The angular momentum of a relative equilibrium

There are two main reasons why relative equilibria of N point masses under the influence of Newton attraction are mathematically more interesting to study when space dimension is at least 4: On the one hand, in a higher dimensional space, a relative equilibrium is determined not only by the initial configuration but also by the choice of a complex structure on the space where the motion takes place; in particular, its angular momentum depends on this choice; On the other hand, relative equilibria are not necessarily periodic: if the configuration is"balanced"but not central, the motion is in general quasi-periodic. In this exploratory paper we address the following question, which touches both aspects: what are the possible frequencies of the angular momentum of a given central (or balanced) configuration and at what values of these frequencies bifurcations from periodic to quasi-periodic relative equilibria do occur ? We give a full answer for relative equilibrium motions in dimension 4 and conjecture that an analogous situation holds true for higher dimensions. A refinement of Horn's problem given by Fomin, Fulton, Li and Poon plays an important role. P.S. The conjecture is now proved (see Alain Chenciner and Hugo Jimenez Perez, Angular momentum and Horn's problem, arXiv:1110.5030v1 [math.DS]).

1 A brief review of the relative equilibrium solutions of the N -body problem Let x = ( r1, r2, · · · , rN ) ∈ E N be a configuration of N point masses m1, m2, · · · , mN , in the euclidean space (E, ) of dimension d. Choosing once and for all an appropriate galilean frame, we shall only consider configurations whose center of mass is fixed at the origin: N k=1 m k r k = 0. We shall identify x with the d×N matrix X whose kth column is composed of the coordinates (r 1k , · · · , r dk ) of r k in some orthonormal basis of E.
Remark. Such a matrix X can be thought of as representing the element x : ξ → N k=1 ξ k r k ∈ Hom(D * , E) ≡ D ⊗ E, where the dispositions space D and its dual D * , D = R N /(1, · · · , 1)R, D * = (ξ1, · · · , ξN ) ∈ R N , ξi = 0 , were introduced in [AC]. Indeed, considered as a linear mapping from (R N ) * ≡ R N to E, it is the unique extension of x : D * → E which vanishes on the line generated by (m1, · · · , mN ).
The equations of the N -body problem may be written the gradient being the one defined by the mass scalar product which, on configurations whose center of mass is at the origin, is given by the formula x · x = ( r 1 , r 2 , · · · , r N ) · ( r 1 , r 2 , · · · , r N ) = N k=1 m k < r k , r k > , A relative equilibrium solution is an equilibrium of the "reduced" equations, obtained by going to the quotient by translations (this was already accomplished by choosing a galilean frame where the center of mass is fixed at the origin) and linear isometries. It is proved in [AC] that these are exactly the rigid motions, where every mutual distance || ri − rj|| stays constant, that is where the N -body configuration behaves as a rigid body. Moreover, the motion is of the form X(t) = e tΩ X0, that is x(t) = (e Ωt r1, e Ωt r2, · · · , e Ωt rN ) if x0 = ( r1, r2, · · · , rN ), where Ω is an -antisymmetric operator on E and, if we call E the actual space of motion (forgetting the non visited dimensions), Ω is non degenerate. In particular, the dimension of E is even: d = 2p. Choosing an orthonormal basis where Ω is normalized, this amounts to saying that there exists a hermitian structure on the space E and an orthogonal decomposition E ≡ C p = C k 1 × · · · × C kr such that x(t) = (x1(t), · · · , xr(t)) = (e iω 1 t x1, · · · , e iωr t xr), where xm is the orthogonal projection on C km of the N -body configuration x and the action of e iωmt on xm is the diagonal action on each body of the projected configuration. Such quasi-periodic motions exist only for very special configurations, called balanced configurations in [AC], which are characterized by the existence of an -symmetric endomorphism Σ of E such that ∇U (x) = Σx. The most degenerate balanced configurations, indeed the only ones occuring in a space of dimension at most 3 (and hence with the actual space of motion E of dimension 2), are the central configurations for which Σ = λId. In this case, Ω = ωJ, with J a hermitian structure on E, and the motion is in the hermitian space E ≡ C 2p ; in particular, it is periodic.

