Regions of Central Configurations in a Symmetric 4 + 1-Body Problem

Abstract

The inverse problem of central configuration of the trapezoidal 5-body problems is investigated. In this 5-body setup, one of the masses is chosen to be stationary at the center of mass of the system and four-point masses are placed on the vertices of an isosceles trapezoid with two equal masses 1 = 4 at positions (∓0.5, ) and 2 = 3 at positions (∓ /2, ). The regions of central configurations where it is possible to choose positive masses are derived both analytically and numerically. It is also shown that in the complement of these regions no central configurations are possible.

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

“…While much is known about trapezoid central configurations in the restricted four-body problem, there is less known about trapezoid central configurations in the restricted five-body problem. Shoaib [19] recently investigated the inverse problem of central configuration in a symmetric 4 + 1-body problem and derived regions of central configuration.…”

Central configurations in the trapezoidal four-body problems

“…is the self-potential, and m i is the mass of the ith body. 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 a central configuration ( [1]- [4], [8] and [9]). A central configuration is a particular configuration of the n-bodies where the acceleration vector of each body is proportional to its position vector, and the constant of proportionality is the same for the n-bodies, therefore…”