On the uniqueness of trapezoidal four-body central configurations

TL;DR

This paper proves that for the Newtonian four-body problem, there is at most one trapezoidal central configuration for each cyclic order of the masses, using a topological approach.

Contribution

It establishes the uniqueness of trapezoidal four-body central configurations for each mass ordering, a new result in celestial mechanics.

Findings

Proves at most one trapezoidal configuration per mass order

Uses topological methods to establish uniqueness

Contributes to understanding of four-body central configurations

Abstract

We study central configurations of the Newtonian four-body problem that form a trapezoid. Using a topological argument we prove that there is at most one trapezoidal central configuration for each cyclic ordering of the masses.

Equations

$$m_i{\ddot{\mathbf{q}}}_i=\frac{\partial \tilde{U}}{\partial {\mathbf{q}}_i}1\le i\le n,$$

$$\tilde{U}(\mathbf{q})=\sum _{i<j}\frac{m_i{m}_j}{\Vert {\mathbf{q}}_i-{\mathbf{q}}_j\Vert },$$

$$\tilde{I}(\mathbf{q})=\frac{1}{2}\sum _{i=1}^n{m}_i\Vert {\mathbf{q}}_i{\Vert }^2$$

$${\nabla }_{\mathbf{q}}\tilde{U}(\mathbf{q})+\lambda {\nabla }_{\mathbf{q}}\tilde{I}(\mathbf{q})=0$$

$$H(\mathbf{r})=288V^2=\left|\begin{array}{ccccc}0& 1& 1& 1& 1\\ 1& 0& r_{12}^2& r_{13}^2& r_{14}^2\\ 1& r_{12}^2& 0& r_{23}^2& r_{24}^2\\ 1& r_{13}^2& r_{23}^2& 0& r_{34}^2\\ 1& r_{14}^2& r_{24}^2& r_{34}^2& 0\end{array}\right|.$$

$$\mathcal{G}=\{\mathbf{r}\in ({\mathbb{R}}^{+}{)}^6|H(\mathbf{r})\ge 0\text{mbox}and{r}_{ij}+r_{jk}>r_{ik}\text{mbox}forall(i,j,k)\text{mbox}wherei\ne j\ne k\}.$$

$$\mathcal{N}=\{\mathbf{r}\in \mathcal{G}|I(\mathbf{r})-1=0,H(\mathbf{r})=0\}$$

$$\frac{\partial U}{\partial r_{ij}^2}=-32{\Delta }_i{\Delta }_j$$

$$F(\mathbf{r})=2r_{12}{r}_{34}-r_{13}^2-r_{24}^2+r_{23}^2+r_{14}^2.$$

$$\mathcal{F}=\{\mathbf{r}\in \mathcal{G}|F(\mathbf{r})=0\}$$

$$\mathcal{M}=\{\mathbf{r}\in {\mathbb{R}}^6|I(\mathbf{r})-1=0,F(\mathbf{r})=0\}$$

$${\mathcal{M}}^{+}=\{\mathbf{r}\in ({\mathbb{R}}^{+}{)}^6|I(\mathbf{r})-1=0,F(\mathbf{r})=0\}.$$

$$\mathcal{D}=\{\mathbf{r}\in {\mathcal{M}}^{+}\cap \mathcal{G}|H(\mathbf{r})=0\},$$

$$2H(\mathbf{r})=F(\mathbf{r})\cdot Q(\mathbf{r})-K^2(\mathbf{r})$$

$$\begin{gathered} Q(\mathbf{r})=-(r_{12}^2{r}_{13}^2-r_{12}^2{r}_{14}^2-r_{12}^2{r}_{23}^2+4r_{14}^2{r}_{23}^2+r_{12}^2{r}_{24}^2-4r_{13}^2{r}_{24}^2+2r_{12}^3{r}_{34}-2r_{12}{r}_{13}^2{r}_{34} \\ -2r_{12}{r}_{14}^2{r}_{34}-2r_{12}{r}_{23}^2{r}_{34}-2r_{12}{r}_{24}^2{r}_{34}+r_{13}^2{r}_{34}^2-r_{14}^2{r}_{34}^2-r_{23}^2{r}_{34}^2+r_{24}^2{r}_{34}^2+2r_{12}{r}_{34}^3) \end{gathered}$$

$$K(\mathbf{r})=r_{12}(r_{13}^2-r_{14}^2+r_{23}^2-r_{24}^2)+r_{34}(-r_{13}^2-r_{14}^2+r_{23}^2+r_{24}^2).$$

$$2H(\mathbf{r})=-K(\mathbf{r}{)}^2\le 0$$

$$U(\mathbf{r})+\lambda M(I(\mathbf{r})-1)+\eta H(\mathbf{r})$$

$${\nabla }_{\mathbf{r}}H(\mathbf{r})=\frac{1}{2}Q(\mathbf{r}){\nabla }_{\mathbf{r}}F(\mathbf{r}),$$