Angular momentum
Given a configuration x = ( r1, · · · , rN ) and a configuration of velocities y =ẋ = ( v1, · · · , vN ), both with center of mass at the origin: N k=1 m k r k = N k=1 m k v k = 0, the angular momentum of (x, y) is the bivector (we use the french convention t x for the transposed of x) (the isomorphism µ : D → D * is the mass scalar product on D, see [AC]). In an orthonormal basis of E where x and y are repectively represented by the d × N matrices X and Y , setting M = diag(m1, · · · , mN ), it can be identified with the antisymmetric matrix The dynamics of a solid body is determined by its inertia tensor (with respect to its center of mass) In the case of an N -body configuration x whose center of mass is at the origin, S is identified with the symmetric matrix S = XM t X with coefficients sij = N k=1 m k r ik r jk , whose trace is the moment of inertia of the configuration x with respect to its center of mass, that is its square norm |x| 2 for the mass scalar product: In particular, the angular momentum of a relative equilibrium solution x(t) = e tΩ x0 is represented, if S0 = X0M t X0, by the antisymmetric matrix C = S0Ω + ΩS0.
Remark. Doing the same computation on the D side instead of the E side leads to the equation of balanced configurations (see [AC]): that is the commutation of A with the intrinsic inertia B = t X0X0.
Corollary 2 Let x0 be an N-body balanced configuration in the euclidean space (E, ). Let x(t) = e tΩ x0 be a relative equilibrium motion of x0. The frequencies of its angular momentum C ∈ Λ 2 E (i.e. the moduli of the eigenvalues of the antisymmetric matrix C = S0Ω + ΩS0 can be written ν 1 1 , · · · , ν k 1 +1 1 , ν 1 2 , · · · , ν k 2 +1 2 , · · · , ν 1 r , · · · , ν kr +1 r so that the following identity holds true: Proof. One chooses an orthonormal basis of E in which the matrices Ω 2 and S0 are both diagonal: To such a basis is associated an orthogonal decomposition E = E1 ⊕ E2 ⊕ · · · ⊕ Er of E into blocks of dimensions 2(k1 + 1), 2(k2 + 1), · · · , 2(kr + 1), on each of which Ω is a multiple of a complex structure: where Ji is a hermitian structure on Ei ≡ R 2k i . It follows that C itself decomposes into r blocks of the form ωi(siJi+Jisi), where s1, · · · , sr, are the diagonal blocks of S0. Hence we are reduced to proving the trace identity in each block and adding them to get I(x0) = traceS0 = r i=1 trace(si). This (trivial) fact will be established in the next section where we study the case when Ω = ωJ is a multiple of a hermitian structure.
Remark. In his thesis [A1], Alain Albouy proved that the rank 2k of the angular momentum of an n-body motion satisfies the following inequalities, generalizing Dziobek's theorem: where d is the dimension of the actual space of motion, that is the dimension of the subspace generated at any instant by the bodies and their velocities. Recall that in the case of a relative equilibrium, d is necessarily even.

The frequency mapping of a central configuration
Given some inertia 2p × 2p matrix S0 (say, the one of a central configuration x0, but it could be any symmetric non-negative matrix), we study the mapping from the space of hermitian structures on E to the set of 2p × 2p antisymmetric real matrices. We shall only be interested in the spectra of the matrices ω −1 C, hence choosing an orientation for J is harmless and we shall consider only those of the form hermitian with spectrum ν1, · · · , νp that we can suppose to be ordered.

