Central Configurations and Mutual Differences
Central configurations are solutions of the equations λmjqj=∂U/∂qj, where U denotes the potential function and each qj is a point in the d-dimensional Euclidean space E≅Rd, for j=1,…,n. We show that the vector of the mutual differences qij=qi−qj satisfies the equation −(λ/α)q=Pm(Ψ(q)), where Pm is t...
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Ferrario, D.L. 2019-02-18T16:25:23Z 2019-02-18T16:25:23Z 2017 Central Configurations and Mutual Differences / D.L. Ferrario // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 17 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37C25; 70F10 DOI:10.3842/SIGMA.2017.021 https://nasplib.isofts.kiev.ua/handle/123456789/148595 Central configurations are solutions of the equations λmjqj=∂U/∂qj, where U denotes the potential function and each qj is a point in the d-dimensional Euclidean space E≅Rd, for j=1,…,n. We show that the vector of the mutual differences qij=qi−qj satisfies the equation −(λ/α)q=Pm(Ψ(q)), where Pm is the orthogonal projection over the spaces of 1-cocycles and Ψ(q)=q/|q|α+2. It is shown that differences qij 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. Work partially supported by the project ERC Advanced Grant 2013 n. 339958 “Complex Patterns for Strongly Interacting Dynamical Systems COMPAT”. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Central Configurations and Mutual Differences Article published earlier |
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Central configurations are solutions of the equations λmjqj=∂U/∂qj, where U denotes the potential function and each qj is a point in the d-dimensional Euclidean space E≅Rd, for j=1,…,n. We show that the vector of the mutual differences qij=qi−qj satisfies the equation −(λ/α)q=Pm(Ψ(q)), where Pm is the orthogonal projection over the spaces of 1-cocycles and Ψ(q)=q/|q|α+2. It is shown that differences qij 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.
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Central Configurations and Mutual Differences / D.L. Ferrario // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 17 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 021, 11 pages
Central Configurations and Mutual Differences
D.L. FERRARIO
Department of Mathematics and Applications, University of Milano-Bicocca,
Via R. Cozzi, 55 20125 Milano, Italy
E-mail: davide.ferrario@unimib.it
URL: http://www.matapp.unimib.it/~ferrario/
Received December 06, 2016, in final form March 27, 2017; Published online March 31, 2017
https://doi.org/10.3842/SIGMA.2017.021
Abstract. Central configurations are solutions of the equations λmjqj = ∂U
∂qj
, where U
denotes the potential function and each qj is a point in the d-dimensional Euclidean space
E ∼= Rd, for j = 1, . . . , n. We show that the vector of the mutual differences qij = qi − qj
satisfies the equation − λ
αq = Pm(Ψ(q)), where Pm is the orthogonal projection over
the spaces of 1-cocycles and Ψ(q) = q
|q|α+2 . It is shown that differences qij 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.
Key words: central configurations; relative equilibria; n-body problem
2010 Mathematics Subject Classification: 37C25; 70F10
1 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 C1 of degree 1 with coefficients in the Euclidean space E. Hence,
we show that central configurations are critical points of a function defined on C1 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 = Rd denote the d-dimensional Euclidean space.
An element of En will be denoted by q = (q1, q2, . . . , qn) where ∀ j, qj ∈ E. Let Fn(E) denote
as in [4] the configuration space of n particles in E:
Fn(E) =
{
q ∈ En : qi 6= qj
}
.
If ∆ is the collision set
∆ =
⋃
i<j
{
q ∈ En : qi = qj
}
,
then Fn(E) = En \∆.
mailto:davide.ferrario@unimib.it
http://www.matapp.unimib.it/~ferrario/
https://doi.org/10.3842/SIGMA.2017.021
2 D.L. Ferrario
For j = 1, . . . , n, let mj > 0 be positive masses. Assume that the masses are normalized, i.e.,
that
n∑
j=1
mj = 1. (1.1)
Let 〈∗, ∗〉M denote the mass-metric on (the tangent vectors of) En, defined as
〈v,w〉M =
n∑
j=1
mjvj ·wj ,
where vj ·wj is the Euclidean scalar product in (the tangent space of) E. Let |vj | denote the
Euclidean norm of a vector vj in E. The norm corresponding to the mass-metric is ‖v‖M =√
〈v,v〉M .
