ISSN: 2641-3086
##### Trends in Computer Science and Information Technology
Research Article       Open Access      Peer-Reviewed

# Central configurations of the circular restricted 4-body problem with three equal 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
*Corresponding author: Jaume Llibre, Department of Mathematics, Autonomous University of Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain, E-mail: jllibre@mat.uab.cat
Received: 05 December, 2020 | Accepted: 29 December, 2020 | Published: 04 January, 2021
Keywords: Central configuration; Circular restricted 4-body problem

Cite this as

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: 10.17352/tcsit.000031

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

The well-known Newtonian n-body problem concerns with the motion of n mass points with positive mass mi moving under their mutual attraction in Rd in accordance with Newton’s law of gravitation.

The equations of the motion of the n-body problem are

${\stackrel{¨}{r}}_{i}=-\sum _{j=1,j\ne i}^{n}\frac{{m}_{j}\left({r}_{i}-{r}_{j}\right)}{{r}_{ij}^{3}},\text{ }1\le i\le n,$

where we have taken the unit of time in such a way that the Newtonian gravitational constant be one, and ri€Rd (i=1…,n) denotes the position vector of the i-body, ${r}_{ij}=|{r}_{i}-{r}_{j}|$ is the Euclidean distance between the i-body and the j-body.

The solutions of the 2-body problem (also called the Kepler problem) has been completely solved, but the solutions for the n-body for n>2, is still an open problem.

For the Newtonian n-body problem the simplest possible motions are such that the configuration formed by the n-bodies is constant up to rotations and scaling, such motions are called the homographic solutions of the n-body problem, and are the unique known explicit solutions of the n-body problem when n>2. Only some special configurations of particles are allowed in the homographic solutions of the n-body problem, called by Wintner  central configurations. Also, central configurations are of utmost importance when studying bifurcations of the hypersurfaces of constant energy and angular momentum, for more details see Meyer  and Smale . These last years some central configurations have been used for different missions of the spacecrafts in the solar system, see for instance [4,5].

More precisely, let

$M={m}_{1}+\cdots +{m}_{n},\text{ }c=\frac{{m}_{1}{r}_{1}+\cdots +{m}_{n}{r}_{n}}{M},$

be the total mass and the center of masses of the n bodies, respectively.

A configuration r=(r1,…,rn) is called a central configuration if the acceleration vectors of the n bodies are proportional to their positions with respect to the center of masses with the same constant λ of proportionality, i.e.

where λ is the constant of proportionality.

Equations (1) are strongly nonlinear and to find the explicit central configurations (r1,…,rn) in function of the masses m1,….mn when n>3 is an unsolved problem.

There is an extensive literature on the study of central configurations, see for instance Euler , Lagrange , Hagihara , Llibre [9,10], Meyer , Moeckel , Moulton , Saari , Smale , ..., and the papers quoted in these references.

In this paper we are interested in the planar central configurations of a circular restricted 4-body problem. Of course, for the central configurations of the 4-body problem there are many partial results, see for instance the papers [13-66].

We note that the set of central configurations is invariant under translations, rotations, and homothecies with respect their center of mass. It is said that two central configurations are equivalent if after having the same center of mass (doing a translation if necessary) we can pass from one to the other through a rotation around its common center of mass and a homothecy. This defines a relation of equivalence in the set of central configurations. From now on when we talk about a central configuration, we are talking on a class of central configurations under this relation of equivalence.

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 .

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 of mass without loss of generality we can assume that the position vector rj of the three primaries with masses m1= m2= m3= 1 are

${r}_{1}=\left({x}_{1},{y}_{1}\right)=\left(-1,0\right),\text{\hspace{0.17em}}{r}_{2}=\left({x}_{2},{y}_{2}\right)=\left(0,0\right),\text{\hspace{0.17em}}{r}_{3}=\left({x}_{3},{y}_{3}\right)=\left(1,0\right).$

We denote the position of the infinitesimal mass m4 = 0 by r4 = (x4,y4) = (x,y). Then our main result is the following one.

