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

Abstract

In this paper, we study the linear stability of the elliptic rhombus solutions, which are the Keplerian homographic solution with the rhombus central configurations in the classical planar four-body problems. Using ω-Maslov index theory and trace formula, we prove the linear instability of elliptic rhombus solutions if the shape parameter u and the eccentricity of the elliptic orbit e satisfy ( u , e ) ∈ ( 1 / 3 , u 2 ) × 0 , f ^ ( 27 4 ) − 1 / 2 ∪ ( u 2 , 1 / u 2 ) × 0 , 1 ∪ ( 1 / u 2 , 3 ) × 0 , f ^ ( 27 4 ) − 1 / 2 where u 2 ≈ 0.6633 and f ^ ( 27 4 ) − 1 / 2 ≈ 0.4454 . Motivated on numerical results of the linear stability to the elliptic Lagrangian solutions in Martínez et al (2006 J. Differ. Equ. 226 619–651), we further analytically prove the linear instability of elliptic rhombus solutions for ( u , e ) ∈ ( 1 / 3 , 3 ) × 0 , 1 .

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

“…When it comes to four-body problems, the research on stability of ERE is quite active in the past decades. 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].…”