Planar Central Configurations of Symmetric Five-Body Problems with Two Pairs of Equal Masses

We study central configuration of a set of symmetric planar five-body problems where (1) the five masses are arranged in such a way that m 1 , m 2 , and m 4 are collinear and m 2 , m 3 , and m 5 are collinear; the two sets of collinear masses form a triangle with m 2 at the intersection of the two sets of collinear masses; (2) four of the bodies are on the vertices of an isosceles trapezoid and the fifth body can take various positions on the axis of symmetry both outside and inside the trapezoid.We form expressions for mass ratios and identify regions in the phase space where it is possible to choose positive masses which will make the configuration central. We also show that the triangular configuration is not possible.


Introduction
The equations of motion for  positive masses subject to Newtonian Gravitation is given by where is the Newtonian potential,   is the position vector of the th body,   is the mass of the th body, and  is the universal constant of gravitation.The central configuration of an body system is obtained, if the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration; that is, →   =   ,  = 1, . . ., , where the constant  ̸ = 0.
Using the Laura/Andoyer/Dziobek equations [1], the condition that ( 1 ,  2 , . . .,   ) form a planar, noncollinear, central configuration is where   = 1/ 3  and Δ  = (  −   ) ∧ (  −   ).The study of the central configurations plays a key role in understanding the Gravitational -body problems (cf.[2][3][4]).Central configurations are useful in providing explicit homographic solutions of the equations of motion and families of periodic solutions [5].They are also useful in understanding the nature of solutions near collisions and the energy level sets that hold the central configuration determine the topology of the integral manifolds.Understanding the four-and five-body problem is very important as it is known that approximately two-thirds of stars in our Galaxy are part of multistellar systems.There is a growing interest in studying spatial central configurations of Newtonian five-body problem with various symmetrical restrictions.Several authors 2 Advances in Astronomy have recently studied central configuration of restricted fourbody and five-body problems [6][7][8][9].According to [10], finiteness problem of -body central configurations is an open problem for the 21st century.Hampton and Moeckel [11] proved that finite central configurations are possible for the Newtonian three-body and four-body problems with positive masses.Finiteness of central configurations of planar fivebody problem is recently established by Albouy and Kaloshin in [12].Hampton and Jensen [13] showed that number of spatial central configurations are finite for positive masses in the Newtonian five-body problem.
Several authors studied the inverse problem of central configurations by finding the mass vectors which makes the configuration central for a given configuration of  bodies.Albouy and Moeckel [14] show that two-parameter family of masses exists for collinear central configurations with  ≤ 6. Xie [15] proved the existence of a singular curve where it is possible to have permutational admissible set of mass vectors in the collinear central configuration fourbody problem.According to [16], families of stacked central configurations can be formed by adding extra bodies to known central configurations of three bodies.Gidea and Llibre [1] studied the stacked symmetric planar central configuration of five bodies with some special symmetries.They have shown that central configuration is possible in rhomboidal arrangement where four masses are kept at the vertices and a fifth mass in the center and a trapezoidal arrangement where four masses are at the vertices and a fifth mass at the midpoint of one of the parallel sides.Shoaib et al. [17] have studied central configuration of the rhomboidal 5-body problem and identified CC regions using similar approaches.
Xie [18] studied central configuration of the planar Newtonian four-body problems, where two equal mass pairs are kept at adjacent vertices of a trapezoid and possible central configuration mass ratios are expressed in terms of the size of quadrilateral.While much is known about trapezoid central configurations in the restricted four-body problem, there is less known about trapezoid central configurations in the restricted five-body problem.Shoaib [19] recently investigated the inverse problem of central configuration in a symmetric 4 + 1-body problem and derived regions of central configuration.
In this work, we study the central configuration of the isosceles trapezoidal five-body problem and identify regions in the phase space where it is possible to choose positive masses which will make the configuration central.We are motivated by the work of [1] and follow similar ideas to study planar symmetric five-body problems.The problems we investigate include two types of trapezoidal five-body problems with four masses on its vertices and a fifth mass on the axis of symmetry which can be both inside and outside the trapezoid.The third type of problem which is investigated here is a triangular problem with two pairs of masses and a fifth mass on the perpendicular bisector of the triangle which is not necessarily equal to any of the four masses.In Section 2, we present the main theorems related to the three five-body problems and their proofs are given in Sections 3, 4, and 5, respectively.
(a) When  = , there is a continuous family of central configurations determined by the region  and the function (, ) = 0, given in Figure 2.There are no central configurations when  > 0.
(b) When  = 1 and  ∈ (0, 0.64) there exists a continuous family of central configurations for  > 0. There are no central configurations when  < 0 or when  > 0.64.
Theorem 2. Consider a 5-body noncollinear configuration ⃗  = (r 1 , r 2 , r 3 , r 4 , r 5 ) with a positive mass vector ( 1 ,  2 ,  3 ,  4 ,  5 ), where r  ,  = 1, 2, . . ., 5 are given by (5).The five masses are arranged in such a way that  1 ,  2 , and  4 are collinear and  2 ,  3 , and  5 are collinear.The two sets of collinear masses form a triangle with  2 at the intersection of the two sets of collinear masses.Then there does not exist any such type of central configuration.

