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

Abstract

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.

Central Configurations and Action Minimizing Orbits in Kite Four-Body Problem

“…Solving the above equations, we obtain 3 . It is proved in Lemmas 2, 3, and 4 that constraint (8) is satisfied by showing the existence of a smooth curve:…”

“…To understand the dynamics presented by a total collision of the masses or the equilibrium state of a rotating system, we are led to the concept of central configurations. A configuration of n bodies is central if the acceleration of each body is a scalar multiple of its position [1][2][3][4]. Let r i ∈ R 2 and m i , i � 1, .…”

“…Consider the function g defined by(3). en, for any a 0 ∈ I � (0.14, 1.32) there exists an interval U ⊂ I containing a 0 and an interval V ⊂ (1, 2) containing b 0 such that there is a unique continuously differentiable function b � ψ(a) defined on U with b ∈ V that satisfies g(a, b) � 0.…”

Orbit classification and networks of periodic orbits in the planar circular restricted five-body problem

“…A previous study on the periodic solutions in this dynamical system has been presented in [36], while in [42] we numerically explored the orbital dy-namics of the five-body system, when all the primaries are sources of radiation. Over the years, the problem of five bodies has been proved a very fertile research ground (e.g., [4,18,21,22,28,30,31,33,34,37,47,49,50]). A generalization of the five-body problem has been performed in [28] where the motion of the massless test particle, under the influence of N − 1 primary bodies in a circular configuration, has been investigated.…”

Numerical Study of the Zero Velocity Surface and Transfer Trajectory of a Circular Restricted Five-Body Problem

“…Bengochea et al [11] studied the necessary and sufficient conditions for periodicity of some doubly symmetric orbits in the planar 1 + 2 -body problem and studied numerically these types of orbits for the case n=2. Shoaib et al [12] considered the central configuration of different types of symmetric fivebody problems that have two pairs of equal masses; the fifth mass can be both inside the trapezoid and outside the trapezoid, but the triangular configuration is impossible. Xu et al [13] discussed the prohibited areas of the Sun-Jupiter-Trojans-Greeks-Spacecraft system and designed a transfer trajectory from Jupiter to Trojans.…”