Central Configurations and Mutual Differences

Central configurations are solutions of the equations $\lambda m_j\boldsymbol{q}_j = \frac{\partial U}{\partial \boldsymbol{q}_j}$, where $U$ denotes the potential function and each $\boldsymbol{q}_j$ is a point in the $d$-dimensional Euclidean space $E\cong {\mathbb R}^d$, for $j=1,\ldots, n$. We show that the vector of the mutual differences $\boldsymbol{q}_{ij} = \boldsymbol{q}_i - \boldsymbol{q}_j$ satisfies the equation $-\frac{\lambda}{\alpha} \boldsymbol{q} = P_m(\Psi(\boldsymbol{q}))$, where $P_m$ is the orthogonal projection over the spaces of $1$-cocycles and $\Psi(\boldsymbol{q}) = \frac{\boldsymbol{q}}{|\boldsymbol{q}|^{\alpha+2}}$. It is shown that differences $\boldsymbol{q}_{ij}$ of central configurations are critical points of an analogue of $U$, defined on the space of $1$-cochains in the Euclidean space $E$, and restricted to the subspace of $1$-cocycles. Some generalizations of well known facts follow almost immediately from this approach.


Introduction
Central configurations play an important role in the (Newtonian) n-body problem: to name two, they arise as configurations yielding homographic solutions, and as rest points in the flow on the McGehee collision manifold. Following the spirit of Albouy and Chenciner [2], in this article we study the problem of central configurations from the point of view of mutual distances; but instead of lengths we consider the space of differences of positions, which turns out to be a suitable group of cochains C 1 of degree 1 with coefficients in the Euclidean space E. Hence, we show that central configurations are critical points of a function defined on C 1 and restricted to the subspace of 1-cocycles, and show some consequences. The technique of embedding the central configurations problem into a suitable space of cocycles was actually already used by Moeckel in [12], in an implicit way, and again by Moeckel and Montgomery in [15]. In this article we study this approach introducing cocycles and cohomology, and show that many calculations can be significantly simplified in this way. For further details and recent remarkable advances we refer to [3,8].
More precisely, assume n ≥ 2, d ≥ 1. Let E = R d denote the d-dimensional Euclidean space. An element of E n will be denoted by q = (q 1 , q 2 , . . . , q n ) where ∀ j, q j ∈ E. Let F n (E) denote as in [4] the configuration space of n particles in E: If ∆ is the collision set ∆ = i<j q ∈ E n : q i = q j , then F n (E) = E n \ ∆.

arXiv:1608.00480v4 [math.DS] 31 Mar 2017
For j = 1, . . . , n, let m j > 0 be positive masses. Assume that the masses are normalized, i.e., that n j=1 m j = 1. (1.1) Let * , * M denote the mass-metric on (the tangent vectors of) E n , defined as where v j · w j is the Euclidean scalar product in (the tangent space of) E. Let |v j | denote the Euclidean norm of a vector v j in E. The norm corresponding to the mass-metric Let α > 0 be a fixed homogeneity parameter, and U : F n (E) → R the potential function defined as A central configuration is a configuration q ∈ F n (E) such that there exists λ ∈ R such that (∀ j = 1, . . . , n) If q is a central configuration, then For an analysis of central configurations for general potential functions U (q), see [6,7]. Also, central configurations can be equivalently seen as: (CC1) Solutions of (1.2) [13].
(CC2) Critical points of the restriction of the potential function U to the inertia ellipsoid S = {q ∈ F n (E) : q 2 M = 1} [14]. (CC3) Fixed points of the map F : S → S defined as F (q) = − ∇ M U (q) , where ∇ M denotes the gradient with respect to the mass-metric on F n (E) [6,7].
(CC4) Critical points on F n (E) of the map q 2 M + U (q) [9]. (CC5) Critical points on F n (E) of the map q 2α M U (q) 2 (or q α M U (q)) [17]. In all these formulations, central configurations appear as O(d)-orbits in F n (E), where the action of the orthogonal group O(d) on F n (E) is diagonal g · q = (gq 1 , . . . , gq n ).
