## Abstract

We study the relationship between the masses and the geometric properties of central configurations. We prove that, in the planar four-body problem, a convex central configuration is symmetric with respect to one diagonal if and only if the masses of the two particles on the other diagonal are equal. If these two masses are unequal, then the less massive one is closer to the former diagonal. Finally, we extend these results to the case of non-planar central configurations of five particles.

## 1. Introduction

Let *q*_{1}, …, *q*_{n} represent the positions in a Euclidean space *E* of *n* particles with respective positive masses *m*_{1}, …, *m*_{n}. Let(1.1)The force acting on the *i*th particle is −*m*_{i}*γ*_{i}. Newton's equations, which define the classical *n*-body motions of the system of particles, are(1.2)In the Newtonian *n*-body problem, the simplest possible motions are such that the configuration is constant up to rotation and scaling, and that each body describes a Keplerian orbit. Only some special configurations of particles are allowed in such motions. Wintner called them ‘central configurations’. Famous authors such as Euler (1764), Lagrange (1772), Laplace (1789), Liouville (1842) and Maxwell (1856) initiated the study of central configurations, while Chazy (1918), Wintner (1941), Smale (1998) and Atiyah & Sutcliffe (2003) drew attention to hard unsolved questions. The latter work relates the three-dimensional central configurations with topics like ‘electrons on a sphere, cages of carbon atoms, rare gas microclusters, soliton models of nuclei, magnetic monopoles scattering’. Chazy, Wintner and Smale conjectured that the number of central configurations of *n* particles with given masses is finite. This was proved recently in the case *n*=4 by Hampton & Moeckel (2006). This reference also provides an excellent review, which describes how central configurations appear in several applications. Let us mention that central configurations are also needed in the application of Ziglin's method or the computation of Kovalevskaya's exponents, in order to get statements of non-existence of additional first integrals in the *n*-body problem (e.g. Tsygvintsev 2007).

In terms of the total mass *M* and the centre of mass *q*_{G}(1.3)a configuration (*q*_{1}, …, *q*_{n}) is called a central configuration if and only if there exists a *λ*∈ such that(1.4)The results in this article are not restricted to the Newtonian case *a*=−3/2. They are true for any *a*<0. In particular, they apply to relative equilibria of Helmholtz vortices in the plane with positive vorticities *m*_{1}, …, *m*_{n} (e.g. O'Neil 1987), for which *a*=−1. While vorticities may be negative, we always assume *m*_{i}>0 for all *i*, 1≤*i*≤*n*. Note that this implies *λ*>0 (for results with some negative masses, see Celli (2007) and O'Neil (2007)). The main result we prove in this article is as follows.

*Let four particles* (*q*_{1}, *q*_{2}, *q*_{3}, *q*_{4}) *form a two-dimensional central configuration, which is a convex quadrilateral having* [*q*_{1}, *q*_{2}] *and* [*q*_{3}, *q*_{4}] *as diagonals (**figure 1**). This configuration is symmetric with respect to the axis* [*q*_{3}, *q*_{4}] *if and only if m*_{1}=*m*_{2}. *It is symmetric with respect to the axis* [*q*_{1}, *q*_{2}] *if and only if m*_{3}=*m*_{4}. *Also m*_{1}<*m*_{2} *if and only if* , *where Δ*_{ijk} *is the oriented area of the triangle* [*q*_{i}, *q*_{j}, *q*_{k}]. *Similarly*, *m*_{3}<*m*_{4} *if and only if* .

In theorem 1.1, each geometric relation between areas has equivalents in terms of distances. For example, is obviously equivalent to that *q*_{1} is closer to the line [*q*_{3}, *q*_{4}] than *q*_{2}. By lemma 4.1 and (2.4), it is also equivalent to or to .

Our hypotheses are not far from optimal. We assume *a*<0 in (1.1) being aware of counter examples if *a*>0, and of course if *a*=0. If the equal masses are *m*_{1} and *m*_{3}, there is no symmetry except if also *m*_{2}=*m*_{4} (but no proof is known of the symmetry in this latter case). In the non-convex case now, assuming, for example, that *q*_{1} is inside the triangle [*q*_{2}, *q*_{3}, *q*_{4}], there is always an axis of symmetry if *m*_{2}=*m*_{3}=*m*_{4} (Long & Sun 2002). One could conjecture that if *m*_{3}=*m*_{4}, [*q*_{1}, *q*_{2}] is a symmetry axis. This conjecture is already disproved by the case *m*_{1}=*m*_{2}=*m*_{3}=*m*_{4}. There are four non-convex solutions (Albouy 1996) of which only two are symmetric with respect to the [*q*_{1},*q*_{2}] axis and the other two are symmetric but with respect to another axis. They become asymmetric if we vary *m*_{2}.

