Stability Study of a Relative Equilibrium in the Planar Circular Restricted Four-Body Problem

We study numerically the problem of four bodies, three of which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle, while the fourth is infinitesimal. The fourth body does not affect the motion of the three bodies (primaries). The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves as well as the positions of the equilibrium points are given. The existence and the number of collinear and noncollinear equilibrium points of the problem depend on the mass parameters of the primaries. For three unequal masses, collinear equilibrium solutions do not exist. Critical masses associated with the existence and the number of equilibrium points, are given. The stability of the relative equilibrium solutions in all cases is also studied. The regions of the basins of attraction for the equilibrium points of the present dynamical model for some values of the mass parameters are illustrated.

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NUMERICAL-SYMBOLIC METHODS FOR SEARCHING RELATIVE EQUILIBRIA IN THE RESTRICTED PROBLEM OF FOUR BODIES

Mathematical Modelling and Analysis ◽

10.3846/mma.2018.030 ◽

2018 ◽

Vol 23 (3) ◽

pp. 507-525 ◽

Cited By ~ 1

Author(s):

Alexander Prokopenya

Keyword(s):

Body Problem ◽

Relative Equilibrium ◽

Numerical Procedure ◽

Relative Equilibria ◽

Algebraic Equations ◽

Three Body Problem ◽

Starting Point ◽

Equilibrium Configurations ◽

Four Body Problem ◽

Three Body

We discuss here the problem of solving the system of two nonlinear algebraic equations determining the relative equilibrium positions in the planar circular restricted four-body problem formulated on the basis of the Euler collinear solution of the three-body problem. The system contains two parameters $\mu_1$, $\mu_2$ and all its solutions coincide with the corresponding solutions in the three-body problem if one of the parameters equals to zero. For small values of one parameter the solutions are found in the form of power series in terms of this parameter, and they are used for separation of different solutions and choosing the starting point in the numerical procedure for the search of equilibria. Combining symbolic and numerical computation, we found all the equilibrium positions and proved that there are 18 different equilibrium configurations of the system for any reasonable values of the two system parameters $\mu_1$, $\mu_2$. All relevant symbolic and numerical calculations are performed with the aid of the computer algebra system Wolfram Mathematica.

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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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