New spatial central configurations in the 5-body problem

In this paper we show the existence of new families of convex and concave spatial central configurations for the 5-body problem. The bodies studied here are arranged as follows: three bodies are at the vertices of an equilateral triangle T , and the other two bodies are on the line passing through the barycenter of T that is perpendicular to the plane that contains T .


INTRODUCTION
Consider n punctual positive masses m 1 , . . ., m n with position vectors r 1 , . . ., r n .Usually r i ∈ R d , d = 2, 3.The Newtonian n-body problem in celestial mechanics consists in studying the motion of theses masses interacting amongst themselves through no other forces than their mutual gravitational attraction according to Newton's gravitational law (Newton 1687).
In this paper we denote the Euclidean distance between the bodies of masses m i and m j by r i j = |r i − r j |.We take the inertial barycentric system, that is the origin of the inertial system is located at the center of mass of the system, which is given by n j=1 m j r j /M, where M = m 1 + . . .+ m n is the total mass.The configuration space is defined by {(r 1 , r 2 , . . ., r n ) ∈ R dn : r i = r j , i = j}.
At a given instant t = t 0 the n bodies make a central configuration if there exists λ = 0 such that ri = λr i , for all i = 1, . . ., n.Two central configurations (r 1 , r 2 , . . ., r n ), (r 1 , r2 , . . ., rn ) of the n bodies are said to be related if we can pass from one to the other through a dilation and a rotation (centered at the center of mass).So we can study the classes of central configurations defined by the above equivalence relation.
In the 3-body problem the collinear solutions of Euler (Euler 1767) and the triangular equilateral solutions of Lagrange (Lagrange 1873) are the first examples in which the bodies are in central configuration at any instant of time.The Euler collinear central configurations were generalized by Moulton in (Moulton 1910) who showed that, to given n masses, the number of collinear central configurations is exactly n!/2.It is know that planar regular n-gons with n equal masses at the vertices are in a central configuration.This is a generalization of Lagrange's result.
The knowledge of central configurations allows us to compute homographic solutions (see Moeckel 1990); there is a relation between central configurations and the bifurcations of the hypersurfaces of constant energy and angular momentum (see Smale 1970); if the n bodies are going to a simultaneous collision, then the bodies tend to a central configuration (see Saari 1980).See also the following references (Hagihara 1970, Moulton 1910, Wintner 1941).
Some examples of spatial central configurations are a regular tetrahedron with arbitrary positive masses at the vertices (Lehmann-Filhés 1891) and a regular octahedron with six equal masses at the vertices (Wintner 1941).
There are also the stacked spatial central configurations, that is central configurations for the n-body problem in which a proper subset of the n bodies is already on a central configuration.Double nested spatial central configurations for 2n bodies were studied for two nested regular polyhedra in (Corbera and Llibre 2008).More recently, the same authors studied central configurations of three regular polyhedra for the spatial 3n-body problem in (Corbera and Llibre 2009).See also (Zhu 2005) in which nested regular tetrahedrons are studied.
Recently Hampton and Santoprete (Hampton and Santoprete 2007) provided new examples of stacked spatial central configurations for the 7-body problem in which the bodies are arranged as concentric three and two dimensional simplexes.New classes of stacked spatial central configurations for the 6-body problem that have four bodies at the vertices of a regular tetrahedron and the other two bodies on a line connecting one vertex of the tetrahedron with the center of the opposite face are studied in (Mello et al. 2009a).
In this paper we study spatial central configurations for the 5-body problem that satisfy (see Fig. 1(a) and 1(b)): 1.The position vectors r 1 , r 2 and r 3 are at the vertices of an equilateral triangle T ; 2. Let ρ be the line passing through the barycenter of T that is perpendicular to the plane that contains T .The position vector r 4 ∈ ρ is fixed and does not belong to the plane that contains T ; 3. The position vector r 5 ∈ ρ, r 5 = r 4 .
This type of configuration is called concave if one body is located in the interior of the convex hull of the other four bodies (see Fig. 1(a)), otherwise the configuration is called convex (see Fig. 1(b)).We say that the configuration is concave of type 1 if the body 5 is in the interior of the convex hull of the bodies 1, 2, 3 and 4. On the other hand, if the body 4 is in the interior of the convex hull of the bodies 1, 2, 3 and 5, we say that the configuration is concave of type 2.
In order to be more precise and without loss of generality, consider a coordinate system such that r 1 = (x, 0, 0), r 2 = (−x/2, − √ 3x/2, 0), r 3 = (−x/2, √ 3x/2, 0), r 4 = (0, 0, √ 6/3) and r 5 = (0, 0, y) with x > 0 and y = √ 6/3.See Fig. 2. Thus, x is the radius of the circumscribed circle that contains r 1 , r 2 , r 3 , and y is the signed height of the body 5 with respect to the plane that contains the triangle An Acad Bras Cienc (2011) 83 (3 T .There is no special reason to take r 4 at (0, 0, √ 6/3).With this choice, the length of the edge of the regular tetrahedron is 1.Note that the body 5 is at the barycenter of T if and only if y = 0. Therefore, the configuration is convex if and only if y ≤ 0, and the configuration is concave of type 1 (type 2, respectively) if and only if 0 < y < √ 6/3 (y > √ 6/3, respectively).As far as we know, the spatial central configurations studied here are new and are, in a certain sense, generalizations of the kite (planar) central configurations (Bernat et al. 2009, Yiming andShanzhong 2002).See also (Mello et al. 2009b). Leandro in (Leandro 2003) also studied the central configurations presented here from another point of view.More precisely, Leandro studied the finiteness and bifurcations of this class of central configurations.