$$2{\nabla }_{\mathbf{r}}H(\mathbf{r})=Q(\mathbf{r}){\nabla }_{\mathbf{r}}F(\mathbf{r})+F(\mathbf{r}){\nabla }_{\mathbf{r}}Q(\mathbf{r})-2K(\mathbf{r}){\nabla }_{\mathbf{r}}K(\mathbf{r}).$$

$${\Delta }_1=\frac{1}{2}{r}_{34}h,{\Delta }_2=-\frac{1}{2}{r}_{34}h,{\Delta }_3=\frac{1}{2}{r}_{12}h,{\Delta }_4=-\frac{1}{2}{r}_{12}h$$

$$\frac{\partial H}{\partial r_{ij}}(\mathbf{r})=\frac{\partial H}{\partial r_{ij}^2}(\mathbf{r})\frac{d{r}_{ij}^2}{d{r}_{ij}}=-64r_{ij}{\Delta }_i{\Delta }_j$$

$${\nabla }_{\mathbf{r}}H(\mathbf{r})=8h^2{r}_{12}{r}_{34}(2r_{34},-2r_{13},2r_{14},2r_{23},-2r_{24},2r_{12}).$$

$${\nabla }_{\mathbf{r}}F(\mathbf{r})=(2r_{34},-2r_{13},2r_{14},2r_{23},-2r_{24},2r_{12}),$$

$$h=\frac{1}{4}\sqrt{\frac{Q(\mathbf{r})}{r_{12}{r}_{34}}}.$$

$$L(\mathbf{r};\lambda ,\sigma )=U(\mathbf{r})+\lambda M(I(\mathbf{r})-1)+\sigma F(\mathbf{r})$$

$$T_{\mathbf{r}}{\mathcal{M}}^{+}=\{\mathbf{v}\in {\mathbb{R}}^6|{\nabla }_{\mathbf{r}}(I(\mathbf{r})-1)\cdot \mathbf{v}=0,{\nabla }_{\mathbf{r}}F(\mathbf{r})\cdot \mathbf{v}=0\}.$$

$$T_{\mathbf{r}}\mathcal{N}=\{\mathbf{v}\in {\mathbb{R}}^6|{\nabla }_{\mathbf{r}}(I(\mathbf{r})-1)\cdot \mathbf{v}=0,{\nabla }_{\mathbf{r}}H(\mathbf{r})\cdot \mathbf{v}=0\}.$$

$$m_1{m}_2(r_{12}^{-3}-\lambda )$$

$$m_1{m}_3(r_{13}^{-3}-\lambda )$$

$$m_1{m}_4(r_{14}^{-3}-\lambda )$$

Full text

1 Introduction

A central configuration (c.c.) of the Newtonian $n$-body problem is a special arrangement of point masses with the property that the gravitational acceleration vector produced on each mass by all the others points toward the center of mass and is proportional to the distance to the center of mass.

The central configurations of the three body problem have been known for a long time. In the three-body problem, up to symmetry, there are exactly five relative equilibria, they are the Eulerian (or collinear) configurations discovered by Euler in 1767, and the Lagrangean configurations discovered by Lagrange in 1772. In the Eulerian configuration all the masses belong to the same line, while in the Lagrangean configurations the masses form an equilateral triangle.

Collinear configurations are also well understood. Moulton [24] provided an exact count of the number of collinear configurations of $n$ bodies: modulo symmetries, there are $n!/2$ central configurations. Also well understood are the $(n-1)$-dimensional configurations of $n$ masses. In this case there is a unique central configuration: the regular simplex. For instance, for four masses the only three-dimensional central configuration is the regular tetrahedron.

If all the masses are equal we have a complete classification of central configuration for $n=4,5,6$ and $7$. For $n=4$ the classification is due to Albouy [2, 3]. In this case the only noncollinear planar central configurations are the square, the equilateral triangle with a mass in the baricenter and an isosceles triangle with a mass on the line of symmetry. For $n=5,6$ and $7$ the classification is given using a computer assisted proof [22]. See also [19] were a complete classification of the isolated central configurations of the 5-body problem was given (note, however, that the approach used in this paper has a numerical component).

As soon as we go to the planar four-body problem, however, there is sufficient complexity to prevent a complete classification of noncollinear central configurations. For general masses we know that there is a finite number of central configurations of four bodies [15], but we don’t even have an exact count of the number of c.c.’s. In a recent paper [10], however, Corbera, Cors and Roberts provided a description of the set of convex central configurations and gave a clear picture of how the special subcases (i.e. trapezoidal, co-circular and kite-shaped, and equidiagonal central configurations) are situated within the broader set. Even less is known for the five-body problem where the finiteness of the number of central configurations was proven for arbitrary positive masses, except for a given codimension 2 subvariety of the mass space [7].

There are several reasons why c.c.’s play an important role in celestial mechanics. Central configurations lead to the only explicit solutions of the $n$-body problem. For instance, if all masses are released from a central configuration with zero initial velocity they accelerate in such a way that the configuration collapses homotethically. The result is a solution in which all the masses collide together after a finite time.

