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 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.

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 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.

scholarly journals Spatial Central Configurations with Two Twisted Regular 4-Gons

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Sen Zhang ◽  
Furong Zhao
Keyword(s):  
Twist Angle ◽  
Central Configurations ◽  
Central Configuration ◽  
Spatial Central Configurations ◽  
The Masses ◽  
Axis Passing

We study the configuration formed by two squares in two parallel layers separated by a distance. We picture the two layers horizontally with thez-axis passing through the centers of the two squares. The masses located on the vertices of each square are equal, but we do not assume that the masses of the top square are equal to the masses of the bottom square. We prove that the above configuration of two squares forms a central configuration if and only if the twist angle is equal tokπ/2or (π/4+kπ/2)(k=1,2,3,4).

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

2020 ◽  
Vol 132 (11-12) ◽  
Author(s):  
Małgorzata Moczurad ◽  
Piotr Zgliczyński
Keyword(s):  
Body Problem ◽  
Central Configurations ◽  
Central Configuration ◽  
Computer Assisted ◽  
Planar Case ◽  
Full List ◽  
Computer Assisted Proof ◽  
N Body Problem

AbstractWe present a computer assisted proof of the full listing of central configurations for spatial n-body problem for $$n=5$$ n = 5 and 6, with equal masses. For each central configuration, we give a full list of its Euclidean symmetries. For all masses sufficiently close to the equal masses case, we give an exact count of configurations in the planar case for $$n=4,5,6,7$$ n = 4 , 5 , 6 , 7 and in the spatial case for $$n=4,5,6$$ n = 4 , 5 , 6 .

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.

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 Dynamical Aspects of an Equilateral Restricted Four-Body Problem

2009 ◽  
Vol 2009 ◽  
pp. 1-23 ◽  
Author(s):  
Martha Álvarez-Ramírez ◽  
Claudio Vidal
Keyword(s):  
Degrees Of Freedom ◽  
Body Problem ◽  
Center Of Mass ◽  
Hamiltonian Function ◽  
First Integral ◽  
Equilibrium Solutions ◽  
Integral Of Motion ◽  
Planar Case ◽  
Four Body Problem ◽  
The Masses

The spatial equilateral restricted four-body problem (ERFBP) is a four body problem where a mass point of negligible mass is moving under the Newtonian gravitational attraction of three positive masses (called the primaries) which move on circular periodic orbits around their center of mass fixed at the origin of the coordinate system such that their configuration is always an equilateral triangle. Since fourth mass is small, it does not affect the motion of the three primaries. In our model we assume that the two masses of the primariesm2andm3are equal toμand the massm1is1−2μ. The Hamiltonian function that governs the motion of the fourth mass is derived and it has three degrees of freedom depending periodically on time. Using a synodical system, we fixed the primaries in order to eliminate the time dependence. Similarly to the circular restricted three-body problem, we obtain a first integral of motion. With the help of the Hamiltonian structure, we characterize the region of the possible motions and the surface of fixed level in the spatial as well as in the planar case. Among other things, we verify that the number of equilibrium solutions depends upon the masses, also we show the existence of periodic solutions by different methods in the planar case.

$4+1$-Moulton Configuration and Positive Mass Deformation

2021 ◽  
pp. 286-300
Author(s):  
Naoko Yoshimi ◽  
Akira Yoshioka
Keyword(s):  
Inverse Problem ◽  
Explicit Formula ◽  
Body Problem ◽  
Central Configuration ◽  
Positive Mass ◽  
Mass Deformation

For given $k$ bodies of collinear central configuration of Newtonian $k$-body problem, we ask whether one can add another body on the line without changing the configuration and motion of the initial bodies so that the total $k+1$ bodies provide a central configuration. The case $k=4$ is analyzed. We study the inverse problem of five bodies and obtain a global explicit formula. Then using the formula we find there are five possible positions of the added body and for each case the mass of the added body is zero. We further consider to deform the position of the added body without changing the positions of the initial four bodies so that the total five bodies are in a state of central configuration and the mass of the added body becomes positive. For each solution above, we find such a deformation of the position of the added body in an explicit manner starting from the solution.

Sign in / Sign up

Export Citation Format

Share Document