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.

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 Dihedral F-Tilings of the Sphere by Equilateral and Scalene Triangles - II

10.37236/815 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
A. M. d'Azevedo Breda ◽  
Patrícia S. Ribeiro ◽  
Altino F. Santos
Keyword(s):  
Equilateral Triangle ◽  
Isosceles Triangle ◽  
Subsequent Paper ◽  
Euclidean Sphere ◽  
Regular Polygons ◽  
Combinatorial Description ◽  
Equilateral Triangles

The study of dihedral f-tilings of the Euclidean sphere $S^2$ by triangles and $r$-sided regular polygons was initiated in 2004 where the case $r=4$ was considered [5]. In a subsequent paper [1], the study of all spherical f-tilings by triangles and $r$-sided regular polygons, for any $r\ge 5$, was described. Later on, in [3], the classification of all f-tilings of $S^2$ whose prototiles are an equilateral triangle and an isosceles triangle is obtained. The algebraic and combinatorial description of spherical f-tilings by equilateral triangles and scalene triangles of angles $\beta$, $\gamma$ and $\delta$ $(\beta>\gamma>\delta)$ whose edge adjacency is performed by the side opposite to $\beta$ was done in [4]. In this paper we extend these results considering the edge adjacency performed by the side opposite to $\delta$.

Collinear relative equilibria of the planarN-body problem

1982 ◽  
Vol 28 (1-2) ◽  
pp. 17-24 ◽  
Author(s):  
Julian I. Palmore
Keyword(s):  
Body Problem ◽  
Relative Equilibria

A Note on the Relative Equilibria Bifurcations in the $$(2N+1)$$ ( 2 N + 1 ) -Body Problem

2014 ◽  
Vol 28 (1) ◽  
pp. 239-251
Author(s):  
Jenică Crînganu ◽  
Daniel Paşca ◽  
Cristina Stoica
Keyword(s):  
Body Problem ◽  
Relative Equilibria
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