This is a preprint of: “Equilic quadrilateral central configurations”, Martha Álvarez-Ramı́rez, Jaume Llibre, Commun. Nonlinear Sci. Numer. Simul., vol. 78, 104872: 1–7, 2019. EQUILIC QUADRILATERAL CENTRAL CONFIGURATIONS MARTHA ALVAREZ–RAMÍREZ1 AND JAUME LLIBRE2 Abstract. An equilic quadrilateral is a quadrilateral with one pair of opposite sides having the same length, which has angles of inclination whose sum is 2π/3. We show that there are some region in which equilic quadrilateral configuration of four bodies is possible for positive masses. 1. Introduction and statement of the main results In the n-body problem a central configuration occurs when the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration vector, and the scalar difference is the same for all particles [36]. There are several aspects of the n-body problem that motivate the study of central configurations, one is that they allow to obtain the homographic solutions of the n–body problem, which are the unique explicit solutions in function of the time known until now of that problem, for such solutions the ratios of the mutual distances between the bodies remain constant. Also, the central configurations appear as a key in the bifurcations of the surfaces of constant energy and angular momentum, for more details see Meyer [29] and Smale [35]. However, although there is an extensive literature concerning the number of classes of planar central configurations of the n-body problem for an ar- bitrary given set of positive masses, it has been only solved for n = 3, where there are three collinear and the two equilateral triangle central configura- tions; see among others [36], [3], [6], [22], [25], [28]. The equations of motion of the planar n–body problem are n X mj (xj − xi ) ẍi = 3 , for i = 1, . . . , n. rij j=1, j6=i 2010 Mathematics Subject Classification. Primary 70F07. Secondary 70F15. Key words and phrases. convex central configuration, four–body problem, equilic quadri- lateral. The second author is partially supported by MINECO grants MTM2013-40998-P and MTM2016-77278-P (FEDER) and the AGAUR grant 2014 SGR568. 1 http://www.gsd.uab.cat 2 M. ALVAREZ–RAMÍREZ AND J. LLIBRE where xi ∈ R2 are the position vectors of the bodies, rij = |xi − xj | are their mutual distances, and mi are their masses. Here the unit of time is taken in order that the Newtonian gravitational constant be equal to one. The configuration of the system formed by the n bodies is given by the vector x = (x1 , . . . , xn ) ∈ R2n . The differential equations of motion are well–defined when there is no collisions between the bodies, namely, when rij 6= 0 for i 6= j. We define the total mass and the center of mass of the n bodies as 1 M = m1 + . . . + mn , c= (m1 x1 + · · · + mn xn ) , M respectively. A configuration x is central when the acceleration vectors of the bodies satisfy Xn mj (xj − xi ) 3 = λ(xi − c), for i = 1, . . . , n, j=1 rij j6=i where λ is a constant. We say that two central configurations are equivalent if we can be pass from one to the other doing a rotation around the center of mass and a homothecy. This defines an equivalent relation in the set of all central configurations. From now on when we say a central configuration, we mean a class of central configurations. The aim of the present article is to improve the knowledge of the pla- nar central configurations of the 4–body problem. First we provide some research cites of planar 4-body central configurations, which shows that it is still an interesting and attracting field of research. A numerical study on the classes of central configurations for the 4–body problem for arbitrary masses was done by Simó in [34]. In 2006 Hampton