New spatial central configurations in the 5-body problem

New spatial central configurations in the 5-body problem

Luis F. Mello; Antonio C. Fernandes

Instituto de Ciências Exatas, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, 37500-903 Itajubá, MG, Brasil

^{Correspondence to}

ABSTRACT

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.

Key words: central configurations, spatial configurations, 5-body problem, convex configuration, concave configuration.

RESUMO

Neste artigo estudamos a existência de novas famílias de configurações centrais espaciais dos tipos côncavas ou convexas para o problema de 5 corpos. Os corpos estudados aqui estão dispostos da seguinte maneira: três corpos estão sobre os vértices de um triângulo equilátero T e os outros dois corpos estão sobre a reta que passa pelo baricentro de T e é perpendicular ao plano que contém T.

Palavras-chave: configurações centrais, configurações espaciais, problema de 5 corpos, configuração convexa, configuração côncava.

INTRODUCTION

Consider n punctual positive masses m₁,..., m_n with position vectors r₁,..., r_n. Usually r_i ∈ , 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_ij = |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 m_jr_j/M, where M = m₁ + ... + m_n is the total mass. The configuration space is defined by .

At a given instant t = t₀ the n bodies make a central configuration if there exists such that , for all i = 1,..., n. Two central configurations (r₁, r₂,..., r_n), 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 Moeckel1990); there is a relation between central configurations and the bifurcations of the hyper surfaces 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 (Hagihara1970, 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-bodyproblem 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 r₁, r₂ and r₃ 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 r₄∈ ρ is fixed and does not belong to the plane that contains T;

3. The position vector .

This type of configuration is called if one body is located in the interior of the convex hull of the 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, 3and 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 r₁ = (x, 0, 0), r₂ = , r₃ = , r₄ = and r₅ = (0, 0, y) with x > 0 and . See Fig. 2. Thus, x is the radius of the circumscribed circle that contains r₁, r₂, r₃, and y is the signed height of the body 5 with respect to the plane that contains the triangle T. There is no special reason to take r₄ at . With this choice, the length of the edge of theregular 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 , 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 and Shanzhong2002). 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.

The main results of this paper are the following ones.

THEOREM 1. Consider the position vectors

r₁ = (x, 0, 0),

at the vertices of an equilateral triangle T, and the position vectors ,r₅ = (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.

1. There exists a minimum positive value x = x_min = such that if 0 < x < x_minthere are no positions r₁,..., r₅and positive masses m₁,..., m₅such that these bodies are in a central configuration according to Fig. 2;

2. There are two open intervals I₁ = and such that for each there is one non-empty segment of possible positions for r₅and positive masses m₁,..., m₅such that these bodies form a 1-parameter family of concave central configurations of type 1;

3. There exists one distinguished and well-determined value for which there is just one position for r₅such 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 such that for each x ∈ I₃there is one non-empty segment of possible positions for r₅and positive masses m₁,..., m₅such that these bodies form a 1-parameter family of convex central configurations;

5. There exists one open and unbounded interval such that for each x ∈ I₄there is one non-empty segment of possible positions for r₅and positive masses m₁,..., m₅such that these bodies form a 1-parameter family of concave central configurations of type 2;

6. There exists one distinguished and well-determined value for which there is just one position for r₅such that these five bodies form a 2-parameter family of concave central configurations of type 2. As in the above item concavethis3, The main results of this paper are the following ones. 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.

PROOF OF THEOREM 1

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_ij = 1/r³_ij and . In fact, 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. 295of (Hampton and Santoprete 2007) and the references therein)

for 1 < i < j < n, h = 1,..., n, h ≠ i, j. Here, again, R_ij = 1/r³_ij and . Thus, 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.

For the proof of Theorem 1 we use Dziobek equations (4). For five bodies, system (4) is a set of 30 equations. From the distances between the bodies it follows that R₁₂ = R₁₃ = R₂₃, R₁₄ = R₂₄ = R₃₄ and R₁₅ = R₂₅ = R₃₅. Taking into account the symmetries, we have the following equalities among others:

Taking these symmetries into the Dziobek equations, the following 9 equations of (4) are trivially satisfied f₁₂₃ = 0, f₁₂₄ = 0, f₁₂₅ = 0, f₁₃₂ = 0, f₁₃₄ = 0, f₁₃₅ = 0, f₂₃₁ = 0, f₂₃₄ = 0 and f₂₃₅ = 0. Again looking for symmetries, we have the following equivalence between the equations:

As our classes of central configurations satisfy , we have . From equations (5), (6) and (7), we have two cases to analyze: the masses m₁ = m₂ = m₃ and r₁, r₂, r₃ and r₄ are at the vertices of a regular tetrahedron.

