Convex Central Configurations of the n-body Problem Which are not Strictly Convex

which are not strictly convex

Kuo-Chang Chen, Jun-Shian Hsiao Department of Mathematics

National Tsing Hua University Hsinchu 30013, Taiwan

Abstract. It is well-known that if a planar central configuration for the Newtonian 4-body problem is convex, then it must be strictly convex. In some literature, same conclu-sion were believed to hold for the case of five or even more bodies but rigorous treatments are absent. In this paper we provide concrete examples of central configurations which are convex but not strictly convex. Our examples include planar central configurations with five bodies and spatial central configurations with seven bodies.

1. Introduction

The classical n-body problem concerns the motion of n mass points moving in space in accordance with Newton’s law of gravitation:

¨ qk = X i6=k mi(qi− qk) |q_i− q_k|3 , k = 1, 2, · · · , n. (1)

Here qk ∈ Rd (1 ≤ d ≤ 3) is the position of mass mk> 0. Alternatively the system (1) can

be written mkq¨k = ∂ ∂qk U (q), k = 1, 2, · · · , n (2) where U (q) := X 1≤i<j≤n mimj |qi− qj|

is the potential of the system. The position vector q = (q1, · · · , qn) ∈ (Rd)nis often referred

to as the configuration of the system, and vectors {qk} are vertices of the configuration q.

Let M = m1+ · · · + mn be the total mass and

ˆ q = 1

M(m1q1+ · · · + mnqn) Key words and phrases. central configuration, n-body problem.

be the mass center of the configuration q. The set ∆ of collision configurations is the algebraic variety defined by

∆ = {q ∈ (Rd)n: qi = qj for some i 6= j}.

A configuration q ∈ (Rd)n\ ∆ is called a central configuration if there exists some positive constant λ, called the multiplier, such that

−λ(q_k− ˆq) =X

i6=k

mi(qi− qk)

|q_i− q_k|3 , k = 1, 2, · · · , n.

(3)

The assumption that λ being positive is redundant as it can be easily verified that there is no negative λ that can possibly satisfy (3). The definition of central configuration can be extended to cases with some zero masses but we shall only consider positive masses throughout this paper.

The set of central configurations are invariant under three classes of transformations on (Rd)n which act equitably on each copy of Rd: translations (qk7→ qk+ a, a ∈ Rd), scalings

(qk7→ µqk, µ 6= 0), and orthogonal transformations (qk 7→ Aqk, A ∈ O(d)). Scaling a central

configuration with multiplier λ by multiplying µ results in a central configuration with multiplier λ/|µ|3. Conventionally two central configurations are considered equivalent if one can be obtained from the other via translations, scalings, and SO(d)-actions. This clearly defines an equivalence relation on central configurations, and two central configurations are different in that aspect if they differ from each other by a reflection with respect to a hyperplane in Rd. The term “central configurations” is often referred to equivalence classes of central configurations.

There are several reasons why central configurations are of special importance in the study of the n-body problem, see [5] for details.

Without loss of generality we consider only central configurations q with mass centers ˆq at the origin. In this case, using the Leibniz formula for the moment of inertia

I(q) := n X k=1 mk|qk|2 = 1 M X i<j mimj|qi− qj|2

the system (3) can be written in a compact form −λ 2∇I(q) = ∇U (q), (4) or equivalently, X i6=k mi  λ M − 1 |q_i− q_k|3  (qi− qk) = 0, k = 1, 2, · · · , n.

By suitable scaling, we may assume without loss of generality that the multiplier λ is equal to the total mass M . Then equations for central configurations become

n X i=1 mi(1 − sik) (qi− qk) = 0, k = 1, 2, · · · , n, (5) where sij = 1 |qi− qj|3

for i 6= j; sii= 1 for each i.

This is the system of equations we will be working with.

The convex hull Conv(q) of the configuration q ∈ (Rd)n is the convex hull in Rd for its vertices {qk}. We say the configuration q is convex if each qk is on the boundary of Conv(q)

(as a subset of Rd); it is strictly convex if no qk is on the convex hull for other vertices.

