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 that is the union of elementary regions in .
Key Words: Central configuration, -body problem, convex central configurations
The study of central configurations in the Newtonian -body problem is an active subfield of celestial mechanics. A configuration is central if the gravitational force on each body is a common scalar multiple of its position vector with respect to the center of mass. Perhaps the most well known example is the equilateral triangle solution of Lagrange, discovered in 1772, consisting of three bodies of arbitrary mass located at the vertices of an equilateral triangle . Released from rest, any central configuration will collapse homothetically toward its center of mass, ending in total collision. In fact, any solution of the -body problem containing a collision must have its colliding bodies asymptotically approaching a central configuration . On the other hand, given the appropriate initial velocities, a central configuration can rotate rigidly about its center of mass, generating a periodic solution known as a relative equilibrium. These are some of the only explicitly known solutions in the -body problem. For more background on central configurations and their special properties, see [4, 24, 25, 27, 29, 32, 34] and the references therein.
In this paper we focus on four-body convex central configurations. A configuration is convex if no body lies inside or on the convex hull of the other three bodies (e.g., a rhombus or a trapezoid); otherwise, it is called concave. Most of the results on four-body central configurations are either for a specific choice of masses or for a particular geometric type of configuration. For instance, Albouy proved that all of the four-body equal mass central configurations possess a line of symmetry. This in turn allows for a complete solution to the equal mass case [1, 2]. Albouy, Fu, and Sun showed that a convex central configuration with two equal masses opposite each other is symmetric, with the equal masses equidistant from the line of symmetry . Recently, Fernandes, Llibre, and Mello proved that a convex central configuration with two pairs of adjacent masses must be an isosceles trapezoid . A numerical study for the number of central configurations in the four-body problem with arbitrary masses was done by Simó in . Other studies have focused on examples with one infinitesimal mass, solutions of the planar restricted four-body problem [7, 8].
In terms of restricting the problem to a particular shape, Cors and Roberts classified the four-body co-circular central configurations in  while Corbera et al. recently studied the trapezoidal solutions  (see also ). Symmetric central configurations are often the easiest to analyze. The regular -gon () is a central configuration as long as the masses are all equal. A kite is a symmetric quadrilateral with two bodies lying on the axis of symmetry and the other two bodies positioned equidistant from it. A kite may either be convex or concave. In the convex case, the diagonals are always perpendicular. A recent investigation of the kite central configurations (both convex and concave) was carried out in .
One of the major results in the study of convex central configurations is that they exist. MacMillan and Bartky showed that for any four masses and any ordering of the bodies, there exists a convex central configuration . This was proven again later in a simpler way by Xia . It is an open question as to whether this solution must be unique. This is problem 10 on a published list of open questions in celestial mechanics . Hampton showed that for any four choices of positive masses there exists a concave central configuration . Uniqueness does not hold in the concave setting as the example of an equilateral triangle with an arbitrary mass at the center illustrates. Finally, Hampton and Moeckel showed that given four positive masses, the number of equivalence classes of central configurations under rotations, translations, and dilations is finite .
Here we study the full space of four-body convex central configurations, focusing on how various geometrically-defined classes fit within the larger set. We introduce simple yet effective coordinates to describe the space up to an isometry, rescaling, or relabeling of the bodies. Three radial coordinates , and represent the distance from three of the bodies, respectively, to the intersection of the diagonals. The remaining coordinate is the angle between the two diagonals. Positivity of the masses imposes various constraints on the coordinates. We find a simply connected domain , the union of four elementary regions, such that for any , there exists a unique angle which makes the configuration central with positive masses. The angle is a differentiable function on the interior of . Thus the set of convex central configurations with positive masses is the graph of a function of three variables. We also prove that , with if and only if the configuration is a kite.
One of the surprising features of our coordinate system is the simple linear and quadratic equations that define various classes of quadrilaterals. The kite configurations lie on two orthogonal planes that intersect in the family of rhombii solutions. These planes form a portion of the boundary of . The co-circular and trapezoidal configurations each lie on saddles in , while the equidiagonal solutions are located on a plane. These three types of configurations intersect in a line corresponding to the isosceles trapezoid family. Our work provides a unifying structure for the set of convex central configurations and a clear picture of how the special sub-cases are situated within the broader set.
