This is a preprint of: “Hjelmslev quadrilateral central configurations”, Martha Álvarez-Ramı́rez, Jaume Llibre, Phys. Lett. A, vol. 383, 103–109, 2019. DOI: [10.1016/j.physleta.2018.08.034] HJELMSLEV QUADRILATERAL CENTRAL CONFIGURATIONS MARTHA ALVAREZ–RAMÍREZ1 AND JAUME LLIBRE2 Abstract. A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices. We classify all planar four–body central configurations where the four bodies are at the vertices of a Hjelmslev quadrilateral. We show that in the positive mass space (m1 , m2 , m3 , m4 ), taking the unit of mass equal to m1 , the set of Hjelmslev quadrilateral central configurations of the four– body problem is an open arc. When the masses tend to the endpoints of this arc three of the masses of the Hjelmslev quadrilateral central configurations tend to an equilateral central configuration of the three–body problem, and the fourth remainder mass tends to zero. 1. Introduction and statement of the main results The n–body problem, i.e. the description of the motion of n particles of positive masses under their mutual Newtonian gravitational forces, is the main problem of the classical Celestial Mechanics. Only the 2–body problem is completely solved, and for n > 2 there are only few partial results on the dynamics of the n–body problem. In R2 the equations of motion of the n–body problem are n X mj (xj − xi ) ẍi = 3 , for i = 1, . . . , n. rij j=1, j6=i where xi ∈ R are the position vectors of the bodies, rij = |xi −xj | are their mutual 2 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 denoted by the vector x = (x1 , . . . , xn ) ∈ R2n . The differential equations of motion are well–defined when rij 6= 0 for i 6= j, i.e. when there is no collisions between the masses. The dimension of the smallest affine subspace of R2 which contains all of the points xi is called the dimension δ(x) of the configuration x. Then the configura- tions with δ(x) = 1, 2 are called collinear and planar, respectively. 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 2010 Mathematics Subject Classification. Primary 70F07. Secondary 70F15. Key words and phrases. convex central configuration, four–body problem, Hjelmslev quadrilateral. 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 respectively. A configuration x is central when the acceleration vectors of the bodies satisfy Xn mj (xj − xi ) (1) 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 there is a rotation of SO(2) with respect to the center of mass and a homothecy which send one into the other. This relation is of equivalency, therefore in what follows when we talk on central configurations we are talking on classes of equivalency of central configura- tions. The investigation of the central configurations begun at the second part of the 18th century. Early results on the central configurations can be found in the books by Wintner [43] and Hagihara [21]. Nowadays, there is an extensive literature on this subject, see among others Abouy and Chenciner [3], Albouy and Kaloshin [6], Hampton and Moeckel [23], Llibre [27], Moeckel [31], Palmore [35], Saari [39], Schmidt [40], Xia [44], ... Central configurations are important for several reasons. First, 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. Second, in the total collisions (i.e. all the bodies collide) or in the total parabolic escape to infinity (i.e. all the bodies escape to infinity with zero radial velocity) the configuration of the n bodies is asymptotic to a central configuration, for more details see Dziobek [16] and [39]. Third, if we fix the first integrals of the total energy h and the angular momentum c of the n–body problem, then some of the bifurcation points (h, c) for the topology of the level