Concave configurations are configurations which are not strictly convex. Strictly concave configurations are configurations which are not convex. The dimension of a configuration q is defined as the dimension dim Conv(q) of Conv(q).

Strictly convex and strictly concave configurations are clearly two open sets in (Rd)nand their boundaries intersect at configurations that are both convex and concave (convex but not strictly convex, in other words). The intersection also includes some collision configura-tions. According to the above definition, a collinear configuration q (i.e. dim Conv(q) ≤ 1) in (Rd)n\ ∆ with n ≥ 3 is concave if d = 1 but is both convex and concave if d ≥ 2. Likewise, a coplanar configuration q (i.e. dim Conv(q) ≤ 2) with n ≥ 4 is convex if d ≥ 3, or d = 2 and no qk is in the interior of the convex hull for other vertices, or d = 2 and q

is collinear. In literature the terms convex and concave configurations are often referred to noncollinear configurations.

Other than the 12 collinear central configurations [6], the planar four-body problem (d = 2) has at least 14 concave central configurations [2, 3], and at least 6 non-collinear convex central configurations, one for each of the 6 cyclic orderings of the four bodies [4]. It follows easily from the perpendicular bisector theorem [5] that all of these convex central configurations are strictly convex. By calculating the derivative of the normalized potential on the boundary of the set of convex configurations, Xia shows that no local minimum can possibly fall on this boundary, following from which he provides a simple alternative proof for the existence of a convex central configuration for each of the 6 cyclic orderings. There it was proposed that such a method may also work for the planar five-body problem, or even planar and spatial n-body problems in general [8, §2]. In [7, §2 and §8-Theorem 8.1] Williams also claimed that convex five-body central configurations cannot have three masses

along the same line. Our main result shows the existence of central configurations which disprove above stated assertions.

Theorem 1. For some positive masses there are 2-dimensional central configurations for the 5-body problem which are convex but not strictly convex. For some positive masses there are 3-dimensional central configurations for the 7-body problem which are convex but not strictly convex.

2. Planar central configurations with five bodies

In this section we construct a 2-dimensional central configuration with five bodies which is convex but not strictly convex.

Consider a planar convex configuration q ∈ (R2)5 whose convex hull Conv(q) is an isosce-les trapezoid. Vertices are

q1 = (−α, −γ1), q2= (α, −γ1), q3 = (−β, γ2), q4= (β, γ2),

where α, β, γ1, γ2 are positive numbers. The fifth vertex q5 is located (0, −γ1), making the

configuration q convex but not strictly convex.

Let m1= m2 = m5 = 1, m3 = m4= µ. The height γ = γ1+ γ2 of the isosceles trapezoid

and γ1, γ2 are related by

γ1=

2µγ

3 + 2µ, γ2= 3γ 3 + 2µ so that the mass center ˆq of the configuration q is at the origin.

Apparently we have s13= s24= (α − β)2+ γ2
−3₂ , s14= s23= (α + β)2+ γ2
−3₂ , s35= s45= β2+ γ2
−3₂ , (6) s15= s25= 8s12= α−3, s34= (2β)−3, sij = sji for any i, j.

From these symmetries, in (5) equations for k = 1, 2 are identical, equations for k = 3, 4 are also identical. Equations (5) for central configurations become

(1 − s12)(q2− q1) + µ(1 − s13)(q3− q1) + µ(1 − s14)(q4− q1) + (1 − s15)(q5− q1) = 0

(1 − s13)(q1− q3) + (1 − s23)(q2− q3) + µ(1 − s34)(q4− q3) + (1 − s35)(q5− q3) = 0

This is actually a system of six equations with positive unknowns α, β, γ, µ, and sij: 2(1 − s12)α + µ(1 − s13)(α − β) + µ(1 − s14)(α + β) + (1 − s15)α = 0 (1 − s13)(−α + β) + (1 − s23)(α + β) + 2µ(1 − s34)β + (1 − s35)β = 0 −(1 − s15)α + (1 − s25)α − µ(1 − s35)β + µ(1 − s45)β = 0 (7) µ(1 − s13)γ + µ(1 − s14)γ = 0 (1 − s13)γ + (1 − s23)γ + (1 − s35)γ = 0 µ(1 − s35)γ + µ(1 − s45)γ = 0

The third identity is obvious as it follows immediately from (6). The fourth and the sixth identities are equivalent to

s13+ s14= 2, s35= 1.