Theorem 1 The circular restricted 4-body problem with three primaries of equal masses m1= m2= m3= 1 with position vectors given in (2), and one infinitessimal mass m4 = 0 with position vector r4 =(x4,y4) =(x,y) have the following six central configurations with r4 = Pj for j=1 being:

(i) ${p}_{1}=\left(x,y\right)=\left(0,1.1394282249562009..\right)$ , where the value of the coordinate is a root of the polynomial $-16-48{y}^{2}+40{y}^{3}-48{y}^{4}+120{y}^{5}+23{y}^{6}+120{y}^{7}-75{y}^{8}+40{y}^{9}-75{y}^{10}-25{y}^{12}$ ;

(ii) ${p}_{2}=\left(x,y\right)=\left(0,-1.1394282249562009..\right)$ ;

(iii) ${p}_{3}=\left(x,y\right)=\left(1.7576799791694022..,0\right)$ , where the value of the coordinate is a root of the polynomial $-4+5{x}^{3}-12{x}^{4}-10{x}^{5}+5{x}^{7}$ ;

(iV) ${p}_{4}=\left(x,y\right)=\left(0.49466649101736443..,0\right)$ , where the value of the coordinate is a root of the polynomial $-4+8{x}^{2}+21{x}^{3}-4{x}^{4}-10{x}^{5}+5{x}^{7}$ ;

(v) ${p}_{5}=\left(x,y\right)=\left(-0.49466649101736443..,0\right)$ ;

(vi) ${p}_{6}=\left(x,y\right)=\left(-1.7576799791694022..,0\right)$ .

Figure 1.

The proof of Theorem 1 in given in the next section.

##### Proof of theorem 1

From (1) we obtain the following eight equations for the central configurations of the 4-body problem in the plane $\begin{array}{rr}\hfill {e}_{j}=& \hfill \sum _{j=1,j\ne i}^{4}\frac{{m}_{j}\left({x}_{i}-{x}_{j}\right)}{{r}_{ij}^{3}}=\lambda \left({x}_{j}-{c}_{1}\right),\text{ }1\le j\le 4,0.2cm\\ \hfill {e}_{j+5}=& \hfill \sum _{j=1,j\ne i}^{4}\frac{{m}_{j}\left({y}_{i}-{y}_{j}\right)}{{r}_{ij}^{3}}=\lambda \left({y}_{j}-{c}_{2}\right),\text{ }1\le j\le 4,\end{array}$ (3)

Where c= (c1, c2). Substituting in (3) the expressions (2), m1= m2= m3= 1, m4 = 0 and r4 =(x4,y4) =(x,y), corresponding to our circular restricted 4-body problem these eight equations reduce to

$\begin{array}{l}{e}_{1}=-{e}_{3}=\frac{5}{4}+\lambda =0,0.2cm\hfill \\ {e}_{2}={e}_{5}={e}_{6}={e}_{7}=0,0.2cm\hfill \\ {e}_{4}=-\lambda x-\frac{x}{{\left({x}^{2}+{y}^{2}\right)}^{3/2}}-\frac{1+x}{{\left({\left(x+1\right)}^{2}+{y}^{2}\right)}^{3/2}}+\frac{1-x}{{\left({\left(x-1\right)}^{2}+{y}^{2}\right)}^{3/2}}=0,0.2cm\hfill \\ {e}_{8}=y\left(-\lambda -\frac{1}{{\left({x}^{2}+{y}^{2}\right)}^{3/2}}-\frac{1}{{\left({\left(x-1\right)}^{2}+{y}^{2}\right)}^{3/2}}-\frac{1}{{\left({\left(x+1\right)}^{2}+{y}^{2}\right)}^{3/2}}\right)=0.\hfill \end{array}$

Therefore $\lambda =-5/4$ , 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

$\begin{array}{l}{e}_{4}=\frac{5}{4}x-\frac{x}{{\left({x}^{2}+{y}^{2}\right)}^{3/2}}-\frac{1+x}{{\left({\left(x+1\right)}^{2}+{y}^{2}\right)}^{3/2}}+\frac{1-x}{{\left({\left(x-1\right)}^{2}+{y}^{2}\right)}^{3/2}}=0,0.2cm\hfill \\ {e}_{8}=y\left(\frac{5}{4}-\frac{1}{{\left({x}^{2}+{y}^{2}\right)}^{3/2}}-\frac{1}{{\left({\left(x-1\right)}^{2}+{y}^{2}\right)}^{3/2}}-\frac{1}{{\left({\left(x+1\right)}^{2}+{y}^{2}\right)}^{3/2}}\right)=0.\hfill \end{array}$

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 $R=\left\{\left(x,y\right)\in {ℝ}^{2}:-2.2\le x\le 2.2,-2.2\le y\le 2.2\right\}$ . Computing the coordenates of these six points numerically using the Newton method (see for instance ), 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 $x\ge 0$ and $y\ge 0$ . Moreover, from Figure 3 we see that all the solutions are of the form (x,0) or (0,y), and since in the origin (0,0) there is one primary, we must look only for the solutions (x,0) or (0,y) with x > 0 and y> 0.