According to MacMillan & Bartky (1932; see Xia (2004) for a simplified proof), for any choice of four positive masses, there exists a convex central configuration with given ordering of the particles, for example the one shown in figure 1.

*Such a central configuration is unique*.

Based on various numerical experiments and other considerations, we think that this conjecture is true in the Newtonian case a =−3/2, which is the main case, and we also guess it is true for any a<0. For more references and related open questions, the reader may consult Albouy & Fu (2007).

Given the presence of some equal masses, it is natural to challenge this conjecture by first proving some symmetry of the central configuration. With this purpose in mind, the first named author took the first step (Albouy 1995, 1996). He studied the equal mass case and gave a complete explanation of the geometric properties and the enumeration of the central configurations whether convex or not. Then Long and the third named author (Long & Sun 2002) studied the convex central configurations with *m*_{1}=*m*_{2} and *m*_{3}=*m*_{4}, and they proved symmetry and uniqueness under some constraints which are left out by Perez-Chavela & Santoprete (2007). Perez-Chavela & Santoprete also generalized further and obtained the symmetry of convex central configurations with *m*_{1}=*m*_{2} only. The uniqueness follows Leandro (2003). Unfortunately, in this situation, Perez-Chavela & Santoprete also need that *m*_{1}=*m*_{2} are not the smallest masses. Our theorem 1.1 gives the expected symmetry assuming simply 0<*m*_{1}=*m*_{2}, 0<*m*_{3} and 0<*m*_{4}, and choosing any *a*<0 in equation (1.1). The uniqueness in the Newtonian case *a*=−3/2 follows Leandro (2003). Thus, theorem 1.1 proves conjecture 1.4 in the particular case where two particles have equal mass and where the segment between them is a diagonal of the quadrilateral. Theorem 1.1 supports conjecture 1.4 in a very convincing way.

## 2. Dziobek's equations

Many results on the planar central configurations of four bodies were obtained or simplified using Dziobek's coordinates (1900). In particular, Meyer & Schmidt (1988) remarked on their effectiveness and extended their use to the non-planar central configurations of five bodies.

Given a configuration of *n* points (*q*_{1}, …, *q*_{n})∈*E*^{n}, we consider the linear system(2.1)with unknown (*Δ*_{1}, …, *Δ*_{n})∈^{n}. If the dimension of the configuration is *n*−2, this system has rank *n*−1. From a non-zero solution (*Δ*_{1}, …, *Δ*_{n}), we deduce all the solutions by multiplying it by any *λ*∈. We may call such a solution a system of *homogeneous barycentric coordinates* of the configuration.

There is a *κ*∈ such that ±*κΔ*_{i} is the (*n*−2)-dimensional volume of the simplex . Dziobek introduced the *Δ*_{i}'s in this way. This gives some geometrical intuition, but at the technical level it is better to use (2.1) as a definition. We rewrite the equations for central configuration (1.4) as(2.2)and compare them with the second equation of (2.1) written as(2.3)By the uniqueness, up to a factor, of (*Δ*_{1}, …, *Δ*_{n}), there exists a real number *θ*_{i} such that . We can write for any (*i*, *k*), 1≤*i*<*k*≤*n*. Thus (*θ*_{1}, …, *θ*_{n}) is proportional to (*Δ*_{1}/*m*_{1}, …, *Δ*_{n}/*m*_{n}). Denoting by *μ* the proportionality factor, we obtain Dziobek's equations(2.4)These equations have a simple aspect and are easy to deduce. Dziobek obtained from them many inequalities between the , that is, between the mutual distances. One first sets one of the *Δ*_{i}'s to zero and observes the consequences. Then one makes different hypotheses on the signs of the *Δ*_{i}'s. In some cases, the configuration is convex and the mutual distances ordered in a certain way. In other cases, it is non-convex and the mutual distances are ordered in another way.