Proof of Theorem 1
3.1.Proof of Theorem 1(a).Consider five bodies of masses  1 ,  2 ,  3 ,  4 , and  5 placed at r 1 , r 2 , r 3 , r 4 , and r 5 :  Four of the masses make a trapezoid (see Figure 1) while the mass  2 is on the line of symmetry.Using the inherent symmetries of the trapezoidal 5-body model we obtain the following system of four equations from (4) which define the central configurations for the model described above:   12 =  1 ℎ 31 +  4 ℎ 33 = 0, where Write ( 7) to ( 10) as a linear homogeneous system in ( 1 ,  2 ,  4 ) given by the matrix ) . ( Using a simple Gaussian elimination approach, matrix  is reduced as in what follows: The above systems can now be written in equation form as in what follows: For the above systems to have a nontrivial solution ℎ 31 ℎ 43 − ℎ 41 ℎ 33 = 0.This will be treated as a geometric constraint on the existence of trapezoidal central configurations.Setting  1 =  1 / 4 and  2 =  2 / 4 , we obtain Such that ℎ 21 ̸ = 0 and ℎ 41 ̸ = 0. Therefore (15) define central configuration for the trapezoidal 5-body problem for all masses subject to the constraint It can be seen from the reduced matrix that ( 8) is not used in deriving  1 and  2 which necessitates a second constraint given as follows: However it is numerically confirmed that  * (, ) = 0 is satisfied everywhere, where (, ) = 0.The constraint  * (, ) = 0 has some additional solutions but that is irrelevant as for nontrivial solution both the constraints have to be satisfied.Hence we will only use (, ) in our analysis.
As only positive solutions are of practical interest, therefore we will now isolate the regions where all the masses are positive.
Proof.To find the central configuration region where  1 is positive, we will need to find regions in the -plane where ℎ 43 and ℎ 41 have opposite signs.We can see that is negative when its two factors have opposite signs.Therefore Similarly As ℎ 41 is a sum of two nonlinear functions of  and , we used some approximation techniques to find the region in the plane, where ℎ 41 > 0 and Thus the CC region where  1 has positive real values is given as follows: This completes the proof of Lemma 3.
Proof.For  2 > 0,   2 and   2 must have the same sign.Therefore central configuration region where  2 > 0 is given by This completes the proof of Lemma 6.
The proof of Theorem 1(a) is a direct consequence of Lemmas 3 and 6.That is, the central configuration region  =   1 ∩  2 , where both  1 and  2 are positive, can be found by taking the intersection of the regions found for  1 and  2 .This region is given in Figure 2 with the geometric constraint (, ) = 0.The continuous family of central configurations is shown by intersection of the bold line with the colored region in Figure 2.This completes the proof of Theorem 1.

