Convex Central Configurations Research Articles

Linear stability of elliptic relative equilibria of four-body problem with two infinitesimal masses

In this paper, we consider the elliptic relative equilibria of four-body problem with two infinitesimal masses. The most interesting case occurs when the two small masses tend to the same Lagrangian point L4 (or L5). In [20], Xia showed that there exist four central configurations: two of them are non-convex, and the other two are convex. We prove that the elliptic relative equilibria raised from the non-convex central configurations are always linearly unstable, while for the elliptic relative equilibria raised from the convex central configurations, the conditions of linear stability with respect to the parameters are provided.

Symmetry and Asymmetry in the $1+N$ Coorbital Problem

The relative equilibria of planar Newtonian $N$-body problem become coorbital around a central mass in the limit when all but one of the masses becomes zero. We prove a variety of results about the coorbital relative equilibria, with an emphasis on the relation between symmetries of the configurations and symmetries in the masses, or lack thereof. We prove that in the $N=4$, $N=6$, and $N=8$ Newtonian coorbital problems there exist symmetric relative equilibria with asymmetric positive masses. This result can be generalized to other homogeneous potentials, and we conjecture similar results hold for larger even numbers of infinitesimal masses. We prove that some equalities of the masses in the $1+4$ and $1+5$ coorbital problems imply symmetry of a class of convex relative equilibria. We also prove there is at most one convex central configuration of the symmetric $1+5$ problem.

On the Convex Central Configurations of the Symmetric (ℓ + 2)-body Problem

For the 4-body problem there is the following conjecture: Given arbitrary positive masses, the planar 4-body problem has a unique convex central configuration for each ordering of the masses on its convex hull. Until now this conjecture has remained open. Our aim is to prove that this conjecture cannot be extended to the (ℓ + 2)-body problem with ℓ ⩾ 3. In particular, we prove that the symmetric (2n + 1)-body problem with masses m1 = … = m2n−1 = 1 and m2n = m2n+1 = m sufficiently small has at least two classes of convex central configuration when n = 2, five when n = 3, and four when n = 4. We conjecture that the (2n + 1)-body problem has at least n classes of convex central configurations for n > 4 and we give some numerical evidence that the conjecture can be true. We also prove that the symmetric (2n + 2)-body problem with masses m1 = … = m2n = 1 and m2n+1 = m2n+2 = m sufficiently small has at least three classes of convex central configuration when n = 3, two when n = 4, and three when n = 5. We also conjecture that the (2n + 2)-body problem has at least [(n +1)/2] classes of convex central configurations for n > 5 and we give some numerical evidences that the conjecture can be true.

4-Vortex Trapezoidal Central Configurations

In this work we examine trapezoidal central configurations in the planar four-vortex problem. More specifically, we consider the convex central configurations in which the convex quadrilateral has two parallel sides. With analytical arguments we classify all possible arrangements. Additionally, we prove the uniqueness of the trapezoidal central configuration for giving four vorticities with the same sign.

Overview and comparison of approaches towards the planar restricted five-body problem with primaries forming an axisymmetric four-body central configuration

The aim of this paper is to present recent results on the restricted five-body problem where the primaries are located at a known central configuration having the $x$-axis as symmetry axis, so that two bodies with equal masses are situated symmetrically with respect to this axis. We firstly give an overview of the axisymmetric central configurations computed by Erdi and Czirjak (2016), as well as that where three bodies with equal masses are situated at the vertices of an equilateral triangle, while the fourth body lies at the center of the triangle. This last one bridges the gap between convex and concave four-body central configurations. After that, the characterization of the geometry of the restricted five-body problem families with these configurations are described, and the number and the evolution positions of equilibrium points, which depend on the mass parameters of the primaries, are compared.

On the uniqueness of the isosceles trapezoidal central configuration in the 4-body problem for power-law potentials

In this article we study the planar 4-body problem under homogeneous power-law potentials where the interaction between the bodies is given by r−a, (the Newtonian case corresponding to a = 3 and the vortex problem corresponding to a = 2). We study convex central configurations assuming two pairs of positive equal masses located at two adjacent vertices of a convex quadrilateral. Under these assumptions we prove that the isosceles trapezoid is the unique central configuration for every . For the case 0 < a < 4/3, which is more difficult to study analytically, we present some numerical considerations.

Spatial Convex but Non-strictly Convex Double-Pyramidal Central Configurations of the $$(2n+2)$$-Body Problem

A configuration of the N bodies is convex if the convex hull of the positions of all the bodies in $$\mathbb {R}^3$$ does not contain in its interior any of these bodies. And a configuration is strictly convex if the convex hull of every subset of the N bodies is convex. Recently some authors have proved the existence of convex but non-strictly convex central configurations for some N-body problems. In this paper we prove the existence of a new family of spatial convex but non-strictly convex central configurations of the $$(2n+2)$$ -body problem.

Classifying four-body convex central configurations

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 kite, trapezoidal, co-circular, equidiagonal, orthodiagonal, and bisecting-diagonal configurations. Good coordinates for describing the set are established. We use them to prove that the set of four-body convex central configurations with positive masses is three-dimensional, a graph over a domain $D$ that is the union of elementary regions in $\mathbb{R}^{+^3}$.