and Moeckel [22] used a computer assisted proof to show that the 4–body problem has finitely many classes of central configurations with any given positive masses. Later on Albouy and Kaloshin [6] proved analytically this result. Llibre [24] showed all the planar central configurations of the 4– body problem with equal masses by studying the intersection points of two planar curves and assuming that such central configurations have an axis of symmetry. In [1, 2] Albouy gave an analytic proof of this result, showing that there are exactly four classes of central configurations with equal masses. Pedersen [30], Gannaway [20], Arenstorf [9] and Barros and Leandro [10, 11] obtained numerically and analytically the central configurations of the 4– body problem when one of the four masses has infinitesimal mas. A central configuration of the 4–body problem having an axis of symmetry passing through two non–adjacent bodies is called kite. The non–collinear classes of EQUILIC QUADRILATERAL CENTRAL CONFIGURATIONS 3 kite central configurations with three equal masses were classified by Bernat et al. in [12], and also by Leandro [23]. A planar configuration of the 4–body problem is convex if none of the bodies is located in the interior of the triangle formed by the others, oth- erwise it is concave. MacMillan and Bartky [27] showed that for any four positive masses and any assigned order, there is a convex planar central configuration of the four bodies. Later on Xia [37] gave a simple proof of this result. In 2010 Piña and Lonngi [32] numerically explored the new properties of symmetric and non-symmetric central configurations of the 4-body problem. Long and Sun [26] showed that any convex non-collinear central configura- tions of the planar 4-body problem with masses m1 = m2 < m3 = m4 , such that the equal masses are located at opposite vertices of a quadrilateral and the diagonal corresponding to the mass m1 is larger than or equal to the one corresponding to the mass m3 , must possess a symmetry and forms a rhombus. Pérez–Chavela and Santoprete [31] extended this result to the case where two of the masses are equal and at most, only one of the remain- ing mass is larger than the equal masses. Moreover, they shown that there exists a unique convex central configuration when the opposite masses are equal and it is a rhombus. Albouy et. al. [5] later on proved that 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. In addition to, if these two masses are unequal, then the less massive one is closer to the former diagonal. Álvarez–Ramı́rez and Llibre [7] discussed the convex and concave central configurations with an axis of symmetry of the 4–body problem when the masses satisfy that m1 = m2 6= m3 = m4 . In the same vein, Érdi and Czirják [18] derived a complete solution in a symmetric case of the planar four-body central configurations, when two bodies are on an axis of symmetry, and the other two bodies have equal masses and are situated symmetrically with respect to the axis of symmetry. Albouy and Fu in [4] (see also [27, 31]) conjectured: For the planar 4–body problem there is a unique convex central configuration for each ordering of the masses in the boundary of the convex hull of their positions. A particular case was considered before by MacMillan and Bartky [27], they proved that there is a unique isosceles trapezoid central configuration of the 4–body when two pairs of equal masses are located at adjacent vertices. Later on Xie in [38] reproved this result. In fact there was also the following subconjecture: For the planar 4–body problem there is a unique convex central configura- tion having two pairs of equal masses located at the adjacent vertices of the configuration and it is an isosceles trapezoid. 