Collinear Central Configuration in Four-Body Problem

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.

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Investigation of Lyapunov stability of a central configuration in the restricted four-body problem

10.1063/1.5034607 ◽

2018 ◽

Author(s):

B. S. Bardin ◽

P. A. Esipov

Keyword(s):

Lyapunov Stability ◽

Body Problem ◽

Central Configuration ◽

Four Body Problem

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Isosceles trapezoid central configurations of the Newtonian four-body problem

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210511000576 ◽

2012 ◽

Vol 142 (3) ◽

pp. 665-672 ◽

Cited By ~ 14

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.

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Central Configurations and Action Minimizing Orbits in Kite Four-Body Problem

Advances in Astronomy ◽

10.1155/2020/5263750 ◽

2020 ◽

Vol 2020 ◽

pp. 1-18

Author(s):

B. Benhammouda ◽

A. Mansur ◽

M. Shoaib ◽

I. Szücs-Csillik ◽

D. Offin

Keyword(s):

Cross Sections ◽

Body Problem ◽

Mass Parameter ◽

Central Configurations ◽

Continuous Family ◽

Classical Action ◽

Dynamical Aspect ◽

Homographic Solutions ◽

Four Body Problem ◽

Current Article

In the current article, we study the kite four-body problems with the goal of identifying global regions in the mass parameter space which admits a corresponding central configuration of the four masses. We consider two different types of symmetrical configurations. In each of the two cases, the existence of a continuous family of central configurations for positive masses is shown. We address the dynamical aspect of periodic solutions in the settings considered and show that the minimizers of the classical action functional restricted to the homographic solutions are the Keplerian elliptical solutions. Finally, we provide numerical explorations via Poincaré cross-sections, to show the existence of periodic and quasiperiodic solutions within the broader dynamical context of the four-body problem.

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