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.
“…, 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.…”
mentioning
confidence: 99%
“…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).…”
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.
“…, 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.…”
mentioning
confidence: 99%
“…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).…”
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.
“…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}.…”
Section: 3mentioning
confidence: 82%
“…, 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].…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…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…”
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.