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

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Muhammad Shoaib ◽  
Abdul Rehman Kashif ◽  
Anoop Sivasankaran
Keyword(s):  
Phase Space ◽  
Central Configurations ◽  
Central Configuration ◽  
Isosceles Trapezoid ◽  
Mass Ratios ◽  
Axis Of Symmetry ◽  
Planar Central Configurations

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.

scholarly journals Symmetry of planar four-body convex central configurations

2008 ◽  
Vol 464 (2093) ◽  
pp. 1355-1365 ◽  
Author(s):  
Alain Albouy ◽  
Yanning Fu ◽  
Shanzhong Sun
Keyword(s):  
Body Problem ◽  
The Other ◽  
Central Configurations ◽  
Central Configuration ◽  
Geometric Properties ◽  
Four Body Problem ◽  
Planar Central Configurations ◽  
The Masses ◽  
The Relationship ◽  
Convex Central Configuration

We study the relationship between the masses and the geometric properties of central configurations. We prove that, in the planar four-body problem, a convex central configuration is symmetric with respect to one diagonal if and only if the masses of the two particles on the other diagonal are equal. If these two masses are unequal, then the less massive one is closer to the former diagonal. Finally, we extend these results to the case of non-planar central configurations of five particles.

Planar central configuration estimates in the n-body problem

1996 ◽  
Vol 16 (5) ◽  
pp. 1059-1070 ◽  
Author(s):  
Christopher K. McCord
Keyword(s):  
Lower Bound ◽  
Potential Function ◽  
Body Problem ◽  
Morse Function ◽  
Central Configurations ◽  
Central Configuration ◽  
N Body Problem ◽  
Planar Central Configurations

AbstractFor all masses, there are at least n − 2, O2-orbits of non-collinear planar central configurations. In particular, this estimate is valid even if the potential function is not a Morse function. If the potential function is a Morse function, then an improved lower bound, of the order of n! ln(n + 1/3)/2, can be given.

Isosceles trapezoid central configurations of the Newtonian four-body problem

2012 ◽  
Vol 142 (3) ◽  
pp. 665-672 ◽  
Author(s):  
Zhifu Xie
Keyword(s):  
Body Problem ◽  
Central Configurations ◽  
Central Configuration ◽  
Basic Method ◽  
One Dimensional ◽  
Isosceles Trapezoid ◽  
Four Body Problem ◽  
Explicit Expressions

We use a simple direct and basic method to prove that there is a unique isosceles trapezoid central configuration of the planar Newtonian four-body problem when two pairs of equal masses are located at adjacent vertices of a trapezoid. Such isosceles trapezoid central configurations are an exactly one-dimensional family. Explicit expressions for masses are given in terms of the size of the quadrilateral.

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

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Muhammad Shoaib
Keyword(s):  
Inverse Problem ◽  
Body Problem ◽  
Center Of Mass ◽  
Central Configurations ◽  
Central Configuration ◽  
Isosceles Trapezoid ◽  
The Masses

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 massesm1=m4at positions∓0.5, rBandm2=m3at positions∓α/2,rA. 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.

scholarly journals Central configurations in planar n-body problem with equal masses for $$n=5,6,7$$

2019 ◽  
Vol 131 (10) ◽  
Author(s):  
Małgorzata Moczurad ◽  
Piotr Zgliczyński
Keyword(s):  
Body Problem ◽  
Central Configurations ◽  
Central Configuration ◽  
Newtonian Potential ◽  
Computer Assisted ◽  
Computer Assisted Proof ◽  
N Body Problem ◽  
Reflective Symmetry

Abstract We give a computer-assisted proof of the full listing of central configuration for n-body problem for Newtonian potential on the plane for $$n=5,6,7$$ n = 5 , 6 , 7 with equal masses. We show all these central configurations have a reflective symmetry with respect to some line. For $$n=8,9,10$$ n = 8 , 9 , 10 , we establish the existence of central configurations without any reflectional symmetry.

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