The paper is organized as follows. In the next section we develop the equations for a four-body central configuration using mutual distance coordinates. In Section 3 we introduce our coordinate system and study the important domain , proving that is a differentiable function on . We also verify the bounds on and show that it increases with . Section 4 focuses on four special cases—kite, trapezoidal, co-circular, and equidiagonal configurations—and how they fit together within .
2 Four-Body Planar Central Configurations
Let and denote the position and mass, respectively, of the th body. We will assume that , while recognizing that the zero-mass case is important for defining certain boundaries of our space. Let represent the distance between the th and th bodies. If is the sum of the masses, then the center of mass is given by . The motion of the bodies is governed by the Newtonian potential function
The moment of inertia with respect to the center of mass is given by
This can be interpreted as a measure of the relative size of the configuration.
There are several ways to describe a central configuration. We follow the topological approach.
A planar central configuration is a critical point of subject to the constraint , where is a constant.
It is important to note that, due to the invariance of and under isometries, any rotation, translation, or scaling of a central configuration still results in a central configuration.
2.1 Mutual distance coordinates
Our derivation of the equations for a four-body central configuration follows the nice exposition of Schmidt . In the case of four bodies, the six mutual distances turn out to be excellent coordinates. They describe a configuration in the plane as long as the Cayley-Menger determinant
vanishes and the triangle inequality holds for any choice of indices with . The constraint is necessary for locating planar central configurations; without it, the only critical points of restricted to are regular tetrahedra (a spatial central configuration for any choice of masses). Therefore, we search for critical points of the function
satisfying and , where and are Lagrange multipliers.
A useful formula involving the Cayley-Menger determinant is
where is the signed area of the triangle whose vertices contain all bodies except for the th body. Formula (2) holds only when restricting to planar configurations.
where and . Arranging the six equations of (3) as
and multiplying together pairwise yields the well-known Dziobek relation 
This assumes that the masses and areas are all non-zero. Eliminating from (5) produces the important equation
In some sense, equation (6) is the defining equation for a four-body central configuration. It or some equivalent variation can be found in many papers and texts (e.g., see p. 278 of .) Equation (6) is clearly necessary given the above derivation. However, it is also sufficient assuming the six mutual distances describe an actual configuration in the plane. The only other restrictions required on the are those that insure solutions to system (4) yield positive masses, as explained in the next section.
2.2 Restrictions on the mutual distances
For the remainder of the paper we will restrict our attention to four-body convex central configurations. We will assume the bodies are ordered consecutively in the counterclockwise direction. This implies that the lengths of the diagonals are and , while the four exterior side lengths are , and . With this choice of labeling, we always have and . We will also assume, without loss of generality, that the largest exterior side length is .
First note that . If this was not the case, then equation (3) and nonzero masses would imply that all are equal, which is the regular tetrahedron solution. If , then system (4) and positive masses implies
This means the two diagonals are strictly longer than any of the exterior sides. On the other hand, if we assume that , then the inequalities in (7) would be reversed. But such a configuration is impossible since it violates geometric properties of convex quadrilaterals such as (see Lemma 2.3 in ).
In addition to (7), further restrictions on the exterior side lengths follow from the Dziobek equation
Since is the largest exterior side length, we have and . It follows that , otherwise equation (8) is violated. We conclude that . A similar argument shows that implies that . Hence, the shortest exterior side is always opposite the longest one, with equality only in the case of a square. In sum, for our particular arrangement of the four bodies, any convex central configuration with positive masses must satisfy
According to the Dziobek equations (5),
These expressions generate nice formulas for the ratios between the masses. From system (4), a short calculation gives
Due to equation (6), these formulas are consistent with each other. They are all well-defined for configurations satisfying the inequalities in (9) unless (and thus ), or (and thus ). For these special cases, which correspond to symmetric kite configurations, we use the alternative formulas
The formulas obtained for the mass ratios explain why equation (6) is also sufficient for obtaining a central configuration. If the mutual distances satisfy both (9) and (6), then the mass ratios (which are positive), are given uniquely by (10), (11), or (12). We can then work backwards and check that system (4) is satisfied so that the configuration is indeed central.