sets with energy h and angular momentum c depend on with the central configurations, for more details see Meyer [33] and Smale [42]. For a fixed mass vector m = (m1 , . . . , mn ) and a fixed ordering of the bodies along the line, Moulton [34] in 1910 proved, up to translation and scaling, that there exists a unique collinear central configuration. This result for the three- body problem was already obtained by Euler [18] in 1767. Also for the three–body problem Lagrange [24] proved that for the unique planar central configuration x with δ(x) = 2 is the equilateral triangle, i.e. the three bodies are at the vertices of an equilateral triangle. Here we want to improve the knowledge of the planar central configurations of the four–body problem. First we provide a brief summary of the results on the central configurations of the four–body problem. A numerical study on the classes of central configurations for the four–body problem with arbitrary masses was done by Simó in [41]. In 2006, Hampton and Moeckel [23] provided a computer assisted proof showing that the four–body problem has finitely many classes of central configurations with any given positive masses. Later on, this result was proved analytically by Albouy and Kaloshin [6]. HJELMSLEV QUADRILATERAL CENTRAL CONFIGURATIONS 3 Llibre in [26] found all the planar central configurations of the four–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. A complete analytic proof of this result was given later on by Albouy in [1, 2], showing that there are exactly four equivalent classes of central of these central configurations. Pedersen [36], Barros and Leandro [10, 11], Gannaway [20] and Arenstorf [9] obtained numerically and analytically the classes of central configurations of the four–body problem when one of the four masses is sufficiently small. A central configuration of the four–body problem having an axis of symmetry passing through two non–adjacent bodies is called kite . The non–collinear classes of kite central configurations with three equal masses where classified by Bernat et al. in [12], see also Leandro [25]. A planar configuration of the four–body problem is convex if none of the bodies is located in the interior of the triangle formed by the others, otherwise it is concave. MacMillan and Bartky [30] shown that for any assigned order of arbitrary four masses there is a convex planar central configuration of the four–body problem where the masses at the vertices of the quadrilateral are located in the given order. Later on Xia [45] gave an easier proof of this result. In 2010, Piña and Lonngi [38] applied a new numerical algorithm to construct general four-body central configurations and described new properties of symmetric and non-symmetric central configurations. Long and Sun [29] proved that any convex central configuration 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 , has an axis of symmetry and the quadrilateral is a rhombus. Pérez–Chavela and Santoprete [37] extended this result to the case where two of the masses are equal and at most, only one of the remaining mass is larger than the equal masses. Furthermore, 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. Álvarez–Ramı́rez and Llibre [7] characterized the convex and concave central configurations with an axis of symmetry of the four–body problem when the masses satisfy that m1 = m2 6= m3 = m4 . On the other hand, Érdi and Czirják [17] 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 [30, 37]) conjectured: For the planar four–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 [30], they proved that there is a unique isosceles trapezoid central configuration of the four–body when two pairs of 4 M. ALVAREZ–RAMÍREZ AND J. LLIBRE Figure 1. Hjelmslev quadrilateral. equal masses are located at adjacent vertices. Later on Xie in [46] reproved this result. There is also the following subconjecture: For the planar four–body problem there is a unique convex central configuration having two pairs of equal masses located at the adjacent vertices of the configuration and it is an isosceles trapezoid. 