New classes of spatial central configurations for the 7-body problem

Abstracts

In this paper we show the existence of new families of spatial central configurations for the 7-body problem. In the studied spatial central configurations, six bodies are at the vertices of two equilateral triangles and one body is located out of the parallel distinct planes containing . The results have simple and analytic proofs., and

Spatial central configuration; 7-body problem; stacked central configuration; central configuration

Neste artigo estudamos a existência de novas famílias de configurações centrais espaciais para o problema de 7 corpos. Nas configurações estudadas aqui seis corpos estão nos vértices de dois triângulos equiláteros e um corpo está localizado fora dos planos paralelos distintos contendo . Os resultados apresentados aqui tem provas simples e analíticas., e

configuração espacial; problema de 7 corpos; configuração central; configuração central empilhada

INTRODUCTION

In this paper we study spatial central configurations for theN-body problem. Before we can address our problem, some definitions are in order. Consider N punctual bodies with massesm_i > 0 located at the points r_i of the Euclidean space i = 1, ..., N. Assume that the origin of the inertial system is the center of mass of the system (inertial barycentric system). The set is called space of configurations. for

For the Newtonian N-body problem a configuration of the system is central if the acceleration of each body is proportional to its position relative to the inertial barycentric system. It is usual to study classes of central configurations modulo dilations and rotations. See Hagihara (1970), Moeckel (1990), Saari (1980),Smale (1970), Wintner (1941) and references therein for more details.

Spatial central configurations give rise to homothetic orbital motions which are the simplest solutions of the N-body problem. However to know the central configurations for a given set of bodies with positive masses is a very hard and unsolved problem even in the case of few bodies. For instance, inLehmann-Filhés (1891) and Wintner (1941) can be found classical examples of spatial central configurations where the bodies with suitable masses are at the vertices of a regular tetrahedron and a regular octahedron, respectively. More recent examples were studied in Corbera and Llibre (2008) and Corbera and Llibre (2009) in which 2N and 3N bodies are arranged at the vertices of two and three nested regular polyhedra, respectively. See also Zhu (2005) in which nested regular tetrahedrons were studied.

A stacked spatial central configuration is defined as a central configuration for the spatial N-body problem where a proper subset of the N bodies is already in a central configuration. See Hampton and Santoprete (2007), Mello and Fernandes (2011a, b), Mello et. al. (2009) and Zhang and Zhou (2001).

Denote by r_ij = |r_i − r_j | the Euclidean distance between the bodies at r_i and r_j . The main results of this paper are the following.

Theorem 1.Consider 7 bodies with masses m ₁ , m ₂ , ..., m ₇ , located according to the following description (see Figure 1):

1. The position vectors r₁, r₂ and r₃ are at the vertices of an equilateral triangle whose sides have length a > 0;

2. The position vectors r₄, r₅and r₆are at the vertices of an equilateral trianglewhose sides have length a > 0;

3. The trianglesandbelong to parallel distinct planes Π₁ and Π₂ , respectively;

4. The triangles and are coincident under translation;

5. The position vector r₇is located between the planes Π₁ and Π₂ . Then the following statements hold.

Then the following statements hold.

1. If r_{i 7} = d > 0 for all i∈ {1, 2, ..., 6}, then in order to have a central configuration the masses must satisfy:

2. If r_{i 7} = d > 0 for all i∈ {1, 2, ..., 6} and m = m ₁ = m ₂ = m ₃ = m ₄ = m ₅ = m ₆, then there is only one class of central configuration.

Furthermore such central configurations are independent of the values of the masses m ₇ and m and are of stacked type.

Figure 1
Illustration of the configurations studied here. The position vectorsr ₁, r ₂ and r ₃ are at the vertices of an equilateral trianglea > 0, r ₄, r ₅ and r ₆ are at the vertices of an equilateral trianglea > 0 and r ₇ is out of the parallel distinct planes Π₁and Π₂ that contain whose sides have length , respectively. whose sides have length and

The proof of Theorem 1 is given in the next section. Concluding remarks are presented in Section 3.