## 3. Routh versus Dziobek

Here we only consider the case *n*=4. Equations (2.4) were proved by Dziobek by considering Cayley's determinant *H*, differentiating *H* on *H*=0 and characterizing the central configurations as critical points of the Newtonian potential restricted to some submanifold of *H*=0. The expression of the derivatives of *H* is quite nice, but the computation giving it is not so easy. Dziobek and many authors after him chose to skip it. An elegant and complete presentation is given in Moeckel (2001).

The ‘vectorial’ deduction of (2.4) we just gave is simpler than Dziobek's approach. It was also published in Moeckel (2001). Previously, many authors presented similar vectorial computations and deduced relations which are easy consequences of Dziobek's equations. A formula such as(3.1)appeared in Routh (1877) and Krediet (1892) before Dziobek's work, and in Laura (1905) and Andoyer (1906) just after. The symbol *Δ*_{ijk} represents the oriented area of the triangle (*i*, *j*, *k*). Setting *Δ*_{4}=*Δ*_{123} and *Δ*_{3}=−*Δ*_{124}, we recognize (3.1) as an easy consequence of (2.4). But (2.4) is not an obvious corollary of (3.1), and none of these four authors wrote it.

In the last edition of Routh's (1905) treatise, 28 years after Routh (1877), he considered again the central configurations. Apparently, he wanted to claim priority over someone, and it is very probable that this someone is Dziobek. Routh uses Dziobek's notation *Δ*, which suggests strongly that he read Dziobek's paper. Routh's priority claim concerns (3.1) and another important formula. There follows a reproduction of the text in Routh (1877), which is not easy to find in the libraries.Ex. 2. If four particles be placed at the corners of a quadrilateral whose sides taken in order are

*a*, *b*, *c*, *d* and diagonals *ρ* and *ρ*′, then the particles could not move under their mutual attractions so as to remain always at the corners of a similar quadrilateral unlesswhere the law of attraction is the inverse (*n*−1)^{th} power of the distance.Show also that the mass at the intersection of

*b*, *c* divided by the mass at intersection of *c*, *d*, is equal to the product of the area formed by *a*, *ρ*′, *d* divided by the area formed by *a*, *b*, *ρ* and the difference divided by the difference .These results may be conveniently arrived at by reducing one angular point as

*A* of the quadrilateral to rest. The resolved part of all the forces which act on each particle perpendicular to the straight line joining it to *A* will then be zero. The case of three particles may be treated in the same manner.Routh (1877)

Routh's priority claim does not concern Dziobek's formula (2.4), which he does not include. But the formulae he numbered (1.1), (1.2), (1.3) and (1.4) in Routh (1905) express the 's of our vectorial proof. Routh could easily prove (2.4) with his vectorial methods.

## 4. More equations for mutual distances

It is clear from equation (2.1) that the quantitydoes not depend on the point *q*. So if we set and , we get immediately(4.1)that is, for any *i*, *j*, and(4.2)Using this expression, we can get a simpler proof of the following lemma proved in Albouy (2003).

*For an n-body central configuration of dimension n*−2, *the inequality**holds for any i and j*, 1≤*i*<*j*≤*n*.

As we assume *a*<0, *s*_{ik}−*s*_{jk} has the sign of that is the sign of according to (2.4). The term with ‘’ in (4.2) has the sign of . We deduce that *Δ*_{j}−*Δ*_{i} has the sign of . As there must be a pair of *Δ*_{i}'s having opposite signs, we deduce that *μ*<0. The required inequality then follows immediately. ▪

We can restrict the study to *normalized central configurations*, choosing the normalization *λ*=*M*. As *a*≠0, equation (1.4) shows that one can obtain a normalized central configuration from any central configuration by changing its scale. Normalized central configurations satisfy equations (4.1) and, for some *μ*∈,(4.3)We said that (*Δ*_{1}, …, *Δ*_{n}) is defined only up to a factor. We can fix this factor and remove the parameter *μ*, which is negative according to the previous proof. To an (*n*−2)-dimensional normalized central configuration, we associate the unique (*Δ*_{1}, …, *Δ*_{n})∈^{n} satisfying (2.1), *Δ*_{i}*Δ*_{j}<*m*_{i}*m*_{j} and(4.4)These coordinates in turn determine the central configuration up to an isometry: from the *Δ*_{i}'s we find the mutual distances through expression (4.4).