CASE 1. Consider r₁, r₂, r₃ and r₄ at the vertices of a regular tetrahedron. We have the following lemma.

LEMMA 3. Consider r₁, r₂, r₃and r₄at the vertices of a regular tetrahedron. Then, there exists just one position for r₅at the center of the tetrahedron and positive masses m₁ = m₂ = m₃ = m₄ = m and m₅such that these bodies form a 2-parameter family of concave central configurations of type 1.

PROOF. Due to symmetries, we have the following equivalences between equations of (4):

Hence the remaining 21 equations are reduced to 15 equations. From f₁₄₂ = 0, f₁₄₃ = 0 and f₂₄₃ = 0 we have m₅ (R₁₅ - R₄₅) = 0, m₅ (R₁₅ - R₄₅) = 0 and m₅ (R₂₅ - R₄₅) = 0, respectively. As , and , it follows that R₁₅ = R₄₅ = R₂₅. This implies that r₅ must be at the center of the tetrahedron. Adding this information into f₁₅₂ = 0, f₁₅₃ = 0 and f₂₅₃ = 0, we have (m₃ - m₄)(R₁₃ - R₃₅) = 0, (m₂ - m₄)(R₁₂ - R₂₅) = 0 and (m₁ - m₄)(R₁₂ - R₁₅) = 0, respectively. These last three equations are verified just when m₁ = m₂ = m₃ = m₄. By another side m₅ can assume any positive value. The remaining equations of (4) are trivially satisfied. The lemma is proved.

Define . Thus, , , , and . From Lemma 3, item 3 of Theorem 1 is proved. In the remainder of the statements we omit this type of concave central configuration.

CASE 2. Consider m₁ = m₂ = m₃ = m. It follows that equations (5), (6), (7), f154 = 0, f₄₅₁ = 0, f₂₅₄ = 0, f₄₅₂ = 0, f₃₅₄ = 0 and f₄₅₃ = 0 are satisfied, and

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 m₄ = m₄(x, y, m) and m₅ = m₅(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 whose ratios of the masses m₄/m and m₅/m are positive. For the study of the signs of the terms that appear in equations (10) and (11), we have R₃₅ - R₁₃ = 0 if and only if (straight lines), R₁₄ - R₄₅ = 0 if and only if (hyperbolas), R₃₄ - R₁₃ = 0 if and only if if and only if (parabola), = 0 if and only if (x, y) if and only if if and only if . See Figs. 3 and 4.

CASE 2.1. Consider x > 0 and y = 0.

LEMMA 4. Consider r₅ = (0, 0, 0), that is y = 0. Then, there is no value x > 0 such that 5 bodies with positive masses form a central configuration according to Fig. 2.

PROOF. From equation (9) we have

m (R₁₃ - R₃₅) + m₄ (R₁₄ - R₄₅) = 0.

By assumption, and >; then, m₄ = 0. This is a contradiction.

CASE 2.2. Consider and . We have the following lemma.

LEMMA 5. Consider . Then, there is no position for the body 5 on the line ρ and positive masses m_i, i = 1,..., 5 such that these bodies form a central configuration according to Fig. 2.

PROOF. There are three cases to analyze: .

Consider y < 0. From equation (8) we have

m (R₁₃ - R₃₄) + m₅ (R₁₅ - R₄₅) = 0.

By assumption R₁₃ - R₃₄ < 0, R₁₅ - R₄₅ < 0, > 0 and > 0. Therefore, the coefficients of the above equation have the same sign. This implies that m and m₅ must have opposite signs.

Consider . By assumption R₁₃ - R₃₄ < 0, R₁₃ - R₃₅ > 0, R₁₄ - R₄₅ < 0, > 0, > 0, > 0, < 0, R₁₅ - R₄₅ < 0, if and R₁₅ - R₄₅ > 0, if .

From equation (8) we have

m (R₁₃ - R₃₄) + m₅ (R₁₅ - R₄₅) = 0.

If , then the coefficients of the above equation have the same sign. Therefore, the masses m and m₅ have opposite signs. Now, from equation (9), we have

m (R₁₃ - R₃₅) + m₄ (R₁₄ - R₄₅) = 0.

