scholarly journals On the relative equilibrium configurations in the planar five-body problem

2010 ◽  
Vol 30 (4) ◽  
pp. 495
Author(s):  
Agnieszka Siluszyk
Keyword(s):  
Body Problem ◽  
Relative Equilibrium ◽  
Equilibrium Configurations

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.

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.

Relative equilibrium configurations of point vortices on a sphere

2013 ◽  
Vol 18 (4) ◽  
pp. 344-355 ◽  
Author(s):  
Maria V. Demina ◽  
Nikolai A. Kudryashov
Keyword(s):  
Relative Equilibrium ◽  
Point Vortices ◽  
Equilibrium Configurations

scholarly journals Bifurcations of relative equilibria in the 4- and 5-body problem

1988 ◽  
Vol 8 (8) ◽  
pp. 215-225 ◽  
Keyword(s):  
Equilateral Triangle ◽  
Body Problem ◽  
Relative Equilibrium ◽  
Isosceles Triangle ◽  
Relative Equilibria ◽  
Arbitrary Mass

AbstractThe equilateral triangle family of relative equilibria of the 4-body problem consists of three particles of mass 1 at the vertices of an equilateral triangle and the fourth particle of arbitrary mass m at the centroid. For one value of the mass m this relative equilibrium is degenerate. We show that families of isosceles triangle relative equilibria bifurcate from the equilateral triangle family as m passes through the degenerate value.The square family of relative equilibria of the 5-body problem consists of four particles of mass 1 at the vertices of a square and the fifth particle of arbitrary mass m at the centroid. For one value of the mass m this relative equilibrium is degenerate. We show that families of kite and isosceles trapezoidal relative equilibria bifurcate from the square family as m passes through the degenerate value.

Sign in / Sign up

Export Citation Format

Share Document