Definition 1 The frequency mapping
We have chosen the notation U (p)\SO(2p) because two rotations R, R define the same hermitian structure if and only if R = U R with U ∈ U (p). The fibers of this mapping correspond to positive hermitian structures J1 = R −1 1 J0R1, J2 = R −1 2 J0R2, which define relative equilibria whose angular momenta are conjugated under complex isomorphisms from (E, J1) to (E, J2), that is: there exists U ∈ U (p) such that Explicit formulae. Let R ∈ SO(2p) be a rotation and where A and B are p × p matrices whose coefficients aij and bij are respectively Considered as a J0-skew-hermitian matrix ω(A + iB), its coefficients are Hence, the coefficients σij of the J0-hermitian matrix Σ = B − iA are given by Lemma 3 The trace of the complex hermitian matrix Σ = F(J) is equal to the trace of the real symmetric matrix S0: Proof. The trace of the real symmetric matrix Σ is equal on the one hand to 2 traceS0, on the other hand to 2(ν1 + · · · + νp) because its spectrum is ν1, ν1, ν2, ν2, · · · , νp, νp. One can also use the explicit formula.
In the sequel we fix an orthonormal basis of E in which S0 is diagonal, which means that the coordinate axes are in the directions of axes of the inertia ellipsoid of x0: Question. Understand the image of F. The question starts being non trivial when p ≥ 2 as, if p = 1, Λ 2 E ≡ R, the hermitian structure is unique up to orientation and any non zero value of the angular momentum is attained with a configuration of the form λx0.
Remarks. 1) Our problem is the one of understanding the periodic motions of a rigid body which moves freely around its center of mass (see [Ar]). Notice that in restriction to this set of periodic motions, the Euler equations become trivial, as they merely state the constancy of the angular momentum in a frame attached to the configuration (the rigid body); equivalently, as S0 commutes with J 2 = −Id, the angular momentum and the angular velocity commute.
2) The space U (p)\SO(2p) of positive hermitian structures on (E, ) is of dimension p(p − 1). It is the same as the isotropic grassmaniann and was identified by Elie Cartan with the space of projective pure spinors (see [LM]). For p = 2 and p = 3, it is respectively diffeomorphic to P1(C) and P3(C). More generally, it is a hermitian symmetric space whose Lusternik-Schnirelman category is 2 p−1 . As the adjoint orbit of J0 in the Lie algebra so(2p), it inherits a Kostant-Souriau symplectic form ωJ (X, Y ) = trace(J[X, Y ]); notice that U (p)\SO(2p) inherits no symplectic structure from its embedding J → (x0, ωJx0) into the phase space of the N -body problem because its image is contained in the fiber of x0 which is a lagrangian submanifold.

The symmetry group Γ
Given two diagonal elements of O(2p), is an involution of SO(2p), which replaces the coefficient ρij of a rotation matrix by uivjρij = ±ρij. If moreover we suppose that which amounts to uiup+i = η for all i = 1, 2, · · · , p, this operation transforms the complex structure J = R −1 J0R into the complex structure ηD JD so that we get a group Γ of involutions of U (p)\SO(2p) by choosing the matrices D and D of the form and noticing that the couples (η, D ) and (η, −D ) define the same involution. The cardinal of Γ is one half of the product of 2 (choices for η) and 2 2p−1 (choices for D with determinant equal to η p ), that is If, moreover, the orthonormal basis of E which is chosen to identify E with R 2p is such that the matrix S0 is diagonal, we have D S0D = S0. It follows that Σ = S0 + J −1 S0J is transformed into D ΣD . and that the mapping F = FS 0 factorizes through a mapping F : U (p)\SO(2p)/Γ R → (ν1, · · · νp) ∈ W + p .

