic Dynamics // Архив

Regul. Chaotic Dyn., 2023, том 28, выпуск 4-5, страницы 731–755 (Mi rcd1230)

Special Issue: On the 80th birthday of professor A. Chenciner

Linear Stability of an Elliptic Relative Equilibrium in the Spatial $n$-Body Problem via Index Theory

Xijun Hu, Yuwei Ou, Xiuting Tang

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

Аннотация: 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 [19], 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.

Ключевые слова: linear stability, elliptic relative equilibrium, Maslov-type index, spatial $n$-body problem.

MSC: 37J25, 70F10, 53D12

Поступила в редакцию: 11.04.2023
Принята в печать: 14.07.2023

Язык публикации: английский

DOI: 10.1134/S1560354723040135



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