ON THE EXISTENCE OF BI–PYRAMIDAL CENTRAL CONFIGURATIONS OF THE n+ 2–BODY PROBLEM WITH AN n–GON BASE

In this paper we prove the existence of central configurations of the n + 2–body problem where n equal masses are located at the vertices of a regular n–gon and the remaining 2 masses, which are not necessarily equal, are located on the straight line orthogonal to the plane containing the n–gon passing through its center. Here this kind of central configurations is called bi–pyramidal central configurations. In particular, we prove that if the masses mn+1 and mn+2 and their positions satisfy convenient relations, then the configuration is central. We give explicitly those relations.

1. Introduction.We consider the spatial N -body problem G m k m j q k − q j |q k − q j | 3 , k = 1, . . ., N , where q k ∈ R 3 is the position vector of the punctual mass m k in an inertial coordinate system, and G is the gravitational constant which can be taken equal to one by choosing conveniently the unit of time.The configuration space of the spatial N -body problem is E = {(q 1 , . . ., q N ) ∈ R 3N : q k = q j , for k = j}.
Given m 1 , . . ., m N a configuration (q 1 , . . ., q N ) ∈ E is central if there exists a positive constant λ such that qk = −λ (q k − c) , k = 1, . . ., N , where c is the center of mass of the system, which is given by Thus a central configuration (q 1 , . . ., q N ) ∈ E of the N -body problem with positive masses m 1 , . . ., m N is a solution of the system of equations k = 1, . . ., N , for some λ.
The first known planar central configuration of the N -body problem for N ≥ 2 consists of N equal masses located at the vertices of a regular N -gon.If we take N equal masses at the vertices of a regular polyhedron with N vertices, then we obtain a spatial central configuration of the N -body problem (see [1]).In addition to regular polyhedron central configurations, the simplest spatial central configurations of the N -body problem are the ones known as pyramidal central configurations.Such configurations consist of N = n + 1 masses, n of which are coplanar and the (n + 1)-th being off the plane (see for instance [2] and [14]).The n positions of the coplanar masses are called the base of the pyramidal central configuration.
The next simplest spatial central configurations are the ones known as double pyramidal central configurations.Such configurations consist of N = n + 2 masses, n of which are coplanar and the other two being off the plane and positioned symmetrically above and below that plane.In the literature we can find some papers related with double pyramidal central configurations with different shapes of basis.For instance [16] studied for all n ≥ 4 the double pyramidal central configurations such that the n coplanar masses are at the vertices of a regular n-gon and the (n + 1)-th and (n + 2)-th masses are equal, under these assumptions the (n + 1)-th and the (n + 2)-th mass are located symmetrically with respect to the n-gon.Liu and his coauthors have some papers related with double pyramidal central configurations of the N body problem for N = 5, 6, 7, 9 for different shapes of their basis (see [4,5,6,7,8,9,10]).Yang and Zhang in [15] also studied double pyramidal central configurations of the 6-body problem.In [12] the authors study central configurations of the N = n + 3 body problem consisting of n masses at the vertices of a regular n-gon, 2 masses on the straight line orthogonal to the plane containing the n-gon passing through its center and positioned symmetrically above and below the n-gon, and a third mass positioned at the center of the n-gon.
In this paper we consider spatial central configurations of the N -body problem, with N = n + 2, n ∈ N and n ≥ 2, consisting of n equal masses m 1 = • • • = m n at the vertices of a (regular) n-gon and 2 masses m n+1 and m n+2 on the straight line orthogonal to the plane containing the n-gon passing through its center.In contrast to what happens in the known double pyramidal central configurations, we do not impose conditions, neither on the positions, nor on the values of the masses m n+1 and m n+2 .In this paper, these configurations are called bi-pyramidal central configurations.Notice that the double pyramidal central configurations studied in [16] are a particular case of our bi-pyramidal central configurations when m n+1 = m n+2 and m n+1 and m n+2 are positioned symmetrically above and below the plane containing the n-gon.The bi-pyramidal central configurations for n = 3 are studied in [3] and [11] from different points of view.In [3] the author proves that, for any pair of positive masses m n+1 and m n+2 , the number of bi-pyramidal central configurations is finite and also provides all the possible numbers of such central configurations.In [11] the authors consider the inverse problem; that is, given a bi-pyramidal configuration, they find the masses which make it central.We do not know any paper considering bi-pyramidal central configurations for n > 3.
Here we analyze the bi-pyramidal central configurations from the inverse problem point of view.
This paper is structured as follows.In Section 2 we give the equations of the bi-pyramidal central configurations.In Section 3 we summarize some results concerning to pyramidal central configurations.Finally, in Section 4, we analyze the 2-pyramidal central configurations.We prove that for all n ≥ 2 we can find positions on the straight line such that convenient masses m n+1 and m n+2 placed at these positions provide central configurations.We also give the explicit relations between the masses and the positions of such central configurations.Moreover we see that for all n < 9 there exists a privileged position and a privileged value of the top mass, which depend on n, so that any arbitrary mass located at the same distance from all other masses provides a central configuration.The precise statement of these results is given in Theorem 4.3.