If then the coefficients of the above equation have the same sign. Therefore, the masses m and m₄ have opposite signs. If , then R₁₅ - R₄₅ = 0 and this implies that the mass m must be zero in equation (8).

Consider . From equation (8) we have

m (R₁₃ - R₃₄) + m₅ (R₁₅ - R₄₅) = 0.

By assumption R₁₃ - R₃₄ < 0, R₁₅ - R₄₅ > 0, > 0 and < 0. Therefore, the coefficients of the above equation have the same sign. This implies that m and m₅ must have opposite signs.

From Lemma 5 item 1 of Theorem 1 is proved. The value is defined as the x-coordinate of an intersection of the curves R₃₅ - R₁₃ = 0 and R₁₅ - R₄₅ = 0 (see Fig. 3).

CASE 2.3. Consider x > 0 and 0 .

Define (see Fig. 3) where

For (x, y) we have R₃₅ - R₁₃ > 0, R₃₄ - R₁₃ < 0, R₁₅ - R₄₅ < 0, R₁₄ - R₄₅ < 0, > 0, < 0, > 0 and > 0. For (x, y) ∈ H₂ we have x > 0, 0 < y < √6/3, R₃₅ - R₁₃ > 0, R₃₄ - R₁₃ > 0, R₁₅ - R₄₅ > 0, R₁₄ - R₄₅ < 0, > 0, < 0, > 0 and > 0.

It is simple to see that for (x, y) the right-hand sides of equations (10) and (11) are positive and, therefore, we have concave central configurations of type 1.

The orthogonal projections of the open sets onto the x-axis give two open intervals by , respectively. For each the straight line x = x_* intersects in a non-empty segment. This proves item 2 of Theorem 1.

CASE 2.4. Consider x > 0 and y < 0.

In order to have convex central configurations, it is necessary that y < 0. Define the open set (see Fig. 3), where

For (x, y) we have x > 0, y < 0, R₃₅ - R₁₃ > 0, R₃₄ - R₁₃ > 0, R₁₅ - R₄₅ > 0, R₁₄ - R₄₅ > 0, < 0, < 0, > 0 and > 0.

It is simple to see that, for (x, y) the right-hand sides of equations (10) and (11) are positive and, therefore, we have convex central configurations. The orthogonal projection of the open set onto the x-axis gives one open interval

For each x_* the straight line x = x_* intersects in a non-empty segment. This proves item 4 of Theorem 1.

CASE 2.5. Consider x > 0 and .

Define the unbounded open set (see Fig. 4)

For (x, y) we have x > 0, y > , R₃₅ - R₁₃ > 0, R₃₄ - R₁₃ > 0, R₁₅ - R₄₅ < 0, R₁₄ - R₄₅ > 0, > 0, > 0, > 0 and < 0.

It is simple to see that, for (x, y) , 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 onto the x-axis gives one open and unbounded interval I₄ = . For each the straight line x = x_* intersects in a non-empty segment. This proves item 5 of Theorem 1.

CASE 2.6. Consider r₁, r₂, r₃ and r₅ at the vertices of a regular tetrahedron. We have the following lemma.

LEMMA 6. If m₁ = m₂ = m₃ = m₅ = m and m₄are at r₁ = , r₂ = , r₃ = , r₄ = and r₅ = , then these bodies form a 2-parameter family of concave central configurations of type 2.

PROOF. By assumption, and . For these values, we have R₁₃ - R₃₅ = 0 and R₁₄ - R₄₅ = 0. Therefore, equation (9) is satisfied for all m₄ > 0. As m₅ = m, equation (8) can be written as

With the above values of x and y, we have r₁₃ = r₁₅ = 4, r₃₄ = r₄₅ = . By a simple calculation, (R₁₃ - R₃₄) + (R₁₅ - R₄₅) = 0. It is simple to see that r₁, r₂, r₃ and r₅ are at the vertices of a regular tetrahedron, and r₄ is at the center of this tetrahedron.

The proof of item 6 of Theorem 1 follows Lemma 6 in which . 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.

ACKNOWLEDGMENTS

The first author is partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico(CNPq), grants 476672/2009-0 and 304926/2009-4. The second author is partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).

Manuscript received on January 4, 2010; accepted for publication on August 9, 2010

AMS Classification: Primary 70F10, 70F15, 37N05.

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Publication Dates

• Publication in this collection
  22 July 2011
• Date of issue
  Sept 2011

History

• Received
  04 Jan 2010
• Accepted
  09 Aug 2010