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.     

The central configurations of four masses x, −x, y, −y

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≠y (a parallelogram and two trapezoids) and two planar non-collinear central configurations for masses x, −x, x, −x (two diamonds). Except the case studied here, the only known case where the four-body central configurations with non-vanishing masses can be listed is the case with equal masses (A. Albouy, 1995-1996), which requires the use of a symbolic computation program. Thanks to a lemma used in the proof of our result, we also show that a co-circular four-body central configuration has non-vanishing total mass or vanishing multiplier.     

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.     

SQUARE AND RHOMBUS CENTRAL CONFIGURATIONS

In this paper, we consider the flat central configurations of bodies using the characteristic set method. We completely solve two special cases of four planets, namely,the square and the rhombus. For the square case, we obtain that a square is a central configuration only in the case where the masses are equal and there exactly are two different square central configurations determined by the mass and the angular velocity; for the rhombus case, we obtain that if a rhombus is a central configuration, then the masses of the diagonal vertices must be equal. Furthermore, there are two or three or four different rhombus central configurations determined by the masses and the angular velocity.     

Some Enumeration Problems on Central Configurations at the Bifurcation Points

In “Counting central configurations at the bifurcation points,” we proposed an algorithm to rigorously count central configurations in some cases that involve one parameter. Here, we improve our algorithm to consider three harder cases: the planar \((3+1)\)-body problem with two equal masses; the planar 4-body problem with two pairs of equal masses which have an axis of symmetry containing one pair of them; the spatial 5-body problem with three equal masses at the vertices of an equilateral triangle and two equal masses on the line passing through the center of the triangle and being perpendicular to the plane containing it.While all three problems have been studied in two parameter cases, numerical observations suggest new results at some points on the bifurcation curves. Applying the improved version of our algorithm, we count at those bifurcation points. As a result, for the \((3+1)\)-body problem, we identify three points on the bifurcation curve where there are 8 central configurations, which adds to the known results of \(8,9,10\) ones. For our 4-body case, at the bifurcation points, there are 3 concave central configurations, which adds to the known results of \(2,4\) ones. For our 5-body case, at the bifurcation point, there is 1 concave central configuration, which adds to the known results of \(0,2\) ones.     

Spatial bi-stacked central configurations formed by two dual regular polyhedra

In this paper we prove the existence of two new families of spatial stacked central configurations, one consisting of eight equal masses on the vertices of a cube and six equal masses on the vertices of a regular octahedron, and the other one consisting of twenty masses at the vertices of a regular dodecahedron and twelve masses at the vertices of a regular icosahedron. The masses on the two different polyhedra are in general different. We note that the cube and the octahedron, the dodecahedron and the icosahedron are dual regular polyhedra. The tetrahedron is itself dual. There are also spatial stacked central configurations formed by two tetrahedra, one and its dual.     

On the upper embedding of symmetric configurations with block size 3

We consider the problem of embedding a symmetric configuration with block size 3 in an orientable surface in such a way that the blocks of the configuration form triangular faces and there is only one extra large face. We develop a sufficient condition for such an embedding to exist given any orientation of the configuration, and show that this condition is satisfied for all configurations on up to 19 points. We also show that there exists a configuration on 21 points which is not embeddable in any orientation. As a by-product, we give a revised table of numbers of configurations, correcting the published figure for 19 points. We give a number of open questions about embeddability of configurations on larger numbers of points.     

Complex geometry of polygonal linkages

We present observations on the complex geometry of polygonal linkages arising in the framework of our approach to extremal problems on configuration spaces. Along with a few general remarks on applications of complex geometry and theory of residues, we present new results obtained in this way. Most of the new results are presented in the case of a planar quadrilateral linkage with generic lengths of the sides. First, we show that, for each configuration of planar quadrilateral linkage Q(a, b, c, d) with pairwise distinct side-lengths (a, b, c, d), the cross-ratio of its vertices belongs to the circle of radius ac/bd centered at the point $ 1\in \mathbb{C} $ . Next, we establish an analog of the Poncelet porism for a discrete dynamical system on a planar moduli space of a 4-bar linkage defined by the product of diagonal involutions and discuss some related issues suggested by a beautiful link to the theory of discrete integrable systems discovered by J. Duistermaat. We also present geometric results concerned with the electrostatic energy of point charges placed at the vertices of a quadrilateral linkage. In particular, we establish that all convex shapes of a quadrilateral linkage arise as the global minima of a system of charges placed at its vertices, and these shapes can be completely controlled by the value of the charge at just one vertex, which suggests a number of interesting problems. In conclusion, we describe a natural connection between certain extremal problems for configurations of linkage and convex polyhedra obtained from its configurations using the Minkowski 1897 theorem and present a few related remarks.     

Linear stability of relative equilibria in the charged three-body problem

A relative equilibrium is a periodic orbit of the n-body problem that rotates uniformly maintaining the same central configuration for all time. In this paper we generalize some results of R. Moeckel and we apply it to study the linear stability of relative equilibria in the charged three-body problem. We find necessary conditions to have relative equilibria linearly stable for the collinear charged three-body problem, for planar relative equilibria we obtain necessary and sufficient conditions for linear stability in terms of the parameters, masses and electrostatic charges. In the last case we obtain a stability inequality which generalizes the Routh condition of celestial mechanics. We also proof the existence of spatial relative equilibria and the existence of planar relative equilibria of any triangular shape.     

The Lifting Model for Reconfiguration

Given a pair of start and target configurations, each consisting of n pairwise disjoint disks in the plane, what is the minimum number of moves that suffice for transforming the start configuration into the target configuration? In one move a disk is lifted from the plane and placed back in the plane at another location, without intersecting any other disk. We discuss efficient algorithms for this task and estimate their number of moves under different assumptions on disk radii. We then extend our results for arbitrary disks to systems of pseudodisks, in particular to sets of homothetic copies of a convex object.     

