# On the existence of central configurations of 2k + 2p + 2/-body problems.

1. Introduction and Main Results

The Newtonian n-body problems [1-3] is concerned with the motions of n particles with masses [m.sub.j] [member of] [R.sup.+] and positions [q.sub.j] [member of] [R.sup.3] (j = 1, 2, ..., n); the motion is governed by Newton's second law and the Universal law:

[m.sub.j] [[??].sub.j] = [partial derivative]U (q)/[partial derivative][q.sub.j], (1)

where q = ([q.sub.1], [q.sub.2], ..., [q.sub.n]) with Newtonian potential

U(q)= [summation over (1[less than or equal to]j<k[less than or equal to]n)] [[m.sub.j][m.sub.k]/[absolute value of [q.sub.j] - [q.sub.k]]]. (2)

Consider the space

X = {q = ([q.sub.1], [q.sub.2], ..., [q.sub.n]) [member of] [R.sup.3n]: [N.summation over (j=1)] [m.sub.j][q.sub.j] = 0}; (3)

that is, suppose that the center of mass is fixed at the origin of the coordinate axis. Because the potential is singular when two particles have same position, it is natural to assume that the configuration avoids the collision set [DELTA] = {q = ([q.sub.1], ..., [q.sub.N]): [q.sub.j] = [q.sub.k] for some k [not equal to] j}. The set X\[DELTA] is called the configuration space.

Definition 1 (see [2,3]). A configuration q = ([q.sub.1], [q.sub.2], ..., [q.sub.n]) [member of] X\[DELTA] is called a central configuration if there exists a constant A such that

[n.summation over (j=1,j[not equal to]k)] [[m.sub.j][m.sub.k]/[[absolute value of [q.sub.j] - [q.sub.k]].sup.3]] ([q.sub.j] - [q.sub.k]) = -[lambda][m.sub.k][q.sub.k], 1 [less than or equal to] k [less than or equal to] n. (4)

The value of constant [lambda] in (4) is uniquely determined by

[lambda] = U/I, (5)

where

I = [n.summation over (k=1)] [m.sub.k] [[absolute value of [q.sub.k]].sup.2]. (6)

Since the general solution of the n-body problem cannot be given, great importance has been attached to search for particular solutions from the very beginning. A homographic solution is a configuration which is self-similar for all time. Central configurations and homographic solutions are linked by the Laplace theorem . Collapse orbits and parabolic orbits have relations with the central configurations [2, 4-6]. So finding central configurations becomes very important.

Long and Sun  studied four-body central configurations with some equal masses; Albouy et al.  took an alternate approach to the symmetry of planar four-body convex central configurations; Llibre and Mello  considered triple and quadruple nested central configurations; Corbera and Llibre  analyzed the existence of central configurations of p nested regular polyhedra. Based on the above works, we study central configuration for Newtonianc 2k + 2p + 2l-body problems. In the 2k + 2p + 2/-body problems, 2k masses are symmetrically located on the z-axis, 2p masses are symmetrically located on the y-axis, and 2l masses are symmetrically located on the x-axis, the masses located on the same axis and symmetrical about the origin are equal (see Figure 1 for k = 1, p = 2, and l = 3).

In this paper, we extend the result of Corbera-Llibre by the following.

Theorem 2. For 2k + 2p + 2l-body problem in [R.sup.3] where k [greater than or equal to] 1, p [greater than or equal to] 1, and l [greater than or equal to] 1, there is at least one central configuration such that 2k points are located along the first axis symmetric with respect to the origin, 2p points are located on the second axis symmetric with respect to the origin, and 2l points are located on the third axis, also symmetric with respect to the origin. Along each axis, the masses symmetric with respect to the origin are equal.

2. The Proof of Theorem 2

2.1. Equations for the Central Configurations of 2+2+2l-Body Problems. To begin, we take a coordinate system which simplifies the analysis. The particles have positions given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where i = 1, ..., l, 0 < [x.sub.1] < [x.sub.2] < ... < [x.sub.1].

The masses are given by [m.sub.1] = [m.sub.2] = [M.sub.1] = 1, [m.sub.3] = [m.sub.4] = [M.sub.2], and [m.sub.4+i] = [m.sub.4+l+i] = [M.sub.2+i], where i = 1, ..., l.

