Linear stability of the elliptic relative equilibrium with (1  +  n)-gon central configurations in planar n-body problem

Abstract

We study the linear stability of (1 + n)-gon elliptic relative equilibrium (ERE for short), that is the Kepler homographic solution with the (1 + n)-gon central configurations. We show that for n ≥ 8 and any eccentricity e ∈ [0, 1), the (1 + n)-gon ERE is stable when the central mass m is large enough. Some linear instability results are given when m is small.

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

“…, β n−2 which defined by (1.14) in [39]. For some special cases of nbody problems, the linear stability of ERE, which is raised from an n-gon or (1 + n)-gon central configurations with n equal masses, was studied by Hu, Long and Ou in [5] recently.…”

“…We here follow the proof of Proposition 3.5 of[5] and obtain following monotonic of the index and nullity of A(α, β, e).…”

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

“…By a similar analysis to the proof of Proposition 6.1 in [6], for every e ∈ [0, 1) and ω ∈ U\{1}, the total multiplicity of ω-degeneracy of γ β,e (2π) for β ∈ [0, 5] is always precisely 2, i.e., β∈ [0,5] v ω (γ β,e (2π)) = 2, ∀ω ∈ U\{1}.…”

“…, β n−2 which defined by (1.14) in [37]. For some special cases of n-body problem, the linear stability of ERE which raised from an n-gon or (1 + n)-gon central configurations with n equal masses was studied by Hu, Long and Ou in [5].…”

“…3 in[6], we can prove the theorem. Specially, for ω = −1, e ∈ [0, 1), we define µ l (e) = min{β 1 (e, −1), β 2 (e, −1)}, µ m (e) = max{β 1 (e, −1), β 2 (e, −1)},(5.39)where β i (e, −1) are the two −1-dgenerate curves as in Theorem 5 5…”