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 classi- fied by Cors and Roberts [15], and also by Álvarez–Ramı́rez et al. [8]. A trapezoid is a convex quadrilateral with at least one pair of parallel sides. Corbera et al. [14] have classified all the trapezoids which are central configurations. A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices, see for more details about these kind of quadrilaterals the book [22], see Figure 1. Since a central configuration is invariant under a homothecy and a rotation around its center of mass, without loss of generality we can assume that (2) x1 = (0, 0), x2 = (1, 0), x3 = (a, b), x4 = (0, c), where a, b and c are non–negative. Let mk > 0 be the mass of the particle located at xk for k = 1, 2, 3, 4. The next lemma states that c can be expressed in terms of a and b parameters. It is proved in Section 3. Lemma 1. If the configuration of four masses is a Hjelmslev quadrilateral with vertices x1 , x2 , x3 and x4 , as is shown in Figure 2, then a2 − a + b2 (3) c= . b HJELMSLEV QUADRILATERAL CENTRAL CONFIGURATIONS 5 Figure 2. Coordinates of the four-body configuration forming a Hjelm- slev quadrilateral for a < 1 and a > 1, respectively We now give a result that allows to characterize all planar four–body central configurations where the four bodies are at the vertices of a Hjelmslev quadrilateral. Theorem 2. We take positive masses for the four–body problem and the unit of mass equal to m1 . Then the points (a, b) of the set of all Hjelmslev quadrilateral central configurations of the four–body problem described by (2) and (3) is formed by the open arc (   √ # ) 1 3 3 p Σ = (a, b) ∈ , × 2 , 1 : b = 1 − (a − 1) . 2 2 2 (a) When the coordinates (a, b) of the Hjelmslev √ quadrilateral central configu- rations tend to the boundary point (1/2, 3/2) of Σ the Hjelmslev quadri- lateral central configurations tend to the equilateral triangle central config- uration of√ the three-body problem formed by the masses m1 = 1, m2 = 4(19 + 21√ 3)/37 and m3 = 1, the position of the remainder four body tends to (0, 1/ 3) and its mass m4 → 0. (b) When the coordinates (a, b) of the Hjelmslev √ quadrilateral central configu- rations tend to the boundary point (3/2, 3/2) of Σ the Hjelmslev quadri- lateral central configurations tend to the equilateral triangle central config- uration of√the three-body problem formed by the masses m1 = 1, m3 = 4(19 + 21 3)/37 and m4 = 1, the position of the remainder four body is (0, 1) and its mass m2 → 0. (c) The coordinates (a, b) = (1, 1) ∈ Σ correspond to the Hjelmslev quadrilat- eral central configuration given by the square with four equal masses at its vertices. √ √ (d) Running the arc Σ starting at (1/2, 3/2) and ending√ at (3/2, 3/2), the mass m2 decreases monontically from 4(19 + 21 3)/37 to zero, the mass m3 remains constant equal√ to one, and the mass m4 increases monontically from zero to 4(19 + 21 3)/37. So every Hjelmslev quadrilateral central 6 M. ALVAREZ–RAMÍREZ AND J. LLIBRE configuration is realized by a unique set of four masses, being two of the masses equal (m1 = m3 = 1), and the other two masses are different except in the square that the four masses are equal. Theorem 2 is proved in Section 3. In Section 2, we recall the Dziobek’s equations for the central configurations of the four–body problem. 