3 The Set of Convex Central Configurations
We now describe the full set of convex central configurations with positive masses, showing it is three-dimensional, the graph of a differentiable function of three variables.
3.1 Good coordinates
We begin by defining simple, but extremely useful coordinates. Since the space of central configurations is invariant under isometries, we may apply a rotation and translation to place bodies 1 and 3 on the horizontal axis, with the origin located at the intersection of the two diagonals. It is also permissible to rescale the configuration so that . This alters the value of the Lagrange multipliers, but preserves the special trait of being central.
Define the remaining three bodies to have positions , and , where are radial variables and is an angular variable (see Figure 1). If one or more of the three radial variables is negative, then the configuration becomes concave or the ordering of the bodies changes. If one or more of the radial variables vanish, then the configuration contains a subset that is collinear or some bodies coalesce (e.g., implies ). Thus, we will assume throughout the paper that and . The coordinates turn out to be remarkably well-suited for describing different classes of quadrilaterals that are also central configurations (see Section 4).
In our coordinates, the six mutual distances are given by
Based on equation (6), define to be the function
where each mutual distance is treated as a function of the variables and .
The previous discussion justifies the following lemma.
Let and denote the sets
Any point in corresponds to a four-body convex central configuration with positive masses. Moreover, up to an isometry, rescaling, or relabeling of the bodies, contains all such configurations.
3.2 Defining the domain
We will find a set such that for each , there exists a unique angle which makes the configuration central. Specifically, we prove that there exists a differentiable function with domain , whose graph is equivalent to . In order to define , we use the mutual distance inequalities in (9) to eliminate the angular variable .
The inequalities in (9) imply the following conditions on the positive variables :
Since and are all positive, and together imply that
Similarly, and imply
Since , the first equation in (21) gives or . Then implies that
which verifies (17).
Similarly, and the second equation in (21) yields . Then implies that
Since and are both positive, inequality (28) clearly holds if . However, for any fixed choice of , the value of must be chosen strictly greater than the largest root of the quadratic . This root is , which verifies implication (18).
which in turn gives
Since and are both positive, inequality (30) clearly holds if . However, for any fixed choice of , the value of must be chosen strictly greater than the largest root of the quadratic . This root is , which proves (19).
Finally, implies that
Since , must be chosen greater than the largest root of the quadratic . This root is , which verifies (20) and completes the proof.
The combined inequalities between the radial variables and given in (15) through (20), along with and , define a bounded set . We will show that this set is the domain of the function and the projection of into -space.
Definition 3.3.Let denote the three-dimensional region, where
Let denote the closure of . A plot of the boundary of is shown in Figure 2. It contains five vertices, six faces, and nine edges (six curved, three straight), in accordance with Poincaré’s generalization of Euler’s formula . The vertices of are
each of which corresponds to a symmetric central configuration with at least two zero masses. and are rhombii with one diagonal congruent to the common side length, while and are kites with horizontal and vertical axes of symmetry, respectively. The point corresponds to an equilateral triangle with bodies 3 and 4 sharing a common vertex.
3.3 Configurations on the boundary of
We now focus on points lying on the boundary of . The next lemma shows that these points correspond to configurations where two or more of the mutual distance inequalities in (9) become equalities. Moreover, the only points for which this is true lie on the boundary of .
Suppose that are chosen so that with , , and . Then
Proof: We first note that under the assumptions of the lemma, the inequalities on and from Lemma 3.2 are still valid, except that the inequalities are no longer strict.
If and , then the left-hand equations in (21) and (22) imply and , respectively. This yields from which it follows that and . Conversely, if , (23) implies that either or . In the former case, and then and follows quickly. In the latter case, inequality (17) and implies that , which contradicts . Thus and and , proving (33).
If and , then the right-hand equations in (21) and (22) imply and , respectively. Thus, . Since and , we must have and hence . Conversely, if , (24) implies that either , or and . In the former case, and then and follows quickly. The latter case is impossible, since and contradicts inequality (20). This proves (34).
If , then the left-hand equation in (25) gives . Likewise, if , then . Thus implies . Conversely, if , then both inequalities in (26) become equalities. From this we deduce that , which verifies (35).