## 5. The main lemmas

*Let ρ*_{1} *and ρ*_{2} *satisfy ρ*_{1}<*ρ*_{2}≤0 *and ρ*_{1}*ρ*_{2}<1. *Choose* *and α*∈]−∞,0[. *Set s*_{12}=(1−*ρ*_{1}*ρ*_{2})^{α}, *s*_{13}=(1−*ρ*_{1}*ρ*)^{α}, *s*_{23}=(1−*ρ*_{2}*ρ*)^{α} *and* . *Then A*(*ρ*_{1}, *ρ*_{2}, *ρ*)>0.

We first minimize *A* with respect to *ρ*. It is enough to minimize . We write *g*′(*ρ*)=0, that is, . Taking the power 1/(*α*−1), the equation becomes linear in *ρ*. There is at most one root, given byTo simplify this expression, we set *u*=1/(*α*−1), that is, *u*+1=*α*/(*α*−1). Since *α*<0, we have −1<*u*<0. We also set *q*=*ρ*_{2}/*ρ*_{1}, then 0<*q*<1 andBy plugging *ρ*_{0} into *g*(*ρ*) and using the definition of *q*, we obtain the minimal value of *s*_{13}−*s*_{23}In the new variables (*q*, *ρ*_{1}, *ρ*), the problem of minimizing *A* can be converted into that of minimizingTaking the minimizer of *B* in *ρ* is the same as replacing *g*(*ρ*) by *g*(*ρ*_{0}) above. We note that the second term in this expression of *B* does not depend on *ρ*_{1}. So, to minimize *B* in *ρ*_{1} it suffices to set *ρ*_{1}=0. Then we haveTo prove *C*>0, we need only to prove that . By raising to the power *u*=1/(*α*−1), the formula changes toTo conclude our proof, we check that this inequality holds for *q*∈]0,1[ and *u*∈]−1,0[. We will proveOne writes and calculates the derivative . By factoring out (1+*q*)^{u}, one gets the factor . Here we used the new variable , which satisfies . This factor is a Laguerre trinomial in *x*. It has at most two positive roots. This implies the same for *f*′(*q*) and excludes the possibility of a root of *f* in ]0,1[. In fact, since *f*(0^{+})=0^{+}, *f*(1)=0 and , *f* having another root between 0 and 1 would imply *f*′ having at least three positive roots. ▪

*Substituting the s*_{ik} *using* (*4.4*), *we consider* *as a function of Δ*_{1}, …, *Δ*_{n} *satisfying Δ*_{1}+⋯+*Δ*_{n}=0 *and of the parameters a*<0, *m*_{1}>0, …, *m*_{n}>0. *Let Δ*_{1}/*m*_{1}<*Δ*_{2}/*m*_{2}≤0, *Δ*_{1}*Δ*_{2}<*m*_{1}*m*_{2}, *Δ*_{i}≥0 *for i*≥3. *Then* .

We consider (4.2) and renumber the particles in such a way that . We get *t*_{1}−*t*_{2}≥*Z*, where . We set *s*_{1}=−*s*_{12}−*s*_{13}+*s*_{23} and *s*_{2}=*s*_{12}−*s*_{13}+*s*_{23} by (4.4) *s*_{12}>1 and *s*_{23}<1. Then *s*_{1}<0 and by lemma 5.1. ▪

Consider the first claim. The ‘only if’ part is easy and does not use the convexity of the configuration. Under the symmetry hypothesis *s*_{13}=*s*_{23} and *s*_{14}=*s*_{24}. Using (4.4), this gives *Δ*_{1}/*m*_{1}=*Δ*_{2}/*m*_{2}. But the symmetry also implies *Δ*_{1}=*Δ*_{2} so *m*_{1}=*m*_{2}. We pass to the ‘if’ part. We assume *m*_{1}=*m*_{2}. The convexity with particles 1 and 2 on a diagonal means, without loss of generality, *Δ*_{1}≤0, *Δ*_{2}≤0, *Δ*_{3}≥0 and *Δ*_{4}≥0. Assuming *Δ*_{1}<*Δ*_{2}, lemma 5.2 applies and gives *t*_{1}>*t*_{2}. This contradicts (4.1): there is no planar central configuration with these *Δ*_{i}'s. The inequality *Δ*_{2}<*Δ*_{1} is excluded in the same way. So *Δ*_{1}=*Δ*_{2} and the configuration is symmetric according to (3.1) or (4.4).