Furthermore, a planar central configuration gives rise to a family of periodic solutions. Given the appropriate initial conditions each particle will follow an elliptical orbit as in the Kepler problem. In this motion the configuration remains similar to the initial configurations, varying only in size. For instance, Eulerian configurations generate a periodic solution where each of the masses follows an elliptical orbit and the masses always lie on a common line, see Figure 1. Similarly, if at the initial moment the masses form an equilateral triangle and if suitable velocities are chosen, then the masses will move periodically on ellipses, as in Figure 2.

Central configurations also play an important role in the study of the topology of the integral manifolds $I_{w}$ of the $n$-body problem. An integral manifold is a subset of phase space obtained by fixing the values of the integrals of motion of the $n$ body problem (e.g. energy, angular momentum). Smale [30, 31] showed that central configurations are associated with changes in the topology of the integral manifolds. Since the integral manifolds have the property that if $v\in I_{w}$ then the orbit through $v$ is contained in $I_{w}$, Smale suggested that the topological type of $I_{w}$ can provide a crude, but important, invariant of the orbits [30, 31] of the planar $n$-body problem. Therefore, an understanding of the central configurations gives information on the topology of the integral manifolds which in turn gives rough information on the orbits of the system. The situation for the spatial $n$-body problem is more complicated and was addressed by Albouy [1].

In this work we will concentrate on the convex planar central configurations of four bodies. A planar configuration is convex if no body lies inside or on the convex hull of the other three bodies; otherwise, it is called concave. MacMillan and Bartky showed that for any four masses and any ordering of the bodies, there exists at least one convex central configuration [21], Xia [33] provided a simpler proof. It is an open question as to whether this solution must be unique. Yoccoz [34] conjectured that this solution is unique

Conjecture 1** (Simó-Yoccoz).**

There is a unique convex planar central configuration of the 4-body problem for each ordering of the masses in the boundary of its convex hull.

This conjecture likely arose from discussions between Yoccoz and Simó, and hence it seems appropriate to call it the Simó-Yoccoz conjecture. This problem is also included on a published list of open questions in celestial mechanics [4], and has often been attributed to Albouy and Fu [5]. The conjecture is known to have a positive answer in the cases all the masses are equal [2, 3], in the case some of masses are equal [20, 26, 6, 13], and in the case three of the masses are small [8]. Some related results were also obtained for point vortices in the case some of the vorticities are equal [16, 27], where it is possible to give a complete classification. Recently, the conjecture was verified also in the case of the co-circular four body problem [29]. In this paper we will show that this conjecture is also true for the trapezoidal four-body problem, which considers the case where the masses form a trapezoid. The central configurations of the trapezoidal four-body problem were studied in detail in [28, 9]. The uniqueness of trapezoidal central configurations was recently proved for the particular case of two pairs of equal masses in the case of power-law potentials [14]. The main goal of this paper is to prove the following theorem:

Theorem 1**.**

There is at most one trapezoidal central configuration of four bodies for each cyclic ordering of the masses.

The method of proof is similar to the one employed for the co-circular four body problem. The main idea of the proof is to use mutual distances as coordinates and replace the Cayley-Menger determinant condition used by Dziobek [12] with a simpler condition which comes from the geometry of trapezoids [17, 28]. It is then possible to show that the critical points of the potential $U$ restricted to a certain subvariety are all minimum points. Knowing the Euler characteristic of the variety one can then use Morse theory to prove the theorem.

The paper is organized as follows. In Section 2 we introduce the n-body problem and define central configurations. In Section 3 we write four-body central configurations in terms of mutual distances between the bodies. In Section 4 we define trapezoidal configurations and find their equations following the approach of [28]. In particular we view such configurations as critical points of the potential restricted to a certain space that we call $\mathcal{M}^{+}$. In Section 5 we prove Theorem 1 using Morse theory. This is done in several steps. In Proposition 3 we show that all the critical points are nondegenerate local minima. In Lemma 5 we obtain the Euler characteristic of the space $\mathcal{M}^{+}$. In Lemma 6 we use Morse theory and the Euler characteristic of $\mathcal{M}^{+}$ to prove that the potential restricted to $\mathcal{M}^{+}$ has a unique critical point. We then use this last result to prove Theorem 1.

2 Central Configurations of the $n$-body problem

The Newtonian $n$-body problem concerns the motion of $n$ point particles of masses $m_{i}>0$ and positions $\mathbf{q}_{i}\in\mathbb{R}^{d}$, where $i=1,\ldots n$. Let $\mathbf{q}=(\mathbf{q}_{1},\ldots,\mathbf{q}_{n})\in\mathbb{R}^{dn}$, let $r_{ij}=\|\mathbf{q}_{i}-\mathbf{q}_{j}\|$ be the Euclidean distance between the masses $m_{i}$ and $m_{j}$, and let $\mathbf{r}=(r_{12},\ldots,r_{n-1\,n})$ be the vector of mutual distances. The equations of motion are given by

[TABLE]

where $\tilde{U}(\mathbf{q})$ is the Newtonian potential

[TABLE]

which we denote by $U(\mathbf{r})$ when viewed as a function of the mutual distances $r_{ij}$. Without any loss of generality we can assume that the center of mass of the particles is at the origin: $\sum_{i=1}^{n}m_{i}\mathbf{q}_{i}=0$. Denote by $\tilde{I}(\mathbf{q})$ the moment of inertia as a function of $\mathbf{q}$

[TABLE]

and by $I(\mathbf{r})=\frac{1}{2M}\sum_{i<j}m_{i}m_{j}r_{ij}^{2}$ the moment of inertia as a function of the distances.