Let α > 0 be a fixed homogeneity parameter, and U : Fn(E) → R the potential function
defined as
U(q) =
∑
1≤i<j≤n
mimj
|qi − qj |α
.
A central configuration is a configuration q ∈ Fn(E) such that there exists λ ∈ R such that
(∀ j = 1, . . . , n)
λmjqj =
∂U
∂qj
= −α
∑
k 6=j
mjmk
qj − qk
|qj − qk|α+2
. (1.2)
If q is a central configuration, then
λ
∑
j
mj |qj |2 = −α
n∑
j=1
n∑
k 6=j,k=1
mjmk
(qj − qk) · qj
|qj − qk|α+2
= −α
∑
j<k
mjmk
(qj − qk) · qj + (qk − qj) · qk
|qj − qk|α+2
= −α
∑
j<k
mjmk
|qj − qk|α
=⇒ λ‖q‖2M = −αU(q),
and hence λ = −α U(q)
‖q‖2M
< 0. By summing equation (1.2) in j
λ
∑
j
mjqj = −α
∑
j<k
mjmk
(qj − qk) + (qk − qj)
|qj − qk|α+2
= 0
and hence∑
j
mjqj = 0.
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 ∈ Fn(E) : ‖q‖2M = 1} [14].
(CC3) Fixed points of the map F : S → S defined as F (q) = − ∇MU(q)
‖∇MU(q)‖M
, where ∇M denotes
the gradient with respect to the mass-metric on Fn(E) [6, 7].
Central Configurations and Mutual Differences 3
(CC4) Critical points on Fn(E) of the map ‖q‖2M + U(q) [9].
(CC5) Critical points on Fn(E) of the map ‖q‖2αMU(q)2 (or ‖q‖αMU(q)) [17].
In all these formulations, central configurations appear as O(d)-orbits in Fn(E), where the
action of the orthogonal group O(d) on Fn(E) is diagonal g · q = (gq1, . . . , gqn).
Define the space X as
X =
q ∈ Fn(E) :
n∑
j=1
mjqj = 0
.
2 Central configurations and mutual differences
Let n be the set n = {1, 2, . . . , n} and C0 the vector space of all maps from n to E: C0 =
{q : n → E}. Let Fn(E) ⊂ C0 denote the inclusion sending q ∈ Fn(E) to the map q : n → E
defined by q(j) = qj for each j ∈ n.
Now, let ñ denote the set of all
(
n
2
)
subsets in n with two elements: ñ = {{1, 2}, {1, 3}, . . .,
{n− 1, n}}. Let C1 denote the vector space of all maps from ñ to E:
C1 = {q : ñ→ E}.
It is isomorphic to Eñ, where ñ =
(
n
2
)
. Note that if En2
denotes that vector space of all maps
q : n2 → E, where n2 = n×n (and hence if q ∈ En2
, we can denote qij = q((i, j)) ∈ E), there
is an embedding C1 ⊂ En2
, by sending an element q ∈ C1 to the map q′ : n2 → E defined by
q′ij =
q({i, j}) if i < j,
−q({i, j}) if i > j,
0 if i = j.
In fact, we are identifying elements in C1 with the skew-symmetric elements in En2
(that is,
maps qij + qji = 0 ∈ E). Given q ∈ C1 with an abuse of notation we will write qij 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
Ck(K;R) is the vector space of all the R-linear combinations of k-dimensional simplexes of K;
the boundary homomorphism ∂k : Ck(K;R)→ Ck−1(K;R) is defined as ∂k(σ) =
k∑
j=0
(−1)jσdj for
each k-simplex σ ofK and 0 otherwise, where dj is the j-th face map. More precisely, all simplices
in K can be ordered, and elements in Ck(K;R) will be linear combinations of ordered k-simplices
in K. An ordered k-simplex with vertices x0, . . . , xk will be denoted either as [x0, . . . , xk] or
simply as x0 . . . xk. With this notation, the j-th face map sends σ = [x0, . . . , xj , . . . , xk] to
σdj = [x0, . . . , x̂j , . . . , xk], where x̂j means that the j-th element is canceled.