First we look for the solutions (0,y) with y>0, then system e4 (x,y) =0 and e8 (x,y) =0 reduce to

$\frac{5y}{4}-\frac{2y}{{\left({y}^{2}+1\right)}^{3/2}}-\frac{1}{{y}^{2}}=0,$

or equivalently to

$8{y}^{3}=\left(1+{y}^{2}{\right)}^{3/2}\left(-4+5{y}^{3}\right).$

Squaring the both sides of the this equation we get the equation

$-16-48{y}^{2}+40{y}^{3}-48{y}^{4}+120{y}^{5}+23{y}^{6}+120{y}^{7}-75{y}^{8}+40{y}^{9}-75{y}^{10}-25{y}^{12}=0.$

This polynomial equation has only two real roots

$0.7625005146027564..\text{ }\text{ }and\text{ }\text{ }1.1394282249562009..,$

but only the second root satisfies equation (4). This provides the solution P1 of Theorem 1, and consequently also the solution P2.

Now we look for the solutions (x,0) with (x>0) of the system e4 (x,y) =0 and e8 (x,y) =0. For these solutions the system reduce to

$\frac{5x}{4}-\frac{1}{{x}^{2}}+\frac{1-x}{|1-x{|}^{3/2}}=\frac{1+x}{|1+x{|}^{3/2}},$

squaring the both sides of the previous equality we obtain

$\frac{1}{{x}^{4}}-\frac{5}{2x}+\frac{25{x}^{2}}{16}+\frac{1}{{\left(x-1\right)}^{4}}-\frac{1}{{\left(x+1\right)}^{4}}-\frac{\left(x-1\right)\left(5{x}^{3}-4\right)}{2{x}^{2}|x-{1|}^{3}}=0.$

Writting this equation with a common denominator, which only vanishes at the positions of the primeries, its numerator equal zero can be written as

$\begin{array}{l}8\left(x-{1\right)}^{5}{x}^{2}{\left(x+1\right)}^{4}\left(5{x}^{3}-4\right)=|x-{1|}^{3}\left(16-64{x}^{2}-40{x}^{3}+96{x}^{4}+288{x}^{5}\hfill \\ -39{x}^{6}-112{x}^{7}-84{x}^{8}+160{x}^{9}+150{x}^{10}-40{x}^{11}-100{x}^{12}+25{x}^{14}\right).\hfill \end{array}$

Squaring again the both sides of the this equality we get

$\begin{array}{l}{\left(x-1\right)}^{6}\left(-4+5{x}^{3}-12{x}^{4}-10{x}^{5}+5{x}^{7}\right)\left(-4+8{x}^{2}-11{x}^{3}-4{x}^{4}-10{x}^{5}+5{x}^{7}\right)\hfill \\ \left(-4+8{x}^{2}+21{x}^{3}-4{x}^{4}-10{x}^{5}+5{x}^{7}\right)\left(-4+16{x}^{2}+5{x}^{3}+4{x}^{4}-10{x}^{5}+5{x}^{7}\right)=0.\hfill \end{array}$

The real zero x = 1 is not good because it correspond to the position of a primary. The unique real root of the polynomial -4+5x3-12x4-10x5+5x7 is 1.7576799791694022.. which also is a zero of equation (5), and consequently provides the central configuration P3, and by the symmetries of the equations of the central configurations also provides the central configuration P6.

The unique real root of the polynomial -4+8x2+21x3-4x4-10x5+5x7 is 0.4946664910171345… which also is a zero of equation (5), and consequently provides the central configuration P4, and due to the symmetries of the equations of the central configurations also provides the central configuration P5.

The real roots of the polynomials -4+8x2-11x3-4x4-10x5+5x7 and 4+16x2+5x3+4x4-10x5+5x7 are not zeros of the equation (5). This completes the proof of Theorem 1.

The first author is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants PID2019-104658GB-I00 (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911.

