Linear stability of elliptic relative equilibria of restricted four-body problem

Abstract

In this paper, we consider the linear stability of the elliptic relative equilibria of the restricted 4-body problems where the three primaries form a Lagrangian triangle. By reduction, the linearized Poincaré map is decomposed to the essential part, the Keplerian part and the elliptic Lagrangian part where the last two parts have been studied in literature. The linear stability of the essential part depends on the masses parameters α, β with α ≥ β > 0 and the eccentricity e ∈ [0, 1). Via ω-Maslov index theory and linear differential operator theory, we obtain the full bifurcation diagram of linearly stable and unstable regions with respect to α, β and e.Especially, two linearly stable sub-regions are found.

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

“…For the Euler collinear case, we still have two cases: the 4 bodies are collinear and the four bodies span R 2 . The case of the four bodies form a collinear configuration has been discussed in [13]. In this paper, we focus on the case of 4 bodies span R 2 .…”

“…One well-studied case is the elliptic rhombus solution [10,12,16] which are linearly instable. For the restricted 4-body problem with three primaries forming Lagrangian equilateral configuration, the full bifurcation diagram of the stability and instability has been obtained for all possible masses and all eccentricity [9,13]. Furthermore, one stable region of the linear stability has been found.…”

Stability Analysis of the Rhomboidal Restricted Six-Body Problem

“…Because of the symmetry of the problem, the equations for m 2 and m 4 are identical to equations ( 7) and (8). e equation for m 0 is identically zero.…”

“…Llibre and Mello [7] classified the central configurations of the four-body problem. Liu and Zhou [8] investigated the four-body problem with three masses forming a Lagrangian triangle, used the bifurcation diagram of linearly stable and unstable regions, and found two linearly stable subregions with respect to α, β, and e. Deng et al [9] investigated the CC of the four-body problem with equal masses and showed that, for the planar Newtonian four-body problem having adjacent equal masses i.e., m 1 � m 2 ≠ m 3 � m 4 and equal lengths for the two diagonals, any convex noncollinear CC must have a symmetry and must be an isosceles trapezoid. ey also showed that when the length between m 1 and m 4 equals the length between m 2 and m 3 , the CC is also an isosceles trapezoid.…”

Qualitative Analysis of Some Libration Points in the Restricted Four-body Problem

“…In [20] the restricted four-body problem was studied in the symmetric case (it agrees with other results) and in the case where A lies outside the Lagrange triangle (6 configurations). In [22] the authors use functional analytic methods and a version of Maslov index to study stability of some situations from [20].…”