By taking homomorphisms valued in an R-vector space E, the chain complex Ck(K;R) yields
the simplicial cochain complex with coefficients in E: the cochain groups are defined as all the
linear homomorsphisms Ck(K;E) = homR(Ck(K;R), E), and the coboundary homomorphisms
δk : Ck(K;E)→ Ck+1(K;E) are defined for each k by
δk(η) = η∂k+1 : Ck+1(K;R)→ E
for each cochain η : Ck(K;R)→ E. The kernel of δk is the group of cocycles, and it is denoted
as Zk(K;E) = ker δk ⊂ Ck(K;E).
4 D.L. Ferrario
Now, let ∆n−1 denote the standard (abstract) simplex with n vertices {1, 2, . . . , n}. Then
the vector spaces Ck 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:
C0 = C0(∆n−1;E) and C1 = C1(∆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 qj , i.e., a map q : n→ E. Furthermore, a 1-di-
mensional 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 qij ∈ E denote the ij-the
component of a vector in Eñ, and for i > j, the variable qij is defined by the property that
∀ i, j, qij + qji = 0.
The coboundary operator δ0 : C0 → C1 is the map defined by δ0q = q∂1 for each q ∈ C0,
and hence
δ0(q)(i, j) = qj − qi ∈ E
for all i, j. In fact, for each q : n→ E, δ0(q)(ij) = q∂1[i, j] = q(j − i) = qj − qi. For k = 1, the
coboundary operator is defined as δ1 : C1 → C2 as
δ1(q)(ijk) = q∂2(ijk) = q(jk − ik + ij) = qij + qjk + qki.
Moreover, since the simplex ∆n−1 is contractible, its cohomology groups are trivial except
for k − 0, and therefore for each k ≥ 0
Zk+1
(
∆n−1;E
)
= ker δk+1 = Im δk.
With an abuse of notation, when not necessary the subscript of ∂k and the supscript in δk will
be omitted.
For each q ∈ C1, let Q be defined as Qjk =
qjk
|qjk|α+2 . Note that Qjk = Ψγ(qjk) with
γ = α+ 2 where Ψγ(x) = x
|x|γ for each x ∈ E. It turns out that the map Ψγ : E \{0} → E \{0}
is a diffeomorphism with inverse Ψγ̂ where γ̂ = γ
γ−1 .
Consider now that if one defines qij = qi−qj , one can read equation (1.2) as (as a consequence
of equation (1.1))
λqj = −α
∑
k 6=j
mk
qjk
|qjk|α+2
= −α
∑
k 6=j
mkQjk,
and hence
−λ
α
qij =
∑
k 6=i
mkQik −
∑
k 6=j
mkQjk =
∑
k 6∈{i,j}
mk(Qik + Qkj) +mjQij +miQij
=
∑
k 6∈{i,j}
mk(Qik + Qkj + Qji)−
∑
k 6∈{i,j}
mkQji +mjQij +miQij
=
∑
k 6∈{i,j}
mk(Qik + Qkj + Qji) +
(∑
k
mk
)
Qij
⇐⇒ −λ
α
qij =
∑
k 6∈{i,j}
mk(Qik + Qkj + Qji) + Qij . (2.1)
Proposition 2.1. The (linear) map Pm : C1 → C1 defined by
(Pm(Q))ij =
∑
k 6∈{i,j}
mk(Qik + Qkj + Qji) + Qij
is a projection from C1 onto Z1 ⊂ C1, where Z1 = ker δ1 : C1 → C2 is the subspace of 1-cocycles.
Central Configurations and Mutual Differences 5
Proof. Consider the homomorphism πm defined on the vector space of simplicial 1-chains
C1(∆
n−1,R) with real coefficient, as
πm([i, j]) = [i, j]−
∑
k 6={i,j}
mk∂2([i, j, k]),
where ∂2 : C2 → C1 is the boundary homomorphism in dimension 2. Then for any Q ∈ C1 and
any i, j with i 6= j
Pm(Q)[i, j] = Q(πm[i, j]).