1. Wintner A (1941) The Analytical Foundations of Celestial Mechanics, Princeton University Press. Link: https://bit.ly/3n7ci8v
2. Meyer KR (1987) Bifurcation of a central configuration. Celestial Mech  40: 273-282. Link: https://bit.ly/3o7RcIl
3. Smale S (1970) Topology and mechanics II: The planar n–body problem. Inventiones math  11: 45-64. Link: https://bit.ly/3rNgltW
4. Gómez G (2001) Dynamics and Mission Design Near Libration Points. Vol. I Fundamentals: The case of collinear libration points, World Scientific Monograph Series in Mathematics 2. Link: https://bit.ly/387zjUc
5. Gómez G (2001) Dynamics and Mission Design Near Libration Points. Vol. I Fundamentals: The case of collinear libration points, World Scientific Monograph Series in Mathematics 2. Link: https://bit.ly/387zjUc
6. Euler L (1767) De moto rectilineo trium corporum se mutuo attahentium. Novi Comm Acad Sci Imp Petrop 11: 144-151. Link: https://bit.ly/38S8Jhg
7. Lagrange JL (1873) Essai surle probleme des trois corps, recueil des pieces qui ont remporte le prix de l’Academie royale des Sciences de Paris,tome IX, 1772, reprinted in Ouvres 6: 229–324.
8. Hagihara Y (1970) Celestial Mechanics. 1. MIT Press, Massachusetts.
9. Llibre J (1991) On the number of central configurations in the n-body problem. Celestial Mech Dynam Astronom  50: 89-96. Link:
10. Llibre J (2017) On the central configurations of the n-body problem. Appl Math Nonlinear Sci  2: 509-518. Link: https://bit.ly/2JHHBZS
11. Moeckel R (1990) On central configurations. Math  Zeitschrift  205: 499-517. Link: https://bit.ly/3hM34xr
12. Saari DG  (1980) On the role and properties of central configurations. Celestial Mech 21: 9-20. Link: https://bit.ly/3862ZRV
13. Albouy A (1995) Symetrie des configurations centrales de quatre corps. CR Acad Sci Paris 320: 217-220. Link: https://www.semanticscholar.org/paper/Sym%C3%A9trie-des-configurations-centrales-de-quatre-Albouy/4408d2a8d129b375cc5eae706f6ddc5583a238e1
14. Albouy A (1995) The symmetric central configurations of four equal masses, Hamiltonian dynamics and celestial mechanics (Seattle, WA, 1995). 131–135, Contemp. Math.  198, Amer. Math. Soc., Providence, RI, 1996. Link:
15. Albouy A, Fu Y, Sun S (2008) Symmetry of planar four body convex central configurations. Proc R Soc Lond Ser A Math Phys Eng Sci 464: 1355-1365. Link: https://hal.archives-ouvertes.fr/hal-00153212
16. Albouy A, Kaloshin V (2012) Finiteness of central configurations of five bodies in the plane. Ann Math 176: 535-588. Link: https://annals.math.princeton.edu/2012/176-1/p10
17. lvarez-RamÃ­rez M, Corbera M, Delgado J, Llibre J (2004) The number of planar central configurations for the 4-body problem is finite when 3 mass positions are fixed. Proc Amer Math Soc  133: 529-536. Link: https://bit.ly/3pLCMOx
18. lvarez-RamÃ­rez M, Delgado J (2003) Central configurations of the symmetric restricted 4-body problem. Celestial Mech Dynam Astronom  87: 371-381. Link: https://bit.ly/3b0YhGM
19. lvarez-RamÃ­rez M, Llibre J (2013) The symmetric central configurations of the 4-body problem with masses. Appl Math Comp  219: 5996-6001. Link: https://bit.ly/3pBge2E
20. lvarez-RamÃ­rez M, Llibre J (2018) Hjelmslev quadrilateral central configurations. Physics Letters A  383: 103-109. Link: https://bit.ly/3pA85eW
21. lvarez-RamÃ­rez M, Llibre J (2019) Equilic quadrilateral central configurations. Commun Nonlinear Sci Numer Simul  78: 104872. Link: https://bit.ly/3hyuiHw
22. Alvarez-Ramirez A, Santos AA, Vidal C (2013) On co-circular central configurations in the four and five body-problem for homogeneous force law. J Dynam Differential Equations 25: 269-290. Link: https://bit.ly/3ob5Tui