Concerning the other claims, what remains to be proved reduces to . This is obtained from lemma 5.2, which gives and . Note that here we have used the first part of the theorem and lemma 4.1. ▪

## 6. Higher dimensional results

*Let five particles* (*q*_{1}, *q*_{2}, *q*_{3}, *q*_{4}, *q*_{5}) *form a central configuration, which is a convex three-dimensional configuration having* [*q*_{1}, *q*_{2}] *as the diagonal. This configuration is symmetric with respect to the plane* [*q*_{3}, *q*_{4}, *q*_{5}] *if and only if m*_{1}=*m*_{2}. *Also m*_{1}<*m*_{2} *if and only if* , *where Δ*_{ijkl} *is the oriented volume of the tetrahedron* [*q*_{i},*q*_{j},*q*_{k},*q*_{l}].

We say ‘the diagonal’ because a generic convex three-dimensional configuration of five points has a unique diagonal. But if four particles are coplanar there are two diagonals in this plane (figure 2). The above theorem is valid for these diagonals. The proof of theorem 6.1 is similar to that of theorem 1.1.

We obtained the symmetry but do not know the existence and the uniqueness of the central configuration. We do not know whether the number of symmetric solutions is finite. With regard to the existence, the nice arguments of Xia (2004) show that there exists a convex central configuration, but Xia does not show that there exists one having [*q*_{1}, *q*_{2}] as the diagonal. In contrast with the four-body case, there are paths in the space of convex configurations going from configurations having [*q*_{1}, *q*_{2}] as the diagonal to, let us say, configurations having [*q*_{3}, *q*_{4}] as the diagonal. The limiting configuration is such that (*q*_{1}, *q*_{2}, *q*_{3}, *q*_{4}) form a convex planar quadrilateral configuration. This shape is possible for a central configuration.

In terms of the *Δ*_{i}'s, such a path starts with, for example, a hexahedron with , passes through a pyramidal configuration, where *Δ*_{5}=0, and finishes with a hexahedron with .

For any given choice of five positive masses, we do not know whether the number of planar or the number of non-planar central configurations is finite. The non-planar case seems easier because Dziobek's coordinates are available. Using them, Moeckel (2001) succeeded in proving the finiteness for almost any choice of masses. But still we do not know, for example, the answer in the case *m*_{1}=*m*_{2}=*m*_{3}=*m*_{4}=*m*_{5}. This state of ignorance surprised Battye *et al*. (2003) and is the motivation for the elegant studies by Santos (2004). Only four central configurations (figure 3) were found by Kotsireas and Lazard (Kotsireas 2001; Kotsireas & Lazard 2002), assuming that non-planar equal mass central configurations of five particles always have several symmetries. This hypothesis is strongly supported by numerical experiments.

*There are only four three-dimensional central configurations of five particles with equal masses* (*up to isometry, rescaling and permutation of the particles*).

The higher dimensional version of theorem 6.1, with *n* particles in dimension *n*−2, is also true, if the convex configuration is obtained by gluing two simplexes by a common hyperface. It should be noted that from *n*=6 this is not the only way to get a convex configuration. We can think of this in terms of the signs of the *Δ*_{i}'s. If only one of them is negative, we are in a non-convex case. If exactly two of them are negative, we are in the convex case where our lemma 5.2 may apply. If there are exactly three negative *Δ*_{i}'s, the configuration is convex but lemma 5.2 does not apply. Another way to obtain some geometrical intuition is to think of the configuration as a simplex plus a point, this point being added nearby but outside a hyperface or nearby but outside a lower dimensional face.

## Acknowledgments

We wish to thank Richard Moeckel for giving us his CC software, which we used to determine the configurations drawn in figure 3. We also thank Jacques Laskar and the referees for their precious comments. Y.F. supported by NSFC (grant nos 10473025 and 10233020). S.S. was partially supported by NSFC (grant nos 10401025 and 10571123) and by NSFBFBEC (grant no. KZ200610028015).

## Footnotes

- Received November 15, 2007.
- Accepted January 28, 2008.

- © 2008 The Royal Society