For each 2-simplex [a, b, c] of ∆n−1 one has ∂2([a, b, k] + [b, c, k] + [c, a, k]) = ∂2[a, b, c] for each
k 6= {a, b, c}, and hence
πm∂2[a, b, c] = πm([a, b] + [b, c] + [c, a]) = [a, b]−
∑
k 6={a,b}
mk∂2[a, b, k]
+ [b, c]−
∑
k 6={b,c}
mk∂2[b, c, k] + [c, a]−
∑
k 6={c,a}
mk∂2[c, a, k]
= ∂2[a, b, c]−
∑
k 6∈{a,b,c}
mk∂2[a, b, c]−mc∂2[a, b, c]−ma∂2[b, c, a]−mb∂2[c, a, b]
= ∂2[a, b, c]−
(∑
k
mk
)
∂2[a, b, c] = 0
=⇒ πm∂2 = 0.
As a consequence, πm is a projection, since for each i, j, i 6= j,
π2m[i, j] = πm
[i, j]−
∑
k 6∈{i,j}
mk∂2([i, j, k])
= πm[i, j]−
∑
k 6∈{i,j}
mkπm∂2[i, j, k] = πm[i, j].
Therefore, also Pm : C1 → C1 is a projection
P 2
m(Q)[i, j] = Pm(Q)(πm[i, j]) = Q(π2m[i, j]) = Q(πm[i, j]) = Pm(Q)[i, j].
Moreover, since
∂1πm = ∂1 (2.2)
it follows that the projection Pm is onto the subspace of all 1-cocycles in C1, denoted in short
by Z1. In fact, since πm∂2 = 0,
δ1PmQ = PmQ∂2 = Qπm∂2 = 0 =⇒ Im(Pm) ⊂ Z1
and, by (2.2), for each cocycle z ∈ Z1 ⇐⇒ z = δ0x one has
Pmz = Pmδ
0x = Pmx∂1 = x∂1πm = x∂1 = δ0x = z
and hence Im(Pm) ⊃ Z1. We can conclude, as claimed, that Im(Pm) = Z1. �
As examples, for d = 1 and n = 3, 4 the matrices of the projection Pm arem1 +m2 m3 −m3
m2 m1 +m3 m2
−m1 m1 m2 +m3
,
6 D.L. Ferrario
m1 +m2 m3 m4 −m3 −m4 0
m2 m1 +m3 m4 m2 0 −m4
m2 m3 m1 +m4 0 m2 m3
−m1 m1 0 m2 +m3 m4 −m4
−m1 0 m1 m3 m2 +m4 m3
0 −m1 m1 −m2 m2 m3 +m4
.
In fact, for n = 3 the space of cochains C1 has standard coordinates Qij for ij ∈ {12, 13, 23},
and by Proposition 2.1 the projection Pm in these coordinates is defined by
(Pm(Q))12 =
∑
k 6∈{1,2}
mk(Q1k + Qk2 + Q21) + Q12 = m3(Q13 + Q32 + Q21) + Q12,
(Pm(Q))13 =
∑
k 6∈{1,3}
mk(Q1k + Qk3 + Q31) + Q13 = m2(Q12 + Q23 + Q31) + Q13,
(Pm(Q))23 =
∑
k 6∈{2,3}
mk(Q2k + Qk3 + Q32) + Q23 = m1(Q21 + Q13 + Q32) + Q23,
from which it follows that
(Pm(Q))12 = (1−m3)Q12 +m3Q13 −m3Q23,
(Pm(Q))13 = m2Q12 + (1−m2)Q13 +m2Q23,
(Pm(Q))23 = −m1Q12 +m1Q13 + (1−m1)Q23.
The same argument yields the matrix for n = 4, in coordinates Qij for ij in the order 12, 13,
14, 23, 24, 34.