If , then the right-hand equation in (25) gives . Likewise, if , then . Thus implies . The quadratic has real roots for , but the smaller root is always negative for these -values. Thus must be taken to be the larger root of . Conversely, if , then and both inequalities in (29) become equalities. From this we deduce that , which verifies (36). The proof of (37) and (38) follows in a similar fashion, using inequalities (27) and (31), respectively.
Lemma 3.4 shows that the six faces on the boundary of , labeled I through VI, are given by the six equations (33) through (38), respectively. The first two faces are the only ones belonging to (positive masses) and contain all of the kite configurations, where . The remaining four faces on the boundary of correspond to cases with one or three zero masses (see Table 1). Points on these faces are interpreted as limiting solutions of a sequence of central configurations with positive masses. The mass values shown in Table 1 follow from formulas (10), (11), and (12). Here we assume that the limiting solution lies in the interior of the given face.
For example, suppose there is a sequence of points in the interior of converging to a point located on the interior of face V. This corresponds to a sequence of central configurations, each with positive masses, that limits on a configuration with . Since does not lie on any of the other faces on the boundary of , the other three limiting mutual distances, and , must be distinct from and each other. Moreover, the limiting values of the areas do not vanish because and are all strictly positive. Using either (10) or (11), it follows that the limiting mass value for must vanish, while the other limiting mass values are strictly positive. A similar argument applied to the other faces determines which masses must vanish in the limit.
Configurations on face IV or V, respectively, correspond to equilibria of the planar, circular, restricted four-body problem with infinitesimal mass or , respectively [7, 8, 20]. Configurations on face III or VI, respectively, correspond to relative equilibria of the -body problem, where a central mass (body 1 or 2, respectively) is equidistant from three infinitesimal masses [9, 16, 26]. Note that we have not made any assumptions on the relative size of the masses. Each of the six faces satisfies either or . Using identity (21), it follows that there is a unique value of for each point on the boundary of , if or if .
The masses at the vertices of are not well-defined because there are more options for the path of a limiting sequence. For example, the point represents a rhombus with one diagonal () congruent to all of the exterior sides. Approaching along the line as (a sequence of rhombii central configurations) yields the limiting mass values and . On the other hand, it is possible to construct a sequence of kite central configurations on face I with masses and that limits on as . The first sequence has two limiting mass values that vanish while the second sequence has three. The difference occurs because the mass ratio at is undefined in either formula (10) or (12).
Regardless of the particular limiting sequence, all five vertices of will have at least two mass values that vanish in the limit. For , this follows from Proposition 2 in . For the other four vertices, this fact is a consequence of formulas (10) and (11). In general, note that a limiting sequence with precisely two zero masses can only occur at vertices or . This somewhat surprising restriction is a consequence of Propositions 3 and 4 in  and the fact that the non-collinear critical points of the restricted three-body problem must form an equilateral triangle with the non-trivial masses.
3.4 The projection of onto the -plane
The set can be written as the union of four elementary regions in -space, where is bounded by functions of and . The projection of onto the -plane is shown in Figure 3. It is determined by and , where is the piecewise function
Here, is the projection of the intersection between faces III and IV, and is the projection of the vertical face V. The edge is the projection of the intersection between faces I and IV, while the edge is the projection of the vertical face II.
The decreasing dashed curve in Figure 3 is the projection of the
intersection of faces I and III, given by .
The increasing dashed curve is the projection of the
intersection of faces IV and VI, given by
These curves divide
the projection into four sub-regions over which is bounded by different functions of and , as indicated below:
3.5 is a graph over
We now prove our main result, showing that for each , there exists a unique angle that makes the configuration central. In general, for any point in the interior of , there is an interval of possible angles for which the mutual distance inequalities (9) hold. According to the identities given in (21), (22), and (25), must be chosen to satisfy
For any point in the interior of , define the constants and by
Proof: On the interior of the first two quantities in the definition of are strictly negative while the two quantities defining are strictly positive. The inequality follows from . The inequality is equivalent to , which is clearly valid for . It also holds for because . Likewise, is equivalent to , which is clearly satisfied for . It also holds for since . Finally, is satisfied because . This verifies that