The masses in a symmetric solution of the four body problem

The planar version of the equilateral restricted four-body problem, with three unequal masses, is numerically investigated. By adopting the grid classification method we locate the coordinates, on the plane [Formula: see text], of the points of equilibrium, for all possible values of the masses of the primaries. The linear stability of the libration points is also determined, as a function of the masses. Our analysis indicates that linearly stable points of equilibrium exist only when one of the primaries has a considerably larger mass, with respect to the other two primary bodies, when the triangular configuration of the primaries is also dynamically stable.

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Symmetry of planar four-body convex central configurations

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2007.0320 ◽

2008 ◽

Vol 464 (2093) ◽

pp. 1355-1365 ◽

Cited By ~ 45

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.

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The Axisymmetric Central Configurations of the Four-Body Problem with Three Equal Masses

Symmetry ◽

10.3390/sym12040648 ◽

2020 ◽

Vol 12 (4) ◽

pp. 648

Author(s):

Emese Kővári ◽

Bálint Érdi

Keyword(s):

Analytical Solution ◽

Body Problem ◽

Equal Mass ◽

The Other ◽

Coordinate Space ◽

Four Body Problem ◽

Axis Of Symmetry ◽

The Masses ◽

Special Case ◽

Two Bodies

In the studied axisymmetric case of the central four-body problem, the axis of symmetry is defined by two unequal-mass bodies, while the other two bodies are situated symmetrically with respect to this axis and have equal masses. Here, we consider a special case of the problem and assume that three of the masses are equal. Using a recently found analytical solution of the general case, we formulate the equations of condition for three equal masses analytically and solve them numerically. A complete description of the problem is given by providing both the coordinates and masses of the bodies. We show furthermore how the three-equal-mass solutions are related to the general case in the coordinate space. The physical aspects of the configurations are also studied and discussed.

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Dynamical Aspects of an Equilateral Restricted Four-Body Problem

Mathematical Problems in Engineering ◽

10.1155/2009/181360 ◽

2009 ◽

Vol 2009 ◽

pp. 1-23 ◽

Cited By ~ 21

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.

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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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