A central configuration of the $n$-body problem is a configuration $\mathbf{q}\in\mathbb{R}^{nd}$ which satisfies the algebraic equation

[TABLE]

where $\lambda$ is a Lagrange multiplier. Hence, a central configuration is simply a critical point of $\tilde{U}$ subject to the constraint $\tilde{I}=\tilde{I}_{0}$.

The central configuration equation (1) is invariant under rotations, reflections and dilations. It is standard to say that two configurations $\mathbf{q}$ and $\mathbf{q}^{\prime}$ are *equivalent * if $\mathbf{q}$ can be transformed to $\mathbf{q}^{\prime}$ by a rotation and a dilation. As a consequence, by convention, central configurations are usually counted up to rotations and dilations. This convention is also used in the statement of Conjecture 1 and of Theorem 1.

We define the dimension of a configuration $\mathbf{q}$, denoted $\operatorname{dim}(\mathbf{q})$, to be the dimension of the subspace spanned by the vectors $\mathbf{q}_{j}$. Then, we say that $\mathbf{q}$ is a Dziobek configurations if $\operatorname{dim}(\mathbf{q})=n-2$ [23].

In the four-body problem $\mathbf{q}$ is a Dziobek central configuration if it is a central configuration with $\operatorname{dim}(\mathbf{q})=2$, that is, in this case, the set of Dziobek configurations coincide with the set of planar, non-collinear, central configurations.

3 Central Configurations in terms of distances

For four bodies it is convenient to recast the equations defining Dziobek central configuration, so that the variables are the distances between the particles rather than their coordinates. Since the mutual distances determine the configuration up to rotation and reflection symmetry, this choice not only reduces the number of variables but also removes the rotational and reflectional degeneracy. The dilational degeneracy can then be eliminated by fixing the size of the configuration with the restriction $I=1$.

Let $\mathbf{r}=(r_{12},r_{13},r_{14},r_{23},r_{24},r_{34})\in(\mathbb{R}^{+})^{6}$ be a vector of non-negative mutual distances, and let the Cayley–Menger determinant of four points $P_{1},\ldots P_{4}$ be

[TABLE]

where $V$ is the volume of the configuration. It is important to note that not all vectors $\mathbf{r}$ realize actual configurations of four bodies in $\mathbb{R}^{3}$. Therefore, we typically want to restrict our attention to configurations that can be realized in $\mathbb{R}^{3}$. For this purpose we consider the sets

[TABLE]

and

[TABLE]

We say that a vector of mutual distances $\mathbf{r}$ is geometrically realizable if $\mathbf{r}\in\mathcal{G}$ and that $\mathbf{r}$ is a normalized Dziobek configuration if $\mathbf{r}\in\mathcal{N}$.

Thus we have the following characterization of planar four body central configurations given by Dziobek:

Proposition 1**.**

*Let $\mathbf{q}$ be a Dziobek configuration, let $\mathbf{r}\in\mathcal{N}$ be its corresponding normalized Dziobek configuration, and let $U|_{\mathcal{N}}:\mathcal{N}\to\mathbb{R}$ be the restriction of the Newtonian potential $U$ to $\mathcal{N}$. Then, $\mathbf{q}$ is a Dziobek central configuration if and only if $\mathbf{r}$ is a critical point of $U|_{\mathcal{N}}$. *

Since equations (1) are invariant under rotations, dilations and reflections in the plane, we can consider two relative equilibria as equivalent if they are related by these symmetry operations. This defines an equivalence relation $\sim$, different from the more standard one introduced in section 2. Let $X$ be the set of equivalence classes with respect to $\sim$, then the set of equivalence classes $X$ is in a one-to-one correspondence with the set $c(U|_{\mathcal{N}})$ of critical points of the function $U(\mathbf{r})|_{\mathcal{N}}$.

To find the equation for the critical points of $U|_{N}$ we need to write the gradient of $U$ restricted to $\mathcal{N}$. The following formula due to Dziobek [12]

[TABLE]

is particularly useful for this purpose. Here, $\Delta_{i}$ denotes the signed area of the triangle whose vertices contain all bodies except for the $i$-th body. This formula is valid when restricting to planar configurations. A generalization of this formula that also works for non planar configurations uses oriented areas and can be found in [18].

4 Trapezoidal Configurations

In this section we study trapezoidal central configurations. Since we use mutual distances as coordinates, we cannot distinguish between bodies ordered counterclockwise and bodies ordered clockwise. Hence, we introduce the following terminology: we say that the bodies are *ordered sequentially * if they are numbered consecutively while traversing the boundary of the quadrilateral in any direction.