PROOF OF THEOREM 1

According to our assumptions, the equations of motion of theN bodies are given by Newton (1687)

(1)

Note that to find central configurations is essentially an algebraic problem. In fact, from the definition of central configuration there exists λ ≠ 0 such that i = 1, ..., N. From equation (1) it follows that, for all

(2)

for i = 1, ... , N. The equations in (2) are called equations of central configurations and are equivalent to the following set of equations (see Hampton and Santoprete 2007)

(3)

for i < j, l≠ i, l ≠ j,i, j, l = 1, ...,N, where R_ij = r_ij ^–3 and Δ_ijlk = (r_i − r_j ) ˆ (r_i − r_l ) · (r_i − r_k ) is six times the oriented volume defined by the tetrahedron with vertices at r_i , r_j , r_l and r_k .

For the 7-body problem there are 105 equations in (3) which are called Andoyer equations. They are a convenient set of equations to study some classes of central configurations, mainly when there exist symmetries in the configurations.

There exist several symmetries in our configurations (see Figure 1). From the hypotheses of Theorem 1 we have

R₁₂ = R ₁₃ = R ₂₃ = R ₄₅ = R ₄₆ = R ₅₆,

R₁₄ = R ₂₅ = R ₃₆,

R_{i 7} = d ^–3 > 0, µi ∈ {1, 2, ..., 6},

2Δ₁₂₃₇ = Δ₁₂₃₄ = Δ₁₂₃₅ = Δ₁₂₃₆ = −Δ₄₅₆₁ = −Δ₄₅₆₂ = −Δ₄₅₆₃ = −2Δ₄₅₆₇, and many others.

Using these symmetries in equations (3), it follows that the equations

f₁₂₄ = 0,f ₁₂₅ = 0,f ₁₃₄ = 0,f ₁₃₆ = 0,f ₂₃₅ = 0,f ₂₃₆ = 0,

f₄₅₁ = 0,f ₄₅₂ = 0,f ₄₆₁ = 0,f ₄₆₃ = 0,f ₅₆₂ = 0,f ₅₆₃ = 0

are already verified. Thus, we still have to study the remaining 93 equations.

Consider the equations f ₁₂₃ = 0 and f ₁₃₅ = 0. Using the above symmetries we have

f₁₂₃ = (m ₄ − m ₅) (R ₁₄ − R ₂₄)Δ₁₂₃₄ = 0

and

f₁₃₅ = (m ₄ − m ₆) (R ₁₄ − R ₃₄)Δ₁₃₅₄ = 0.

By the hypotheses of Theorem 1, R ₂₄ ≠ R ₁₄ ≠ R ₃₄, Δ₁₂₃₄ ≠ 0 and Δ₁₃₅₄ ≠ 0. So, such equations are satisfied if and only if

m₄ = m ₅ = m ₆.

Consider also the equations f ₄₅₆ = 0 and f ₄₆₂ = 0. Using the above symmetries we have

f₄₅₆ = (m ₁ − m ₂) (R ₄₁ − R ₅₁) Δ₄₅₆₁ = 0

and

f₄₆₂ = (m ₁ − m ₃) (R ₄₁ − R ₆₁) Δ₄₆₂₁ = 0.

By the hypotheses of Theorem 1, R ₅₁ ≠ R ₄₁ ≠ R ₆₁, Δ₄₅₆₁ ≠ 0 and Δ₄₆₂₁ ≠ 0. So, such equations are satisfied if and only if

m₁ = m ₂ = m ₃.

Consider now the equation f ₁₄₂ = 0. Using the above symmetries we have

f₁₄₂ = (m ₃ − m ₆) (R ₁₃ − R ₄₃) Δ₁₄₂₃ = 0.

By the hypotheses of Theorem 1, R ₁₃ ≠ R ₄₃, Δ₁₄₂₃ ≠ 0. So, such equation is satisfied if and only if

m₃ = m ₆.

Thus, in order to have a central configuration with the symmetries imposed in Theorem 1, the masses m ₁, m ₂, m ₃, m ₄, m ₅ and m ₆ must be equal. Item a) of Theorem 1 is proved.