On a Class of Central Configurations in the Planar $${\varvec{}}$$-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.     

On the Order Dimension of Convex Geometries

We study the order dimension of the lattice of closed sets for a convex geometry. We show that the lattice of closed subsets of the planar point set of Erd?s and Szekeres from 1961, which is a set of 2 ^n???2 points and contains no vertex set of a convex n-gon, has order dimension n???1 and any larger set of points has order dimension at least n.     

Morse index of a cyclic polygon

It is known that cyclic configurations of a planar polygonal linkage are critical points of the signed area function. In the paper we give an explicit formula of the Morse index for the signed area of a cyclic configuration. We show that it depends not only on the combinatorics of a cyclic configuration, but also on its metric properties.     

Configuration spaces and signature formulas

We show that nontrivial topological and geometric information about configuration spaces of linkages and tensegrities can be obtained using the signature formulas for the mapping degree and Euler characteristic. In particular, we prove that the Euler characteristics of such configuration spaces can be effectively calculated using signature formulas. We also investigate the critical points of signed area function on the configuration space of a planar polygon. We show that our approach enables one to effectively count the critical points in question and discuss a few related problems. One of them is concerned with the so-called cyclic polygons and formulas of Brahmagupta type. We describe an effective method of counting cyclic configurations of a given polygon and formulate four general conjectures about the critical points of the signed area function on the configuration space of a generic planar polygon. Several concrete results for planar quadrilaterals and pentagons are also presented. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 59, Algebra and Geometry, 2008.     

On the classification of pyramidal central configurations

We present some results associated with the existence of central configurations (c.c.'s) in the classical gravitational -body problem of Newton. We call a central configuration of five bodies, four of which are coplanar, a central configuration (p.c.c). It can be shown that there are only three types of p.c.c.'s, admitting one or more planes of symmetry, viz. (i) the case where the planar bodies lie at the vertices of a regular trapezoid, (ii) the case where the bodies lie at the vertices of a kite-shaped quadrilateral, and (iii) the case where the bodies lie at the vertices of a rectangle. In this paper we classify all p.c.c.'s with a rectangular base and, in fact, prove that there is only one such c.c., namely, the square-based pyramid with equal masses at the corners of the square. The classification of all p.c.c.'s satisfying either (i) or (ii) will be discussed in subsequent papers.

     

关于4-体问题中心构型的一点研究

N-体问题的中心构型是应用数学领域广泛研究的问题．关于N-体问题的中心构型已有许多研究结果．但是对于n≥4,其中心构型解的计算是比较困难的．作者运用Wu-Ritt零点分解方法和子结式序列研究了一般的平面4体中心构型问题,给出了这类4体中心构型问题的解析解,从而证明了一类平面牛顿4-体问题的中心构型个数是有限的．     

Action minimizers under topological constraints in the planar equal-mass four-body problem

It is shown that in the planar equal-mass four-body problem, there exist two sets of action minimizers connecting two planar boundary configurations with fixed symmetry axes and specific order constraints: a double isosceles configuration and an isosceles trapezoid configuration, while order constraints are introduced on the boundary configurations. By applying the level estimate method, these minimizers are shown to be collision-free and they can be extended to two new sets of periodic or quasi-periodic orbits.     

正四面体套中心构型的存在唯一性

Two cases of the nested configurations in R^3 consisting of two regular quadrilaterals are discussed. One case of them do not form central configuration, the other case can be central configuration. In the second case the existence and uniqueness of the central configuration are studied. If the configuration is a central configuration, then all masses of outside layer are equivalent, similar to the masses of inside layer. At the same time the following relation between r(the ratio of the sizes) and mass ratio b = m/m must be satisfied b=24(3的立方根)（3r^2 2r 3）^-3/2-8（1-r）|1-r|^-3-3(6r的立方根)/24(3的立方根)（3 r）（3r^2 2r 3）^-3/2-8r（1-r）|1-r|^-3-3(6r^-2的立方根)in which the masses at outside layer are not less than the masses at inside layer, and the solution of this kind of central configuration is unique for the given ratio (b) of masses.     

Symmetry of the restricted 4 + 1 body problem with equal masses

We consider the problem of symmetry of the central configurations in the restricted 4 + 1 body problem when the four positive masses are equal and disposed in symmetric configurations, namely, on a line, at the vertices of a square, at the vertices of a equilateral triangle with a mass at the barycenter, and finally, at the vertices of a regular tetrahedron [1–3]. In these situations, we show that in order to form a non collinear central configuration of the restricted 4 + 1 body problem, the null mass must be on an axis of symmetry. In our approach, we will use as the main tool the quadratic forms introduced by A. Albouy and A. Chenciner [4]. Our arguments are general enough, so that we can consider the generalized Newtonian potential and even the logarithmic case. To get our results, we identify some properties of the Newtonian potential (in fact, of the function ϕ(s) = −s ^k, with k < 0) which are crucial in the proof of the symmetry.     

Diagonal Flips in Labelled Planar Triangulations

 A classical result of Wagner states that any two (unlabelled) planar triangulations with the same number of vertices can be transformed into each other by a finite sequence of diagonal flips. Recently Komuro gives a linear bound on the maximum number of diagonal flips needed for such a transformation. In this paper we show that any two labelled triangulations can be transformed into each other using at most O(nlogn) diagonal flips. We will also show that any planar triangulation with n>4 vertices has at least n− 2 flippable edges. Finally, we prove that if the minimum degree of a triangulation is at least 4, then it contains at least 2n + 3 flippable edges. These bounds can be achieved by an infinite class of triangulations. Received: June 3, 1998 Final version received: January 26, 2001