Adapted hermitian structures
Among positive hermitian structures, a special role is played by structures which are fixed by elements of Γ of a special type: Definition 2 In an orthonormal basis of E where the inertia matrix S0 is diagonal, an "adapted hermitian structure" is a positive hermitian structure which can be written (not uniquely) where ρ ∈ SO(p) and P ∈ SO(2p) is a signed permutation. When ρ can be chosen equal to Id, that is when R is a signed permutation, one speaks of a "basic hermitian structure".
A filtration of the set of adapted structures is obtained by assigning the rotations ρ to belong to the subgroup SO(k1 + 1) × · · · × SO(kr + 1) of SO(p) formed by the elements which respect the factors of a direct sum decomposition R p = R k 1 +1 ⊕ · · · ⊕ R kr +1 where each factor is generated by elements of the canonical basis of R p . We shall speak in this case of (k1, · · · , kr)-adapted structures.
Lemma 4 The signed permutation P being given, the adapted hermitian structures Jρ,P are precisely the ones which are fixed by the involution The basic hermitian structures are the ones fixed by a subgroup of Γ of order 2 p consisting of elements (η, D ) of the following form: there exists a signed permutation P such that if theṽi are the coefficients of the diagonal matrix P D P −1 , one has ηṽiṽp+i = +1 for i = 1, · · · , p.
More generally, the (k1, · · · , kr)-adapted structures are fixed by a subgroup of Γ of order 2 r .
6 The image of the frequency mapping
In particular, if J Id,P is a basic hermitian structure, the frequencies νi of the associated angular momentum are, up to reordering, the σ π(i) + σ π(p+i) .
Notations. Let Aσ − ,σ + ⊂ W + p be the set of ordered spectra of a sum of real symmetric p × p matrices with spectra σ− and σ+ respectively (here we identify the diagonal matrices σ− and σ+ with the set of their elements, let Bourbaki forgive us). Let Bσ − ,σ + ∈ W + p be the ordered spectrum of the diagonal matrix σ− + σ+. The images A and B under the frequency mapping F of the adapted and the basic hermitian structures are respectively A = ∪Aσ − ,σ + , B = ∪Bσ − ,σ + , the union being on all the couples (σ−, σ+) such that σ−∪σ+ = {σ1, · · · , σ2p}.
Finally, let A k 1 ,··· ,kr be the image under F of the (k1, · · · , kr)-adapted hermitian structures (in particular, A = Ap and B = A0,··· ,0). The tableau which follows shows the orders of the subgroups γ of Γ, the dimensions of the corresponding invariant subsets of hermitian structures and the generic dimension of their images under F: Proposition 5 The set A is a convex polytope. The subsets A k 1 k 2 ···kr (resp. B) lie in the faces of A (resp. the vertices of A located in the interior of W + p ). Proof. Thanks to the works of Weyl, Horn, Atiyah, Guillemin & Sternberg, Kirwan, Klyachko, Knutson & Tao, Fulton,. . ., the subsets Aσ − ,σ + , which can be described as the images of moment maps, are well understood. In particular they are convex polytopes. More precisely, we have: Theorem 6 (see [K]) Let O λ , Oµ be the spaces of Hermitian matrices with spectrum λ, µ. Let e : O λ × Oµ → R n take a pair of matrices to the spectrum of their sum, listed in decreasing order. Then the image of e is a convex polytope. Also, if e(H λ , Hµ) is on a face of the image polytope and is also a strictly decreasing list, then H λ , Hµ are simultaneously block diagonalisable.
We need a version of this theorem which applies to the real symmetric matrices. Such a version exists, at least for the first part asserting that the image is a convex polytope, indeed the same as in the hermitian case (see [F2] Theorem 3). The characterization of the faces, nevertheless, holds only in the sense that the matrices need be simultanously block diagonalisable only in the complex domain ( [F2] Theorem 5 and the counter-example which follows the statement of the theorem).
It remains to notice that the union A of the convex polytopes Aσ − ,σ + corresponding to the various partitions {σ1, · · · , σ2p} = σ− ∪ σ+ forms itself a convex polytope whose extremal points not on the boundary of the Weyl chamber (i.e. corresponding to strictly decreasing sets of eigenvalues) belong to B. Indeed, the following is true: Theorem 7 ( [FFLP] Proposition 2.2) Let A and B be p × p Hermitian matrices. Let σ1 ≥ σ2 ≥ · · · ≥ σ2p be the eigenvalues of A and B arranged in descending order. Then there exist Hermitian matricesÃ and B with eigenvalues σ1 ≥ σ3 ≥ · · · ≥ σ2p−1 and σ2 ≥ σ4 ≥ · · · ≥ σ2p respectively, such thatÃ +B = A + B.
Hence, if σ1 ≥ σ2 ≥ · · · ≥ σ2p, the set A coincides with Aσ − ,σ + , where 6.2 Some questions and a conjecture QUESTION 1: Is there a big polytope in some product space whose different projections give the various Aσ − ,σ + ? In a sense, the conjecture below would give a possible answer.
QUESTION 2: Is the image of F itself the same as the image of some moment map ? (the answer is probably "no") QUESTION 3: What is the union over all the central configurations x0 with moment of inertia I(x0) = trace S0 = 1 of the images of FS 0 ? (here, one can either fix the number N of bodies or let it be arbitrary).
CONJECTURE ImF coincides with the convex polytope A. This is equivalent to asserting that every level of F meets some adapted hermitian structure i.e. to proving that for every such level, the action of at least one of the involutions (−1, D P ) has a fixed point inside this level.
In what follows, we prove the conjecture in the simple case p = 2 (the case p = 1 is trivial) and give a picture of A in a case with p = 3. 7 The case of N bodies in E = R 4 7.1 The trivial case E = R 2 As SO(2) = U (1), the image of the frequency map is the single point σ1 + σ2.
Finally, one checks that the symmetry (−1, D P ) correspond in each of the three cases to the invariance of f (ϕ, θ) under the symmetry with respect to the corresponding great circle, that is (ϕ, θ) → (ϕ, −θ) in the first case, (ϕ, θ) → (ϕ, π − θ) in the second case, (ϕ, θ) → (−ϕ, θ) in the third case. Each critical point is a complex structure invariant under the action of a subgroup of order 4 of Γ, generated by the symmetries with respect to two orthogonal great circles, and hence is forced by the symmetries of f . This ends the proof of the lemma.

