Linear stability of elliptic relative equilibria of four-body problem with two infinitesimal masses

Abstract

In this paper, we consider the elliptic relative equilibria of four-body problem with two infinitesimal masses. The most interesting case is when the two small masses tend to the same Lagrangian point $L_4$ (or $L_5$). In \cite{Xia}, Z. Xia showed that there exist four central configurations: two of them are non-convex, and the other two are convex. We prove that the elliptic relative equilibria raised from the non-convex central configurations are always linearly unstable; while for the elliptic relative equilibria raised from the convex central configurations, the conditions of linear stability with respect to the parameters are given.

60 pages, 5 figures. arXiv admin note: text overlap with arXiv:1510.06822 and arXiv:1907.13475

References in corpus (4)

• Collision index and stability of elliptic relative equilibria in planar n-body problem
• Maslov-type indices and linear stability of elliptic Euler solutions of the three-body problem
• Linear stability of the elliptic relative equilibrium with $(1+n)$-gon central configurations in planar $n$-body problem
• The reduction on the linear stability of elliptic Euler-Moulton solutions of the n-body problem to those of 3-body problems

Cited by in corpus (2)

• Linear Instability of Elliptic Rhombus Solutions to the Planar Four-body Problem
• Linear stability of the elliptic relative equilibria for the restricted N-body problem: two special cases