On Strictly Convex Central Configurations of the 2n-Body Problem

We consider planar central configurations of the Newtonian 2n-body problem consisting in two twisted regular n-gons of equal masses. We prove the conjecture that for $$n\ge 5$$ all convex central configurations of two twisted regular n-gons are strictly convex.

Strictly convex central configurations of the planar five-body problem

In this paper we investigate strictly convex central configurations of the planar five-body problem, and prove some necessary conditions for such configurations. In particular, given such a central configuration with multiplier λ \lambda and total mass M M , we show that all exterior edges are less than r 0 = ( M / λ ) 1 / 3 r_0=(M/\lambda )^{1/3} , at most two interior edges are less than or equal to r 0 r_0 , and its subsystem with four masses cannot be a central configuration. We also obtain some other necessary conditions for strictly convex central configurations with five bodies, and show numerical examples of strictly convex central configurations with five bodies that have either one or two interior edges less than or equal to r 0 r_0 . Our work develops some formulae in a classic work by W. L. Williams in 1938 and we rectify some unsupported assumptions there.

Convex Central Configurations of the 4-Body Problem with Two Pairs of Equal Adjacent Masses

We study the convex central configurations of the 4-body problem assuming that they have two pairs of equal masses located at two adjacent vertices of a convex quadrilateral. Under these assumptions we prove that the isosceles trapezoid is the unique central configuration.

Convex but not Strictly Convex Central Configurations

Central configurations of the n-body problem have been studied for more than 200 years since the pioneer works of Euler and Lagrange. In this article we study convex central configurations which are not strictly convex. We give explicit examples of such configurations in both planar and spatial n-body problems. Particularly, in the spatial case, we consider regular polyhedra with bodies of same mass m at the vertices and bodies of same mass M at the middle points of each edge. In this setting we prove that the cube is the unique regular polyhedron such that this construction leads to a convex central configuration which is not strictly convex.

A Four-Body Convex Central Configuration with Perpendicular Diagonals is Necessarily a Kite

We prove that any four-body convex central configuration with perpendicular diagonals must be a kite configuration. The result extends to general power-law potential functions, including the planar four-vortex problem.

Relative Equilibria in the Four-Vortex Problem with Two Pairs of Equal Vorticities

We examine in detail the relative equilibria in the four-vortex problem where two pairs of vortices have equal strength, that is, \Gamma_1 = \Gamma_2 = 1 and \Gamma_3 = \Gamma_4 = m where m is a nonzero real parameter. One main result is that for m > 0, the convex configurations all contain a line of symmetry, forming a rhombus or an isosceles trapezoid. The rhombus solutions exist for all m but the isosceles trapezoid case exists only when m is positive. In fact, there exist asymmetric convex configurations when m < 0. In contrast to the Newtonian four-body problem with two equal pairs of masses, where the symmetry of all convex central configurations is unproven, the equations in the vortex case are easier to handle, allowing for a complete classification of all solutions. Precise counts on the number and type of solutions (equivalence classes) for different values of m, as well as a description of some of the bifurcations that occur, are provided. Our techniques involve a combination of analysis and modern and computational algebraic geometry.

The symmetric central configurations of the 4-body problem with masses [formula omitted

We characterize the planar Central configurations of the 4-body problem with masses m1=m2≠m3=m4 which have an axis of symmetry.It is known that this problem has exactly two classes of convex Central configurations, one with the shape of a rhombus and the other with the shape of an isosceles trapezoid.We show that this 4-body problem also has exactly two classes of concave Central configurations with the shape of a kite, this proof is assisted by computer.

Convex Central Configurations of the n-Body Problem Which are not Strictly Convex

It is well-known that if a planar central configuration for the Newtonian 4-body problem is convex, then it must be strictly convex. In some literature, same conclusion was believed to hold for the case of five or even more bodies but rigorous treatments are absent. With the help of some numerical calculations, in this paper we provide concrete examples of central configurations which are convex but not strictly convex. Our examples include planar central configurations with five bodies and spatial central configurations with seven bodies.

New spatial central configurations in the 5-body problem

In this paper we show the existence of new families of convex and concave spatial central configurations for the 5-body problem. The bodies studied here are arranged as follows: three bodies are at the vertices of an equilateral triangle T, and the other two bodies are on the line passing through the barycenter of T that is perpendicular to the plane that contains T.

Classification of four-body central configurations with three equal masses

It is known that a central configuration of the planar four body problem consisting of three particles of equal mass possesses a symmetry if the configuration is convex or is concave with the unequal mass in the interior. We use analytic methods to show that besides the family of equilateral triangle configurations, there are exactly one family of concave and one family of convex central configurations, which completely classifies such central configurations.

Symmetry of planar four-body convex central configurations

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 masses of the two particles on the other diagonal are equal. If these two masses are unequal, then the less massive one is closer to the former diagonal. Finally, we extend these results to the case of non-planar central configurations of five particles.

On the central configurations of the planar restricted four-body problem

This article is devoted to answering several questions about the central configurations of the planar ( 3 + 1 ) -body problem. Firstly, we study bifurcations of central configurations, proving the uniqueness of convex central configurations up to symmetry. Secondly, we settle the finiteness problem in the case of two nonzero equal masses. Lastly, we provide all the possibilities for the number of symmetrical central configurations, and discuss their bifurcations and spectral stability. Our proofs are based on applications of rational parametrizations and computer algebra.