Consider the following scalar product on C1, similar to the mass-metric on the configuration
space: for v,w ∈ C1,
〈v,w〉M =
∑
i<j
mimj (vij ·wij) , (2.3)
where as above the dot denotes the standard d-dimensional scalar product in E. It is the
mass-metric on C1, and as above ‖v‖2M = 〈v,v〉M . It follows that
〈v, Pm(w)〉M =
∑
i<j
mimj
(
vij · (Pm(w))ij
)
=
∑
i<j
mimj
vij ·
∑
k 6∈{i,j}
mk(wik + wkj + wji) + wij
=
∑
i<k
∑
k 6∈{i,j}
mimjmk(vij · (wik + wkj + wji)) +
∑
i<j
mimjvij ·wij
=
∑
a<b<c
mambmc
(
vab · (wac + wcb + wba)
+ vac · (wab + wbc + wca) + vbc · (wba + wac + wcb)
)
+
∑
i<j
mimjvij ·wij
= −
∑
a<b<c
mambmc ((vab + vbc + vca) · (wab + wbc + wca))
+
∑
i<j
mimjvij ·wij
Central Configurations and Mutual Differences 7
=
∑
i<k
∑
k 6∈{i,j}
mimjmk((vik + vkj + vji) ·wij) +
∑
i<j
mimjvij ·wij
=
∑
i<j
mimj
∑
k 6∈{i,j}
mk(vik + vkj + vji) + vij
·wij
= 〈Pm(v),w〉M ,
hence the following proposition holds.
Proposition 2.2. The projection Pm : C1 → Z1 ⊂ C1 is orthogonal (self-adjoint) with respect
to the scalar product 〈−,−〉M in (2.3) defined on C1.
Now, consider the subspace X ⊂ Fn(E) of all configurations with center of mass in 0:
X = {q ∈ Fn(E) :
∑
jmjqj = 0}, i.e., of all q ∈ C0 such that q
∑
jmj [j] = 0. The coboundary
morphism δ0|X : X ⊂ C0 → C1 induces an isomorphism δ0|X : X → Z1. Moreover, since if q ∈ X
then
2
∑
i<j
mimj |qi − qj |2 =
∑
i,j
mimj |qi − qj |2 =
∑
i,j
mimj
(
|qi|2 − 2qi · qj + |qj |2
)
= 2
∑
i
mi|qi|2 − 2
∑
j
mjqj
2
= 2
∑
i
mi|qi|2
the isomorphism δ0|X : X → Z1 is an isometry, where X and Z1 have the mass-metrics. Explicitly,
for each q ∈ X,
‖q‖M = ‖δ0(q)‖M ,
where the two norms with the same symbol, with an abuse of notation, are actually different
norms in C0 and C1 respectively.
Furthermore, the potential U is the composition of the restriction to X of the coboundary
map δ0 with the map Ũ : C1 → R (partially) defined by
Ũ(q) =
∑
i<j
mimj |qij |−α,
as illustrated in the following diagram
C0 δ0 // C1 Ũ //
Pm
��
R
X
OO
OO
δ0|X
∼= // Z1.
GG
XX
Now, recall that (condition (CC4)) a configuration q ∈ Fn(E) ⊂ C0 is a central configuration
is and only if it is a critical points of the map ‖q‖2M +U(q), defined on Fn(E). It is easy to see
that this is equivalent to say that q is a critical point of the map ‖q‖2M +U(q) restricted to X.
But this means that δ0|X sends central configurations in X to all the critical points of the map
‖q̃‖2M + Ũ(q̃) (defined on C1) restricted to the space of 1-cocycles Z1.
Hence, the following theorem holds.
8 D.L. Ferrario
Theorem 2.3. Central configurations are critical points of the function partially defined as
C1 → R
q 7→
∑
i<j
mimj
(
|qij |−α + |qij |2
)
restricted to the space of 1-cocycles Z1 ⊂ C1.
A co-chain q ∈ C1 is a central configuration if and only if there exists λ ∈ R such that
λq = Pm(Ψ(Q)), where Qij =
qij
|qij |α+2 for each i, j and Pm : C1 → Z1 ⊂ C1 is the orthogonal
projection defined in Proposition 2.1, which sends C1 onto the space of 1-cocycles.
Remark 2.4. Since the function r−α + r2 is convex on (0,∞), Theorem 2.3 implies that the
restriction of Ũ to each component of Z1 minus collisions is convex for d = 1, and hence one
derives the existence (and uniqueness) of Moulton collinear central configurations.
