ENGINEERING GROUP ISSN: 2641-3086 DOI: https://dx.doi.org/10.17352/tcsit Received: 05 December, 2020 Research Article Accepted: 29 December, 2020 Published: 04 January, 2021 Central configurations of the *Corresponding author: Jaume Llibre, Department of Mathematics, Autonomous University of Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain, circular restricted 4-body E-mail: Keywords: Central configuration; Circular restricted problem with three equal 4-body problem https://www.peertechz.com primaries in the collinear central configuration of the-3 body problem Jaume Llibre* Department of Mathematics, Autonomous University of Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Abstract In this paper we classify the central configurations of the circular restricted 4-body problem with three primaries with equal masses at the collinear configuration of the 3-body problem and an infinitisimal mass. Introduction and results For the Newtonian n-body problem the simplest possible motions are such that the configuration formed by the n-bodies The well-known Newtonian n-body problem concerns is constant up to rotations and scaling, such motions are called with the motion of n mass points with positive mass mi the homographic solutions of the n-body problem, and are the moving under their mutual attraction in Rd in accordance with unique known explicit solutions of the n-body problem when Newton’s law of gravitation. n>2. Only some special configurations of particles are allowed in the homographic solutions of the n-body problem, called by The equations of the motion of the n-body problem are Wintner [1] central configurations. Also, central configurations n m j (ri  rj ) are of utmost importance when studying bifurcations of the ri =    j =1, j  i rij3 , 1  i  n, hypersurfaces of constant energy and angular momentum, for more details see Meyer [2] and Smale [3]. These last years some central configurations have been used for different missions of where we have taken the unit of time in such a way that the the spacecrafts in the solar system, see for instance [4,5]. Newtonian gravitational constant be one, and ri€Rd (i=1…,n) More precisely, let denotes the position vector of the i-body, rij =| ri  rj | is the m1r1    mn rn Euclidean distance between the i-body and the j-body. M = m1    mn , c = , M The solutions of the 2-body problem (also called the Kepler problem) has been completely solved, but the solutions for the be the total mass and the center of masses of the n bodies, n-body for n>2, is still an open problem. respectively. 001 Citation: Llibre J (2021) Central configurations of the circular restricted 4-body problem with three equal primaries in the collinear central configuration of the 3-body problem. Trends Comput Sci Inf Technol 6(1): 001-006. DOI: https://dx.doi.org/10.17352/tcsit.000031 https://www.peertechz.com/journals/trends-in-computer-science-and-information-technology A configuration r=(r1,…,rn) is called a central configuration if r1 = ( x1 , y1 ) = (1, 0), r2 = ( x2 , y2 ) = (0, 0), r3 = ( x3 , y3 ) = (1, 0). the acceleration vectors of the n bodies are proportional to their (2) positions with respect to the center of masses with the same constant  of proportionality, i.e. We denote the position of the infinitesimal mass m4 = 0 by n m j (ri  rj ) r4 = (x4,y4) = (x,y). Then our main result is the following one.  j =1, j  i rij3 =  (rj  c), 1  j  n, Theorem 1 The circular restricted 4-body problem with three (1) primaries of equal masses m1= m2= m3= 1 with position vectors given where λ is the constant of proportionality. in (2), and one infinitessimal mass m4 = 0 with position vector r4 =(x4,y4) =(x,y) have the following six central configurations with r4 Equations (1) are strongly nonlinear and to find the explicit = Pj for j=1 being: central configurations (r1,…,rn) in function of the masses m1,…. mn when n>3 is an unsolved problem. (i) p1 = ( x, y ) = (0,1.1394282249562009..) , where the value of the coordinate y is a root of the polynomial There is an extensive literature on the study of central configurations, see for instance Euler [6], Lagrange [7], 16  48 y 2  40 y 3  48 y 4  120 y 5  