Equations of the central configurations. We consider
the vertices of a (regular) n-gon and 2 masses m n+1 and m n+2 on the straight line orthogonal to the plane containing the n-gon passing through its center.Without loss of generality we can choose the unit of mass so that m 1 = • • • = m n = 1, and we take the unit of length in order that the radius of the circle containing the n-gon be one.By using complex coordinates in the plane that contains the regular n-gon, the positions of the vertices of the n-gon can be written as q k = (e iα k , 0) ∈ C × R with α k = 2πk/n for k = 1, . . ., n.Let m n+1 = µ 1 , m n+2 = µ 2 , q n+1 = (0, z 1 ), and q n+2 = (0, z 2 ), with z 1 > z 2 .We note that this last condition is not restrictive.
Using these notations the center of mass of the system is given by The first n equations of (1) become for k = 1, . . ., n, and the last 2 equations of (1) are It is easy to check that |q k − q j | = |e iα k − e iαj | for j = k, and j, k = 1, . . ., n, So the first components of the n vectorial equations ( 2) are for k = 1, . . ., n.By dividing the k-th equation by e iα k we get for k = 1, . . ., n. Defining after some simplifications we get So the first components of the n vectorial equations (4) can be reduced to the single equation Using the fact that n j=1 e iαj = 0, we can see easily that the first components of the 2 vectorial equations (3) are always satisfied.
The second components of the n equations (2) become for all k = 1, . . ., n.Finally, the second components of the 2 vectorial equations (3) become In short, system (1) can be reduced to the 4 equations ( 5), ( 6) and (7) with the 3 unknowns λ, z 1 , z 2 .We note that one of these equations is redundant.Indeed, if we multiply the first equation of ( 7) by µ 1 and the second one by µ 2 and we add the resulting equations, then we get ¿From (6), the left hand of (8) becomes n λ c z .On the other hand, from the definition of c z , we have that µ 1 z 1 + µ 2 z 2 = (n + µ 1 + µ 2 )c z .Therefore (8) becomes which is always satisfied.We add equation ( 6) to each equation of ( 7) and we substitute the value of λ by the value given by ( 5) into the resulting equations.Then the system formed by equations ( 5), ( 6) and ( 7) reduces to a system of 2 equations that is linear in the variables µ 1 and µ 2 , and its matrix form is where . for all ℓ, j = 1, 2 and ℓ = j.
We note that if in (9) we replace z ℓ by −z ℓ for all ℓ = 1, 2, we obtain the same system of equations.So if In order to simplify our computations, in what follows we assume that z 1 > 0.
Since we have assumed that z 1 > z 2 , we have that 3. Pyramidal central configurations.We note that the formulation (9) allows us to find the pyramidal central configurations obtained in [14] in an easy way.
Indeed, the pyramidal central configurations consists of placing only one mass on the straight line orthogonal to the plane containing the n-gon passing through its center.For these configurations equation ( 9) becomes the single equation The solutions of equation t 1 = 0 are We note that the values z β are defined only for n such that n/β n ≥ 1.In [13] it is proved that this condition is satisfied for n < 473.In short we have proved the following known result.A different proof of Theorem 3.1 can be found in [14].
4. Bi-pyramidal central configurations.¿From elementary algebra we get the following result.
4.1.The curves c 2,1 = 0, c 1,2 = 0, t 1 = 0 and t 2 = 0.It is easy to see that equation c 1,2 = 0 has a unique solution which is given by and c 2,1 = 0 has the unique solution Analyzing the properties of the functions f and g we see that both functions are continuous for z 1 > 0. Since they are increasing.Moreover We note that this means that f tends to infinity assymptotically to the straight line z 2 = z 1 /2 when z 1 → +∞.Finally f crosses the positive semiaxis z 1 at the point z 1 = 1.The plot of the curves z 2 = f (z 1 ) and z 2 = g(z 1 ) is given in Figure 1.
We observe that when z 1 > 0 the curves z 2 = f (z 1 ) and z 2 = g(z 1 ) intersect at the unique point ).On the other hand in Section 3 we have seen that equation t ℓ = 0 for ℓ = 1, 2 has a unique solution z ℓ = 0 when n ≥ 473 and it has three solutions z ℓ = 0 and z ℓ = ±z β when 2 ≤ n < 473.
Finally, we analyze the sign of the functions c 1,2 , c 2,1 , and t ℓ for ℓ = 1, 2. The results that we have obtained are summarized in the next result.
By means of Lemma 4.2, we compute the signs of µ 1 and µ 2 on all the regions delimited by the curves z 2 = g(z 1 ), z 2 = 0, z 2 = ±z β , z 2 = f (z 1 ) and z 1 = z β for each one of the four shapes depending on n.The regions on which µ 1 > 0, µ 2 > 0 and µ 1 , µ 2 > 0 are shaded in Figures 2 and 3.
Notice that the intersection point p 1 of the curves c 2,1 = 0 and t 2 = 0 does not belong to the region where µ 2 > 0. Therefore there are no solutions of system (9) of type (C) (see Lemma 4.1) providing central configurations.
The intersection point p 2 of the curves c 1,2 = 0 and t 1 = 0 belongs to the region with µ 1 > 0 only when 2 ≤ n < 9. Therefore system (9) has solutions of type (B) (see Lemma 4.1) that provide central configurations only when 2 ≤ n < 9.It is easy to check that the value of µ 1 at the point (z 1 , z 2 ) = p 2 is In short, we have proved the following theorem.

Theorem 3 . 1 .
The following statements hold.(a) For all µ 1 > 0 and 2 ≤ n < 473 the problem has three central configurations, one with µ 1 at the center of the n-gon (i.e.z 1 = 0), and two with µ 1 at the positions z 1 = z β and z 1 = −z β respectively.(b) For all µ 1 > 0 and n ≥ 473 the problem has a unique central configuration, the one with µ 1 at the center or the n-gon.