3 Hessians and indices
Let P ∈ Fn(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 ∈ Fn(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 TPS. The Hessian of f = U |S at a critical
point P ∈ S can be computed in general as D2f(P )[u(P ),v(P )] = ((DuDvf −DDuv)f)(P ) =
(Du(Dvf))(P ), where u and v are vector fields on S (see, e.g., [10, formula (1), p. 343]). This
yields the well-known formula of the Hessian in terms of second derivatives with respect to
a local chart (see also [14, Proposition 2.8.8, p. 136]). Given the mass-metric, the Hessian can
be written as D2f [u,v] = 〈Du∇̂Mf,v〉M or as the self-adjoint endomorphism TPS → TPS
defined by u ∈ TPS 7→ Du∇̂Mf(P ) ∈ TPS, where ∇̂M denotes the gradient induced by the
mass-metric restricted to S (hence ∇̂Mf(P ) is the projection of ∇MU(P ) to TPS, orthogonal
with respect to 〈−,−〉M ).
If N denotes a vector normal to the tangent space TPS (such as P − O, where O denotes
the origin of the Euclidean space E), the projection ∇̂Mf(P ) is equal to ∇MU −
〈∇MU,N〉M
‖N‖2M
N
evaluated at P . If N = P − O, by Euler formula 〈∇MU,N〉M = −αU , and because P is
a critical point of f = U |S and u ∈ TPS the derivative Du
αU(q)
‖q‖2M
vanishes at P , and hence
Du∇̂Mf(P ) = Du
(
∇MU(q) +
αU(q)
‖q‖2M
N
)
(P ) = Du(∇MU)(P ) +
αU(P )
‖P‖2M
Du(q)
= Du
(
∇MU − λ∇M
‖q‖2M
2
)
,
where λ is as above the constant −αU(P )
‖P‖2M
. Hence the following lemma holds.
Lemma 3.1. If P is a critical point of the restriction U |Sr , with the inertia ellipsoid Sr = {q ∈
Fn(E) : ‖q‖M = r} and with λ defined as λ = −αU(P )
‖P‖2M
= −αU(P )
r2
, 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 TPSr of the Hessian of the map U(q)− λ
2‖q‖
2
M defined on Fn(E), evaluated P .
Proposition 3.2. If P ∈ Fn(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 Sr 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.
Central Configurations and Mutual Differences 9
Proof. Since∇MU(q) is homogeneous of degree −α−1, if N = P−O one has DN (∇MU)(P ) =
−(α+ 1)∇MU(P ) = −λ(α+ 1)N . Therefore
DN
(
∇MU − λ∇M
‖q‖2M
2
)
(P ) = −λ(α+ 1)N − λN = −λ(α+ 2)N . �
Now, consider the function f : C1 → R defined on cochains in Theorem 2.3 as
f(q) =
∑
i<j
mimj
(
|qij |−α + |qij |2
)
.
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 ∈ Fn(E) be a central configuration which is a critical point of the
function F (q) = U(q) + ‖q‖2M in C0 (and hence q ∈ X). Let H be the Hessian of F at q
(with respect to the mass-metric in C0), and H̃ the Hessian matrix of the composition f ◦ Pm
at δ0(q) ∈ Z1 ⊂ C1 (with respect to the mass-metric in C1). Then the non-zero eigenvalues
of H̃ are the same as the non-zero eigenvalues of H, except for the eigenvalue 2 occurring in H
with multiplicity dimE (which corresponds to the group of translations in E, or equivalently the
orthogonal complement of X in C0).
Proof. Since U is invariant with respect to translations in C0, H has the autospace q1 = q2 =
· · · = qn ⊂ C0 (which is the tangent space of the group of translations acting on Fn(E), and is
orthogonal to X with respect to the mass-metric), over which D2U vanishes and D2‖q‖2M = 2;
hence it is an eigenspace with eigenvalue 2. The rest of eigenvalues of H correspond via the
isometric embedding δ0|X to eigenvalues of the restriction of f to Z1, and hence to the eigenvalues
in Z1 of the composition f ◦ Pm. The orthogonal complement of Z1, which is the kernel of Pm,
yields zero eigenvalues to H̃. �
4 Simple proofs of some corollaries
Equations (2.1) can be written as the following:
λ
α
qij + Qij =
∑
k 6∈{i,j}
mk(Qij + Qjk + Qki). (4.1)
Now, consider for each triple i, j, k the corresponding term Qijk = Qij + Qjk + Qki. 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.
