Related papers: On the Uniqueness of Co-circular Four Body Central…
We study central configurations of the Newtonian four-body problem that form a trapezoid. Using a topological argument we prove that there is at most one trapezoidal central configuration for each cyclic ordering of the masses.
The plane case of central configurations with four different masses is analyzed theoretically and is computed numerically. We follow Dziobek's approach to four body central configurations with a direct implicit method of our own in which…
This research investigates centered co-circular central configurations in the general power-law potential $n$-body problem. Firstly, there are no such configurations when all masses are equal, except for two; secondly, unless all masses are…
We study the bifurcations of central configurations of the Newtonian four-body problem when some of the masses are equal. First, we continue numerically the solutions for the equal mass case, and we find values of the mass parameter at…
For the power-law potential $n$-body problem, we study a special kind of central configurations where all the masses lie on a circle and the center of mass coincides with the center of the circle. It is also called the centered co-circular…
In this paper,we study spatial central configurations where N bodies are at the vertices of a regular N-gon $T$ and the other 4 bodies are symmetrically located on the straight line that is perpendicular to the plane that contains $T$ and…
We classify the full set of convex central configurations in the Newtonian four-body problem. Particular attention is given to configurations possessing some type of symmetry or defining geometric property. Special cases considered include…
This paper examines the existence of centered co-circular central configurations in the general power-law potential n-body problem. We prove the nonexistence of such configurations when the system consists of n-3 equal masses and three…
The conjecture of the existence and the uniqueness of the strictly convex quadrilateral central configuration for the Newtonian 4-body problem is one of the most-talked open problems in the study of the classical n-body problems in…
We prove that there is a unique convex non-collinear central configuration of the planar Newtonian four-body problem when two equal masses are located at opposite vertices of a quadrilateral and, at most, only one of the remaining masses is…
We discuss several conditions for four points to lie on a plane, and we use them to find new equations for four-body central configurations that use angles as variables. We use these equations to give novel proofs of some results for…
In this work we are interested in the central configurations of the spatial seven-body problem where six of them are at vertices of two congruents equilateral triangles belong to parallel planes and one triangle is a rotation by the angle…
We study central configurations in the four body problem, i.e., configurations in which the forces on all the bodies point to a fixed, single point in space. The newly formulated pair-space formalism yields a set of vectorial equations that…
The configuration of a homothetic motion in the N-body problem is called a central configuration. In this paper, we prove that there are exactly three planar non-collinear central configurations for masses x, -x, y, -y with x different from…
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…
We consider the $N$-body problem of celestial mechanics in spaces of nonzero constant curvature. Using the concept of locked inertia tensor, we compute the moment of inertia for systems moving on spheres and hyperbolic spheres and show that…
In this paper, we consider the problem: given a symmetric concave configuration of four bodies, under what conditions is it possible to choose positive masses which make it central. We show that there are some regions in which no central…
We study four-body central configurations with one pair of opposite sides parallel. We use a novel constraint to write the central configuration equations in this special case, using distances as variables. We prove that, for a given…
We establish the existence of a single-parameter family of the concave kite central configurations in the 4-body problem with two pairs of equal masses. In such configurations, one pair of the masses must lie on the base of an isosceles…
The main result of this paper is the existence of a new family of central configurations in the Newtonian spatial seven-body problem. This family is unusual in that it is a simplex stacked central configuration, i.e the bodies are arranged…