Without loss of generality, we may assume that any trapezoid is ordered sequentially so that $r_{13}$ and $r_{24}$ are the lengths of the diagonals. This is justified because we can always relabel the bodies so that they are ordered sequentially. Denote

[TABLE]

Let $\mathcal{F}$ be the set of geometrically realizable $\mathbf{r}$ satisfying $F(\mathbf{r})=0$, that is

[TABLE]

Moreover, we define $\mathcal{M}$ and $\mathcal{M}^{+}$ as follows:

[TABLE]

and

[TABLE]

Let us denote by $\mathcal{M}_{0}$ and by $\mathcal{M}^{+}_{0}$ the sets $\mathcal{M}$ and $\mathcal{M}^{+}$ in the case $m_{1}=m_{2}=m_{3}=m_{4}$. We can also define the set

[TABLE]

which will play an important role, in this paper.

There is an interesting relationship between the conditions $F(\mathbf{r})=0$ and $H(\mathbf{r})=0$, which is outlined in the following lemma

Lemma 1**.**

If $\mathbf{r}\in\mathcal{F}$, then $H(\mathbf{r})=0$. In other words on the set of geometrically realizable vectors for which $F=0$ the configuration of four bodies is coplanar.

Proof.

A computation shows that

[TABLE]

where

[TABLE]

and

[TABLE]

Note that equation (3) is the analogue of equation (12) in [25] for cyclic quadrilaterals. If $F=0$ we have

[TABLE]

Since $\mathbf{r}\in\mathcal{G}$ implies that $H(\mathbf{r})\geq 0$, it follows that we must have $H(\mathbf{r})=0$, which concludes the proof. ∎

Since trapezoidal central configurations are Dziobek configuration we can give the following definition

Definition 1**.**

The configuration vector $\mathbf{q}$ is a sequentially ordered trapezoidal central configuration if and only if its corresponding distance vector $\mathbf{r}$ belongs to $\mathcal{D}$ and it is a critical point of $U|_{\mathcal{N}}$ with respect to $\mathbf{r}$.

In terms of Lagrange multipliers this means that $\mathbf{r}\in\mathcal{D}$ is a sequentially ordered trapezoidal four body central configuration if and only if it is a critical point of the function

[TABLE]

satisfying $I-1=0$, $F=0$ and $H=0$, where $\lambda$, and $\eta$ are Lagrange multipliers. The following lemma shows that $\nabla_{\mathbf{r}}F(\mathbf{r})$ and $\nabla_{\mathbf{r}}H(\mathbf{r})$ are parallel on the set of geometrically realizable configurations with $H=F=0$. See [28] for a different proof. A similar result was obtained by Cors and Roberts for the co-circular four body problem [11].

Lemma 2**.**

For any $\mathbf{r}\in\mathcal{F}$

[TABLE]

where $Q(\mathbf{r})=16h^{2}r_{12}r_{34}$, with $h$ the height of the trapezoid. In other words, on the set of geometrically realizable vectors for which $F$ vanish, the gradients of $H$ and $F$ are parallel.

Proof.

Since $2H(\mathbf{r})=F(\mathbf{r})\cdot Q(\mathbf{r})-K^{2}(\mathbf{r})$, we have that

[TABLE]

Since $\mathbf{r}\in\mathcal{F}$, then $H=F=0$. It follows that $K=0$ as well. Hence, $2\nabla_{\mathbf{r}}H(\mathbf{r})=Q(\mathbf{r})\nabla_{\mathbf{r}}F(\mathbf{r})$.

We now want to show that, in this case, $Q(\mathbf{r})$ has a meaningful geometric interpretation and can be written in terms of the height of the trapezoid. For a convex quadrilateral ordered sequentially we can choose the signed areas so that $\Delta_{1},\Delta_{3}>0$ and $\Delta_{2},\Delta_{4}<0$. In a trapezoid these signed areas are

[TABLE]

where $h$ is the height of the trapezoid, namely the distance between the opposite parallel sides. From (2) we get

[TABLE]

and hence, at a trapezoidal central configuration, we have

[TABLE]

On the other hand, the gradient of $F$ at a trapezoidal configuration is

[TABLE]

from which it follows that $Q(\mathbf{r})=16h^{2}r_{12}r_{34}$. ∎

Remark**.**

In the previous lemma we showed that $Q(\mathbf{r})=16h^{2}r_{12}r_{34}$. Note that this equality is not trivial. In fact, solving for $h$ we find the following formula for the height of a trapezoid as a function of the mutual distances:

[TABLE]

This formula is different from the well known one given in [17, 32, 28], and has the advantage of working even when the bases of the trapezoid have the same length.

We now have the following characterization of trapezoidal configurations [28]:

Proposition 2**.**

Let $\mathbf{r}\in\mathcal{D}$. Then $\mathbf{r}$ is a critical point of $U|_{\mathcal{N}}$, the restriction of $U$ to $\mathcal{N}$, if and only if $\mathbf{r}$ is a critical point of the function $U|_{\mathcal{M}^{+}}:\mathcal{M}^{+}\to\mathbb{R}$. Therefore the vector $\mathbf{q}$ is a sequentially ordered trapezoidal four-body c.c. if and only if the corresponding distance vector $\mathbf{r}\in\mathcal{D}$ is a critical point of the Lagrangian function

[TABLE]

satisfying $I-1=0$, $F=0$ and $H=0$, where $\lambda$ and $\sigma$ are Lagrange multipliers.

