New Classes of Spatial Central Configurations for N + N + 2-Body Problem

and Applied Analysis 3 Recently, Hampton and Santoprete 17 provided new examples of stacked spatial central configurations—central configurations for theN-body problemwhere a proper subset of the N bodies are already on a central configuration—for the 7-body problem where the bodies are arranged as concentric threeand two-dimensional simplex. New classes of stacked spatial central configurations for the 6-body problem which have four bodies at the vertices of a regular tetrahedron and the other two bodies on a straight line connecting one vertex of the tetrahedron with the center of the opposite face were studied in 18 . In this paper, we study new classes of spatial double pyramidical central configurations d.p.c.c for the N N 2-body that satisfy the following. 1 The position vectors r1, r2, . . . , rN of masses m1, m2, . . . , mN are at the vertices of a regular N-gon Γ1, whose sides have length ι1. The position vectors rN 1, rN 2, . . . , r2N of masses mN 1, mN 2, . . . , m2N are at the vertices of another regular N-gon Γ2, whose sides have length ι2. Two N-gons have a same geometric center and form a affine nested N-gons base plane Π . 2 Let L be the straight line perpendicular to the base plane Π that contains Γ1 and Γ2 and passes through the geometric center of Γ1 ∪ Γ2. The position vectors r2N 1 and r2N 2 of massesm2N 1 andm2N 2 are on L and on opposite sides with respect to the plane Π. The central configurations studied in this paper are in some measure related to the double pyramidal central configurations d.p.c.c studied in 19 and the paper in 20 . The configuration in 19 consists of n masses on a plane that are located at the vertices of a regular n-gon and two equal masses located on the line perpendicular to passing through the geometric center of the N-gon. And the authors in 20 assumed that the center of the N-gon is at the origin, that is, Σnj 1mjrj 0, and more that the origin of the inertial system is the center of mass of the system, that is, Σ 3 j 1mjrj 0 and rn 3 0. In fact the origin is the geometric center of the N-gon. Hence the configuration in 20 is only to append a mass at the geometric center of the N-gon in 19 . As far as we know, the spatial central configurations studied here are very new. The number of bodies masses is increased to 2N 2, it is not to suppose the origin of the inertial system, and the proofs are more difficult than those in 19, 20 . The main results of this paper are the following. Theorem 1.2. Consider N N 2 bodies with masses m1, m2, . . . , m2N,m2N 1, m2N 2 located according to the following. i r1, r2, . . . , rN are at the vertices of a regular n-gon Γ1 inscribed on a circle of radius α. ii rN 1, rN 2, . . . , r2N are at the vertices of another regular n-gon Γ2 inscribed on a circle of radius a. iii r2N 1 and r2N 2 are on the straight line L, on opposite sides with respect to the plane Π, where L is the straight line that is perpendicular to Π and passes through the geometric center of Γ1 ∪ Γ2. Let h1 distance r2N 1,Π and h2 distance r2N 2,Π . In order that the N N 2 bodies can be in a central configuration (c.c.), the following statements hold. 1 If h1 h2 : h, then there ism2N 1 m2N 2. 4 Abstract and Applied Analysis 2 If h1 h2 : h, then not only m2N 1 m2N 2, but also m1 m2 · · · mN : m, mN 1 mN 2 · · · m2N : m̃. 1.8 3 The origin is the mass center of m1, m2, . . . , m2N and also the mass center of m1, m2, . . . , m2N 2, that is,


Main Results
The Newtonian n-body problem see 1-7 concerns the motion of n point particles with masses m j ∈ R and positions q j ∈ R 3 j 1, . . . , n . The motion is governed by Newton's law: where q q 1 , . . . , q n and U q is the Newtonian potential: that is, suppose that the center of mass is fixed at the origin of the space. Because the potential is singular when two particles have the same position, it is natural to assume that the configuration avoids the set Δ {q : q k q j for some k / j}. The set X \ Δ is called the configuration space.  9 ; if the N bodies are moving towards a simultaneous collision, then the bodies tend to a central configuration see 10 . See also 11, 12 . Some examples of spatial central configurations are a regular tetrahedron with arbitrary positive masses at the vertices 13 and a regular octahedron with six equal masses at the vertices 12 . Double nested spatial central configurations for 2N bodies were studied for two nested regular polyhedra in 14 . More recently, the same authors studied central configurations of three regular polyhedra for the spatial 3N-body problem in 15 . See also 16 , where nested regular tetrahedrons are studied. Santoprete 17 provided new examples of stacked spatial  central configurations-central configurations for the N-body problem where a proper subset  of the N bodies are already on a central configuration-for the 7-body problem where the bodies are arranged as concentric three-and two-dimensional simplex. New classes of stacked spatial central configurations for the 6-body problem which have four bodies at the vertices of a regular tetrahedron and the other two bodies on a straight line connecting one vertex of the tetrahedron with the center of the opposite face were studied in 18 .