Lemma 4.1. Let q1, q2 and q3 be three non-collinear points in E. Then Q123 = 0 if and only
if q1, q2 and q3 are vertices of an equilateral triangle.
Furthermore, there exists c ∈ R such that Q123 = cq12 if and only if |q13| = |q23|, that is, if
and only if the triangle with vertices in q1, q2 and q3 is isosceles in q3.
Proof. If q1, q2 and q3 are not collinear (in E), then the differences q12, q13 and q23 generate
a plane. Since q12 + q23 + q31 = 0, Q123 = 0 implies
Q123 = Q12 + Q23 + Q31 =
q12
|q12|α+2
+
q23
|q23|α+2
+
q31
|q31|α+2
= 0 = q12 + q23 + q31.
10 D.L. Ferrario
By taking barycentric coordinates in the plane generated by the three points, it follows that
Q123 = 0 if and only if |q12|α+2 = |q23|α+2 = |q31|α+2, that is, if and only if the three points are
vertices of an equilateral triangle.
If c1, c2 and c3 are three non-zero real numbers such that c3q12 + c1q23 + c2q31 = cq12, then
(c3− c)q12 + c1q23 + c2q31 = 0, and as above this implies c3− c = c1 = c2. Hence if Q123 = cq12,
it holds that |q23|α+2 = |q13|α+2 as claimed. �
Corollary 4.2. For n = 3, d ≥ 2 and α > 0, the only non-collinear central configuration is the
equilateral configuration.
Proof. Equation (4.1) implies that, if Qijk 6= 0, then the configuration is collinear (since (4.1)
implies there exist three real numbers c12, c23, c31 such that c12q12 = m3Q123, c23q23 = m1Q231,
c31q31 = m2Q312, and it is easy to see that Q123 = Q231 = Q312). Therefore, if the configuration
is not collinear, Qijk = 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).
Corollary 4.3. For n = 4, d ≥ 2 and α > 0, if a central configuration has three collinear bodies,
then it is a collinear configuration.
Proof. Assume that q1, q2 and q3 are collinear, and q4 is not. Then, equation (4.1) implies
that for suitable real numbers c12, c23 and c31 the following equations hold:
c12q12 = m3Q123 +m4Q124, c23q23 = m1Q231 +m4Q234,
c31q31 = m2Q312 +m4Q314.
This implies that there are c̃12, c̃23 and c̃31 such that
Q124 = c̃12q12, Q234 = c̃23q23, Q314 = c̃31q31,
and by Lemma 4.1 this implies that |q14| = |q24| = |q34|, which is not possible since q1, q2
and q3 are collinear. �
Corollary 4.3 can be easily generalized to arbitrary n as follows:
Corollary 4.4. For n ≥ 4, d ≥ 2 and α > 0, if n− 1 of the bodies in the central configuration
are collinear, then all of them are.
Corollary 4.5. For n ≥ 4, d ≥ 3 and α > 0, if the first n − 1 bodies q1, . . . , qn−1 in a central
configuration belong to a plane π ⊂ E, and the n-th body qn does not belong to the plane π, then
the distance between qn and any qj does not depend on j = 1, . . . , n− 1, i.e., there exists c > 0
such that |qn − qj | = c for all j < n. Hence the n− 1 coplanar bodies are cocircular.
Proof. For each i, j ≤ n− 1 there exists cij ∈ R such that
cijqij =
∑
k 6∈{i,j}
mkQijk =
∑
k 6∈{i,j,n}
mkQijk +mnQijn.
The term
∑
k 6∈{i,j,n}
mkQijk is parallel to the plane π, while the sum cijqij −mnQijn is a vector
parallel to the plane containing qi, qj and qn. Being equal, they both need to be parallel to
both planes, and hence they are multiples of qij . Therefore, by Lemma 4.1, there exists c̃ij ∈ R
such that Qijn = c̃ijqij , and as above this implies that |qin| = |qjn|. �
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.
Central Configurations and Mutual Differences 11
Acknowledgements
Work partially supported by the project ERC Advanced Grant 2013 n. 339958 “Complex Pat-
terns for Strongly Interacting Dynamical Systems COMPAT”.
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1 Introduction
2 Central configurations and mutual differences
3 Hessians and indices
4 Simple proofs of some corollaries
References
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