Proof.

Recall that $\nabla_{\mathbf{r}}U|_{\mathcal{M}^{+}}$ is the orthogonal projection of $\nabla_{\mathbf{r}}U(\mathbf{r})$ onto the tangent space $T_{\mathbf{r}}\mathcal{M}^{+}$, which is given by

[TABLE]

Similarly, $\nabla_{\mathbf{r}}U|_{\mathcal{N}}$ is the orthogonal projection of $\nabla_{\mathbf{r}}U(\mathbf{r})$ onto the tangent space $T_{\mathbf{r}}\mathcal{N}$, which is given by

[TABLE]

Since $\mathbf{r}\in\mathcal{D}$, by Lemma 2, $\nabla_{\mathbf{r}}H(\mathbf{r})=\frac{1}{2}Q(\mathbf{r})\nabla_{\mathbf{r}}F(\mathbf{r})$. It follows that, if $\mathbf{r}\in\mathcal{D}$, then $T_{\mathbf{r}}\mathcal{M}^{+}=T_{\mathbf{r}}\mathcal{N}$, and hence $\nabla_{\mathbf{r}}U|_{\mathcal{N}}=\nabla_{\mathbf{r}}U|_{\mathcal{M}^{+}}$ for any $\mathbf{r}\in\mathcal{D}$. Then $\nabla_{\mathbf{r}}U|_{\mathcal{M}^{+}}=0$ if and only if $\nabla_{\mathbf{r}}U|_{\mathcal{N}}=0$, that is, $\mathbf{r}$ is a critical point of $U|_{\mathcal{N}}$ if and only if $\mathbf{r}$ is a critical point of the function $U|_{\mathcal{M}^{+}}$. ∎

By Proposition 2, we can find the critical points of $U|_{\mathcal{N}}$ that lie in $\mathcal{D}$ by finding the critical points of $U|_{\mathcal{M}^{+}}$ which lie in $\mathcal{D}$. The equations of the critical points of $U|_{\mathcal{M}^{+}}:\,{\mathcal{M}^{+}}\to\mathbb{R}$, are given by $\nabla_{\mathbf{r}}L(\mathbf{r};\lambda,\sigma)=\nabla_{\mathbf{r}}U+\lambda M\nabla_{\mathbf{r}}I+\sigma\nabla_{\mathbf{r}}F$, the gradient of the Lagrangian $L$. Explicitly, we have

[TABLE]

Note that these equations hold for $\mathbf{r}\in\mathcal{M}^{+}$, and not just for $\mathbf{r}\in\mathcal{D}$. When $\mathbf{r}\in D\subset\mathcal{M}^{+}$, the solutions of these equations give trapezoidal central configurations.

The equations have been grouped in pairs so that when they are multiplied together the product of the right-hand sides is $\sigma^{2}$. Consequently, from equations (5),(6) and (7) we obtain three equations for $\sigma^{2}$:

[TABLE]

5 Uniqueness of Trapezoidal configurations

In this section we prove Theorem 1. The strategy of the proof is as follows.

We first show that if $\mathbf{r}\in\mathcal{M}^{+}$ is a critical point of $U|_{\mathcal{M}^{+}}$, then it is necessarily a nondegenerate local minimum. This is proved in Proposition 3. Lemma 3 is a technical lemma required to prove Proposition 3.

We then study the topology of $\mathcal{M}_{0}$ and $\mathcal{M}^{+}$. In Lemma 4 we show that $\mathcal{M}_{0}\approx S^{2}\times S^{2}$. In Lemma 5 we show that the Euler characteristic $\chi(\mathcal{M}^{+})$ of $\mathcal{M}^{+}$ is $1$.

Finally we use Morse theory to prove that the function $U|_{\mathcal{M}^{+}}$ has a unique critical point on $\mathcal{M}^{+}$. This is done in Lemma 6. The proof of the theorem follows immediately.

We start with the following technical lemma which is needed in the proof of Proposition 3.

Lemma 3**.**

If $\mathbf{r}^{\ast}\in\mathcal{M}^{+}$ is a critical point of $U|_{\mathcal{M}^{+}}$ then $\lambda>0$.

Proof.

Suppose, for the sake of contradiction, that $\lambda\leq 0$. By the first of the two equation in (8) we find that

[TABLE]

and hence $\sigma>0$, since $r_{12},r_{34}>0$ in $\mathcal{M}^{+}$. By the first of the two equation in (9) we find that

[TABLE]

and hence $\sigma<0$, which contradicts the fact that $\lambda\leq 0$. It follows that $\lambda>0$.

∎

Note that the second derivative of $D^{2}L(\mathbf{r};\lambda,\sigma)$ of $L(\cdot;\lambda,\sigma)$ with respect to the variable $\mathbf{r}$ is the matrix

[TABLE]

This second derivative, with appropriate choices of $\lambda$ and $\sigma$ is the second derivative of $U|_{\mathcal{M}^{+}}$, at the critical points. We can now prove the following proposition

Proposition 3**.**

If $\mathbf{r}^{\ast}\in\mathcal{M}^{+}$ is a critical point of $U|_{\mathcal{M}^{+}}$ then $\mathbf{r}^{\ast}$ is a nondegenerate minimum point for $U|_{\mathcal{M}^{+}}$.

Proof.

The second derivative of $L$ is the matrix

[TABLE]

where $f_{ij}(\mathbf{r})=m_{i}m_{j}(2r_{ij}^{-3}+\lambda)$, $\operatorname{diag}$ denotes a $6\times 6$ diagonal matrix, and $\operatorname{adiag}(2\sigma,0,0,0,0,2\sigma)$ denotes the $6\times 6$ anti-diagonal matrix whose entries on the anti-diagonal are $2\sigma,0,0,0,0,2\sigma$. As we observed earlier, $D^{2}L(\mathbf{r}^{\ast};\lambda,\sigma)$ coincides with $D^{2}(U|_{\mathcal{M}^{+}})$ evaluated at the critical point $\mathbf{r}^{\ast}$.

Let $P_{k}(\mathbf{r})$ be the principal minor of order $k$ of $D^{2}L(\mathbf{r};\lambda,\sigma)$. We first prove that if $\mathbf{r}^{\ast}$ satisfies equations (5-7), then $P_{k}(\mathbf{r}^{\ast})>0$ for $k=1,\ldots 6$.

Let

[TABLE]

Since $\lambda>0$, eliminating $\sigma^{2}$ using equations (5-7) yields

[TABLE]

Furthermore, eliminating $\sigma^{2}$ from $A_{5}(\mathbf{r})$ using (8) gives

[TABLE]

Since $\lambda>0$ by Lemma 3, it is easy to see that all the principal minors are positive:

[TABLE]

It follows that $D^{2}L(\mathbf{r}^{\ast},\lambda,\sigma)$ is positive definite, and $\mathbf{r}^{\ast}$ is a nondegenerate local minimum of $U|_{\mathcal{M}^{+}}$. ∎

Remark**.**

Note that using the condition $F=0$ instead of $H=0$ in this problem does not make a big difference when computing the gradient, but it leads to much simpler computations when computing the second derivative. This can be seen from the following computation. Recall that if $f:\mathbb{R}^{n}\to\mathbb{R}$, then $\nabla_{\mathbf{x}}f(p)$ is an $n\times 1$ matrix whose entries are the partial derivatives of $f$ at $p$, while $D_{\mathbf{x}}f(p)$ is a $1\times n$ matrix whose entries are the partial derivatives of $f$ at $p$. We compute the Hessian $D^{2}H(\mathbf{r})=D_{\mathbf{r}}\nabla_{\mathbf{r}}H$ of $H(\mathbf{r})$ by computing the derivative of equation (4) and we obtain

[TABLE]

where the dot represents matrix multiplication. Since at a trapezoidal c.c. we have that $F=0$ and $K=0$, it follows that

[TABLE]

which is much more complicated than $D^{2}F$.

We now turn to study the topology of $\mathcal{M}$ and $\mathcal{M}^{+}$.

Lemma 4**.**

$\mathcal{M}_{0}\approx S^{2}\times S^{2}$.

Proof.

Since in this case $m_{1}=m_{2}=m_{3}=m_{4}=1$, the equation for the moment of inertia, reduces to

[TABLE]

which defines a sphere. Adding $F=0$ to this equation gives

[TABLE]

subtracting $F=0$ from it gives

[TABLE]

which shows that the manifold $\mathcal{M}^{+}$ is diffeomorphic to $S^{2}\times S^{2}$, provided that $m_{1}=m_{2}=m_{3}=m_{4}=1$.

∎

We can now better understand the topology of $\mathcal{M}^{+}$.

Lemma 5**.**

*The Euler characteristic $\chi(\mathcal{M}^{+})$ of $\mathcal{M}^{+}$ is $1$. *

Proof.

Suppose $m_{1}=m_{2}=m_{3}=m_{4}=1$, and consider the change of variables

[TABLE]

equations (12) and (13) can be rewritten in the form

[TABLE]

Clearly the set $\mathcal{M}^{+}_{0}$ is homeomorphic to $E$, the subset of $S_{1}\times S_{2}$ defined by the following inequalities

[TABLE]

The inequalities for $r_{12}$ and $r_{34}$ can be expressed more compactly as $v_{1}\geq|w_{1}|$, which clearly implies $v_{1}\geq 0$. The inequalities $v_{1},v_{2},v_{3}\geq 0$ select a spherical triangle $T$ corresponding to one octant of the sphere $S_{1}$. Such spherical triangle is homeomorphic to a closed disk, and can be represented with coordinates $(v_{1},v_{2})$ in the set $B=\{(v_{1},v_{2})\in\mathbb{R}^{2}|\,v_{1}\geq 0,v_{2}\geq 0\}$. Corresponding to each point $(v_{1},v_{2})\in B$ there is a region $R$ on the sphere $S_{2}$ defined by the inequalities $|w_{1}|\leq v_{1}$, $w_{2}\geq 0$ and $w_{3}\geq 0$. Clearly we have