Recently, Hampton and
In this paper, we study new classes of spatial double pyramidical central configurations d.p.c.c for the N N 2-body that satisfy the following.
1 The position vectors r 1 , r 2 , . . . , r N of masses m 1 , m 2 , . . . , m N are at the vertices of a regular N-gon Γ 1 , whose sides have length ι 1 . The position vectors r N 1 , r N 2 , . . . , r 2N of masses m N 1 , m N 2 , . . . , m 2N are at the vertices of another regular N-gon Γ 2 , whose sides have length ι 2 . Two N-gons have a same geometric center and form a affine nested N-gons base plane Π .
2 Let L be the straight line perpendicular to the base plane Π that contains Γ 1 and Γ 2 and passes through the geometric center of Γ 1 ∪ Γ 2 . The position vectors r 2N 1 and r 2N 2 of masses m 2N 1 and m 2N 2 are on L and on opposite sides with respect to the plane Π.
The central configurations studied in this paper are in some measure related to the double pyramidal central configurations d.p.c.c studied in 19 and the paper in 20 . The configuration in 19 consists of n masses on a plane that are located at the vertices of a regular n-gon and two equal masses located on the line perpendicular to passing through the geometric center of the N-gon. And the authors in 20 assumed that the center of the N-gon is at the origin, that is, Σ n j 1 m j r j 0, and more that the origin of the inertial system is the center of mass of the system, that is, Σ n 3 j 1 m j r j 0 and r n 3 0. In fact the origin is the geometric center of the N-gon. Hence the configuration in 20 is only to append a mass at the geometric center of the N-gon in 19 .
As far as we know, the spatial central configurations studied here are very new. The number of bodies masses is increased to 2N 2, it is not to suppose the origin of the inertial system, and the proofs are more difficult than those in 19, 20 .
The main results of this paper are the following.

1.10
where b m/m, c m 2N 1 /m and ρ j exp i 2πj/N .

Some Lemmas
where we assume a i,0 a i,N and a 0,j a N,j , then one calls that A is a circular matrix. ii Let A a i,j be an N × N circular matrix; the eigenvalues λ k and the eigenvectors v k of A are

iii Let A and B be circular matrices, and let λ k A and λ k B be eigenvalues of A and B. Then the eigenvalues of
It is clear that.
Abstract and Applied Analysis

2.5
Lemma 2.6. The complex subspace L of C N generated by X where γ stands for the Euler-Mascheroni constant and B 2k stands for the Bernoulli numbers. Lemma 2.9 see 5 . Let Φ ν x j 1/d ν j , where ν > 0 and d j 1 x 2 − 2x cos 2πj/N ; then, for 0 < x < 1, Φ ν x and all of its any order derivatives are positive. Moreover, the same is thus forΨ ν x j cos 2πj/N /d ν j .

The Proof of Theorem
Proof. From central configuration equation 1.4 or 1.5 , we easily prove.
Abstract and Applied Analysis 7 Denote r k x k , y k , z k ∈ R 3 , and z 0, 0, 1 . Observing 1.5 , one could have a free choice of the origin for a configuration. Without loss of generality, consider that the origin is at the geometric center of Γ 1 ∪ Γ 2 , and let z k 0, 1 ≤ k ≤ 2N.
Proof. In 3.1 , considering the equations along the direction z, and k 1, N 1, we have

3.6
By the h 1 h 2 , we have

3.11
Proof. Because if r → εr is a transformation in a central configuration, then λ → 1/ε 2 λ can be a new parameter of a central configuration. We say that the old and the new are equivalent. Hence without loss of generality, we may let α 1, 0 < a < 1. Then the vectors of positions based on the previous assumptions can be interpreted by the following: where ρ k exp 2πk/N i , ρ k aρ k , a > 0, and i √ −1, ρ k denote the N complex kth roots of unity, that is, that m k 1 ≤ k ≤ N each locates at the vertices r k of the one regular N-gon Γ 1 , m N k 1 ≤ k ≤ N each locates at the vertices r N k of the other regular N-gon Γ 2 , and m 2N 1 and m 2N 2 lie on the vertices of r 2N 1 , r 2N

3.16
where k 1, 2, . . . , N. Now we define the N × N circular matrices C c k,j , A a k,j , B b k,j , and D d k,j as follows:

3.17
Also define

3.18
We see that 3.16 holds if and only if the matrix equation

3.23
By 3.21 and 3.22 we have

3.24
From Lemma 2.2 we see that 1 and CD − AB are circular matrix, and we know that they must have positive real eigenvector 1. By the properties of circular matrix, 3.24 can be written as Hence one has the following.
Abstract and Applied Analysis 13 1 If γ 1 0, then γ 2 0. By 3.25 , there are We notice that 3.29 or 3.30 must have positive real solutions, which is equivalent to that CD − AB has positive real eigenvectors corresponding to eigenvalue 0. But we notice that 3.29 and 3.30 must hold, and for k 1, we have an eigenvalue λ 1 0, and an matching eigenvector v 1 1, 1, . . . , 1 T of CD − AB. Noticing that A, B, C, and D are Hermite circular matrices, from the properties of circular and Hermite matrix in Lemmas 2.2 and 2.4, then CD − AB G is also a Hermite circular matrix. We may denote G by cir a 0 , g 1 , g 2 , . . . , g m−1 , g m , g m−1 , . . . , g 2 , g 1 when N 2m, where a 0 , g m ∈ R. We also denote G by cir a 0 , g 1 , g 2 , . . . , g m−1 , g m , g m , . . . , g 2 , g 1 when N 2m 1, where a 0 ∈ R. Using Lemmas 2.4 and 2.5, after complex computation, we may prove that the kernel of G is at most a subspace in L when N 2m > 2, and a subspace in L when N 2m 1 > 3 see 22 , where the meanings of L and L are in Lemma 2.6. Hence the kernel of G does not contain any positive real vectors other than multiplication of 1 1, 1, . . . , 1 . When N 2, 3, we may easily prove the conclusion.

3.46
It suffices to prove that K x, y 0, and L x, y 0 have a unique positive solution x, y for any given ratios b, c > 0 in x ∈ 0, 1 , and y ∈ 0, ∞ .  where x a ∈ 0, 1 . From Lemmas 2.8 and 2.9 and their proofs, and with implicit function theory, after some complex calculation some ideas partially see 23 , we can prove that K 0, and L 0 have only one solution for any given ratios of masses b > 0 and c > 0 such that 0 < x < 1, and 0 < h < ∞.