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 [33], 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.

Linear stability of the elliptic relative equilibria for the restricted $4$-body problem: the Euler case

“…Furthermore, one stable region of the linear stability has been found. Regarding the linear stability of other ERE to the four-body problem, readers may refer to [4,24] In this paper, we focus on the restricted 4-body problem with three primaries forming Euller collinear configurations and the four bodies span R 2 . Since the ERE of the three primaries is always linearly unstable [25], it is reasonable to study the stability problem of the massless particle.…”

Linear instability of elliptic rhombus solutions to the planar four-body problem

“…For circular rhombus solutions of a homogeneous potential with degree a, Leandro in [9] obtained the condition for stability and instability with respect to a. Regarding the linear stability of other ERE to the four-body problem, readers may refer to [2,25], and [11].…”