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Anais da Academia Brasileira de Ciências

versão impressa ISSN 0001-3765

An. Acad. Bras. Ciênc. vol.86 no.1 Rio de Janeiro mar. 2014

https://doi.org/10.1590/0001-37652014107412 

Mathematical Sciences

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

ANTONIO CARLOS FERNANDES1 

LUIS FERNANDO MELLO1 

1Instituto de Matemática e Computação, Universidade Federal de Itajubá, Avenida BPS, 1303, Pinheirinho, 37500-903 Itajubá, MG, Brasil


ABSTRACT

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 and . The results have simple and analytic proofs.

Key words: Spatial central configuration; 7-body problem; stacked central configuration; central configuration

RESUMO

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 e . Os resultados apresentados aqui tem provas simples e analíticas.

Palavras-Chave: 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 massesmi > 0 located at the points ri of the Euclidean space for 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 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 rij = |ri rj | the Euclidean distance between the bodies at ri and rj . 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 r 1 , r 2 and r 3 are at the vertices of an equilateral triangle whose sides have length a > 0;

  2. The position vectors r 4 , r 5 and r 6 are at the vertices of an equilateral triangle whose sides have length a > 0;

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

  4. The triangles and are coincident under translation;

  5. The position vector r 7 is located between the planes Π1 and Π2 . Then the following statements hold.

Then the following statements hold.

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

  2. If ri 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 triangle whose sides have length a > 0, r 4, r 5 and r 6 are at the vertices of an equilateral triangle whose sides have length a > 0 and r 7 is out of the parallel distinct planes Π1and Π2 that contain and , respectively. 

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 , for all i = 1, ..., N. From equation (1) it follows that

(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, li, lj,i, j, l = 1, ...,N, where Rij = rij –3 and Δ ijlk = (ri rj ) ˆ (ri rl ) · (ri rk ) is six times the oriented volume defined by the tetrahedron with vertices at ri , rj , rl and rk .

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

R 12 = R 13 = R 23 = R 45 = R 46 = R 56,

R 14 = R 25 = R 36,

Ri 7 = d –3 > 0, µi ∈ {1, 2, ..., 6},

1237 = Δ1234 = Δ1235 = Δ1236 = −Δ4561 = −Δ4562 = −Δ4563 = −2Δ4567, and many others.

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

f 124 = 0,f 125 = 0,f 134 = 0,f 136 = 0,f 235 = 0,f 236 = 0,

f 451 = 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

f 123 = (m 4m 5) (R 14R 241234 = 0

and

f 135 = (m 4m 6) (R 14R 341354 = 0.

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

m 4 = m 5 = m 6.

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

f 456 = (m 1m 2) (R 41R 51) Δ4561 = 0

and

f 462 = (m 1m 3) (R 41R 61) Δ4621 = 0.

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

m 1 = m 2 = m 3.

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

f 142 = (m 3m 6) (R 13R 43) Δ1423 = 0.

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

m 3 = 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:

f 123 = 0,f 126 = 0,f 127 = 0,f 132 = 0,f 135 = 0,

f 137 = 0,f 142 = 0,f 143 = 0,f 145 = 0,f 146 = 0,

f 147 = 0,f 152 = 0,f 154 = 0,f 163 = 0,f 164 = 0,

f 174 = 0,f 231 = 0,f 234 = 0,f 237 = 0,f 241 = 0,

f 245 = 0,f 251 = 0,f 253 = 0,f 254 = 0,f 256 = 0,

f 257 = 0,f 263 = 0,f 265 = 0,f 275 = 0,f 341 = 0,

f 346 = 0,f 352 = 0,f 356 = 0,f 361 = 0,f 362 = 0,

f 364 = 0,f 365 = 0,f 367 = 0,f 376 = 0,f 453 = 0,

f 456 = 0,f 457 = 0,f 462 = 0,f 465 = 0,f 467 = 0,

f 471 = 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:

f 153 = 0,f 156 = 0,f 162 = 0,f 165 = 0,f 243 = 0,f 246 = 0,

f 261 = 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 52R 53) Δ1532 = 0.(4)

Case 2. The following 6 equations are equivalent:

f 157 = 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 52R 53) Δ1572 = 0.(5)

Case 3. The following 24 equations are equivalent

f 172 = 0,f 173 = 0,f 175 = 0,f 176 = 0,f 271 = 0,f 273 = 0,

f 274 = 0,f 276 = 0,f 371 = 0,f 372 = 0,f 374 = 0,f 375 = 0,

f 472 = 0,f 473 = 0,f 475 = 0,f 476 = 0,f 571 = 0,f 573 = 0,

f 574 = 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 52R 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 52R 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

r 7 = (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.

Acknowledgements

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.

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

Received: July 24, 2012; Accepted: May 6, 2013

Correspondence to: Luis Fernando Mello E-mail: lfmelo@unifei.edu.br

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