23 y 6  120 y 7  75 y 8  40 y 9  75 y10  25 y12 ; Hagihara [8], Llibre [9,10], Meyer [2], Moeckel [11], Moulton [11], Saari [12], Smale [3], ..., and the papers quoted in these references. (ii) p2 = ( x, y ) = (0, 1.1394282249562009..) ; In this paper we are interested in the planar central (iii) p3 = ( x, y ) = (1.7576799791694022.., 0) , where the configurations of a circular restricted 4-body problem. Of value of the coordinate y is a root of the polynomial course, for the central configurations of the 4-body problem ; there are many partial results, see for instance the papers [13- 4  5 x 3  12 x 4  10 x 5  5 x 7 66]. (iV) p4 = ( x, y ) = (0.49466649101736443.., 0) , where We note that the set of central configurations is invariant the value of the coordinate y is a root of the polynomial under translations, rotations, and homothecies with respect 7; their center of mass. It is said that two central configurations 4  8 x  21x  4 x  10 x  5 x 2 3 4 5 are equivalent if after having the same center of mass (doing a translation if necessary) we can pass from one to the other (v) p5 = ( x, y ) = (0.49466649101736443.., 0) ; through a rotation around its common center of mass and a (vi) p6 = ( x, y ) = (1.7576799791694022.., 0) . homothecy. This defines a relation of equivalence in the set of central configurations. From now on when we talk about Figure 1. a central configuration, we are talking on a class of central configurations under this relation of equivalence. The proof of Theorem 1 in given in the next section. The objective of the present article is to study the central configurations of the circular restricted 4-body problem with three equal primaries in the collinear central configuration of the 3-body problem. We recall that for the 3-body problem when the three masses are equal there is a unique collinear central configuration, where the mass in the middle equidistant from the other two, of course the equal masses can be permuted in the positions of this configuration. As in any circular restricted problem the objective is to describe the motion of the infinitesimal mass with respect to the primaries. Usually this problem is studied in a rotating system of coordinates where the positions of the primaries remain fixed, see for more details on the restricted problems the book of Szebehely [67]. More precisely, taking the unit of mass equal to the masses of the three primaries and since a central configuration is invarinat under rotations and homothecies through its center Figure 1: The six central configurations of the circular restricted 4-body probl of mass without loss of generality we can assume that the qem with three equal primaries in the collinear central configuration of the 3-body position vector rj of the three primaries with masses m1= m2= problem. The three primaries are indicated with the big circles, and the position of m3= 1 are the infinitesimal mass in the corresponding six central configurations is indicated with a small circle. 002 Citation: Llibre J (2021) Central configurations of the circular restricted 4-body problem with three equal primaries in the collinear central configuration of the 3-body problem. Trends Comput Sci Inf Technol 6(1): 001-006. DOI: https://dx.doi.org/10.17352/tcsit.000031 https://www.peertechz.com/journals/trends-in-computer-science-and-information-technology Proof of theorem 1 First we look for the solutions (0,y) with y>0, then system e4 (x,y) =0 and e8 (x,y) =0 reduce to From (1) we obtain the following eight equations for the central configurations of the 4-body problem in the plane 5y 2y 1   2 = 0, 4 m j ( xi  x j ) 4  y 2  1 3/2 y ej =  j =1, j  i rij3 =  ( x j  c1 ), 1  j  4, (4) 4 m j ( yi  y j ) or equivalently to e j 5 =  j =1, j  i rij3 =  ( y j  c2 ), 1  j  4, 8 y 3 = (1  y 2 )3/2 (4  5 y 3 ). (3) Where c= (c1, c2). Substituting in (3) the expressions (2), Squaring the both sides of the this equation we get the m1= m2= m3= 1, m4 = 0 and r4 =(x4,y4) =(x,y), corresponding to equation our circular restricted 4-body problem these eight equations 16  48 y 2  40 y 3  48 y 4  120 y 5  23 y 6  120 y 7  75 y 8  40 y 9  75 y10  25 y12 = 0. reduce to This polynomial equation has only two real roots 5 e1 = e3 =   = 0, 4 0.7625005146027564.. and 1.1394282249562009.., e2 = e5 = e6 = e7 = 0, x 1 x 1 x but only the second root satisfies equation (4). This e4 =  x    = 0,    2 3/ 2   2 3/ 2  y2  3/ 2 x 2  y ( x  1) 2  y ( x  1) 2 provides the solution P1 of Theorem 1, and consequently also the solution P2.   