4 M. ALVAREZ–RAMÍREZ AND J. LLIBRE Using these previous results on the symmetries Corbera and Llibre [13] gave a complete description of the families of central configurations with two pairs of equals masses and two equal masses sufficiently small, proving for these masses the convex conjecture and the subconjecture. Recently Fernandes et al. [19] have proved the subconjecture for arbitrary masses. The central configurations when the four masses are on a circle have been classified by Cors and Roberts [16], and also by Álvarez–Ramı́rez et al. [8]. Moreover, Corbera et al. [15] have shown that any 4-body convex central configuration with perpendicular diagonals must be a kite configuration. Recently, Santoprete [33] proved that for a given ordering of the mutual distances, a trapezoidal central configuration must have a certain partial ordering of the masses. He also showed that if opposite masses of a four- body trapezoidal central configuration are equal, then the configuration has a line of symmetry and it must be a kite. Later on, Corbera et al. [14] classified all planar 4-body central configurations where two pairs of the bodies are on parallel lines, that is, all trapezoidal central configurations. The paper is organized as follows. In Section 2 we present the system of equations for Dziobek’s equations of the equilic central configurations of the 4-body problem here studied. Finally, in Section 3 we prove our main result, see Theorem 1. 2. Dziobek’s equations of the equilic central configurations A quadrilateral ABCD is equilic if AD = BC and if angle A + angle B = α+β = 2π/3, see Figure 1. If α or β is equal to π/2, the equilic quadrilateral is called right equilic. For more details about these kind of quadrilaterals consult [21]. (a) (b) Figure 1. Equilic quadrilateral: (a) α + β = 2π/3 and (b) right equilic. In this section we give a derivation of the 4-body central configuration where four particles are at the vertices of an equilic quadrilateral, with m1 , m2 , m3 and m4 in clockwise order. Since we are interested in convex EQUILIC QUADRILATERAL CENTRAL CONFIGURATIONS 5 quadrilateral central configurations, we will not be considering the right equilic. It is well-known that the set of central configurations are invariant under rotations and homotheties. So without loss of generality, we can assume that the positions of the vertices of the equilic quadrilateral are x1 = (0, 0), x2 = (1, 0), x3 = (1 − a cos(2π/3 − α), a sin(2π/3 − α)), x4 = (a cos α, a sin α), where a is a positive constant and α ∈ (0, π/3). Note that α < π/3, because for α = π/3 we should have a triangle instead of a quadrilateral. Let mk > 0 be the mass of the particle located at xk for k = 1, 2, 3, 4. See Figure 2. Figure 2. Coordinates of the 4-body configuration forming an equilic quadrilateral. Let x = (x1 , . . . , x4 ) ∈ (R2 )4 be the configuration vector and we associate to each x a 4 × 4 matrix   1 ··· 1 X =  x1 · · · x4  . 0 ··· 0 We define the 3 × 3 matrix Xk as the matrix obtained from the matrix X by deleting the k–th column and the last row. Then let Dk = (−1)k+1 det(Xk ) be for k = 1, . . . , 4. Dk is twice the signed area of the triangle whose vertices contain all bodies except for the k-th body. It is easy to check that D1 , D3 > 0 and D2 , D4 < 0, satisfying the equation D1 + D2 + D3 + D4 = 0. In [17] Dziobek studied shapes of central configurations for the planar 4-body problem using mutual distances as variables and obtained algebraic 6 M. ALVAREZ–RAMÍREZ AND J. LLIBRE equations for determining the central configurations. By restricting the cen- ter of mass to the origin of coordinates, the Dziobek’s equations are 1 Di Dj 3 = c1 + c2 , rij mi mj ti − tj = 0, for 1 ≤ i < j ≤ 4, with 4 X 2 ti = Dj rij , j=1, j6=i There are 12 unknowns, namely, 6 mutual distances rij , the 4 variables Di , and the 2 constants c1 and c2 . Writing the first six Dziobek’s equations (2) explicitly we obtain −3  m1 m2 r12 − c1 = c2 D1 D2 , −3  m1 m3 r13 − c1 = c2 D1 D3 , −3  m2 m3 r23 − c1 = c2 D2 D3 , −3  m1 m4 r14 − c1 = c2 D1 D4 , −3  m2 m4 r24 − c1 = c2 D2 D4 , −3  m3 m4 r34 − c1 = c2 D3 D4 . We multiply the equations (2) by row in order that the product of the right– hand side be simply c22 D1 D2 D3 D4 , and considering the fact that the masses are positive, we get the so called Dziobeck relation −3 −3 −3 −3 −3 −3 (r12 − c1 )(r34 − c1 ) = (r13 − c1 )(r24 − c1 ) = (r14 − c1 )(r23 − c1 ). Combining any two of (2) we get −3 −3 −3 −3 −3 −3 −3 −3 −3 −3 −3 −3 r12 r34 − r13 r24 r13 r24 − r14 r23 r14 r23 − r12 r34 c1 = −3 −3 −3 −3 = −3 −3 −3 −3 = −3 −3 −3 −3 . r12 + r34 − r13 − r24 r13 + r24 − r14 − r23 r14 + r23 − r12 − r34 If we set −3 −3 −3 −3 −3 −3 s1 = r12 + r34 , p1 = r12 r34 , s2 = r13 + r24 , −3 −3 −3 −3 −3 −3 p2 = r13 r24 , s3 = r14 + r23 , p3 = r14 r23 . Then equation (2) can be written as p1 − p2 p2 − p3 p3 − p1 c1 = = = , s1 − s2 s2 − s3 s3 − s1 EQUILIC QUADRILATERAL CENTRAL CONFIGURATIONS 7 which means that (s1 , p1 ), (s2 , p2 ), (s3 , p3 ) viewed as points in (R+ )2 , must lie on the same line with slope c1 , or equivalently 1 1 1 s1 s2 s3 = 0. p1 p2 p3 Following Dziobek [17] it is customary to use the following planarity con- dition of the 4–body problem: 3 3 3 3 3 3 3 3 3 3 3 3 D = (r13 − r12 )(r23 − r34 )(r24 − r14 ) − (r12 − r14 )(r24 − r34 )(r13 − r23 ) = 0. Now we are able to establish the main result of our article. Theorem 1. All points (α, a) of the open arc of the curve (2) √ in the variables (α, a) with endpoints (α, a) = (1, 0) and (α, a) = (π/6, 1/ 3) inside the region   1 M = (α, a) : 0 < √ < a < 1 with α ∈ (0, π/3) . cos α + 3 sin α correspond to all equilic quadrilateral central configurations. 3. Proof of Theorem 1 By convenience, but without loss of generality, from now on we will assume that m1 = 1 changing the unit of mass. From (2) the masses m2 , m3 and m4 can be expressed in terms of the rij and the Dk as follows 3 − r 3 )r 3 r 3 D2 (r13 14 23 24 m2 = 3 − r 3 )r 3 r 3 , D1 (r23 24 13 14 3 − r 3 )r 3 r 3 D3 (r12 14 23 34 m3 = 3 − r 3 )r 3 r 3 , D1 (r23 34 12 14 3 − r 3 )r 3 r 3 D4 (r12 13 24 34 m4 = 3 − r 3 )r 3 r 3 . D1 (r24 34 12 13 According to MacMillan and Bartky [27] the longest and smaller sides of the quadrilateral correspond to opposite sides. Moreover, each side of the quadrilateral is shorter in length than both diagonals, i.e. r13 , r24 > r12 , r14 , r23 , r34 . It follows immediately from the geometry of the equilic quadrilateral, Figure 2, that the mutual distances and the areas can be written in terms 8 M. ALVAREZ–RAMÍREZ AND J. LLIBRE of a and α as follows: p √ r12 = 1, r13 = 1 + a2 + a cos α − 3a sin α, √ r14 = a, r24 = 1 + a2 − 2a cos α, p √ r23 = a, r34 = 1 + a2 − a cos α − 3a sin α, and √ √ √ D1 = a(− 3a + 3 cos α + sin α)/2, D2 = a( 3a − 2 sin α)/2, D3 = a sin α, D4 = −a cos((π − 6α)/6) . Now it was already known that D1 , D3 > 0 and D2 , D4 < 0, and from (3) we get r14 < r13 , r23 < r24 , r12 < r13 , r34 < r24 . In order to become an equilic central configuration, the equilic quadri- lateral must satisfy (2) and the masses (??) must be positive. From (3), it follows that m2 and m4 are positive for all α ∈ (0, π/3) and a > 0. Conse- quently we only need to check conditions for which m3 is positive. This can happen only if r12 > r14 and r23 > r34 hold. We remark that r12 > r14 implies that 1 > a. In addition, m3 = 0 when r12 = r14 , and corresponds to a = 1. In fact m2 and m4 are also zero when α = 0 and a = 1. Moreover, the condition that m3 = ∞ is r23 = r34 , which can be expressed as 1 a= √ . cos α + 3 sin α This curve intersects the line a = 1 at the point (α, a) = (0, 1). Putting these observations together, we find that the inequalities (3) hold in the region M. This region is the one shown in Figure 3, in which α is the abscissa and a is the ordinate. For solving system (2) we replace the expressions of the masses (??), and taking only the numerators of these six equations because the denominators do not vanish, we obtain 3 3 3 3 3 3 3 3 e1 = −D2 (c1 r12 r13 r23 r24 − c1 r12 r14 r23 r24 + c2 D12 r12 3 3 3 3 r13 r14 r23 − c2 D12 r12 3 3 3 3 3 3 3 r13 r14 r24 − r13 3 3 3 r23 r24 + r14 r23 r24 ) = 0, 3 3 3 3 3 3 3 3 e2 = −D3 (c1 r12 r13 r23 r34 − c1 r13 r14 r23 r34 + c2 D12 r12 3 3 3 3 r13 r14 r23 − c2 D12 r12 3 3 3 3 3 3 3 r13 r14 r34 − r12 3 3 3 r23 r34 + r14 r23 r34 ) = 0, EQUILIC QUADRILATERAL CENTRAL CONFIGURATIONS 9 Figure 3. The gray shaded indicate the region in the (α, a)–plane where m3 is positive. 3 3 6 3 3 3 3 6 3 3 3 3 6 3 3 e3 = −D2 D3 (c1 r12 r13 r23 r24 r34 − c1 r12 r14 r23 r24 r34 − c1 r13 r14 r23 r24 r34 6 6 3 3 + c1 r14 r23 r24 r34 + c2 D12 r12 3 3 6 6 r13 r14 r23 − c2 D12 r12 3 3 6 3 3 r13 r14 r23 r24 − c2 D12 r12 3 3 6 3 3 r13 r14 r23 r34 + c2 D12 r12 3 3 6 3 3 3 3 3 3 3 r13 r14 r24 r34 − r12 r13 r23 r24 r34 3 3 3 3 3 3 3 3 3 3 6 3 3 3 + r12 r14 r23 r24 r34 + r13 r14 r23 r24 r34 − r14 r23 r24 r34 ) = 0, 3 3 3 3 3 3 3 3 e4 = −D4 (c1 r12 r14 r24 r34 − c1 r13 r14 r24 r34 + c2 D12 r12 3 3 3 3 r13 r14 r24 − c2 D12 r12 3 3 3 3 3 3 3 r13 r14 r34 − r12 3 3 3 r24 r34 + r13 r24 r34 ) = 0, 3 3 3 6 3 3 3 3 6 3 6 3 6 3 e5 = −D2 D4 (c1 r12 r13 r23 r24 r34 − c1 r12 r14 r23 r24 r34 − c1 r13 r23 r24 r34 3 3 3 6 3 + c1 r13 r14 r23 r24 r34 + c2 D12 r12 3 6 3 3 3 r13 r14 r23 r24 − c2 D12 r12 3 6 3 3 3 r13 r14 r23 r34 − c2 D12 r12 3 6 3 6 r13 r14 r24 + c2 D12 r12 3 6 3 3 3 3 3 3 3 3 r13 r14 r24 r34 − r12 r13 r23 r24 r34 3 3 3 3 3 6 3 3 3 3 3 3 3 3 + r12 r14 r23 r24 r34 + r13 r23 r24 r34 − r13 r14 r23 r24 r34 ) = 0, 6 3 3 6 3 3 3 3 6 3 3 3 3 6 e6 = −D3 D4 (c1 r12 r23 r24 r34 − c1 r12 r13 r23 r24 r34 − c1 r12 r14 r23 r24 r34 3 3 3 3 6 + c1 r13 r14 r23 r24 r34 + c2 D12 r12 6 3 3 3 3 r13 r14 r23 r24 − c2 D12 r12 6 3 3 3 3 r13 r14 r23 r34 − c2 D12 r12 6 3 3 3 3 r13 r14 r24 r34 + c2 D12 r12 6 3 3 6 6 3 3 3 r13 r14 r34 − r12 r23 r24 r34 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 + r12 r13 r23 r24 r34 + r12 r14 r23 r24 r34 − r13 r14 r23 r24 r34 ) = 0. 10 M. ALVAREZ–RAMÍREZ AND J. LLIBRE We known that all the D` are non-zero, from e1 = 0 and e2 = 0 we isolate c1 and c2 , and we substitute these two values into the other four equations obtaining the system D 6 3 3 2 2 2 2 e31 = D2 D3 r23 r24 r34 (r12 − r14 )(r12 + r12 r14 + r14 )(r14 − r13 )(r13 + r13 r14 + r14 ) = 0, d e41 = 0, D 3 6 3 2 2 2 2 e51 = D2 D4 r23 r24 r34 (r12 − r13 )(r12 + r12 r13 + r13 )(r13 − r14 )(r13 + r13 r14 + r14 ) = 0, d D 3 3 6 2 2 2 2 e61 = D3 D4 r23 r24 r34 (r12 − r13 )(r12 + r12 r13 + r13 )(r12 − r14 )(r12 + r12 r14 + r14 )=0 d where D = 0 is the Dziobek equation (2), and 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 d = −r12 r13 r23 r24 +r12 r13 r23 r34 +r12 r14 r23 r24 −r12 r14 r24 r34 −r13 r14 r23 r34 +r13 r14 r24 r34 From the geometry of the equilic quadrilateral configuration we have that the areas D1 , D2 , D3 , D4 are different from zero, as well as r12 = r13 and r13 = r14 are not allowed, because binary collisions have been excluded, while r12 = r14 corresponds to square configuration. Thus, e31 = 0, e51 = 0 and e61 = 0 are satisfied if and only if D = 0. On substituting the values (3) and (3) in Dziobek equation (2), this is expressed in the variables (α, a) as √ D = (1 − a3 )((a2 − 2a cos α + 1)3/2 − (a2 − 3a sin α − a cos α + 1)3/2 ) √ (a3 − (a2 − 3a sin α + a cos α + 1)3/2 ) √ + (a3 − (a2 − 2a cos α + 1)3/2 )(1 − (a2 − 3a sin α + a cos α + 1)3/2 ) √ (a3 − (a2 − a( 3 sin α + cos α) + 1)3/2 ) = 0. We have only found some necessary conditions for the existence of 4-body central configurations of the prescribed type. We now have to see if such configurations actually exist. In Figure 4 we provide numerically the graphic of the curve D = 0. This figure shows that the curve D = 0 has a non-empty intersection with the region M of positive masses. From the previous analysis, a = 1 is the unique solution of m3 = 0 for α ∈ [0, π/3]. Then substituting a = 1 in D = 0 we obtain    √ 3/2  1 − (2 − 2 cos α)3/2 1 − − 3 sin α − cos α + 2   √ 3/2  1 − − 3 sin α + cos α + 2 = 0, which has α = 0 as the unique solution in (0, π/3). 1 Now we note that when a = cos α+√ 3 sin α we have D = 0 only at α = π/6. Thus the curves m3 = ∞ and D = 0 intersect at one point, namely (α, a) = EQUILIC QUADRILATERAL CENTRAL CONFIGURATIONS 11 √ (π/6, 1/ 3). Hence we have an equilic quadrilateral central configuration for each point (α, a) of the curve D = 0 contained in the region M defined √ in the statement of Theorem 1, excluding the endpoints (0, 1) and (π/6, 1/ 3) = 0 contained in the boundary of the region M. This completes the proof of Theorem 1. Figure 4. Curve D = 0 and the positive masses region M. In Figure 5 we show the degenerate quadrilateral configuration corre- spoinding to the values (α, a) = (0, 1) and ( π6 , √13 ). 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Figure 5. Degenerate configurations associated with (α, a) = (0, 1) and ( π6 , √13 ), respectively. References [1] A. Albouy, Symétrie des configurations centrales de quarte corps, C. R. Acad. Sci. Paris 320 (1995), 217–220. [2] A. Albouy, Recherches sur le problème des n corps, Notes Scientifiques et Techniques du Bureau des Longitudes, Paris (1997), 78 pp. [3] A. Albouy and A. Chenciner, Le problème des n corps et les distances mutuelles, Invent. math. 131 (1998), 151–184. 12 M. ALVAREZ–RAMÍREZ AND J. LLIBRE [4] A. Albouy and Y. Fu, Euler configurations and quasi polynomial systems, Regul. Chaotic Dyn. 12 (2007), 39–55. [5] A. Albouy, Y. Fu and S. Sun, Symmetry of planar four-body convex central config- urations, Proc. R. Soc. Lond. Ser. A 464 (2008), 1355–1365. [6] A. Albouy and V. Kaloshin, Finiteness of central configurations of five bodies in the plane, Annals of Math. (2) 176 (2012), 535–588. [7] M. Álvarez-Ramı́rez and J. Llibre, The symmetric central configurations of the 4-body problem with masses m1 = m2 6= m3 = m4 , Appl. Math. Comput. 219 (2013), no. 11, 5996–6001. [8] M. Álvarez-Ramı́rez, A. A. Santos and C. Vidal, On co-circular central con- figurations in the four and five body-problems for homogeneous force law, J. Dynam. Differential Equations 25 (2013), no. 2, 269–290. [9] R.F. Arenstorf, Central configurations of four bodies with one inferior mass, Cel. Mech. 28 (1982), 9–15. [10] J.F. Barros and E.S.G. Leandro, The set of degenerate central configurations in the planar restricted four–body problem, SIAM Journal on Mathematical Analysis 43 (2011), 634–661. [11] J.F. Barros and E.S.G. Leandro, Bifurcations and enumeration of classes of rel- ative equilibria in the planar restricted four–body problem, SIAM Journal on Mathe- matical Analysis 46 (2014), 1185–1203. [12] J. Bernat, J. Llibre and E. Perez–Chavela, On the planar central configurations of the 4–body problem with three equal masses, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 16 (2009), 1–13. [13] M. Corbera and J. Llibre, Central configurations of the 4–body problem with masses m1 = m2 > m3 = m4 = m > 0 and m small, Appl. Math. Comput. 