Related papers: Planarity conditions and four-body central configu…
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…
We study central configurations lying on a common circle in the Newtonian four-body problem. Using a topological argument we prove that there is at most one co-circular central configuration for each cyclic ordering of the masses on the…
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…
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…
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 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 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…
A new coordinate system on the tangent space to planar configurations is introduced to simplify some calculations on central configurations and relative equilibria in the $N$-body problem with a homogeneous potential, which includes the…
An interesting description of a collinear configuration of four particles is found in terms of two spherical coordinates. An algorithm to compute the four coordinates of particles of a collinear Four-Body central configuration is presented…
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 consider the geometry of four spatial displacements, arranged in cyclic order, such that the relative motion between neighbouring displacements is a pure rotation. We compute the locus of points whose homologous images lie on a circle,…
We classify all planar four-body central configurations where two pairs of the bodies are on parallel lines. Using Cartesian coordinates, we show that the set of four-body trapezoid central configurations with positive masses forms a…
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 consider a symmetric five-body problem with three unequal collinear masses on the axis of symmetry. The remaining two masses are symmetrically placed on both sides of the axis of symmetry. Regions of possible central configurations are…
An algorithm to compute the six distances between particles of a planar Four-Body central configuration is presented according to the following schema. An orthocentric tetrahedron is computed as a function of given masses. Each mass is…
Central configurations are fundamental equilibrium solutions of the Newtonian $n$-body problem and play a key role in understanding the structure and dynamics of gravitational systems. However, the classification and enumeration of such…
A previous work introduced pair space, which is spanned by the center of mass of a system and the relative positions (pair positions) of its constituent bodies. Here, I show that in the $N$-body Newtonian problem, a configuration that does…
Planar central configurations can be seen as critical points of the reduced potential or solutions of a system of equations. By the homogeneity and invariance of the potential with respect to SO(2), it is possible to see that the…
A formula for the radii and positions of four circles in the plane for an arbitrary linearly independent circle configuration is found. Among special cases is the recent extended Descartes Theorem on the Descartes configuration and an…
For any convex non-collinear central configuration of the planar Newtonian 4-body problem with adjacent equal masses $m_1=m_2\neq m_3=m_4$, with equal lengths for the two diagonals, we prove it must possess a symmetry and must be an…