On the libration collinear points in the restricted three – body problem

Abstract

Abstract:In the restricted problem of three bodies when the primaries are triaxial rigid bodies, the necessary and sufficient conditions to find the locations of the three libration collinear points are stated. In addition, the Linear stability of these points is studied for the case of the Euler angles of rotational motion being θ i = 0, ψ i + φ i = π/2, i = 1, 2 accordingly. We underline that the model studied in this paper has special importance in space dynamics when the third body moves in gravitational fields of planetary systems and particularly in a Jupiter model or a problem including an irregular asteroid.

Libration points in the restricted three-body problem: Euler angles, existence and stability

“…The zeros of (12) represent the locations of the collinear points in the case of Euler angles of rotational motion be equal to θ i = π/2, ψ i = 0, ϕ i = π/2. We note that this result agrees with the result obtained when the Euler angles are equal to θ i = 0, ψ i + ϕ i = π/2, see [16]. 4.…”

“…Substituting (16) into (1) and since we are studying the the linear stability of the motion, we do not consider the effect of the terms of high order of ξ and η. Then, considering only the linear terms the variational equations are equal tö…”

On the perturbed photogravitational restricted five-body problem: the analysis of fractal basins of convergence

“…For describing the problem in very realistic manner, the dynamics of the test particles has been studied by adding several perturbing terms in the effective potential of the circular restricted five-body problem. The study of the five-body problem with several modification is motivated by the various previous study of the restricted three or four-body problem where the same modifications are proposed and significant results are obtained (e.g., [4], [20], [7], [17], [1], [3], [10], [2]).…”

The perturbed restricted three-body problem with angular velocity: Analysis of basins of convergence linked to the libration points

“…Many researchers and scientists are attracted towards it due to its applications in various other fields (e.g. Abouelmagd and Abdullah (2019a), Abouelmagd et al (2019b), Abouelmagd (2012), Abouelmagd et al (2016), Alzahrani et al (2017), Goździewski and Maciejewski (1987), Sano (2007), Pathak et al (2015), Selim et al (2019)). In addition, several modifications have been proposed by various researchers to be more realistic in the classical R3BP which make the applications of this problem in wider sense.…”