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

Muhammad Shoaib,Abdul Rehman Kashif,Anoop Sivasankaran

doi:10.1155/2016/9897681

Muhammad Shoaib, Abdul Rehman Kashif + Show 1 more

Open Access

https://doi.org/10.1155/2016/9897681

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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.

Highlights

The equations of motion for n positive masses subject to Newtonian Gravitation is given by mi→󳨀rï = ∇iUi, i = 1, . . . , n, (1) where U = N −G∑ i=1 j
We study the central configuration of the isosceles trapezoidal five-body problem and identify regions in the phase space where it is possible to choose positive masses which will make the configuration central
We studied the central configuration of different types of symmetric 5-body problems

Summary

Introduction

Hampton and Moeckel [11] proved that finite central configurations are possible for the Newtonian three-body and four-body problems with positive masses. Gidea and Llibre [1] studied the stacked symmetric planar central configuration of five bodies with some special symmetries 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. We study the central configuration of the isosceles trapezoidal five-body problem and identify regions in the phase space where it is possible to choose positive masses which will make the configuration central.

Main Results

Proof of Theorem 1

Proof of Theorem 2

Conclusions