LUIS F. MELLO and ANTONIO C. FERNANDES
The main results of this paper are the following ones.THEOREM 1.Consider the position vectors at the vertices of an equilateral triangle T , and the position vectors r 4 = 0, 0, √ 6/3 , r 5 = (0, 0, y) on ρ, where ρ is the line passing through the barycenter of T and perpendicular to the plane that contains T , according to Fig. 2. In this way, the following statements hold.

There exists a minimum positive value x
x min there are no positions r 1 , . . ., r 5 and positive masses m 1 , . . ., m 5 such that these bodies are in a central configuration according to Fig. 2;

There are two open intervals I
/3 such that for each x ∈ I 1 ∪ I 2 there is one non-empty segment of possible positions for r 5 and positive masses m 1 , . . ., m 5 such that these bodies form a 1-parameter family of concave central configurations of type 1; 3. There exists one distinguished and well-determined value x = x for which there is just one position for r 5 such that these five bodies form a 2-parameter family of concave central configurations of type 1. Indeed this concave central configuration is exactly the well-known central configuration whose four equal masses are at the vertices of a regular tetrahedron and the fifth mass is at the center of the tetrahedron; 4.
There exists one open interval I 3 = √ 3/3, (2 √ 3 + 3 √ 2)/3 such that for each x ∈ I 3 there is one non-empty segment of possible positions for r 5 and positive masses m 1 , . . ., m 5 such that these bodies form a 1-parameter family of convex central configurations;