23. Arribas M, Abad A, Elipe A, Palacios M (2016) Equilibria of the symmetric collinear restricted four-body problem with radiation pressure, Astrophys. Space Sci  361: 12. Link: https://bit.ly/38UNfAe
24. Arenstorf RF (1982) Central configurations of four bodies with one inferior mass. Cel Mechanics 28: 9-15. Link: https://bit.ly/3aWsRBv
25. Barros JF, Leandro ESG (2011) The set of degenerate central configurations in the planar restricted four-body problem. SIAM Journal on Mathematical Analysis 43: 634-661. Link: https://bit.ly/3hyufvk
26. Barros JF, Leandro ESG (2014) Bifurcations and enumeration of classes of relative equilibria in the planar restricted four-body problem. SIAM Joural on Mathematical Analysis 46: 1185-1203. Link: https://bit.ly/3pK1aQi
27. Bernat J, Llibre J, Perez-Chavela E (2009) On the planar central configurations of the 4-body problem with three equal masses. Dyn Contin Discrete Impuls. Syst Ser A Math Anal 16: 1-13.
28. Chenciner A (2017) Are nonsymmetric balanced configurations of four equal masses virtual or real?. Regul Chaotic Dyn  22: 677-687. Link: https://bit.ly/3rIlWBQ
29. Corbera M, Cors JM, Llibre J, Perez-Chavela E (2019) Trapezoid central configurations. Appl Math Comput 346: 127-142. Link: https://bit.ly/3n5WCSW
30. Corbera M, Llibre J (2014) Central configurations of the 4-body problem with masses m1=m2>m3=m4=m>0 and m small. Appl Math Comput 246: 121-147. Link: https://bit.ly/2MpY0Dc
31. Corbera M, Cors JM, Llibre J (2011) On the central configurations of the planar -body problem. Celestial Mech Dynam Astronom  109: 27-43.
32. Corbera M, Cors JM, Roberts GE (2018) A four-body convex central configurationGwith perpendicular diagonals is necessarily a kite. Qual Theory Dyn Syst  17: 367-374. Link: https://bit.ly/3bdf8GL
33. Corbera M, Cors JM, Roberts GE (2019) Classifying four-body convex central configurations. Celestial Mech Dynam Astronom 131: 34. Link: https://bit.ly/3hykFst
34. Cors JM, Roberts GE (2012) Four-body co-circular central configurations. Nonlinearity 25: 343-370. Link: https://bit.ly/2MrdKG0
35. Cors JM, Llibre J, Ollé M (2004) Central configurations of the planar coorbital satellite problem. Celestial Mech Dynam Astronom  89: 319-342. Link: https://bit.ly/2LhG1hO
36. Deng Y, Li B, Zhang S (2017) Four-body central configurations with adjacent equal masses. J Geom Phys  114: 329-335. Link: https://bit.ly/38LAI1S
37. Deng Y, Li B, Zhang S (2017) Some notes on four-body co-circular central configurations. J Math Anal Appl  453: 398-409. Link: https://bit.ly/3hxqo1I
38. Deng C, Zhang S (2014) Planar symmetric concave central configurations in Newtonian four-body problems. J Geom Phys  83: 43-52. Link: https://bit.ly/3pMVwgF
39. Rdi B, Czirj  KZ  (2016) Central configuration of four bodies with an axis of symmetry. Celestial Mech Dynam Astronom 125: 33-70. Link: https://bit.ly/2MqBEkZ
40. Fernandes AC, Llibre J, Mello LF (2017) Convex central configurations of the 4-body problem with two pairs of equal masses. Arch Rational Mech Anal 226: 303-320. Link: https://bit.ly/38O6FXw
41. Gannaway JR (1981) Determination of all central configurations in the planar 4-body problem with one inferior mass, Ph. D., Vanderbilt University, Nashville, USA.
42. Fernandes AC, Garcia BA, Llibre J, Mello LF (2018) New central configurations of the (n+1) body problem. J Geom Phys  124: 199-207. Link: https://bit.ly/3rL2K6m
43. Grebenikov EA, Ikhsanov EV, Prokopenya AN (2006) Numeric-symbolic computations in the study of central configurations in the planar Newtonian four-body problem, Computer algebra in scientific computing, 192–204. Lecture Notes Comput Sci. Link:
44. Hampton M (2003) Co-circular central configurations in the four-body problem. EQUADIFF 993–998. Link: https://bit.ly/38Wj63F