(8)

These two identities together with s14 = s23 imply the fifth identity in (7). The second

identity in (7) can therefore be simplified to

µ(1 − s34)β = (1 − s13)α.

(9)

By (6), (8), (9), the first identity in (7) can be written

(3 − 10s12)(s34− 1) + 2(s13− 1)2 = 0.

(10)

The system (7) is now reduced to (8), (9), (10).

We will soon see that every variable can be expressed in terms of s13= θ. Observe that

(2 − θ)−23 = s− 2 3 14 = (α + β) 2_{+ γ}2_{, θ}−2_{3 = s}−23 13 = (α − β) 2_{+ γ}2_. Thus α = θ −2 3 + (2 − θ)− 2 3 − 2 2 !1₂ , β = (2 − θ) −2 3 − θ− 2 3 4α ,

both of which are positive since

Every sij can now be expressed in terms of θ via (6) and (8), and so is γ = p 1 − β2_{. In} terms of θ, (10) becomes  3√2θ−23 + (2 − θ)− 2 3 − 2 3₂ − 5    1 −  (2 − θ)−23 − θ− 2 3 3 2√2 θ−23 + (2 − θ)− 2 3 − 2
3 2    (11) +  (2 − θ)−23 − θ− 2 3 3 (θ − 1)2 = 0.

Let R(θ) be the function in (11). Then it increases on [1.4, 1.6] from R(1.4) ≈ −0.647 to R(1.6) ≈ 0.866. The unique root θ∗ in there is approximately 1.506654. One can easily check that the corresponding γ, µ are both strictly positive. This proves the existence of a 2-dimensional central configuration for the five-body problem with suitable masses.

Numerical data accurate to the 24th decimal places are given in table 1. Figure 1 shows a homographic solution for this particular central configuration with period 2π and eccen-tricity 0.6.

From (4), central configurations are exactly the set of critical points of the normalized potential ˜U = √IU . The central configuration we found is of local minimum type as one can easily check that eigenvalues of D2U are all positive, except the two zero eigenvalues˜ due to the invariance under rotations and scalings. Therefore, the idea of perturbing the intermediate mass q5 downward so as to decrease the normalized potential (see [8]) is not

applicable to this case.

-4 -2 0 2 4

-4 -2 0 2

θ = 1.506654068878065383203023, µ = 11.23156072828415553841745, α = 0.425756342462430700206462, γ1 = 0.767184777048876570481630,

β = 0.493679334448944202819131, γ2 = 0.102459239050841954854995.

Table 1. Numerical data for the example of planar five-body central configuration.

Same calculations can be carried out for the more general case with m5 = ν as a new

parameter. The system is reduced to (8), (9), and an identity similar to (10): (2 + ν − (2 + 8ν)s12)(s34− 1) + 2(s13− 1)2 = 0.

(12)

Following the procedure demonstrated above we may reduce (12) to an equation in θ = s13

with an additional parameter ν. By solving the equation numerically, as ν increases the mass µ increases and the shape of the isosceles trapezoid deforms closer and closer to a rectangle. Figure 2 shows one central configuration with ν = 10−4 and the other one with ν = 104. The corresponding µ are approximately 2.758 and 81952.332, respectively. The one with ν = 104 is of local minimum type but the one with ν = 10−4 is not. A degenerate central configuration is found when ν reaches a threshold value around 0.518085751, for which case µ is approximately 7.215.

-0.4 -0.2 0.0 0.2 0.4 -0.6 -0.4 -0.2 0.0 0.2 -0.4 -0.2 0.0 0.2 0.4 -0.8 -0.6 -0.4 -0.2 0.0

Figure 2. Central configurations with ν = 10−4 (left) and ν = 104 (right).