Taking into account m = m ₁ = m ₂ = m ₃ = m ₄ = m ₅ = m ₆ and the symmetries in the hypotheses of Theorem 1, the following equations are already satisfied:

f₁₂₃ = 0,f ₁₂₆ = 0,f ₁₂₇ = 0,f ₁₃₂ = 0,f ₁₃₅ = 0,

f₁₃₇ = 0,f ₁₄₂ = 0,f ₁₄₃ = 0,f ₁₄₅ = 0,f ₁₄₆ = 0,

f₁₄₇ = 0,f ₁₅₂ = 0,f ₁₅₄ = 0,f ₁₆₃ = 0,f ₁₆₄ = 0,

f₁₇₄ = 0,f ₂₃₁ = 0,f ₂₃₄ = 0,f ₂₃₇ = 0,f ₂₄₁ = 0,

f₂₄₅ = 0,f ₂₅₁ = 0,f ₂₅₃ = 0,f ₂₅₄ = 0,f ₂₅₆ = 0,

f₂₅₇ = 0,f ₂₆₃ = 0,f ₂₆₅ = 0,f ₂₇₅ = 0,f ₃₄₁ = 0,

f₃₄₆ = 0,f ₃₅₂ = 0,f ₃₅₆ = 0,f ₃₆₁ = 0,f ₃₆₂ = 0,

f₃₆₄ = 0,f ₃₆₅ = 0,f ₃₆₇ = 0,f ₃₇₆ = 0,f ₄₅₃ = 0,

f₄₅₆ = 0,f ₄₅₇ = 0,f ₄₆₂ = 0,f ₄₆₅ = 0,f ₄₆₇ = 0,

f₄₇₁ = 0,f ₅₆₁ = 0,f ₅₆₄ = 0,f ₅₆₇ = 0,f ₅₇₂ = 0,

and f ₆₇₃ = 0. Thus, we have 42 equations remaining to analyze.

The 42 remaining equations can be divided into three sets of equivalent equations.

Case 1. The following 12 equations are equivalent:

f₁₅₃ = 0,f ₁₅₆ = 0,f ₁₆₂ = 0,f ₁₆₅ = 0,f ₂₄₃ = 0,f ₂₄₆ = 0,

f₂₆₁ = 0,f ₂₆₄ = 0,f ₃₄₂ = 0,f ₃₄₅ = 0,f ₃₅₁ = 0,f ₃₅₄ = 0.

Thus it is sufficient to study only one of these equations, for instance, the equation f ₁₅₃ = 0 which can be written as

(3R ₁₂ − 2R ₅₂ − R ₅₃) Δ₁₅₃₂ = 0.(4)

Case 2. The following 6 equations are equivalent:

f₁₅₇ = 0,f ₁₆₇ = 0,f ₂₄₇ = 0,f ₂₆₇ = 0,f ₃₄₇ = 0,f ₃₅₇ = 0.

Thus it is sufficient to study only one of these equations, for instance, the equation f ₁₅₇ = 0 which can be written as

(3R ₁₂ − 2R ₅₂ − R ₅₃) Δ₁₅₇₂ = 0.(5)

Case 3. The following 24 equations are equivalent

f₁₇₂ = 0,f ₁₇₃ = 0,f ₁₇₅ = 0,f ₁₇₆ = 0,f ₂₇₁ = 0,f ₂₇₃ = 0,

f₂₇₄ = 0,f ₂₇₆ = 0,f ₃₇₁ = 0,f ₃₇₂ = 0,f ₃₇₄ = 0,f ₃₇₅ = 0,

f₄₇₂ = 0,f ₄₇₃ = 0,f ₄₇₅ = 0,f ₄₇₆ = 0,f ₅₇₁ = 0,f ₅₇₃ = 0,

f₅₇₄ = 0,f ₅₇₆ = 0,f ₆₇₁ = 0,f ₆₇₂ = 0,f ₆₇₄ = 0,f ₆₇₅ = 0.

Thus it is sufficient to study only one of these equations, for instance, the equation f ₁₇₂ = 0 which can be written as

(3R ₁₂ − 2R ₅₂ − R ₅₃) Δ₁₇₂₃ = 0.(6)

Under our hypotheses the terms Δ₁₅₃₂, Δ₁₅₇₂ and Δ₁₇₂₃ do not vanish, so equations (4), (5) and (6) are satisfied if and only if

3R ₁₂ − 2R ₅₂ − R ₅₃ = 0.(7)

Equation (7) implies that the central configurations studied here do not depend on the value of the mass m ₇.

In order to simplify our analysis and without loss of generality, take a system of coordinates in which , where

r₇ = (0, 0, 0),x > 0,y> 0.

With these coordinates x = a> 0 and 2y > 0 is the distance between the planes Π₁ and Π₂. It follows that equation (7) is written as

(8)

with x > 0 and y > 0.