5.
There exists one open and unbounded interval I 4 = 4 √ 3/3, +∞ such that for each x ∈ I 4 there is one non-empty segment of possible positions for r 5 and positive masses m 1 , . . ., m 5 such that these bodies form a 1-parameter family of concave central configurations of type 2; 6.There exists one distinguished and well-determined value x = x for which there is just one position for r 5 such that these five bodies form a 2-parameter family of concave central configurations of type 2. As in the above item 3, this concave central configuration is exactly the one whose four equal masses are at the vertices of a regular tetrahedron and the fifth mass is at the center of the tetrahedron.
REMARK 2. Items 2, 3 and 4 of Theorem 1 are closely related with some results obtained by Leandro in (Leandro 2003).However, the assumptions, statements and proofs presented here are different and much simpler than those that appear in (Leandro 2003).Items 3 and 6 of Theorem 1 are not new.We have included them here for completeness.
The proof of Theorem 1 is given in the next section.Concluding comments are presented in Section 3. The equations of motion of the n-body problem are given by for i = 1, 2, . . ., n.In (1) the gravitational constant is taken equal to one.
From the definition of central configuration, equation ( 1) can be written as for i = 1, 2, . . ., n.For the planar case, that is d = 2, simple computations allow us to write equation ( 2) in the following form for 1 ≤ i < j ≤ n, where R i j = 1/r 3 i j and i jk = (r i − r j ) ∧ (r i − r k ).In fact, i jk is twice the oriented area of the triangle formed by the bodies of masses m i , m j and m k .See the references (Hagihara 1970, Mello et al. 2009b).These n(n − 1)/2 equations are called Dziobek or Laura-Andoyer equations.
The computation of spatial central configuration is very difficult if we begin with equation (2).Instead of working with equation (2) we shall use another equivalent system of equations (see equation ( 6), p. 295 of (Hampton and Santoprete 2007) and the references therein) for 1 ≤ i < j ≤ n, h = 1, . . ., n, h = i, j.Here, again, R i j = 1/r 3 i j and i j hk = (r i − r j ) ∧ (r j − r h ) • (r h − r k ).Thus, i j hk gives six times the signed volume of the tetrahedron formed by the bodies of masses m i , m j , m h and m k .These n(n − 1)(n − 2)/2 equations are also called here Dziobek equations.
Again looking for symmetries, we have the following equivalence between the equations: As our classes of central configurations satisfy r 4 = r 5 , we have 1452 = 0, 2451 0 and 3451 = 0.
From equations ( 5), ( 6) and ( 7), we have two cases to analyze: the masses m 1 = m 2 = m 3 and r 1 , r 2 , r 3 and r 4 are at the vertices of a regular tetrahedron.
CASE 1.Consider r 1 , r 2 , r 3 and r 4 at the vertices of a regular tetrahedron.We have the following lemma.
LEMMA 3. Consider r 1 , r 2 , r 3 and r 4 at the vertices of a regular tetrahedron.Then, there exists just one position for r 5 at the center of the tetrahedron and positive masses m 1 = m 2 = m 3 = m 4 = m and m 5 such that these bodies form a 2-parameter family of concave central configurations of type 1.

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In other words, the initial 30 equations were reduced to 2 equations, which are the following: Equations ( 8) and ( 9) can be explicitly solved in the form m 4 = m 4 (x, y, m) and m 5 = m 5 (x, y, m), respectively.In these equations, m can be understood as a parameter for the central configurations.
From equations ( 8) and ( 9) we have We wish to find subsets of D = {x > 0, y ∈ R, y = √ 6/3} whose ratios of the masses m 4 /m and m 5 /m are positive.For the study of the signs of the terms that appear in equations ( 10) and ( 11), we have  Define the unbounded open set (see Fig. 4) 1523 > 0, 1524 > 0, 1423 > 0 and 1425 < 0.
It is simple to see that, for (x, y) ∈ U , the right-hand sides of equations ( 10) and ( 11) are positive and, therefore, we have concave central configurations of type 2.
The orthogonal projection of the open set U onto the x-axis gives one open and unbounded interval I 4 = (4 √ 3/3, +∞).For each x * ∈ I 4 the straight line x = x * intersects U in a non-empty segment.This proves item 5 of Theorem 1. CASE 2.6.Consider r 1 , r 2 , r 3 and r 5 at the vertices of a regular tetrahedron.We have the following lemma.LEMMA The proof of item 6 of Theorem 1 follows Lemma 6 in which x = 4 √ 3/3.In short we have proved Theorem 1.

CONCLUDING COMMENTS
In this paper it was shown the existence (and the nonexistence) of concave/convex spatial central configurations in the 5-body problem.The spatial central configurations studied here are generalizations of the kite planar central configurations.
Some of the results presented here (see Remark 2) are closely related with the results presented by Leandro in (Leandro 2003).However, the methods used in the two articles are different.The techniques used here are very simple, and the use of the computer is not necessary in the proofs of the results presented (see Theorem 1 and its proof).In this sense, the two articles are complementary.