Central configurations in the five-body problem: rhombus plus one
type: Item
instance of: scholarly article
title: Central configurations in the five-body problem: rhombus plus one
zbMATH Open document ID: 1498.70018
author: J. Lino Cornelio
author: Martha Alvarez-Ramírez
author: Josep M. Cors
published in: Qualitative Theory of Dynamical Systems
publication date: 2021-08-19T00:00:00Z
full work available at URL: http://hdl.handle.net/2117/352053
review text: The paper demonstrates the existence of a particular central configuration of five bodies. Specifically, four masses lie at the vertices of a rhombus with the remaining fifth body lying on one of the diagonals of the rhombus. The authors consider both the case where the fifth body lies inside and outside the rhombus. Positions and masses of the five bodies are determined by examination of the Dziobek/Laura/Andoyer equations. These give a system of equations in terms of the positions and masses of the bodies and the signed areas of the triangles formed by sets of three bodies. Under these conditions, it can be shown that on the diagonal containing two bodies, both must have equal mass. With suitable rescaling, these are given unit mass, and the length of the diagonal is set to one unit as well. A numerical plot shows a 2-dimensional region of possible values for the positions of the bodies in terms of the position of the non-vertex body and the length of the non-unit diagonal. The remainder of the paper is dedicated to showing that for a given set of positions, the mass values are unique. This is done by writing the remaining unsolved Dziobek/Laura/Andoyer equations in terms of a matrix equation with the non-unit masses as variables. The proof is given by establishing the invertibility of the coefficient matrix. This is done by reducing the determinant of the matrix to a polynomial equation and proving that there are no zeros within particular suitable intervals. For those who research central configurations, the results are worth noting.
Mathematics Subject Classification ID: 70F10
Mathematics Subject Classification ID: 37N05
Mathematics Subject Classification ID: 70F15
zbMATH DE Number: 7383378
zbMATH Keywords: five-body problem
zbMATH Keywords: central configuration
zbMATH Keywords: rhombus configuration
zbMATH Keywords: square configuration
reviewed by: Skyler C. Simmons
MaRDI profile type: Publication
OpenAlex ID: W3164467166
cites work: Finiteness of central configurations of five bodies in the plane
cites work: Strictly convex central configurations of the planar five-body problem
cites work: Classifying four-body convex central configurations
cites work: A family of stacked central configurations in the planar five-body problem
cites work: A special family of stacked central configurations: Lagrange plus Euler in one
cites work: On stacked planar central configurations with five bodies when one body is removed
cites work: Symmetric planar central configurations of five bodies: Euler plus two
cites work: Q5653612
cites work: Stacked central configurations: new examples in the planar five-body problem
cites work: Finiteness of relative equilibria of the four-body problem
cites work: Seven-body central configurations: a family of central configurations in the spatial seven-body problem
cites work: Finiteness of spatial central configurations in the five-body problem
cites work: New central configurations for the planar 5-body problem
cites work: New stacked central configurations for the planar 5-body problem
cites work: On the role and the properties ofn body central configurations
cites work: Topology and mechanics. I
cites work: Topology and mechanics. II: The planar \(n\)-body problem
cites work: Mathematical problems for the next century
cites work: Permanent Configurations in the Problem of Five Bodies
DOI: 10.1007/S12346-021-00486-6
description: scientific article; zbMATH DE number 7383378