Corollary 9
The conjecture is true when p = 2: the image of the frequency map F coincides with the image A of the adapted hermitian structures.
Remarks. 1) As the product of the symmetries with respect to the three planes xOy, yOz, zOx induces the antipody on S 2 , the function f is defined on P2(R).
The critical values of det Σ are ordered as follows: or, as σ1 + σ2 + σ3 = I(x0), The maximum value of ν1ν2 = det Σ is attained when σ1 = I(x0)/2 maximizes the function (I(x0)−σ)σ. The frequencies ν1, ν2 of the angular momentum are then equal, which means that the angular momentum is conjugated to a multiple of J0.
An example of this is given by an equal mass equilateral triangle in the plane generated by e1 and e2 rotating under any complex structure of the form J = (sin ϕ)J0 + (cos ϕ)J2, that is such that J e1 belongs to the plane generated by e3 and e4.
Notice that nothing changes qualitatively in the first figure if σ4 > 0 is small.
3) The round cases. If the ellipsoid of inertia is a round ball, i.e. if S0 = σId, the angular momentum of all the rotational motions in R 4 will be the same up to rotation. Indeed, in this case, C0 = 2ωσJ0. In the same vein, we have the Lemma 10 Let x0 be a 3 dimensional central configuration of N bodies and µ be masses such that S0 = diag(σ, σ, σ, 0). Whatever be the relative equilibrium motion in R 4 of x0 with such masses, the angular momentum endomorphism C has eigenvalues ± 2iω 3 I(x0), ± iω 3 I(x0). Proof. The proof is based on the simple remark that, under the hypotheses, if a = (a1, a2, a3, a4) and b = (b1, b2, b3, b4) are two vectors in R 4 , we have < a, b >S 0 = σ(< a, b > −a4b4) Hence the endomorphism Σ0 depends only on the last column of the matrix R, but every element of SO(4)/U (2) has a representative whose last column is (0, 0, 0, 1), hence Σ0 = σ 2 0 0 1 , which proves the lemma.
Examples satisfying this lemma are the regular tetrahedron with four equal masses or sufficiently symmetric N -body configurations.
8 Picture of A in an example with p = 3 The 6-dimensional space of positive hermitian structures is diffeomorphic to the complex projective space P3(C). There are 15 basic hermitian structures Jπ, corresponding to the 15 partitions π of the set {1, 2, 3, 4, 5, 6} into 3 pairs. Each Jπ is invariant under a subgroup of order 8 of Γ (described in 5) and belongs to three circles, each one invariant under a subgroup of order 4 of Γ and containing 4 basic hermitian structures ±Jπ and ±J π . Each two such circles are contained in a 3-dimensional subspace formed by the adapted structures invariant under a subgroup of Γ of order 2.

Bifurcation values of the frequency map
What values taken by the frequency map correspond to periodic relative equilibrium motions of x0 from which stems a family of quasi-periodic relative equilibria obtained by deforming x0 through balanced configurations ? It results from the description we gave in section 2 that, the ordered frequencies at such a bifurcation point lie in A and even in the codimension one subset of the image of F formed by the union of the A k 1 ,k 2 ,··· ,kr such that r ≥ 2.
Remarks. 1) The above statement does not imply that the bifurcation values lie on the boundary of the image of F.
Symmetric balanced configurations of 3 or 4 bodies and their associated relative equilibrium motions in R 3 or R 4 are studied in [C2].