[TABLE]

The region $R$ is homeomorphic to a region $\bar{R}$ on the plane $(w_{2},w_{3})$ defined by the inequalities

[TABLE]

If $v_{1}=0$, then $w_{2}^{2}+w_{3}^{2}=1$ and the region is an arc of the unit circle. If $v_{1}=1$, then $\bar{R}$ is a quarter unit disk. In all other cases $\bar{R}$ is a quarter of an annular ring, see 3.

It follows that $\bar{R}$ is always contractible, and so is $R$.

The restriction of the projection $\tilde{p}:(v_{1},v_{2},v_{3},w_{1},w_{2},w_{3})\to(v_{1},v_{2},v_{3})$, induces a fibration $p:E\to T$ with base space $T$ and fibers given by $R$. Hence, the projection $p$ is a fibration with contractible fibers. Since $T$ is also contractible, it follows that $E$ is contractible, and hence $\chi(\mathcal{M}^{+}_{0})=\chi(E)=1$ when $m_{1}=m_{2}=m_{3}=m_{4}=1$.

Consider the rays having the origin as a initial point. Each of these rays intersect $S_{0}$, the region of the sphere defined by equation (11) satisfying the inequalities $r_{ij}\geq 0$, in exactly one point. Each ray also intersects $E_{0}$, the region of the ellipsoid of inertia $I(\mathbf{r})=1$ such that $r_{ij}\geq 0$, in one point. Thus the points of $E_{0}$ are in one-to-one correspondence with the points of $S_{0}$. Let $f:S_{0}\to E_{0}$ be the homeomorphism defined by the rays having the origin as initial point. Since $F=0$ defines a cone, and $\mathcal{M}^{+}\subset S_{0}$, then $f(\mathcal{M}^{+}_{0})=\mathcal{M}^{+}$. Since the restriction of an homeomorphism to a subset is still a homeomorphism, it follows that $\mathcal{M}^{+}_{0}\approx\mathcal{M}^{+}$. Hence, $\chi(\mathcal{M}^{+})=\chi(\mathcal{M}_{0}^{+})=1$, which concludes the proof.

∎

Since we have determined the topology of $\mathcal{M}^{+}$ we can now use Morse theory to prove the following Lemma

Lemma 6**.**

The function $U|_{\mathcal{M}^{+}}$ has a unique critical point on $\mathcal{M}^{+}$.

Proof.

The proof is analogous to the proof of Lemma 6 in [29], and to Smale’s proof of Moulton’s theorem for the collinear $n$-body problem [31] (which however, is presented without details). We repeat it here for convenience of the reader. By Proposition 3 any critical point $\mathbf{r}\in\mathcal{M}^{+}$ is a nondegenerate local minimum of the function $U|_{\mathcal{M}^{+}}$, and hence $U|_{\mathcal{M}^{+}}$ is a Morse function that tends to $+\infty$ as $\mathbf{r}$ nears $\partial\mathcal{M}^{+}$, the boundary of $\mathcal{M}^{+}$. Therefore, the function $U|_{\mathcal{M}^{+}}$ admits a global minimum value in the interior of $\mathcal{M}^{+}$. Suppose there are several global minimum points where the function obtains its least possible value. By Proposition 3 any of such point must be a non-degenerate local minimum point. By Lemma 5, the Euler characteristic of $\mathcal{M}^{+}$ is $\chi(\mathcal{M}^{+})=1$. By Morse theory we have

[TABLE]

where the sum is over the critical points, $\gamma$ is the Morse index of the critical points and $C^{\gamma}$ is the number of critical points of index $\gamma$. We know that there is at least one local minimum, and that all the critical points of $U|_{\mathcal{M}^{+}}$ are local minimum points and hence have index [math]. However, this function cannot have more than one minimum point since otherwise, equation (14) would imply the existence of at least one non-minimum critical point, contradicting Proposition 3. ∎

We are finally in a position to prove Theorem 1, our main result

Proof of Theorem 1.

Recall that, by Proposition 3, trapezoidal central configurations correspond to distance vectors $\mathbf{r}\in\mathcal{D}$ that are critical points of the function $U|_{\mathcal{M}^{+}}$. Lemma 6 shows that $U|_{\mathcal{M}^{+}}$ has a unique critical point on $\mathcal{M}^{+}$. Since $\mathcal{D}\subset\mathcal{M}^{+}$, there is at most one critical point of $U|_{\mathcal{M}^{+}}$ on $\mathcal{D}$. Hence, we have shown that there is a most one trapezoidal central configurations for each ordering of the masses, and the theorem follows. ∎

Acknowledgments

I would like to thank Alessandro Portaluri and Shengda Hu, for interesting discussions related to this work.