Proof of Theorem 1(b).
Consider five bodies of masses  1 ,  2 ,  3 ,  4 , and  5 placed at r 1 = (−1, 0), r 2 = (0, −), r 3 = (1, 0), r 4 = (−, 1), and r 5 = (, 1), respectively.Similar to Theorem 1, four of the masses make a trapezoid (see Figure 1) while the mass  2 is on the line of symmetry.In Theorem 1, the trapezoid considered had  = .In this case, the height of trapezoid is fixed at 1 with variable width.The general CC equations are the same as in Theorem 1 with different   's.The   which have different values are The general equations for  1 ,  2 and the constraints  and  * are the same as in Theorem 1 but with different values of all the ℎ  functions: Advances in Astronomy Similar to Theorem 1(a), the constraints  and  * have identical solutions; therefore we use only .
Proof.The method of proof is the same as in Theorem 1.For  1 > 0, ℎ 43 and ℎ 41 must have opposite signs.The functions ℎ 43 and ℎ 41 are positive in the following regions, respectively: Therefore the CC region where  1 > 0 is determined by This completes the proof of Lemma 7.
Proof.For the mass ratio  2 to be positive the functions   2 and   2 must have the same signs.The denominator of  2 ,   2 = ℎ 41 ℎ 22 < 0 when ℎ 41 and ℎ 22 have opposite sign.The sign analysis of ℎ 41 is given in (38).The second factor of   2 , that is, ℎ 22 , is positive in  ℎ 22 + and negative in ( ℎ 22 + )  : Therefore   2 < 0 in For the numerator of  2 , that is,   2 = (ℎ 21 ℎ 43 − ℎ 23 ℎ 41 ), to be negative we have the following possibilities with the corresponding regions mentioned alongside them.We will only list the possibilities where we get a nonempty region for   2 < 0 and will ignore all other possibilities: ( Therefore   2 < 0 in Thus the CC region where  2 > 0 is given by This completes the proof of Lemma 8.
The intersection of   1 and   2 will give the region which is the central configuration region where both  1 and  2 are positive.Region  will satisfy (3) subject to the constraint (, ) = 0.The intersection of the constraints (, ) = 0,  * (, ) = 0 and the region where all the masses are positive is represented by a line marked  = 1 in Figure 3.This confirms the existence of central configuration for the trapezoidal 5-body problem at  = 1.There is a continuous family of central configuration when 0 <  < 0.64 and 0.2 <  < 1.8.There is no central configuration when 0.64 <  < 1.This completes the proof of Theorem 1(b).
To complement the analysis given in Theorem 1 a number of graphs are given in Figure 3 for fixed values of  ∈ (0, 1) alongside  = 1.The general expressions for  1 ,  2 and the constraints can be derived in the same way as in Theorem 1.
In each case there is a continuous family of central configurations for both positive and negative values of .In Figures 4(a  2 are positive, respectively.Figure 5 gives the intersection of regions given in Figure 4; this will provide the central configuration region subject to the two constraints (, ) = 0,  * (, ) = 0.The part of this region is given in Figure 6 where both the constraints are satisfied which is the central configuration for the trapezoidal 5-body problem where one of the masses is allowed to move symmetrically on the axis of symmetry.

Proof of Theorem 2
Consider masses  1 ,  2 , and  4 and masses  2 ,  3 , and  5 to be collinear.The two sets of collinear masses form a triangle with  2 at the intersection of the two sets of collinear masses; see Figure 7.To enforce collinearity take  = (1 − ) in (5).We derive the CC equations from (4) as follows: We repeat the same procedure as was done in the previous case to obtain the following expressions for the two mass ratios  1 =  1 / 4 ,  2 =  2 / 4 and the geometric constraints (, ) = 0 and  * (, ) = 0 * (, ) = ℎ 43 ℎ 11 − ℎ 13 ℎ * 41 .
Equation (50) determine CC regions in the -plane subject to the constraint given by  in (52).First we will analyse  1 and  2 and will then discuss  to find CC regions where ( 4) is satisfied.
Proof.The function ℎ 22 is given below as function of  and : Using the same technique as in Theorem 1, we find that ℎ 22 (, ) is positive in the following region: Similarly  ℎ 11 + ∩  ℎ 22 + can be found as follows: It is clear from Figure 8 that the constraint (, ) = 0 does not intersect the region where all the masses are positive.Hence no central configurations of this particular type are possible.

Conclusions
We studied the central configuration of different types of symmetric 5-body problems.All the problems investigated have two pairs of equal masses.In one case two pairs of equal masses are placed at the vertices of an isosceles trapezoid and the fifth body not necessarily equal to the other masses was allowed to take various positions on the axis of symmetry.Initially we formed expressions for mass ratios and derived regions of central configuration for positive masses.The existence of continuous family of central configurations is shown to exist for trapezoids of various sizes.It is also shown that it is possible for the fifth mass to be both inside the trapezoid and outside the trapezoid.In the second case the five masses are arranged in two sets of three collinear masses such that one of the masses is at the intersection of the two collinear sets, hence making a triangle; similarly expressions are formed for mass ratios to derive regions of central configuration for positive masses.It was shown that no such type of central configuration is possible.

8 Figure 3 :
Figure 3: Families of central configuration in the case of isosceles trapezoid for values of  ∈ (0, 1).The various lines correspond to values of  as labelled.

Figure 8 :
Figure8: Region   is given by the colored portion of the graph and the constraints (, ) = 0,  * (, ) = 0 are given by the purple lines.There is no intersection between the CC region and the constraints.