1 1 1 e8 = y       = 0.     2 3/ 2   2 3/ 2  y2   3/ 2  x 2  y ( x  1) 2  y ( x  1) 2  Therefore  = 5 / 4 , and the position vector of r4 =(x4,y4) in order to have a central configuration of the circular restricted 4-body problem must be a real solution of the system 5 x 1 x 1 x e4 = x   = 0,    2 3/ 2   2 3/ 2  y2  3/ 2 4 x 2  y ( x  1) 2  y ( x  1) 2   5 1 1 1 e8 = y      = 0.  4  x 2  y 2 3/ 2  ( x  1) 2  y 2 3/ 2  ( x  1) 2  y 2 3/ 2  (a) The curve e4 ( x, y ) = 0 (b) The curve e8 ( x, y ) = 0   Figure 2: The curves e4(x,y) = 0 and e8(x,y) = 0 in the rectangle (x,y)  [−2.2,2.2] × [−2.2,2.2]. In Figure 2 we have shwon the curves e4 (x,y) =0 and e8 (x,y) =0, and in Figure 3 the intersection of these two curves. We see that these two curves intersect in six points inside the rectangle R = {( x, y )   2 : 2.2  x  2.2, 2.2  y  2.2} . Computing the coordenates of these six points numerically using the Newton method (see for instance [68]), we get the six points Pj which appear in the statement of Theorem 1. Of course we have omitted the three points where are located the three primaries in the intersections of the two curves e4 (x,y) =0 and e8 (x,y) =0, because there really these two curves are not defined. Now we shall prove that these six points obtained numerically really are solutions of the system e4 (x,y) =0 and e8 (x,y) =0. We note that equations e4 (x,y) =0 and e8 (x,y) =0 are invariant if we change x by -x, and y by -y, so if (x,y) is a solution of the system e4 (x,y) =0 and e8 (x,y) =0, also (-x,y), (x, -y) and (-x,-y) are solutions. So in order to prove Theorem 1 we only need to study the solutions of system e4 (x,y) =0 and e8 (x,y) =0 satisfying x  0 and y  0 . Moreover, from Figure 3 Figure 3: In this picture we can see the six intersection points between the two we see that all the solutions are of the form (x,0) or (0,y), and curves e4(x,y) = 0 and e8(x,y) = 0 different from the positions of the primaries, which since in the origin (0,0) there is one primary, we must look provide the six central configurations of the circular restricted 4-body problem with three primaries of equal masses at the collinear configuration of the 3body problem only for the solutions (x,0) or (0,y) with x > 0 and y> 0. and an infinitisimal mass. 003 Citation: Llibre J (2021) Central configurations of the circular restricted 4-body problem with three equal primaries in the collinear central configuration of the 3-body problem. Trends Comput Sci Inf Technol 6(1): 001-006. DOI: https://dx.doi.org/10.17352/tcsit.000031 https://www.peertechz.com/journals/trends-in-computer-science-and-information-technology Now we look for the solutions (x,0) with (x>0) of the system 4. Gómez G (2001) Dynamics and Mission Design Near Libration Points. Vol. I Fundamentals: The case of collinear libration points, World Scientific e4 (x,y) =0 and e8 (x,y) =0. For these solutions the system reduce Monograph Series in Mathematics 2. Link: https://bit.ly/387zjUc to 5x 1 1 x 1 x 5. Gómez G (2001) Dynamics and Mission Design Near Libration Points.  2 = , Vol. I Fundamentals: The case of collinear libration points, World Scientific 4 x |1  x |3/2 |1  x |3/2 (5) Monograph Series in Mathematics 2. Link: https://bit.ly/387zjUc 6. Euler L (1767) De moto rectilineo trium corporum se mutuo attahentium. Novi squaring the both sides of the previous equality we obtain Comm Acad Sci Imp Petrop 11: 144-151. Link: https://bit.ly/38S8Jhg 7. 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This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 006 Citation: Llibre J (2021) Central configurations of the circular restricted 4-body problem with three equal primaries in the collinear central configuration of the 3-body problem. Trends Comput Sci Inf Technol 6(1): 001-006. DOI: https://dx.doi.org/10.17352/tcsit.000031