2. Dziobek’s equations Let x = (x1 , . . . , x4 ) ∈ (R2 )4 be the configuration vector and we associate to each one the 4 × 4 matrix   1 ··· 1 X =  x1 ··· x4  . 0 ··· 0 We define the 3×3 matrix Xk as the matrix obtained deleting from the matrix X its k–th column and its last row. Then let Dk = (−1)k+1 det(Xk ) be for k = 1, . . . , 4. The equations for the central configurations (1) of the four–body problem were written by Dziobek [16] (see also equations (8) and (16) of Moeckel [32], or [21]) as the following 12 equations with 12 unknowns 1 Di Dj 3 = c1 + c2 m m , rij (4) i j ti − tj = 0, for 1 ≤ i < j ≤ 4, with 4 X 2 ti = Dj rij . j=1, j6=i In equations (4) the 12 unknowns are the 6 mutual distances rij , the 4 variables Di , and the 2 constants ck . The first six Dziobek’s equations (4) are −3  m1 m2 r12 − c1 = c2 D1 D2 , −3  m1 m3 r13 − c1 = c2 D1 D3 , −3  m2 m3 r23 − c1 = c2 D2 D3 , (5) −3  m1 m4 r14 − c1 = c2 D1 D4 , −3  m2 m4 r24 − c1 = c2 D2 D4 , −3  m3 m4 r34 − c1 = c2 D3 D4 . Multiplying the above equations by row in order that the product of the right– hand side be simply c22 D1 D2 D3 D4 , and since the masses are positive we get the so called Dziobeck relation −3 −3 −3 −3 −3 −3 (6) (r12 − c1 )(r34 − c1 ) = (r13 − c1 )(r24 − c1 ) = (r14 − c1 )(r23 − c1 ), HJELMSLEV QUADRILATERAL CENTRAL CONFIGURATIONS 7 which holds for every planar central configuration of the 4–body problem. Solving with respect to c1 any two of these equations we have (7) r−3 r−3 − r13 −3 −3 r24 −3 −3 r13 −3 −3 r24 − r14 r23 −3 −3 r14 −3 −3 r23 − r12 r34 c1 = −312 34 −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 s1 = r12 + r34 , p1 = r12 r34 , −3 −3 −3 −3 s2 = r13 + r24 , p2 = r13 r24 , −3 −3 −3 −3 s3 = r14 + r23 , p3 = r14 r23 , then equation (7) can be written as p1 − p2 p2 − p3 p3 − p1 (8) c1 = = = , s1 − s2 s2 − s3 s3 − s1 which means that (s1 , p1 ), (s2 , p2 ), (s3 , p3 ) viewed as points in (R+ )2 , must lie on the same line with slope c1 . This in turn, is equivalent to 1 1 1 s1 s2 s3 = 0, p1 p2 p3 which allows to write Dziobeck relation (6) as the equation 3 3 3 3 3 3 3 3 3 3 3 3 (9) D = (r13 − r12 )(r23 − r34 )(r24 − r14 ) − (r12 − r14 )(r24 − r34 )(r13 − r23 ) = 0, that we call the Dziobeck equation D = 0. Of course, the equation D = 0 is satisfied for every planar central configuration of the four–body problem. 3. Proof of Theorem 2 Firstly we prove the Lemma 1. Proof. Since the configuration is a Hjelmslev quadrilateral, using Figure 2 we have that c2 + 1 = b2 + (a − 1)2 + a2 + (c − b)2 . Then a straightforward computation shows that 2b2 + 2a2 − 2a − 2bc = 0. So, we have a2 − a + b2 c= . b  8 M. ALVAREZ–RAMÍREZ AND J. LLIBRE From (2) and (3) we have √ r12 = 1, r13 = a2 + b2 , (a − 1)a + b2 p r14 = , r23 = (a − 1)2 + b2 , b s 2 (a − 1)a + b2 ap (10) r24 = 2 + 1, r34 = (a − 1)2 + b2 , b b a3 − 2a2 + ab2 + a −a3 + a2 − ab2 D1 = , D2 = , b b a2 − a + b2 D3 = , D4 = −b. b By substituting these expression into last six Dziobek’s equations (5), it is easy to see that are identically zero for a Hjelmslev quadrilateral configuration. The following three equations −3 −3 −3 m2 (r23 − c1 ) D2 m3 (r23 − c1 ) D3 m4 (r24 − c1 ) D4 (11) −3 = , −3 = , −3 = , m1 (r13 − c1 ) D1 m1 (r12 − c1 ) D1 m1 (r12 − c1 ) D1 are obtained dividing the third equation of (5) by the second one, dividing the third equation of (5) by the first one, and dividing the fifth equation of (5)by the first one, respectively. Taking the value of the mass m1 as the unit of mass and using (7), we obtain that the three equations of (11) become (12) 3 3 3 3 3 3 3 3 3 3 3 3 D2 r23 r24 (r13 − r14 ) D3 r23 r34 (r12 − r14 ) D4 r24 r34 (r12 − r13 ) m2 = 3 3 3 3 , m3 = 3 3 3 3 , m4 = 3 3 3 3 . D1 r13 r14 (r23 − r24 ) D1 r12 r14 (r23 − r34 ) D1 r12 r13 (r24 − r34 ) Substituting these values of m2 , m3 and m4 into the first six Dziobek’s equations (4), and taking only the numerators of these six equations because the denominators do not vanish, we obtain   3  e1 = D2 c2 D12 r12 3 3 3 r13 r14 r243 − r233 − r233 3 r24 c1 r12 3 − 1 r13 3 − r14 , 2 3 3 3 3 3  3 3 3  3 3  e2 = D3 c2 D1 r12 r13 r14 r34 − r23 − r23 r34 c1 r13 − 1 r12 − r14 , 3 3 3 3  3 3  3 3  e3 = D2 D3 r23 r24 r34 c1 r23 − 1 r12 − r14  r14 − r13 3 3 6 −c2 D12 r12 3 r13 r14 r23 − r24 3 3 r23 3 − r34 ,   3  (13) 2 3 3 3 3 3 3 3 e4 = D4 c2 D1 r12 r13 r14 r34 − r24 − r24 r34 c1 r14 3 − 1 r12 3 − r13 , 3 3 3 3  3 3  3 3  e5 = D2 D4 −r23 r24 r34 c1 r24 −1 r12 − r 13 r13 − r14 3 6 3 −c2 D12 r12 3 r13 r14 r23 − r24 3 3 r24 3 − r34 ,  3  e6 = D3 D4 c2 D12 r12 6 3 3 3 r13 r14  r23 3 − r34  r34 − r24 3  3 3 3 3 3 3 3 3 −r23 r24 r34 c1 r34 − 1 r12 − r13 r12 − r14 . Now, we eliminate the factors D` from the equations (13), and they are non–zero. After we solve the first two equations with respect c1 and c2 , and we substituted c1 and c2 in the last four equations, and then the system (13) is reduced to four HJELMSLEV QUADRILATERAL CENTRAL CONFIGURATIONS 9 equations, one of which is identically zero, (14) D 6 3 3 2 2  2 2  e3 = r23 r24 r34 (r14 − r12 ) r12 + r12 r14 + r14 (r14 − r13 ) r13 + r13 r14 + r14 , d e4 = 0, D 3 6 3 2 2  2 2  e5 = r23 r24 r34 (r13 − r12 ) r12 + r12 r13 + r13 (r13 − r14 ) r13 + r13 r14 + r14 , d D 3 3 6 2 2  2 2  e6 = r23 r24 r34 (r13 − r12 ) r12 + r12 r13 + r13 (r12 − r14 ) r12 + r12 r14 + r14 , d where D = 0 is the Dziobek equation (9), and 3 3 3 3 3  3 3 3 3  3 3 3 3 3  d = r12 r13 r23 r24 − r34 + r14 r24 r34 − r23 + r13 r14 r34 r23 − r24 is the denominator which comes from the denominator of c1 and c2 . In summary, the Hjelmslev quadrilateral central configurations must satisfy the three equations (e3 , e5 , e6 ) = (0, 0, 0) of (14). Substituting (10) into e3 = 0, e5 = 0 and e6 = 0, we get that these three equations are satisfied if and only if the next three equations have solutions    (a − 1)a + b2 (a − 1)a + b2 √ 2 E3 = D 1 − − a + b2 = 0, b b   √  (a − 1)a + b2 √ (15) E5 = D 1 − a2 + b2 − a2 + b2 = 0, b   (a − 1)a + b 2 √  E6 = D 1 − 1 − a2 + b2 = 0, b respectively, where again D = 0 is the Dziobek equation (9). Hence the unique solutions (a, b) satisfying system (15) are the ones satisfying D = 0. To compute the values of a and b for D = 0, we used Mathematica to obtain a graphical solution as shown in Figure 3. Then, the set of solutions of system (15) is formed by two open upper half–cercles given by  s    2  1 1 p Ω = (a, b) : either b = − a− , or b = 1 − (a − 1)2 .  4 2  It only remains to find values for (a, b) ∈ Ω for which the bodies in the Hjelmslev quadrilateral central configurations have positive masses. Next, by assuming that (a, b) ∈ Ω and taking into account (12), from Fig. 3pwe have numerical evidence that m2 , m3 and m4 are positive only for b = 1 − (a − 1)2 with a √∈ (1/2, 3/2). Furthermore, the masses are positive for (a, b) ∈ (1, 2/, 3/2) × ( 23 , 1] and we have the following conclusion: (i) m4 = −1 on the arc of Ω contained into the circle of radius 1/2; (ii) m3 = 1 on the arc of Ω contained into the circle of radius 1; (iii) the mass m2 only is positive on the subarc of Ω contained into the circle of radius 1 when a ∈ (1, 3/2), in this subarc the mass m2 decreases from 10 M. ALVAREZ–RAMÍREZ AND J. 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A, Math. 142 (2012), 665–672. 1 Departamento de Matemáticas, UAM–Iztapalapa, San Rafael Atlixco 186, Col. Vi- centina, 09340 Iztapalapa, México, City, México. E-mail address: mar@xanum.uam.mx 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat