Planar Central Configurations Research Articles

Linear stability of the elliptic relative equilibrium with $ (1 + 7) $-gon central configuration in planar $ N $-body problem

The $ (1+n) $-gon elliptic relative equilibrium (ERE for short) is the planar central configuration solution consisting of $ n $ unit masses at the vertices of a regular $ n $-gon with a body of mass $ m $ at the center, and each particle moves on a Keplerian orbit with a common eccentricity $ e\in [0,1) $. Maxwell first considered this model in his study on the stability of the Saturn's rings. Moeckel [10] proves that for $ e = 0 $, the $ (1+n) $-gon is linearly stable for sufficiently large $ m $ for $ n\geq7 $. A natural question is that whether Moeckel's stability result holds for $ e>0 $. In the recent paper [2], Hu, Long and Ou give an affirmative answer for $ n\geq8 $, but the case $ n = 7 $ is difficult and still open. In this paper, we show that it is also true for $ n = 7 $, that is the $ (1+7) $-gon ERE is still linearly stable for any $ e\in(0,1) $ when $ m $ is large enough.

Planar central configurations of some restricted (4 + 1)-body problems

We start with the 13 central configurations of the restricted (4 + 1) body problem having four primaries with equal masses at the vertices of a square. Then, we describe the evolution of these central configurations when some of the masses of the four primaries tend to zero and the remainder ones keep constant. More precisely, we consider the cases where one of the masses tends to zero, where either two adjacent or two opposite equal masses tend to zero simultaneously, and where three equal masses tend to zero simultaneously. Here, simultaneously means that the masses that go to zero take the same value at any moment.

Planar central configurations with five different positive masses

We study planar five-body central configurations with different positive masses. From Laura–Andoyer equations, we recover the variational formulation in terms of the potential energy and the total inertia moment, including constraints on the distances. We use coordinates to prove the permanence of the central configuration by Newton equations of motion. The masses appear in our coordinates, forming a rigid orthocentric simplex in four dimensions, with the masses at the vertices. The linear dimensions are determined by a distance. We use different angles, with one angle measuring the rotation around the center of mass. The rest of the coordinates are angles that are constants of motion for central configurations. One angle measures the ratio of the two principal moments of inertia. The rest of the coordinates determine the rotation around the center of mass of the rigid simplex of masses with respect to the plane and to the direction of projection from four to two dimensions. We transform the Laura–Andoyer equations into different expressions in 10 and 5 dimensions, including the properties of the areas formed by five point bodies in the plane. Furthermore, we presented three numerical computed configurations of planar five-body central configurations, one convex and two concave, which give numerical confirmation of our equations with a precision better than 10−12.

A stochastic optimization algorithm for analyzing planar central and balanced configurations in the n-body problem

A stochastic optimization algorithm for analyzing planar central and balanced configurations in the n-body problem is presented. We find a comprehensive list of equal mass central configurations satisfying the Morse equality up to n=12. We show some exemplary balanced configurations in the case n=5, as well as some balanced configurations without any axis of symmetry in the cases n=4 and n=10.

Planar central configurations of six bodies

Central configurations of the n-body problem play an important role in the study of celestial mechanics. In this paper, we study six-body planar central configurations having certain symmetries. Special attention is given to the existence results of such configurations. With analytic proofs, we show the existence of a new central configuration, which is convex but not strictly convex, a non-symmetric concave case of central configuration, and some cases of stacked central configurations, which are central configurations with subsets of the bodies also forming central configurations.

Central configurations on the plane with [formula omitted] heavy and [formula omitted] light bodies

We study the problem of planar central configurations with N heavy bodies and k bodies with arbitrary small masses. We derive the equation describing the limit of light masses going to zero, which can be seen as the equation for central configurations in the anisotropic plane. Using computer rigorous computations we compute all central configurations for N=2 and k=3,4 and for the derived limit problem. We show that the results are consistent.

Tangential Trapezoid Central Configurations

A tangential trapezoid, also called a circumscribed trapezoid, is a trapezoid whose four sides are all tangent to a circle within the trapezoid: the in-circle or inscribed circle. In this paper we classify all planar four-body central configurations, where the four bodies are at the vertices of a tangential trapezoid.

Bicentric quadrilateral central configurations

A bicentric quadrilateral is a tangential cyclic quadrilateral. In a tangential quadrilateral, the four sides are tangents to an inscribed circle, and in a cyclic quadrilateral the four vertices lie on a circumscribed circle. In this paper, we classify all planar central configurations of the 4-body problem, where the four bodies are at the vertices of a bicentric quadrilateral.

Generic Finiteness for a Class of Symmetric Planar Central Configurations of the Six-Body Problem and the Six-Vortex Problem

A symmetric planar central configuration of the Newtonian six-body problem $x$ is called cross central configuration if there are precisely four bodies on a symmetry line of $x$. We use complex algebraic geometry and Groebner basis theory to prove that for a generic choice of positive real masses $m_1,m_2,m_3,m_4,m_5=m_6$ there is a finite number of cross central configurations. We also show one explicit example of a configuration in this class. A part of our approach is based on relaxing the output of the Groebner basis computations. This procedure allows us to obtain upper bounds for the dimension of an algebraic variety. We get the same results considering cross central configurations of the six-vortex problem.

Planar N-body central configurations with a homogeneous potential

Central configurations give rise to self-similar solutions to the Newtonian N-body problem and play important roles in understanding its complicated dynamics. Even the simple question of whether or not there are finitely many planar central configurations for N positive masses remains unsolved in most cases. Considering central configurations as critical points of a function f, we explicitly compute the eigenvalues of the Hessian of f for all N for the point vortex potential for the regular polygon with equal masses. For homogeneous potentials including the Newtonian case, we compute bounds on the eigenvalues for the regular polygon with equal masses and give estimates on where bifurcations occur. These eigenvalue computations imply results on the Morse indices of f for the regular polygon. Explicit formulae for the eigenvalues of the Hessian are also given for all central configurations of the equal-mass four-body problem with a homogeneous potential. Classic results on collinear central configurations are also generalized to the homogeneous potential case. Numerical results, conjectures, and suggestions for the future work in the context of a homogeneous potential are given.

On central configurations of the κn-body problem

We consider planar central configurations of the Newtonian κn-body problem consisting in κ groups of regular n-gons of equal masses, called (κ,n)-crown. We derive the equations of central configurations for a general (κ,n)-crown. When κ=2 we prove the existence of a twisted (2,n)-crown for any value of the mass ratio. Moreover, for n=3,4 and any value of the mass ratio, we give the exact number of twisted (2,n)-crowns, and describe their location. Finally, we conjecture that for any value of the mass ratio there exist exactly three (2,n)-crowns for n≥5.

On the Symmetric Central Configurations for the Planar 1 + 4‐Body Problem

We study the planar symmetric central configurations of the 1 + 4‐body problem where the symmetry axis does not contain any infinitesimal masses. Under certain assumptions, we find analytically some central configurations for suitable positive masses and also get some numerical results of symmetric central configurations where infinitesimal masses are not necessarily equal.

On the Sectional Curvature Along Central Configurations

In this paper we characterize planar central configurations in terms of a sectional curvature value of the Jacobi-Maupertuis metric. This characterization works for the $N$-body problem with general masses and any $1/r^{\alpha}$ potential with $\alpha> 0$. We also observe dynamical consequences of these curvature values for relative equilibrium solutions. These curvature methods work well for strong forces ($\alpha \ge 2$).

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.

Correction to: On Stacked Planar Central Configurations with Five Bodies when One Body is Removed

We correct an error in the proof of Lemma 2.7 in the article mentioned in the title. This correction does not affect the main results of the article.

On the planar central configurations of rhomboidal and triangular four- and five-body problems

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 derived for the four- and five-body problems. These regions are determined analytically and explored numerically. The equations of motion are regularized using Levi-Civita type transformations and then the phase space is investigated for chaotic and periodic orbits by means of Poincar\'e surface of sections.

A family of stacked central configurations in the planar five-body problem

We study planar central configurations of the five-body problem where three bodies, $$m_1, m_2$$ and $$m_3$$ , are collinear and ordered from left to right, while the other two, $$m_4$$ and $$m_5$$ , are placed symmetrically with respect to the line containing the three collinear bodies. We prove that when the collinear bodies form an Euler central configuration of the three-body problem with $$m_1=m_3$$ , there exists a new family, missed by Gidea and Llibre (Celest Mech Dyn Astron 106:89–107, 2010), of stacked five-body central configuration where the segments $$m_4m_5$$ and $$m_1m_3$$ do not intersect.

On a Class of Central Configurations in the Planar {varvec{3n}}-Body Problem

The existence of a central configuration of 2n bodies located on two concentric regular n-gons with the polygons which are homotetic or similar with an angle equal to frac{pi }{n} and the masses on the same polygon, are equal, has proved by Elmabsout (C R Acad Sci 312(5):467–472, 1991). Moreover, the existence of a planar central configuration which consists of 3n bodies, also situated on two regular polygons, the interior n-gon with equal masses and the exterior 2n-gon with masses on the 2n-gon alternating, has shown by author. Following Smale (Invent Math 11:45-64, 1970), we reduce this problem to one, concerning the critical points of some effective-type potential. Using computer assisted methods of proof we show the existence of ten classes of such critical points which corresponds to ten classes of central configurations in the planar six-body problem.

Andoyer equations for noncollinear planar central configurations

In this article we obtain the Andoyer equations for noncollinear planar central configurations taking into account the center of mass of the system. We apply these equations to study two configurations. In the first one we prove that it is not possible to put a square central configuration and an equilateral triangle central configuration as a cocircular central configuration. In the second one we give the central configurations for the noncollinear planar $4$-body problem with one pair of equal positive masses and two null masses.

Collision Index and Stability of Elliptic Relative Equilibria in Planar $${n}$$ n -body Problem

It is well known that a planar central configuration of the $n$-body problem gives rise to solutions where each particle moves on a specific Keplerian orbit while the totality of the particles move on a homographic motion. When the eccentricity $e$ of the Keplerian orbit belongs in $[0,1)$, following Meyer and Schmidt, we call such solutions elliptic relative equilibria (shortly, ERE). In order to study the linear stability of ERE in the near-collision case, namely when $1-e$ is small enough, we introduce the collision index for planar central configurations. The collision index is a Maslov-type index for heteroclinic orbits and orbits parametrised by half-lines that, according to the Definition given by authors in [16], we shall refer to as half-clinic orbits and whose Definition in this context, is essentially based on a blow up technique in the case $e=1$. We get the fundamental properties of collision index and approximation theorems. As applications, we give some new hyperbolic criteria and prove that, generically, the ERE of minimal central configurations are hyperbolic in the near-collision case, and we give detailed analysis of Euler collinear orbits in the near-collision case.