Define the space X as

Central conf igurations and mutual dif ferences
Let n be the set n = {1, 2, . . . , n} and C 0 the vector space of all maps from n to E: C 0 = {q : n → E}. Let F n (E) ⊂ C 0 denote the inclusion sending q ∈ F n (E) to the map q : n → E defined by q(j) = q j for each j ∈ n. Now, letñ denote the set of all n 2 subsets in n with two elements:ñ = {{1, 2}, {1, 3}, . . ., {n − 1, n}}. Let C 1 denote the vector space of all maps fromñ to E: It is isomorphic to Eñ, whereñ = n 2 . Note that if E n 2 denotes that vector space of all maps q : n 2 → E, where n 2 = n × n (and hence if q ∈ E n 2 , we can denote q ij = q((i, j)) ∈ E), there is an embedding C 1 ⊂ E n 2 , by sending an element q ∈ C 1 to the map q : n 2 → E defined by In fact, we are identifying elements in C 1 with the skew-symmetric elements in E n 2 (that is, maps q ij + q ji = 0 ∈ E). Given q ∈ C 1 with an abuse of notation we will write q ij instead of q ij , and ij instead (i, j). If K is an abstract simplicial complex, recall that the simplicial chain complex of K with real coefficients, denoted by C * (K; R), is defined as follows: for each k ∈ Z, the chain group C k (K; R) is the vector space of all the R-linear combinations of k-dimensional simplexes of K; (−1) j σd j for each k-simplex σ of K and 0 otherwise, where d j is the j-th face map. More precisely, all simplices in K can be ordered, and elements in C k (K; R) will be linear combinations of ordered k-simplices in K. An ordered k-simplex with vertices x 0 , . . . , x k will be denoted either as [x 0 , . . . , x k ] or simply as x 0 . . . x k . With this notation, the j-th face map sends σ = [x 0 , . . . , x j , . . . , x k ] to σd j = [x 0 , . . . , x j , . . . , x k ], where x j means that the j-th element is canceled.
By taking homomorphisms valued in an R-vector space E, the chain complex C k (K; R) yields the simplicial cochain complex with coefficients in E: the cochain groups are defined as all the linear homomorsphisms C k (K; E) = hom R (C k (K; R), E), and the coboundary homomorphisms The kernel of δ k is the group of cocycles, and it is denoted Now, let ∆ n−1 denote the standard (abstract) simplex with n vertices {1, 2, . . . , n}. Then the vector spaces C k defined above for k = 0, 1 are exactly the groups of k-dimensional simplicial cochains (with coefficients in the vector space E) of the simplicial complex ∆ n−1 : C 0 = C 0 (∆ n−1 ; E) and C 1 = C 1 (∆ n−1 ; E). A 0-simplex of ∆ n−1 is simply an element j ∈ n, and hence a 0-dimensional cochain is an n-tuple q j , i.e., a map q : n → E. Furthermore, a 1-dimensional cochain is a map q defined with values in E and as domain the set of 1-dimensional simplices of ∆ n−1 , i.e., pairs ij with 1 ≤ i < j ≤ n.
In simpler terms, for each i, j such that 1 ≤ i < j ≤ n, let q ij ∈ E denote the ij-the component of a vector in Eñ, and for i > j, the variable q ij is defined by the property that ∀ i, j, q ij + q ji = 0.
The coboundary operator δ 0 : C 0 → C 1 is the map defined by δ 0 q = q∂ 1 for each q ∈ C 0 , and hence Moreover, since the simplex ∆ n−1 is contractible, its cohomology groups are trivial except for k − 0, and therefore for each k ≥ 0 With an abuse of notation, when not necessary the subscript of ∂ k and the supscript in δ k will be omitted. For Proof . Consider the homomorphism π m defined on the vector space of simplicial 1-chains where ∂ 2 : C 2 → C 1 is the boundary homomorphism in dimension 2. Then for any Q ∈ C 1 and any i, .
As a consequence, π m is a projection, since for each i, j, i = j, Therefore, also P m : Moreover, since it follows that the projection P m is onto the subspace of all 1-cocycles in C 1 , denoted in short by Z 1 . In fact, since π m ∂ 2 = 0, and, by (2.2), for each cocycle z ∈ Z 1 ⇐⇒ z = δ 0 x one has P m z = P m δ 0 x = P m x∂ 1 = x∂ 1 π m = x∂ 1 = δ 0 x = z and hence Im(P m ) ⊃ Z 1 . We can conclude, as claimed, that Im(P m ) = Z 1 .