By the symmetries of the system, (4) is equivalent to the following equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

for i = 2, ..., l - 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

In order to simplify the equations, we defined the following: [a.sub.i,1] = 1, where i = + l; [a.sub.1,2] = -2/[[absolute value of 1 + [y.sup.2.sub.1]].sup.3/2]; [a.sub.1,2+j] = -2/[[absolute value of 1 + [x.sup.2.sub.j]].sup.3/2], where j = 1, ..., l; [a.sub.2,2] = -1/4 [y.sup.3.sub.1], [a.sub.2,2+j] = -2/[[absolute value of [x.sup.2.sub.j] + [y.sup.2.sub.1]].sup.3/2], where j = 1, ..., l; [a.sub.j+2,j+2] = -1/4[x.sup.3.sub.j], where j = 1, ..., l; [a.sub.i+2,2] = -2/[[absolute value of [y.sup.2.sub.1] + [x.sup.2.sub.i]].sup.3/2], where i = 1, ..., l; [a.sub.i+2,j+2] = -(1/[[absolute value of [x.sub.j] - [x.sub.i]].sup.2] [x.sub.i]) - (1/[[absolute value of [x.sub.j] + [x.sub.i]].sub.2] [x.sub.i]), where 1 [less than or equal to] j < i [less than or equal to] 1; [a.sub.i+2,j+2] = 1/[[absolute value of [x.sub.j] - [x.sub.i]].sup.2] [x.sub.i] - 1/[[absolute value of [x.sub.j] + [x.sub.i]].sup.2] [x.sub.i], where 1 [less than or equal to] i < j [less than or equal to] l; [b.sub.1] = 1/4, [b.sub.2] = 2/[[absolute value of 1 + [y.sup.2.sub.1]].sup.3/2], [b.sub.i+2] = 2/[[absolute value of 1 + [x.sup.2.sub.i]].sup.3/2], where i = 1, ..., l.

Equations (8) and 9) can be written as a linear system of the form AX = b given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

The column vector is given by the variables X =[ ([lambda], [M.sub.2], [M.sub.2+1], ..., [M.sub.2+l]).sup.T]. Since the are function of [y.sub.1], [x.sub.1], ..., [x.sub.l] we write the coefficient matrix as [A.sub.2+l] ([y.sub.1], [x.sub.1], ..., [x.sub.l]).

2.2. For k = 1, p = 1, and Z = 1

Lemma 3. When p = 1, k = 1, and Z = 1, there exists [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that (10) has a unique solution ([lambda], [M.sub.2], [M.sub.3]) satisfying [lambda], [M.sub.2] > 0 and [M.sub.3] > 0.

Proof. If we consider [absolute value of [A.sub.2+1]([x.sub.1], [y.sub.1])] as a function of [x.sub.1] and [y.sub.1], then [absolute value of [A.sub.2+1] ([x.sub.1], [y.sub.1])] is an analytic function and nonconstant. When p = 1, k = 1, and Z = 1, 10) is equivalent to

[F.sub.1] ([x.sub.1], [y.sub.1], [M.sub.2], [M.sub.3]) = 0, [F.sub.2] ([x.sub.1], [y.sub.1], [M.sub.2], [M.sub.3]) = 0, (11)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

It is obvious that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

By implicit function theorem, there exists [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that (10) has a unique solution ([lambda], [M.sub.2], [M.sub.3]) satisfying [lambda], [M.sub.2] > 0 and [M.sub.3] > 0.

2.3. For k = 1, p = 1, and l > 1. The proof for k = 1, p = 1, and l [greater than or equal to] 1 is done by induction. At each step, we assume we have inductively shown the existence of a central configuration symmetrically located at the v-axis, and we add two particles with zero masses. Using continuity with respect to the masses of the above defined equations, we prove the existence of a central configuration when the mass of the added particles is zero and the mass of the remaining particles are changed suitably. We then verify that the Hessian of this new system of masses has no kernel and we conclude that the two new particles can have nonzero masses which still maintain a central configuration.

For k = 1, p = 1, and l [greater than or equal to] 1, we claim that there exists 0 < [y.sub.1], 0 < [x.sub.1] < [x.sub.2] < ... < [x.sub.l] such that system (10) have a unique solution [lambda] = [lambda]([y.sub.1], [x.sub.1], ..., [x.sub.l]) > 0, [M.sub.i] = [M.sub.i]([y.sub.1], [x.sub.1],..., [x.sub.l]) > 0 for i = 2, ..., l + 2. We have seen that the claim is true for k = 1, p = 1, and l = 1.By induction, we assume the claim is true for k = 1, p = 1, and l - 1, and we will prove it for k = 1, p = 1, and l.

Assume by induction hypothesis that there exists [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that system (10) with l - 1 instead of l has a unique solution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We need the next Lemma.