3. Spatial central configurations with seven bodies

In this section we construct a 3-dimensional central configuration with seven bodies which is convex but not strictly convex. Its convex hull is an octahedron with two planes of symmetry. The idea is to add two mass points along the vertical line through the circumcenter of a horizontal five-body isosceles trapezoidal configuration similar to the one in the previous section.

Consider a 3-dimensional configuration q ∈ (R3)7 with vertices

q1 = (−α, −ζ, 0), q2= (α, −ζ, 0), q3= (−β, γ − ζ, 0),

q4 = (β, γ − ζ, 0), q5= (0, −ζ, 0), q6= (0, δ − ζ, η),

q7 = (0, δ − ζ, −η).

Here α, β, γ, η are all assumed to be positive. The term δ is given by

δ = −α

2_{+ β}2_{+ γ}2

2γ

so that q6 and q7 are equally distant from the vertices q1, q2, q3, q4 of a trapezoid. The

configuration is clearly convex but not strictly convex as q5 is the midpoint of the edge q1q2.

Let m1 = m2 = m5= 1, m3 = m4 = µ, m6 = m7= ν. By setting

ζ = 2(µγ + νδ) 3 + 2µ + 2ν

the mass center ˆq of the configuration q is exactly the origin. We will show that q is indeed a central configuration for some positive α, β, γ, η, µ, and ν.

The main idea is the same as the planar case. First we write down equations of central configurations in terms of α, β, γ, η, µ, ν, and sij’s, and then look for solutions with all

of these variables positive. Each sij can be easily expressed in terms of α, β, γ, η and

are subject to some symmetry constraints. By straightforward reductions, variables µ and ν can be also easily expressed in terms of these four variables, and we are left with four equations with four variables.

We begin with some observations on sij’s:

s13= s24= ((α − β)2+ γ2)− 3 2, s14= s23= ((α + β)2+ γ2)− 3 2, s35= s45= (β2+ γ2)− 3 2, s15= s25= 8s12= α−3, s34= (2β)−3, (13) s17= s27= s37= s47= s16= s26= s36= s46= (α2+ δ2+ η2)− 3 2, s56= s57= (δ2+ η2)− 3 2, s67= (2η)−3, sij = sij for any i, j.

Using merely symmetries among sij’s, equations (5) are reduced to cases k = 1, 3, 5, 7 and

third component of (5) is clearly zero when k = 1, 3, 5, and the first component of (5) is also zero when k = 5, 7. There are seven equations left:

α(3 − 10s12) + µα(2 − s13− s14) + 2να(1 − s17) + µβ(s13− s14) = 0 µγ(2 − s13− s14) + 2νδ(1 − s17) = 0 α(s13− s14) + β(3 − s13− s14− s35) + 2µβ(1 − s34) + 2νβ(1 − s17) = 0 γ(3 − s13− s14− s35) + 2ν(γ − δ)(1 − s17) = 0 (14) µγ(1 − s35) + νδ(1 − s57) = 0 2(µ(γ − δ) − δ)(1 − s17) − δ(1 − s57) = 0 2(1 + µ)(1 − s17) + (1 − s57) + 2ν(1 − s67) = 0.

For brevity we keep the notation δ in these equations. One can easily check linear depen-dence of the second, fourth, fifth, and the sixth equations. By the substitutions

s35 = νδ µγ(1 − s57) + 1 s57 = 1 − 2(µγ − µδ − δ) δ (1 − s17) (15) s17 = µγ 2νδ(2 − s13− s14) + 1

the system is reduced to three equations, two of which can be used to determine masses µ and ν: µ = α(s14− s13) β(s13+ s14− 2s34) (16) ν = µγ δ s 2 − s13− s14 2(1 − s67) .

In particular we have s13+ s14− 2s34< 0, since s14< s13. The remaining equation can be

written (3 − 10s12)(s13+ s14− 2s34) (17) = (s13− s14)2+ α(δ − γ) βδ (s13− s14)(2 − s13− s14) .

