Relative equilibrium solutions in the four body problem

1978 ◽  
Vol 18 (2) ◽  
pp. 165-184 ◽  
Author(s):  
Carles Sim�
Keyword(s):  
Body Problem ◽  
Relative Equilibrium ◽  
Equilibrium Solutions ◽  
Four Body Problem

EQUILIBRIUM POINTS AND THEIR STABILITY IN THE RESTRICTED FOUR-BODY PROBLEM

2011 ◽  
Vol 21 (08) ◽  
pp. 2179-2193 ◽  
Author(s):  
A. N. BALTAGIANNIS ◽  
K. E. PAPADAKIS
Keyword(s):  
Body Problem ◽  
Equilibrium Points ◽  
Relative Equilibrium ◽  
Center Of Mass ◽  
Basins Of Attraction ◽  
Equilibrium Solutions ◽  
Velocity Surface ◽  
Four Body Problem ◽  
The Stability ◽  
Zero Velocity Surface

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.

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.

Collinear Equilibrium Solutions of Four-body Problem

2011 ◽  
Vol 32 (3) ◽  
pp. 411-423 ◽  
Author(s):  
Muhammad Shoaib ◽  
Ibrahima Faye
Keyword(s):  
Body Problem ◽  
Equilibrium Solutions ◽  
Four Body Problem

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

2018 ◽  
Vol 23 (3) ◽  
pp. 507-525 ◽  
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.

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

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.

11Li-12C scattering as a four-body problem

1995 ◽  
Vol 21 (1) ◽  
pp. 87-100 ◽  
Author(s):  
J Formanek ◽  
R J Lombard
Keyword(s):  
Body Problem ◽  
Four Body Problem
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