45. Hampton M, Moeckel R (2006) Finiteness of relative equilibria of the four-body problem. Invent Math 163: 289-312. Link: https://bit.ly/34YmsSj
46. Hassan MR, Ullah MS, Aminul HM, Prasad U (2017) Applications of planar Newtonian four-body problem to the central configurations. Appl Appl Math  12: 1088-1108. Link: https://bit.ly/3hycLiP
47. Leandro ESG (2006) On the central configurations of the planar restricted four-body problem. J Differential Equations  226: 323-351. Link: https://bit.ly/3o8Wovk
48. Llibre J (1976) Posiciones de equilibrio relativo del problema de 4 cuerpos. Publicacions Matemàtiques UAB 3: 73-88. Link: https://bit.ly/2KMCZCi
49. Llibre J, Yuan P (2019) Bicentric quadrilateral central configurations. Appl Math Comput  362: 124507. Link: https://bit.ly/3rLKyK0
50. Llibre J, Yuan P (2020) Tangential trapezoid central configurations. Regul Chaotic Dyn 25: 651-661. Link: https://bit.ly/2X3F6nG
51. Long Y (2003) Admissible shapes of 4-body non-collinear relative equilibria. Adv Nonlinear Stud 3: 495-509. Link: https://bit.ly/387q5rc
52. Long Y, Sun S (2002) Four-Body Central Configurations with some Equal Masses. Arch Ration Mech Anal 162: 25-44. Link: https://bit.ly/3b89Zzk
53. MacMillan WD, Bartky W (1932) Permanent Configurations in the Problem of Four Bodies. Trans Amer Math Soc 34: 838-875. Link: https://bit.ly/2X3EYVe
54. Ouyang T, Xie Z (2005) Collinear central configuration in four-body problem. Celestial Mech Dynam Astronom  93: 147-166. Link: https://bit.ly/3hzJPqA
55. Pedersen P (1944) Librationspunkte im restringierten Vierkörperproblem. Danske Vid Selsk Math Fys 21: 1-80. Link: https://bit.ly/3b1ftMA
56. Perez-Chavela E, Santoprete M (2007) Convex four-body central configurations with some equal masses. Arch Rational Mech Anal 185: 481-494. Link: https://bit.ly/3naQzMB
57. Pina E (2013) Computing collinear 4-body problem central configurations with given masse. Discrete Contin. Dyn Syst  33: 1215–1230. Link: https://bit.ly/38UbE9e
58. Pina E, Lonngi P (2010) Central configuration for the planar Newtonian four-body problem. Celest Mech Dyn Astron 108: 73-93. Link: https://bit.ly/2LdRAWW
59. Rusu D, Santoprete M (2016) Bifurcations of central configurations in the four-body problem with some equal masses. SIAM J Appl Dyn Syst  15: 440-458. Link: https://bit.ly/2KWIGgK
60. Shi J, Xie Z (2010) Classification of four-body central configurations with three equal masses. J Math Anal Appl  363:  512-524. Link: https://bit.ly/2LdKiTn
61. Shoaib M, Kashif AR, Szücs-Csillik I (2017) On the planar central configurations of rhomboidal and triangular four- and five-body problems. Astrophys Space Sci  362: 182. Link: https://bit.ly/2L7l4G5
62. Simo C (1978) Relative equilibrium solutions in the four-body problem. Cel Mechanics 18: 165-184. Link: https://bit.ly/3pFMbXJ
63. Tang JL (2006) A study on the central configuration in the Newtonian 4-body problem of celestial mechanics (Chinese). J Systems Sci Math Sci  26: 647-650.
64. Xie Z (2012) Isosceles trapezoid central configurations of the Newtonian four-body problem. Proc Roy Soc Edinburgh Sect A  142: 665-672. Link: https://bit.ly/3o6LlD5
65. Yoshimi N, Yoshioka A (2018) 3+1 Moulton configuration. SUT J Math  54: 173-190.
66. Herget P (1967) Theory of orbits, The restricted problem of three bodies, Academic Press, New York. Link: https://bit.ly/3n7ki9a
67. Bernat J, Llibre J, Perez-Chavela E (2009) On the planar central configurations of the 4-body problem with three equal masses. Dyn Contin Discrete Impuls. Syst Ser A Math Anal 16: 1-13.
68. Stoer J, Bulirsch R (1980) Introduction to numerical analysis, Springer-Verlag, New York. Link: https://bit.ly/3521NNi
© 2021 Llibre J. 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.  