As examples, for d = 1 and n = 3, 4 the matrices of the projection P m are In fact, for n = 3 the space of cochains C 1 has standard coordinates Q ij for ij ∈ {12, 13, 23}, and by Proposition 2.1 the projection P m in these coordinates is defined by from which it follows that The same argument yields the matrix for n = 4, in coordinates Q ij for ij in the order 12, 13, 14, 23, 24, 34. Consider the following scalar product on C 1 , similar to the mass-metric on the configuration space: where as above the dot denotes the standard d-dimensional scalar product in E. It is the mass-metric on C 1 , and as above v 2 hence the following proposition holds.
Proposition 2.2. The projection P m : Now, consider the subspace X ⊂ F n (E) of all configurations with center of mass in 0: the isomorphism δ 0 |X : X → Z 1 is an isometry, where X and Z 1 have the mass-metrics. Explicitly, for each q ∈ X, where the two norms with the same symbol, with an abuse of notation, are actually different norms in C 0 and C 1 respectively. Furthermore, the potential U is the composition of the restriction to X of the coboundary map δ 0 with the mapŨ : C 1 → R (partially) defined bỹ as illustrated in the following diagram Now, recall that (condition (CC4)) a configuration q ∈ F n (E) ⊂ C 0 is a central configuration is and only if it is a critical points of the map q 2 M + U (q), defined on F n (E). It is easy to see that this is equivalent to say that q is a critical point of the map q 2 M + U (q) restricted to X. But this means that δ 0 |X sends central configurations in X to all the critical points of the map q 2 M +Ũ (q) (defined on C 1 ) restricted to the space of 1-cocycles Z 1 . Hence, the following theorem holds. Theorem 2.3. Central configurations are critical points of the function partially defined as restricted to the space of 1-cocycles Z 1 ⊂ C 1 . A co-chain q ∈ C 1 is a central configuration if and only if there exists λ ∈ R such that λq = P m (Ψ(Q)), where Q ij = q ij |q ij | α+2 for each i, j and P m : C 1 → Z 1 ⊂ C 1 is the orthogonal projection defined in Proposition 2.1, which sends C 1 onto the space of 1-cocycles.
Remark 2.4. Since the function r −α + r 2 is convex on (0, ∞), Theorem 2.3 implies that the restriction ofŨ to each component of Z 1 minus collisions is convex for d = 1, and hence one derives the existence (and uniqueness) of Moulton collinear central configurations.

Hessians and indices
Let P ∈ F n (E) be a central configuration, with mass-norm r = P M . As in the case r = 1, seen in (CC1), it is a critical point of the restriction of the potential function U to the inertia ellipsoid S = {q ∈ F n (E) : q M = r}. As such, its Morse index is the Hessian of the restriction U | S , which is a bilinear form defined on the tangent space T P S. The Hessian of f = U | S at a critical point P ∈ S can be computed in general as where λ is as above the constant − αU (P ) . Hence the following lemma holds. = − αU (P ) r 2 , then P is a critical point of the function U (q) − λ 2 q 2 M ; moreover, the Hessian of U | Sr at P is the restriction to the tangent space T P S r of the Hessian of the map U (q) − λ 2 q 2 M defined on F n (E), evaluated P . Proposition 3.2. If P ∈ F n (E) is a central configuration, then the Morse index at P of the restriction f = U | Sr is equal to the Morse index at P of the function F (q) = U (q) − λ 2 q 2 M , where λ and S r are as above. Furthermore, the direction parallel to P − O is an eigenvector of the Hessian of F , with (positive) eigenvalue equal to −λ(α + 2) > 0.
Now, consider the function f : C 1 → R defined on cochains in Theorem 2.3 as The following proposition links its Hessian with the Hessian of the function F of Proposition 3.2, for λ = −2.
