Stacked Central Configurations for the Spatial Nine-Body Problem

We show the existence of the twisted stacked central configurations for the 9-body problem. More precisely, the position vectors x 1, x 2, x 3, x 4, and x 5 are at the vertices of a square pyramid Σ; the position vectors x 6, x 7, x 8, and x 9 are at the vertices of a square Π.


Introduction and Main Results
The classical -body problem [1,2] concerns the motion of mass points moving in space according to Newton's law: Here, ∈ R is the position of mass > 0, the gravitational constant is taken equal to 1, and = | − | is the Euclidean distance between and . The space of configuration is defined by while the center of mass is given by where = 1 + ⋅ ⋅ ⋅ + is the total mass. A configuration = ( 1 , . . . , ) ∈ is called a central configuration [2,3] if there exists a constant , called the multiplier, such that It is easy to see that a central configuration remains a central configuration after a rotation in R and a scalar multiplication. More precisely, let ∈ SO( ) and > 0, if = ( 1 , . . . , ) is a central configuration, so are = ( 1 , . . . , ) and = ( 1 , . . . , ). Two central configurations are said to be equivalent if one can be transformed to the other by a scalar multiplication and a rotation. In this paper, when we say a central configuration, we mean a class of central configurations as defined by the above equivalent relation.
Central configurations of the -body problem are important because they allow the computation of homographic solutions; if the bodies are heading for a simultaneous collision, then the bodies tend to a central configuration (see [3,4]); there is a relation between central configurations and the bifurcations of the hypersurfaces of constant energy and angular momentum (see [5]).
In this paper, we are interested in spatial central configurations, that is, = 3. In 2005, Hampton [6] provides a new family of planar central configurations for the 5body problem with an interesting property: the central configuration has a subset of three bodies forming a central configuration of the 3-body problem. The authors [7] find new classes of central configurations of the 5-body problem which are the ones studied by Hampton [6] having three bodies in the vertices of an equilateral triangle, but the other two, instead of being located symmetrically with respect to a perpendicular bisector, are on the perpendicular bisector. The The Scientific World Journal stacked central configurations studied by Hampton [6] were completed by Llibre et al. [8] (see also [9]).
Zhang and Zhou [10] showed the existence of double pyramidal central configurations of + 2-body problem. The authors [11][12][13] provided new examples of stacked central configurations for the spatial 7-body problem where four bodies are at the vertices of a regular tetrahedron and the other three bodies are located at the vertices of an equilateral triangle.
In this paper, we find new classes of stacked spatial central configurations for the 9-body problem which have five bodies at the vertices of a square pyramid, and the other four bodies are located at the vertices of a square. More precisely, the spatial central configurations considered here satisfy the following (see Figure 1): the position vectors 1 , 2 , 3 , 4 , and 5 are at the vertices of a square pyramid Σ; the position vectors 6 , 7 , 8 , and 9 are at the vertices of a square Π.
Without loss of generality, we can assume that where > 0, ∈ R, and ̸ = 0; the positive constant ℎ satisfies the equation (see [10] and the references therein); that is, ℎ = 1.26276522.
The main results of this paper are the following.

Theorem 1.
Consider the spatial configurations according to Figure 1, in order that the nine mass points are in a central configuration, the following statements are necessary: (1) the masses 1 , 2 , 3 , and 4 must be equal; (2) the masses 6 , 7 , 8 , and 9 must be equal. Figure 2) such that the nine bodies take the coordinates Then, there are positive solutions of 1 , 5 , 6 such that these bodies form a spatial central configuration according to Figure 1.
The proofs of the theorems are given in the next sections.
x 1 x 4 x 6 x 9 Π x 5 x 7 x 8 x 2 x 3 Σ Figure 1: The configuration for the 9-body problem.

Proof of Theorem 1
For the spatial central configurations, instead of working with (4), we consider the Dziobek-Laura-Andoyer equations (see [9,[11][12][13] and the references therein): Due to assumption (5) and the definition of Δ , we have several symmetries in the signed volumes. By using the symmetries and the properties of Δ , we obtain the following results.
From Lemma 5, in order to study central configurations according to Figure 1 in the set −1 (0), it is sufficient to study the following 2 equations: Denote by = ( ) the matrix of the coefficients of the homogeneous linear system in the variables , 4 , defined by (22). Thus,

Let
= ( 4 ). Then in order to get the spatial central configuration as Figure 1, we need to find a positive solution , 4 , of the following system: where = 0.

The Existence of Spatial Central Configurations
In order to prove the existence of positive solutions of (24) in the set −1 (0), it is sufficient to prove that the entries in each row of change the signs. So if the entries of some row of have the same signs, there are no admissible masses such that the bodies are in a central configuration according to Figure 1. Proof of Theorem 2. Since the rank of matrix is two in the set −1 (0), there are nontrivial solutions of (24) in the set −1 (0). Now we prove the existence of spatial central configurations according to Figure 1 for some points in the set (see Figure 2). In order to prove the existence of positive solutions of (24) in the set −1 (0), the entries 21 , 23 of the second line in the matrix should have opposite signs. Thus, we consider the following set , where is surrounded by curves = 0, = 0, 21 = 0, and 23 = 0.
In the set , the entries of matrix have the following signs: 21 > 0, 23 < 0 (see Figures 3 and 4); 11 > 0, 12 < 0, surrounded by curves = 0, = 0, and = ℎ(1 − ) (see Figures 5,6,7,and 8). In short, the signs of the entries of the matrix restricted to the set are the following: In the rest of the proof, we show that the set −1 (0) has intersection with the set . We consider the subset of : where 1 ∈ (0, 1). Obviously is a segment with endpoints (see Figure 9), and the point ( 1 , 1 ) satisfies the equation 21 = 0. Evaluating the function at these points, we have Thus, there exists a point 0 = ( 0 , 0 ) ∈ , such that ( 0 ) = 0. So at the point 0 we have nontrivial positive solutions of (24), since the signs of the entries of the matrix at this point are the following: Thus, the proof of Theorem 2 is completed.