Equivalence of linear stabilities of elliptic triangle solutions of the planar charged and classical three-body problems

Zhou, Qinglong; Ye, Long

doi:10.1016/j.jde.2015.01.045

Cited by 3 publications

(3 citation statements)

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“…37) and(3.38), in the plane e = 0, A(α, β, 0) is degenerate when for α = −(n 2 + 1) + 9β 2 + 4n and ker A(α, β, 0) = span R(t)(a n sin nt, cos nt) T , R(t)(a n sin nt, − cos nt) T ,(7.46) with a n ∈ R. By (3.38), the equation system A(α, β, 0)R(t)(a n sin nt, cos nt) T = 0 readsn 2 a n − 2n + (1 + α + 3β)a n = 0, n 2 − 2na n − (1 + α − 3β) = 0. (7.47) Then by direct computations, a n = n 2 +1+α−3β 2n and α = −(n 2 + 1) + 9β 2 + 4n.…”

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

Liu

Zhou

2020

Journal of Differential Equations

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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.

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

Liu

Zhou

2020

Journal of Differential Equations

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“…then (1.13) can be written as ẇ = JB(t)w, (1.15) where 14) coincides with B(t) of (2.35) in [15]. Thus a lot of results which developed in [15] can be applied to this paper.…”

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confidence: 99%

“…Inspired by the analytic method, Q. Zhou and Y. Long in [14] studied the linear stability of elliptic triangle solutions of the charged three-body problem.…”

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confidence: 99%

Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem

Zhou¹

2017

Discrete &Amp; Continuous Dynamical Systems - A

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We study the Robe's restricted three-body problem. Such a motion was firstly studied by A. G. Robe in [11], which is used to model small oscillations of the earth's inner core taking into account the moon attraction. For the linear stability of elliptic equilibrium points of the Robe's restricted three-body problem, earlier results of such linear stability problem depend on a lot of numerical computations, while we give an analytic approach to it. The linearized Hamiltonian system near the elliptic relative equilibrium point in our problem coincides with the linearized system near the Euler elliptic relative equilibria in the classical three-body problem except for the rang of the mass parameter. We first establish some relations from the linear stability problem to symplectic paths and its corresponding linear operators. Then using the Maslov-type ω-index theory of symplectic paths and the theory of linear operators, we compute ω-indices and obtain certain properties of the linear stability of elliptic equilibrium points of the Robe's restricted three-body problem. Figure 1: The Robe's restricted three-body problem considered: m 1 is a spherical shell filled with a fluid of density ρ 1 ; m 2 a mass point outside the shell and m 3 a small solid sphere of density ρ 3 inside the shell. Later on, A. R. Plastino and A. Plastino ([10]) studied the linear stability of the equilibrium points and the connection between the effect of the buoyancy forces and a perturbation of a Coriolis force. In 2001, P. P. Hallen and N. Rana ([1]) found other new equilibrium points of the restricted problem and discussed their linear stabilities. K. T. Singh, B. S. Kushvah and B. Ishwar ([ 13]) examined the stability of triangular equilibrium points in Robe's generalized restricted three body problem where the problem is generalized in the sense that a more massive primary has been taken as an oblate spheroid.However, in [11], for the elliptic equilibrium points, the bifurcation diagram of linear stability was obtained by numerical methods. In [10,1,12,13], the authors studied the linear stability of some kinds of equilibrium points, but their studies did not contain the elliptic case.On the other hand, in [4, 5] of 2009-2010, X. Hu and S. Sun found a new way to relate the stability problem to the iterated Morse indices. Recently, by observing new phenomenons and discovering new properties of elliptic Lagrangian solutions, in the joint paper [2] of X. Hu, Y. Long and S. Sun, the linear stability of elliptic Lagrangian solutions is completely solved analytically by index theory (cf. [6]) and the new results are related directly to (β, e) in the full parameter rectangle. Inspired by the analytic method, Q. Zhou and Y. Long in [14] studied the linear stability of elliptic triangle solutions of the charged three-body problem.Recently, in [15,16], Q. Zhou and Y. Long studied the linear stability of elliptic Euler-Moulton solutions of n-body problem for n = 3 and for general n ≥ 4, respectively. Also, the linear stability of Euler collision so...

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Geometric characterizations for variational minimizing solutions of charged 3-body problems

Kuang

2016

Front. Math. China

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Equivalence of linear stabilities of elliptic triangle solutions of the planar charged and classical three-body problems

Cited by 3 publications

References 18 publications

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

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

Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem

Geometric characterizations for variational minimizing solutions of charged 3-body problems

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