Proposition 3.3. Let q ∈ F n (E) be a central configuration which is a critical point of the function F (q) = U (q) + q 2 M in C 0 (and hence q ∈ X). Let H be the Hessian of F at q (with respect to the mass-metric in C 0 ), andH the Hessian matrix of the composition f • P m at δ 0 (q) ∈ Z 1 ⊂ C 1 (with respect to the mass-metric in C 1 ). Then the non-zero eigenvalues ofH are the same as the non-zero eigenvalues of H, except for the eigenvalue 2 occurring in H with multiplicity dim E (which corresponds to the group of translations in E, or equivalently the orthogonal complement of X in C 0 ).
Proof . Since U is invariant with respect to translations in C 0 , H has the autospace q 1 = q 2 = · · · = q n ⊂ C 0 (which is the tangent space of the group of translations acting on F n (E), and is orthogonal to X with respect to the mass-metric), over which D 2 U vanishes and D 2 q 2 M = 2; hence it is an eigenspace with eigenvalue 2. The rest of eigenvalues of H correspond via the isometric embedding δ 0 |X to eigenvalues of the restriction of f to Z 1 , and hence to the eigenvalues in Z 1 of the composition f • P m . The orthogonal complement of Z 1 , which is the kernel of P m , yields zero eigenvalues toH.

Simple proofs of some corollaries
Equations (2.1) can be written as the following: Now, consider for each triple i, j, k the corresponding term Q ijk = Q ij + Q jk + Q ki . We give some very simple proofs to some well-known propositions (actually generalizing them to any homogeneity α), that follow from the following simple geometric lemma. Furthermore, there exists c ∈ R such that Q 123 = cq 12 if and only if |q 13 | = |q 23 |, that is, if and only if the triangle with vertices in q 1 , q 2 and q 3 is isosceles in q 3 .
By taking barycentric coordinates in the plane generated by the three points, it follows that Q 123 = 0 if and only if |q 12 | α+2 = |q 23 | α+2 = |q 31 | α+2 , that is, if and only if the three points are vertices of an equilateral triangle.
Proof . Equation (4.1) implies that, if Q ijk = 0, then the configuration is collinear (since (4.1) implies there exist three real numbers c 12 , c 23 , c 31 such that c 12 q 12 = m 3 Q 123 , c 23 q 23 = m 1 Q 231 , c 31 q 31 = m 2 Q 312 , and it is easy to see that Q 123 = Q 231 = Q 312 ). Therefore, if the configuration is not collinear, Q ijk = 0, and by Lemma 4.1 the configuration is an equilateral triangle.
Another easy consequence of Lemma 4.1 is the following proposition (see [1,11,17] for its importance in estimating the number non-degenerate planar central configurations of four bodies). Proof . Assume that q 1 , q 2 and q 3 are collinear, and q 4 is not. Then, equation (4.1) implies that for suitable real numbers c 12 , c 23 and c 31 the following equations hold: This implies that there arec 12 ,c 23 andc 31 such that Q 124 =c 12 q 12 , Q 234 =c 23 q 23 , Q 314 =c 31 q 31 , and by Lemma 4.1 this implies that |q 14 | = |q 24 | = |q 34 |, which is not possible since q 1 , q 2 and q 3 are collinear.  . . , q n−1 in a central configuration belong to a plane π ⊂ E, and the n-th body q n does not belong to the plane π, then the distance between q n and any q j does not depend on j = 1, . . . , n − 1, i.e., there exists c > 0 such that |q n − q j | = c for all j < n. Hence the n − 1 coplanar bodies are cocircular.
Proof . For each i, j ≤ n − 1 there exists c ij ∈ R such that c ij q ij = k ∈{i,j} m k Q ijk = k ∈{i,j,n} m k Q ijk + m n Q ijn .
The term k ∈{i,j,n} m k Q ijk is parallel to the plane π, while the sum c ij q ij − m n Q ijn is a vector parallel to the plane containing q i , q j and q n . Being equal, they both need to be parallel to both planes, and hence they are multiples of q ij . Therefore, by Lemma 4.1, there existsc ij ∈ R such that Q ijn =c ij q ij , and as above this implies that |q in | = |q jn |.
Pyramidal configurations for d = 3 and α = 1 were studied in first [5]; see also [16] for applications to perverse solutions and for the value of the constant c.