Therefore, to complete the proof of Theorem 1 we need to study the zero level of the function F in (8). We claim that the zero level of F is contained in a straight line passing through the origin. In fact, consider the change of variables defined by

With these new variables the function F in (8) is written as

Now, taking polar coordinates

u = r cos θ,v = r sin θ,

the function F is given by

(10)

From (10) the zero level of F is obtained from the zeros of the function

3sin³ θ − 2cos³ θ − sin³ θcos³ θ = cos³ θ (3tan³ θ − sin³ θ −2),

that is, from the function

(11)

From elementary calculations we have

for all θ ∈ (0,π/2). Thus, f is an increasing function that changes sign only once. Therefore, there is only one θ ₀ ∈ (0, π/2) such thatf(θ ₀) = 0. This implies that the zero level of Fis contained in the set {θ ₀, r > 0}, that is the zero level ofF is given by the following set

(12)

for some α > 0 in the original coordinates. Simple numerical computations give the approximated value α≃ 0.7935817272.

The uniqueness of the class of central configuration studied here follows from the set in (12). This end the proof of Theorem 1.

CONCLUSIONS

An interesting fact about the configuration studied here is that it does not depend on the values of the masses m and m ₇. So we have a unique two parameter class of central configurations. Also, if we remove the body of mass m ₇ the remaining six bodies are already in a central configuration (seeCedó and Llibre 1989). Thus the central configuration studied here is an example of spatial stacked central configuration with seven bodies (see Hampton and Santoprete 2007, Mello et al. 2009).

The results obtained in this paper also work for other regularn-gons instead the equilateral triangle, but this is a subject of a future work. At the moment we have just numerical results.

We believe that the results obtained are true for the case where one triangle is rotated by an angle of π/3 with respect to the other one. Rotations by other angles require an approach different of the presented here and new techniques must be found.

We also believe that similar results can be obtained taking the same structure with two equal co-circular central configurations (see Cors and Roberts 2012), instead of two equilateral triangles.

The authors were partially supported by Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) grant APQ-00015/12. The second author was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) grant 301758/2012-3 and by FAPEMIG grant PPM-00092-13.

REFERENCES

• Cedó F and Llibre J. 1989. Symmetric central configurations of the spatial n-body problem. J Geom Phys 6: 367-394.
• Corbera M and Llibre J. 2008. Central configurations of nested regular polyhedra for the spatial 2n-body problem. J Geom Phys 58: 1241-1252.
• Corbera M and Llibre J. 2009. Central configurations of three regular polyhedra for the spatial 3n-body problem. J Geom Phys 59: 321-339.
• Cors JM and Roberts G. 2012. Four-body co-circular central configurations. Nonlinearity 25: 343-370.
• Hagihara Y. 1970. Celestial Mechanics, vol. 1, Massachusetts: MIT Press, 720 p.
• Hampton M and Santoprete M. 2007. Seven-body central configurations: a family of central configurations in the spatial seven-body problem. Celestial Mech Dyn Astr 99: 293-305.
• Lehmann-Filhés R. 1891. Ueber zwei Fälle des Vielkörperproblems. Astr Naschr 127: 137-144.
• Mello LF, Chaves FE, Fernandes AC and Garcia BA. 2009. Stacked central configurations for the spatial 6-body problem. J Geom Phys 59: 1216-1226.
• Mello LF and Fernandes AC. 2011a. New classes of spatial central configurations for the n+3-body problem. Nonlinear Anal Real World Appl 12: 723-730.
• Mello LF and Fernandes AC. 2011b. New spatial central configurations in the 5-body problem. An Acad Bras Cienc 83: 763-774.
• Moeckel R. 1990. On central configurations. Math Z 205: 499-517.
• Newton I. 1687. Philosophi Naturalis Principia Mathematica. London: Royal Society, 512 p.
• Saari D. 1980. On the role and properties of central configurations. Celestial Mech 21: 9-20.
• Smale S. 1970. Topology and mechanics II: The planar n-body problem. Invent Math 11: 45-64.
• Wintner A. 1941. The Analytical Foundations of Celestial Mechanics. Princeton: Princeton University Press, 448 p.
• Zhang S and Zhou Q. 2001. Double pyramidal central configurations. Phys Lett A 281: 240-248.
• Zhu C. 2005. Central configurations of nested regular tetrahedrons. J Math Anal Appl 312: 83-92.

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

Publication Dates

• Publication in this collection
  Mar 2014

History

• Received
  24 July 2012
• Accepted
  6 May 2013