With the help of (13) and (16), equations (15) and (17) are now four equations in four unknowns α, β, γ, η. The Newtonian method converges rapidly near (0.4, 0.5, 0.87, 0.77), and resulting masses µ and ν are both positive. This shows the existence of a central configuration with seven bodies which is convex but not strictly convex, and finishes the proof of Theorem 1.

α = 0.396619093609962801426587, γ = 0.871188462806049795533515, β = 0.495555327813506194775772, η = 0.772004237900502489476959, δ = 0.486253980284487121746795, ζ = 0.777294735425231051490145, µ = 14.09290585257097996322574, ν = 0.540451308652400689509046. Table 2. Numerical data for the example of spatial seven-body central configuration

Numerical data accurate to the 24th decimal places are given in table 2. Figure 3 shows the corresponding central configuration. This central configuration is not of local mini-mum type since the second derivative of the normalized potential D2U has two negative˜ eigenvalues that are approximately −134.616 and −66.881.

As in the previous section, we may consider the more general case with m5 = σ as

a parameter, then obtain a family of convex spatial central configurations which are not strictly convex. The spectrum of D2U evaluated at the central configuration varies with σ˜ and, among those we computed, the number of negative eigenvalues are two or three. A degenerate central configuration can be obtained when σ ≈ 0.504336299.

-0.5 0.0 0.5 -0.6 -0.4 -0.2 0.0 -0.5 0.0 0.5

Figure 3. A convex but not strictly convex central configuration with seven bodies. Convex central configurations which are not strictly convex should be rare; we have searched miscellaneous seemly possible candidates and the one in this section is the only one we found for the spatial seven-body central configuration.

Now we finish this paper with some remarks on spatial central configurations with less bodies. There is no 3-dimensional convex central configuration with five bodies which is

not strictly convex. If there were, then there would be a plane π which contains exactly four masses. One way of excluding this possibility is by using a spatial version of the perpendicular bisector theorem [5, pp.511]. Another simple approach is by using a result in [1, Proposition 5] which implies in this case that the four masses on π must be cocircular, thus forming a strictly convex polygon. Consequently, since the fifth body is not on π, the configuration is strictly convex.

It would be interesting to know if there exists any 3-dimensional convex central config-uration with six bodies which is not strictly convex. We don’t know the answer. A case we find difficult to exclude is a hexahedron with an isosceles triangular base and with three collinear bodies on its top which fall on the perpendicular bisecting plane of the base.

Acknowledgement.

We are grateful to Alain Albouy and Rick Moeckel for helpful conversations during prepa-ration of this work. Our research is partly supported by the National Science Council and the National Center for Theoretical Sciences in Taiwan.

References

[1] Albouy, A., On a paper of Moeckel on central configurations. Regul. Chaotic Dyn. 8 (2003), 133–142. [2] Hampton, M., Convex central configurations in the four body problem. Thesis, University of

Wash-ington (2002).

[3] Hampton, M., Moeckel, R., Finiteness of relative equilibria of the four-body problem, Invent. Math. 163 (2006), 289–312.

[4] MacMillan, W. D.; Bartky, W., Permanent configurations in the problem of four bodies. Trans. Amer. Math. Soc. 34 (1932), 838–875.

[5] Moeckel, R., On central configurations. Math Z. 205 (1990), 499–517.

[6] Moulton, F. R., The straight line solutions of the problem of N bodies. Ann. of Math. (2) 12 (1910), 1–17.

[7] Williams, W. L., Permanent configurations in the problem of five bodies. Trans. Amer. Math. Soc. 44 (1938), 563–579.

[8] Xia, Z., Convex central configurations for the n-body problem. J. Differential Equations 200 (2004), 185–190.

Kuo-Chang Chen, Department of Mathematics, National Tsing Hua University, Taiwan E-mail address: kchen@math.nthu.edu.tw

Jun-Shian Hsiao, Department of Mathematics, National Tsing Hua University, Taiwan E-mail address: d9621804@oz.nthu.edu.tw