Lemma 4. There exists [[??].sub.l] > [[??].sub.l-1] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for i = 2, ..., l + 1, and [[??].sub.l+2] = 0 is a solution of (10).

Proof. Since [[??].sub.l+2] = 0, we have that the first l + 1 equations of (10) are satisfied when [lambda] = [??] and [M.sub.i] = [[??].sub.i] for i = 2, ..., l + 1 and [M.sub.l+2] = [[??].sub.l+2] = 0. Substituting this solution into the last equations of (10), we get the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

We have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Therefore, there exists at least a value [x.sub.l] = [[??].sub.l] > [[??].sub.l-1] satisfying equation f([x.sub.l]) = 0. This completes the proof of Lemma 4.

By using the implicit function theorem we will prove that the solution of (10) given in Lemma 4 can be continued to a solution with [M.sub.l+2] > 0.

Let s = ([lambda], [M.sub.2], ..., [M.sub.l+1], [M.sub.l+2], [y.sub.1], [x.sub.1], ..., [x.sub.l]), we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

It is not difficult to see that the system (10) is equivalent to [g.sub.i](s) = 0 for i = 1, ..., l + 2. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the solution of system (10) given in Lemma 4. The differential of (16) with respect to the variables ([lambda], [M.sub.2], [M.sub.2+1], ..., [M.sub.2+l-1], [x.sub.l]) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

We have assumed that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] makes the system (10) with l - 1 instead of l have a unique solution, so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; therefore, D([??]) [not equal to] 0. Applying the implicit function theorem, there exists a neighborhood U of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and unique analytic functions [lambda] = [lambda]([M.sub.l+2], [y.sub.1], [x.sub.1], ..., [x.sub.l-1]), [M.sub.i] = [M.sub.i]([M.sub.l+2], [y.sub.1], [x.sub.1], ..., [x.sub.l-1]) for i = 2, ..., l + 1 and [x.sub.l] = [x.sub.l]([M.sub.l+2], [y.sub.1], [x.sub.1], ..., [x.sub.l-1]), such that ([lambda], [M.sub.2], [M.sub.2+1], ..., [M.sub.2+l]) is the solution of the system (10) for all ([M.sub.l+2], [y.sub.1], [x.sub.1], ..., [x.sub.l-1]) [member of] U. The determinant is calculated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (19)

where [B.sub.2+l,j] is the algebraic cofactor of [a.sub.2+l,j].

We see that [absolute value of [A.sub.2+l-1] ([y.sub.1], [x.sub.1], ..., [x.sub.l-1])], [B.sub.2+l,j], and [a.sub.2+l,j] for j = 1, ..., 2 + l - 1 do not contain the factor 1/[x.sup.3.sub.l]. If we consider [absolute value of [A.sub.2+l] ([y.sub.1], [x.sub.1], ..., [x.sub.l])] as a function of [x.sub.l], then [absolute value of [A.sub.2+l] ([y.sub.1], [x.sub.1], ..., [x.sub.l])] is analytic and nonconstant. It is obvious that [partial derivative][g.sub.i]([??])/[partial derivative][M.sub.l+2] = [a.sub.i,l+2] [not equal to] 0 for i = 1, 2, ..., 1 + 2. We can find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] sufficiently close to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and, therefore, a solution ([lambda], [M.sub.2], ..., [M.sub.2+l]) of system (10) satisfying [lambda] > 0, [M.sub.i] > 0 for i = 2, ..., l + 2.

2.4. For k [greater than or equal to] 1, p [greater than or equal to] 1, and l [greater than or equal to] 1. The proof for k = 1, p [greater than or equal to] 1, and l [greater than or equal to] 1 is also done by induction. At each step, two particles are symmetrically located at the y-axis. Using the similar arguments above we prove Theorem 2 for k = 1, p [greater than or equal to] 1, and l [greater than or equal to] 1. At each step, two particles are symmetrically located at the z-axis. Using the similar arguments above we prove Theorem 2 for k [greater than or equal to] 1, p [greater than or equal to] 1, and l [greater than or equal to] 1.

The proof of Theorem 2 is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by Youth Found of Mianyang Normal University. The authors express their gratitude to the reviewers for their useful suggestions and constructive criticism.

References

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Yueyong Jiang (1) and Furong Zhao (1,2)

(1) Department of Mathematics and Computer Science, Mianyang Normal University, Mianyang, Sichuan 621000, China

(2) Department of Mathematics, Sichuan University, Chengdu 610064, China

Correspondence should be addressed to Furong Zhao; zhaofurong2006@163.com

Received 24 March 2014; Accepted 25 June 2014; Published 9 July 2014

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