Some special solutions of the rhomboidal five-body problem

Shoaib, Muhammad; Faye, Ibrahima; Sivasankaran, Anoop

doi:10.1063/1.4757521

Cited by 7 publications

(6 citation statements)

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“…e mass m 0 is at the center of the configuration, and the equilibria obtained in this case were all shown to be unstable. Shoaib et al [11] established the central configuration for the rhomboidal 5-body problem and highlighted the regions in the phase plane where it is possible to have central configuration. On the axis of symmetry, Shoaib et al [12] considered a symmetric five-body problem with three unequal collinear masses.…”

Section: Introductionmentioning

confidence: 99%

Stability Analysis of the Rhomboidal Restricted Six-Body Problem

Siddique

Kashif

Shoaib³

et al. 2021

Advances in Astronomy

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We discuss the restricted rhomboidal six-body problem (RR6BP), which has four positive masses at the vertices of the rhombus, and the fifth mass is at the intersection of the two diagonals. These masses always move in rhomboidal CC with diagonals 2 a and 2 b . The sixth body, having a very small mass, does not influence the motion of the five masses, also called primaries. The masses of the primaries are m 1 = m 2 = m 0 = m and m 3 = m 4 = m ˜ . The masses m and m ˜ are written as functions of parameters a and b such that they always form a rhomboidal central configuration. The evolution of zero velocity curves is discussed for fixed values of positive masses. Using the first integral of motion, we derive the region of possible motion of test particle m 5 and identify the value of Jacobian constant C for different energy intervals at which these regions become disconnected. Using semianalytical techniques, we show the existence and uniqueness of equilibrium solutions on the axes and off the axes. We show that, for b ∈ 1 / 3 , 1.1394282249562009 , there always exist 12 equilibrium points. We also show that all 12 equilibrium points are unstable.

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Section: Introductionmentioning

confidence: 99%

Stability Analysis of the Rhomboidal Restricted Six-Body Problem

Siddique

Kashif

Shoaib³

et al. 2021

Advances in Astronomy

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“…They have shown that central configuration is possible in rhomboidal arrangement where four masses are kept at the vertices and a fifth mass in the center and a trapezoidal arrangement where four masses are at the vertices and a fifth mass at the midpoint of one of the parallel sides. Shoaib et al [17] have studied central configuration of the rhomboidal 5-body problem and identified CC regions using similar approaches.…”

Section: Introductionmentioning

confidence: 99%

Planar Central Configurations of Symmetric Five-Body Problems with Two Pairs of Equal Masses

Shoaib

Kashif

Sivasankaran

2016

Advances in Astronomy

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We study central configuration of a set of symmetric planar five-body problems where(1)the five masses are arranged in such a way thatm1,m2, andm4are collinear andm2,m3, andm5are collinear; the two sets of collinear masses form a triangle withm2at the intersection of the two sets of collinear masses;(2)four of the bodies are on the vertices of an isosceles trapezoid and the fifth body can take various positions on the axis of symmetry both outside and inside the trapezoid. We form expressions for mass ratios and identify regions in the phase space where it is possible to choose positive masses which will make the configuration central. We also show that the triangular configuration is not possible.

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“…Theoretically, these conditions correspond to the Caledonian symmetric five-body problem (CS5BP) in which one body is fixed at the origin, and the other four bodies are always configured in a parallelogram (Shoaib et al 2013). We define the functions r t min ( ) and r t max ( ) as the minimum and maximum of q i j 1 5 ij | |( )   < at a real time t, respectively.…”

Section: Reversibility Checkmentioning

confidence: 99%

An Efficient Conservative Integrator With a Chain Regularization for the Few-Body Problem

Minesaki

2015

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We design an efficient orbital integration scheme for the general N-body problem that preserves all the conserved quantities except the angular momentum. This scheme is based on the chain concept and is regarded as an extension of a d'Alembert-type scheme for constrained Hamiltonian systems. It also coincides with the discretetime general three-body problem for particle number N = 3. Although the proposed scheme is only second-order accurate, it can accurately reproduce some periodic orbits, which cannot be done with generic geometric numerical integrators.

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Some special solutions of the rhomboidal five-body problem

Cited by 7 publications

References 0 publications

Stability Analysis of the Rhomboidal Restricted Six-Body Problem

Stability Analysis of the Rhomboidal Restricted Six-Body Problem

Planar Central Configurations of Symmetric Five-Body Problems with Two Pairs of Equal Masses

An Efficient Conservative Integrator With a Chain Regularization for the Few-Body Problem

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