246 (2014), 121–147. [14] M. Corbera, J.M. Cors, J. Llibre and E. Perez–Chavela, Trapezoid central configurations, arXiv:1712.07155 [math.DS], 2017. [15] M. Corbera, J.M. Cors and G.E. Roberts, A four-body convex central config- uration with perpendicular diagonals is necessarily a kite, to appear in Qual. Theory Dyn. Syst. (2018). [16] J.M. Cors and G.E. Roberts, Four-body co-circular central configurations, Non- linearity 25 (2012), 343–370. [17] O. Dziobek, Ueber einen merkwürdigen Fall des Vielkörperproblems, Astron. Nach. 152 (1900), 32–46. [18] B. Érdi and Z. Czirják, Central configurations of four bodies with an axis of sym- metry, Celestial Mech. Dynam. Astronom. 125 (2016), no. 1, 33-70. [19] A.C. Fernandes, J. Llibre and L.F. Mello, Convex central configurations of the 4–body problem with two pairs of equal masses, Arch. Rational Mech. Anal. 226 (2017), 303–320. [20] J.R. Gannaway, Determination of all central configurations in the planar 4–body problem with one inferior mass, Ph. D., Vanderbilt University, Nashville, USA, 1981. [21] J. Garfunkel, The Equilic Quadrilateral, Pi Mu Epsilon J. 7 (1981), 317-329. [22] H. Hampton and R. Moeckel, Finiteness of relative equilibria of the four–body problem, Inventiones math. 163 (2006), 289–312. [23] E.S.G. Leandro, Finiteness and bifurcation of some symmetrical classes of central configurations, Arch. Rational Mech. Anal. 167 (2003), 147–177. [24] J. Llibre, Posiciones de equilibrio realtivo del problema de 4 cuerpos, Publicacions Matemàtiques UAB 3 (1976), 73–88. [25] J. Llibre, On the number of central configurations in the N -body problem, Cel. Mech. Dynam. Astronom. 50 (1991), 89–96. [26] Y. Long and S. Sun, Four–body central configurations with some equal masses, Arch. Rational Mech. Anal. 162 (2002), 24–44. EQUILIC QUADRILATERAL CENTRAL CONFIGURATIONS 13 [27] W.D. MacMillan and W. Bartky, Permanent Configurations in the Problem of Four Bodies, Trans. Amer. Math. Soc. 34 (1932), 838–875. [28] R. Moeckel, On central configurations, Mah. Z. 205 (1990), 499–517. [29] K.R. Meyer, Bifurcation of a central configuration, Cel. Mech. 40 (1987), 273–282. [30] P. Pedersen, Librationspunkte im restringierten Vierk” orperproblem, Danske Vid. Selsk. Math.–Fys. 21 (1944), 1–80. [31] E. Pérez-Chavela and M. Santoprete, Convex four-body central configurations with Some equal masses, Arch. Rational Mech. Anal. 185 (2007), 481–494. [32] E. Piña and P. Lonngi, Central configurations for the planar Newtonian four-body problem, Celest. Mech. Dyn. Astron. 108, 73-93 (2010). [33] M. Santoprete, Four-body central configurations with one pair of opposite sides parallel, J. Math. Anal. Appl. 464 (2018), 421–434. [34] C. Simó, Relative equilibrium solutions in the four-body problem, Cel. Mech. 18 (1978), 165–184. [35] S. Smale, Topology and mechanics II: The planar n–body problem, Inventiones math. 11 (1970), 45–64. [36] A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Univer- sity Press, 1941. [37] Z. Xia, Convex central configurations for the n-body problem, J. Differential Equations 200 (2004), 185–190. [38] Z. Xie, Isosceles trapezoid central configurations of the Newtonian four-body problem, Proc. R. Soc. Edinb., Sect. A, Math. 142 (2012), 665–672. 1 Departamento de Matemáticas, UAM–Iztapalapa, San Rafael Atlixco 186, Col. Vicentina, 09340 Iztapalapa, México, City, México. E-mail address: mar@xanum.uam.mx 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat