Non-avoided Crossings for $n$-Body Balanced Configurations in ${\mathbb{R}}^{3}$ Near a Central Configuration

An $n$-body configuration $x=(\overrightarrow{r}_{1},\cdots,\overrightarrow{r}_{n})$ up to translation in the Euclidean space¹¹The Euclidean structure is identified with an isomorphism $\epsilon:E\to E^{*}$ from $E$ to its dual. $(E,\epsilon)$ is a mapping

equivalently an element of ${\cal D}\otimes E$, where

is the dispositions space and ${\cal D}^{*}=\{(\xi_{1},\ldots,\xi_{n})|\sum_{i=1}^{n}\xi_{i}=0\}$ is its dual. Fixing the masses naturally endows ${\mathcal{D}}$ (resp. ${\mathcal{D}}^{*}$) with the mass Euclidean structure $\mu:{\cal D}\to{\cal D}^{*}$ defined by

where $x_{G}=(m_{1}x_{1}+\cdots+m_{n}x_{n})/\sum{m_{i}}$ is the center of mass of the $x_{i}$ (resp. $\mu^{-1}:{\cal D}^{*}\to{\cal D}$ defined by $\mu^{-1}(\xi_{1},\dots,\xi_{n})=(\frac{\xi_{1}}{m_{1}},\dots,\frac{\xi_{n}}{m_{n}})$).

The intrinsic inertia form (resp. the (dual) inertia form) of the configuration $x$ is the quadratic form $\beta$ on ${\cal D}^{*}$ (resp. the quadratic form $b$ on $E^{*}$), defined by

The form $\beta$ defines the configuration up to a rigid motion (translation and rotation) in $E$. The intrinsic inertia endomorphism (resp. the (dual) inertia endomorphism) are respectively the $\mu^{-1}$-symmetric (resp. $\epsilon$-symmetric) endomorphisms

Let

be the potential²²Nothing essential would change for a more general homogeneous $\Phi$. of the $n$-body configuration $x$. Its invariance under isometries implies the factorization $U(x)=\hat{U}(\beta)=\tilde{U}(B)$. The Wintner–Conley endomorphism associated to $x_{0}$ is the ($\mu^{-1}-symmetric$) endomorphism of ${\mathcal{D}}^{*}$ defined by

It is characterized (see [Albouy and Chenciner1998]; [Chenciner2013b]) by the fact that the equations of motion are

In some $\mu^{-1}$-orthonormal basis, the two $\mu^{-1}$-symmetric endomorphisms $A$ and $B$ of ${\mathcal{D}}^{*}$, are represented by symmetric matrices. Giving any one of these two matrices is equivalent to giving the squared mutual distances $s_{ij}=r_{ij}^{2}$, that is defining the configuration up to isometry (see [Albouy and Chenciner1998]), hence the mapping $F:B\mapsto A$ is well defined outside of the collisions (that is when all the $r_{ij}$ are strictly positive) and bijective on its image: the shape of the configuration determines the forces and the forces determine the shape. Moreover,

Indeed, in well chosen bases of the space of symmetric matrices, the coordinates of $B$ are the squared mutual distances $s_{ij}=r_{ij}^{2},\;1\leq i<j<n$ while the coordinates of $A$ are the $\varphi(s_{ij}):=\Phi^{\prime}(s_{ij})=-\frac{1}{2}\mathcal{G}s_{ij}^{-\frac{3 }{2}}$.

A rigid motion is a solution of the equations of motion along which the mutual distances $r_{ij}$ remain constant. Such a motion is necessarily a relative equilibrium, that is an equilibrium of the equations after reduction of their natural symmetry under isometries ([Albouy and Chenciner1998], Proposition 2.5). Moreover, relative equilibria are of the following form ([Albouy and Chenciner1998], Propositions 2.8 and 2.9):

where $\Omega$ is a constant $\epsilon$-antisymmetric³³I.e. such that $\varpi=\epsilon\circ\Omega=\Omega^{tr}\circ\epsilon=\varpi^{tr}\in\Lambda^{2}E^{*}$. isomorphism⁴⁴It is assumed that $E$ is the space effectively visited by the motion. of the (necessarily even dimensional) Euclidean space $(E,\epsilon)$ and $x$ belongs to the very special class of balanced configurations which we now characterize: it follows from the equations of motion that

From now on, we shall identify ${\mathcal{D}}$ and $E$ with their respective duals ${\mathcal{D}}^{*}$ and $E^{*}$ using their Euclidean structures $\mu$ and $\epsilon$. Moreover, we shall choose a $\mu^{-1}$-orthonormal basis of ${\cal D}^{*}$ and an $\epsilon$-orthonormal basis of $E$ and represent endomorphisms of ${\cal D}^{*}$ and endomorphisms of $E$ by matrices in such bases.

If $X$ is the $2p\times(n-1)$ matrix representing $x_{0}$ in these bases, we have:

From the symmetry of the matrices $2BA=X^{tr}\Omega^{2}X$ and $\Omega^{2}S=2XAX^{tr}$, we deduce the vanishing of the following commutators:

Of course, as $\Omega^{2}$ commutes with $e^{\Omega t}$, one can replace $S$ by the inertia endomorphism $S(t)=X(t)X(t)^{tr}$. The first equation, independent of the dimension of the ambient space, was shown in [Albouy and Chenciner1998] to define the balanced configurations. The following diagram summarizes these relations (on the left, the “side of the bodies”, on the right the “side of ambient space”):

In terms of mutual distances, the equations of balanced configurations are ([Albouy and Chenciner1998])

with (recalling that $\varphi(s)=-\frac{1}{2}s^{-\frac{3}{2}}$)

As soon as the number $n$ of bodies is greater than three, these equations are not independent: they are obtained by taking the exterior product by $(1,1,\ldots,1)$ of the equation $[A,B]=0$, that is by embedding the space $\wedge^{2}\mathcal{D}$, whose dimension is ${n\choose 2}=\frac{(n-1)(n-2)}{2}$, into the space $\wedge^{3}\mathbb{R}^{n}$, whose dimension is ${n\choose 3}=\frac{n(n-1)(n-2)}{6}$.

Finally, let us recall (see [Albouy and Chenciner1998]) that the balanced (resp. central) configurations are those configurations whose intrinsic inertia endomorphism $B$ is a critical point of the potential function $\tilde{U}$ restricted to its isospectral submanifold, consisting in all the $\mu^{-1}$-symmetric endomorphisms with the same spectrum (resp. restricted to the submanifold of symmetric $\mu^{-1}$-symmetric endomorphisms which have the same trace $I$).

Each $n$-body balanced configuration $x$ admits relative equilibrium motions in a Euclidean space $E$ of high enough even dimension (twice the rank of $x$, that is twice the rank of $B$, suffices, see [Albouy and Chenciner1998], Proposition 2.8). The smallest possible dimension allowing such a motion depends on the multiplicities of the eigenvalues of the Wintner–Conley endomorphism $A$ ([Albouy and Chenciner1998], Remark 2.11). In order to explain this, we study the invariant subspaces of the instantaneous rotation matrix $\Omega$.

Fixing the balanced configuration $x$, we choose a $\mu^{-1}$-orthonormal basis of ${\cal D}^{*}$ and an $\epsilon$-orthonormal basis of $E$ such that the matrices $B$ and $S$ representing the inertia endomorphisms of $x$ be diagonal:

From the above mentioned commutations, it follows that such bases can be chosen so as to satisfy also

Note that all the eigenvalues of $2A$ are strictly negative because the Newton force is attractive. The non-zero eigenvalues of $S=XX^{tr}$ and $B=X^{tr}X$ are the same; their number is the dimension of the configuration, that is the rank $d=\hbox{dim}\hbox{Im}X$ of $X$. Moreover, as $\hbox{Im}\,X=\hbox{Im}\,S$ is generated by vectors of the basis of $E$, we can suppose, after a possible reordering of the bases of ${\cal D}^{*}$ and $E$, that $X$ is of the form $X=\left(\begin{array}[]{ll}V&\quad W\\ 0&\quad 0\end{array}\right),$ where the $d\times d$ upper left block $V$ is invertible. Moreover, as $B=X^{tr}X$ is diagonal, $W=0$, that is

and $\;VV^{tr}=diag(\sigma_{1},\ldots,\sigma_{d}),\quad V^{tr}V=diag(b_{1},\ldots,b _{d})$, hence

Finally, the equation $\Omega^{2}X=2XA$, is equivalent to

hence, after possibly replacing $V$ by its product $VP$ with a permutation matrix, which amounts to permuting the first $d$ vectors of the basis of $\mathcal{D}^{*}$, which generate $\hbox{Im}B$, one can suppose that

while the $b_{i},\;i=1,\ldots,d,$ are a permutation of the $\sigma_{i},\;i=1,\ldots,d$.

This shows in particular, and this comes as no surprise, that it is only the restriction $A_{B}$ of $A$ to the image of $B$ which plays a role. Recall that a necessary and sufficient condition for $x_{0}$ to be a central configuration is that the restriction of $A$ to this subspace be proportional to the Identity, that is $\lambda_{1}=\lambda_{2}=\cdots=\lambda_{d}$.

The real invariant planes of $\Omega$ can be generated either by the couple formed by a vector $\rho_{k}\in\hbox{Im}\,x$ and a vector $\rho_{d+l}$ in the orthogonal $(\hbox{Im}\,x)^{\perp}$, or by the couple formed by two vectors $\rho_{k},\rho_{l}\in\hbox{Im}\,x$, both associated with the same eigenvalue $\lambda_{k}=\lambda_{l}$ of $A$. Similar descriptions hold for higher dimensional invariant subspaces. It follows that, writing $\tilde{\omega}_{1}^{2},\ldots,\tilde{\omega}_{r}^{2}$ the distinct values taken by the $\omega_{k}^{2}=\lambda_{k},k=1,\ldots,d$, a space $E$ of minimal dimension where a relative equilibrium motion with such a configuration may take place decomposes into a direct sum $E_{1}\oplus\cdots\oplus E_{r}$ of eigenspaces of $\Omega$, which are complex⁵⁵More precisely “hermitian”, that is such that each $J_{l}$ is an isometry. spaces $(E_{l},J_{l})$, and the motion is quasi-periodic of the form

When $r=1$, that is when $\lambda_{1}=\cdots=\lambda_{d}=\tilde{\omega}^{2}$, which means that the configuration $x$ is central, the motion becomes periodic, of the form $x(t)=e^{\tilde{\omega}Jt}x$, with $J$ a complex (hermitian) structure on $E$.

Let us denote by

the instantaneous rotation bivector. The main possibilities of invariant spaces for $\Omega$ can be read on the inverse image of $\varpi$ by $x$:

which is nothing (up to sign) but the antisymmetric part of $x^{tr}\circ\epsilon\circ y$:

which was introduced in [Lagrange1772] in the case of three bodies and in [Albouy and Chenciner1998] in the general case of $n$ bodies. In term of matrices, the endomorphism $R=\mu\circ\rho$ of ${\mathcal{D}}^{*}$ is represented by

Indeed, only contribute to $\rho$ the invariant subspaces of $\Omega$ entirely contained in $\hbox{Im}\,x_{0}$. In particular, $\rho$ vanishes in the generic case where the eigenvalues $\lambda_{i}$ of $A|_{\hbox{Im}B}$ are all distinct, that is when the motion takes place in a space of dimension twice the one of $\hbox{Im}\,x$ (compare [Albouy and Chenciner1998], Remark 2.11).

The equalities $\Omega^{2}X=2XA$ and $R=-X^{tr}\Omega X$ imply the commutation of $R$ with the Wintner–Conley matrix: $[A,R]=0$ (compare to the equations of relative equilibrium in [Albouy and Chenciner1998]). This looks quite natural in view of the following lemma, where “degenerate” means “possess a multiple eigenvalue”:

The proof is obvious in an orthonormal basis where $A$ is diagonal.

Of course, the existence of a double eigenvalue is also equivalent to the vanishing of the discriminant, that is the resultant of the characteristic polynomial and its derivative. But already for $3\times 3$ symmetric matrices, the discriminant is a quite long homogeneous degree six polynomial in the six coefficients of the matrix, which can be written as a sum of five squares see [Domokos2011]; and it is not even clear on this expression that, as was already known to Von Neuman and Wigner, its regular part (corresponding to the existence of exactly one pair of equal eigenvalues) defines a codimension 2 submanifold (this is the classical phenomenon of avoided crossings in quantum mechanics, the obvious geometric proof of which can be found in [Arnold1989]).

One is tempted to use this criterion in the study of degeneracies of the Wintner–Conley matrix $A$ but this appears to be impracticable in the general case. Fortunately, another approach succeeds as will be seen in the next section. It is only in the case of four different masses that, using computer algebra, this criterion could be used to find the tangents to the three branches of degenerate balanced configurations stemming from the regular tetrahedron (see Sects. 3.2 and 3.7). We shall also use it when characterizing in Proposition 9 the configurations of four bodies of general type (which are defined in Sect. 1.5).

If for $s>0$ small the first $d$ eigenvalues of the Wintner–Conley matrices $A_{s}$ of the balanced configuration $x_{s}(0)$ are all distinct, that is if the quasi-periodic relative equilibria $x_{s}(t)$ which bifurcate from the periodic relative equilibrium $x_{0}(t)$ have $d$ frequencies, the eigenspaces of $\Omega_{s}$ are necessarily generated by an eigenvector of $\Omega_{s}^{2}$ (and hence of $S_{s}$) contained in $\hbox{Im}\,x_{s}$ and a vector orthogonal to $\hbox{Im}\,x_{s}$. Going to the limit when $s$ tends to 0, we get that $\hbox{dim}\,E=2\,\hbox{dim}\,\hbox{Im}\,x_{0}$ and that the complex structure $J$ sends each eigenvector $\rho_{k}$ of $S_{0}$ onto a vector $k$ orthogonal to $\hbox{Im}\,x_{0}=\hbox{Im}\,S_{0}$.

Given a central configuration of dimension $d$ in an euclidean space of dimension $2d$, a basic hermitian structure is one for which there exists a partition of an eigenbasis of $S_{0}$ into $d$ pairs such that the planes generated by the two members of each pair are complex lines (see [Chenciner2013a]).

From the above discussion, we get

Let $x_{0}\in\mathcal{D}\otimes E$ be a central configuration of dimension $d$ of $n$ bodies in the Euclidean space $E$ of dimension $2p=d$ if $d$ is even, $2p=d+1$ if $d$ is odd. Let $x(t)=e^{\omega Jt}x_{0}$ be a relative equilibrium of $x_{0}$ in $E$ directed by the complex structure $J$. From Sect. 1.3, we know that a family of quasi-periodic relative equilibria of balanced configurations can bifurcate in the same space $E$ from this periodic relative equilibrium if and only in two conditions are satisfied:

1. (1)

   $E$ admits a direct sum decomposition $E=E_{1}\oplus\cdots\oplus E_{r}$ into at least two $J$-complex subspaces generated (over $\mathbb{R}$) by eigenvectors of the inertia $S_{0}$;

2. (2)

   The non-zero eigenvalues of the Wintner–Conley matrix of $x_{0}$ corresponding to eigenvectors generating any of these subspaces are equal.

Thanks to Proposition 6, one deduces the codimension of the corresponding bifurcations; in particular :

One should not forget that because of the projective invariance, the pertinent dimensions are respectively $p-1$ and 1.

I leave to the reader the pleasure of describing the intermediate strata.

(2) Three-dimensional four-body central configurations of general type are characterized in Corollary 10.

(3) The three-dimensional balanced configurations of four bodies farthest from being of general type—the ones with $B$ proportional to Identity—are the orthocentric tetrahedra (i.e. the tetrahedra which possess an orthocenter, which is the intersection of the four heights) such that the orthocenter coincides with the center of mass; this is equivalent to the mutual distances being given by $r_{ij}^{2}=\hbox{constant}(\frac{1}{m_{i}}+\frac{1}{m_{j}})$.

Expressed in the basis $\{u_{1},u_{2},u_{3}\}$, $B=\begin{pmatrix}u&\quad z&\quad y\\ z&\quad v&\quad x\\ y&\quad x&\quad w\end{pmatrix},$ where

with the following notations ($X=MYZ=TU/V$ is for future use in Sect. 3.3):

For the regular tetrahedron $(a=b^{\prime}=b^{\prime\prime}=d^{\prime}=d^{\prime\prime}=f=1)$, $B$ becomes the matrix

which reduces to $B=\frac{1}{2}Id$ when all the masses are equal to 1.

As explained in Sect. 1.4, we shall use the criterion given by Lemma 2, which leads to much more transparent equations than the resultant: $B$ is degenerate if and only if there exists a non trivial antisymmetric matrix

which commutes with $B$, that is such that $[B,R]=BR-RB=0$. Writing the six coefficients of this symmetric matrix in the order $(u,v,w,x,y,z)$, this is equivalent to the following linear equation:

This equation has a non trivial solution if and only if the above $6\times 3$ matrix is not of maximal rank, that is if its rows generate a subspace of dimension 2 or less. We shall distinguish two cases:

(1) two of the off-diagonal coefficients $x,y,z$ of $B$ are equal to zero. Up to a reordering of the basis, we may suppose that $x=y=0$. Then the condition is reduced to the vanishing of two minors: $(u-v)[z^{2}+(v-w)(w-u)]$ and $z[z^{2}+(v-w)(w-u)]$, that is

(2) at most one of the off-diagonal coefficients is equal to zero. Then the first two lines are linearly independent while the sum of the first three is equal to zero (because the trace of a commutator vanishes). Writing that the last three lines belong to the plane generated by the first two leads to the following equations

As expected, these equations are not independent: multiplying the first by $x$, the second by $y$, the third by $z$ and adding, one gets 0. Recall that they are not valid if two off-diagonal coefficients vanish.

If, for example, $m_{2}=m_{3}=m_{4}$, we have

whose eigenvalues are

From Proposition 4 on then deduces

In what follows we give a direct proof of Corollary 11, computing in particular the curves of degenerate balanced configurations, first in case 2 masses are equal (Sect. 3.5), then, at first order, in the general case (Sect. 3.7).

A tedious but straightforward computation gives the following expression of the Wintner–Conley endomorphism in this basis:

For the regular tetrahedron with unit sides, we find $A=M\varphi(1)Id=-\frac{M}{2}Id$.

With the above notations, the four equations of balanced configurations take the following form, where one checks that, in accordance with Sect. 1.2, they satisfy the relation $P_{123}-P_{124}+P_{134}-P_{234}\equiv 0.$

Let us use these equations to give a direct proof of the first part of Corollary 11: linearized at the regular tetrahedron whose sides have length 1, the equations of balanced configurations take a particularly simple form, independent of the precise form of $\varphi$:

The rank of the matrix $K$ is 0 if the four masses are equal, 2 if three of the masses are equal and 3 otherwise. In this last case, its kernel is generated by the three vectors

As the rank at neighboring points cannot be higher than three because the equations we have used of the set of balanced configurations are not independent, the rank of the matrix is locally constant if no three of the masses are equal.

In this section, we suppose that at least two masses are equal, say $m_{2}=m_{4}$, which makes pertinent the use of the basis $\{u_{1},u_{2},u_{3}\}$ introduced at the beginning of Sect. 3. We check directly that in this case the degeneracy of the Wintner–Conley matrix becomes a codimension 1 condition (Corollary 11) among balanced configurations close to the regular tetrahedron. We start with a nice corollary of the fact that the intrinsic inertia matrices of these balanced configurations form a three-dimensional manifold:

To distinguish from a purely geometric symmetry, we shall call dynamically symmetric such a symmetric configuration for which symmetric masses are equal.

With the same proof, based on a dimension count, we get

Corollary 12 makes it easy to get a good understanding of the degenerate balanced configurations near the regular tetrahedron when two masses and not three are equal. Indeed, suppose as above that $m_{2}=m_{4},\;b^{\prime}=b^{\prime\prime}=b,\;d^{\prime}=d^{\prime\prime}=d$. In the $\mu^{-1}$-orthonormal basis $\{u_{1},u_{2},u_{3}\}$ of $\mathcal{D}^{*}$ defined at the beginning of Sect. 3, the Wintner–Conley matrix decomposes into two blocks:

Degeneracy occurs if (compare to Sect. 3.2, condition $(C_{1})$) either the upper block degenerates, i.e.

or the lower right coefficient is equal to an eigenvalue of the upper block, i.e.

(i) The first case is equivalent to $a=b=d$ and the equation of balanced configurations is automatically satisfied.

(ii) In order to study the second case, we introduce the following coordinates:

that is

In these coordinates, the degenerate matrices other than the ones defined by $\psi=\phi=0$ are defined by the equation

On the other hand, the Wintner–Conley matrices of balanced configurations are defined by the equation $P_{124}=P_{234}$, where $a,b,c,d$ are expressed in terms of the new coordinates $\alpha,\theta,\psi,\phi$ via the inverse equations (where for saving space we have noted $s_{13}=m_{1}+m_{3}$; thanks to Jacques Féjoz for the computation of the inverse matrix),

We have seen that, if no three masses are equal, the balanced configurations near the regular tetrahedron ($a=b=d=f=l$) form a three-dimensional manifold whose tangent space at this point is defined by the equation

On the side of the Wintner–Conley matrices, that is setting

this equation becomes

It defines a linear susbspace “transversal” to the cone defined by the degenerate matrices. Indeed, It is enough to look in the space of coordinates $(\theta,\psi,\phi)$ obtained by going to the quotient by the $\alpha$ axis : in the coordinates $(\alpha,\beta,\gamma,\phi)$, this corresponds to going to the quotient by the line generated by $(1,1,1,0)$, that is by the addition to the Wintner–Conley matrix of a multiple of the identity, which does not change its degeneracy type and leaves invariant the linearized equations of balanced configurations (see Sect. 3.7). Hence we have identified the three (projective) directions of bifurcation to relative equilibria in $\mathbb{R}^{4}$ guaranteed by Corollary 11 (Fig. 3).

Suppose for example that $m_{2}=m_{3}=m_{4}$; we have the following explicit (projective) curves of degenerate symmetric balanced configurations:

(1) The ($\mathbb{Z}/2\mathbb{Z}$-symmetric) configurations: we have seen in Sect. 3.5 that, supposing only $m_{2}=m_{4}$, the configurations such that $a=b^{\prime}=b^{\prime\prime}=d^{\prime}=d^{\prime\prime}$ are balanced and degenerate. In the basis $\{u_{1},u_{2},u_{3}\}$, their Wintner–Conley matrix is

From the two other equalities $m_{2}=m_{3}$ (resp. $m_{3}=m_{4}$), one gets two other families of $\mathbb{Z}/2\mathbb{Z}$-symmetric degenerate balanced configurations, namely the configurations such that $a=b^{\prime}=b^{\prime\prime}=d^{\prime}=f$ (resp. $a=b^{\prime}=b^{\prime\prime}=d^{\prime\prime}=f$).

We have seen in Sect. 3.4 that, as soon as $\varphi^{\prime}(1)\not=0$, the linearization at the regular tetrahedron of the equations of balanced configurations takes the form

We suppose that the masses are all different, hence the rank of $K$ equals 3 and its kernel is generated by the three vectors $E_{1},E_{2},E_{3}$ defined in Sect. 3.4.

Computing a Taylor expansion of $A$ in the neighborhood of the regular tetrahedron with unit sides, one gets

where we have used the notations of Sect. 3.3. Hence, the tangent space at the regular tetrahedron of the manifold of Wintner–Conley matrices of balanced configurations is generated by the three vectors in $\mathbb{R}^{6}$ (coordinates $(\alpha,\beta,\gamma,\delta,\epsilon,\phi)$)

As the equations of degenerate quadratic forms are also invariant by the addition of a multiple of the identity matrix, that is a multiple of $L(E_{1})$ in the $(\alpha,\beta,\gamma,\delta,\epsilon,\phi)$ space, in order to understand the tangent spaces at the regular tetrahedron to the degenerate balanced configurations, it is enough to substitute $L(E_{2})+xL(E_{3})$ to $(\alpha,\beta,\gamma,\delta,\epsilon,\phi)$ in the three equations $(C_{2})$ of Sect. 3.2 (after replacing $u,v,w,x,y,z$ respectively by $\alpha,\beta,\gamma,\delta,\epsilon,\phi$). and this is what I had asked Jacques Laskar to do.

Using the computer algebra software TRIP developed by [Gastineau and Laskar2011], he discovered that the three equations became proportional, namely he obtained

where

The equation $E$ has three real roots, all negative. Indeed, if we set

we have

and hence

One concludes because $F$ has the four real roots $-1/m_{1},-1/m_{2},-1/m_{3},-1/m_{4}$, all negative. This computation gives the tangents to the three (projective) curves of degenerate balanced configurations which intersect at the regular tetrahedron. As I already said in the abstract, this came as a surprise as I had asked Jacques to show that, for masses without any symmetry, no other solution than the regular tetrahedron existed locally, in accordance with the generic crossing of eigenvalues of symmetric matrices being of codimension two. This surprise was the incentive to prove Proposition 4 and Corollary 11.

In the even more special case of rhombus relative equilibria, that is when

the three roots

of the equation $E(x)=0$ give back the three directions of degeneracy of the Wintner–Conley matrix; indeed:

The first two cases correspond to actual lines of degeneracy of the Wintner–Conley matrix, while in the third case, $(m_{1}-m_{2})b=m_{1}a-m_{2}f$ gives only the tangent to the actual degeneracy curve $(m_{1}-m_{2})\varphi(b)=m_{1}\varphi(a)-m_{2}\varphi(f)$.

In case 3 masses are equal, say $m_{2}=m_{3}=m_{4}:=m,$ one has $E_{3}=mE_{1}+\frac{1}{m}E_{2}$ and, indeed, $E_{2}$ is the line of degenerate $\mathbb{Z}/3\mathbb{Z}$-symmetric balanced configurations described in Sect. 3.6.

Finally, when the four masses are equal, the three vectors $E_{1},E_{2},E_{3}$ are proportional and hence they yield only trivial information.

We now fix a central configuration $x_{0}$. In this case, the relative equilibria are all periodic, of the form $x(t)=e^{\tilde{\omega}Jt}x_{0}$, where $J$is a complex structure on $E\equiv\mathbb{R}^{2p}$ (see Sect. 1.3). By scaling the configuration, one may even assume that the frequency $\tilde{\omega}$ is equal to 1. As soon as the dimension of the ambient space is 4 or more, the same central configuration admits a whole family of relative equilibrium motions parametrized by the $J$’s. In case the intrinsic inertia $B$ of the configuration is generic, these motions can be characterized by their angular momentum.

To the central configuration $x_{0}=\{\overrightarrow{r}_{1},\ldots,\overrightarrow{r}_{n}\}$ is naturally attached (see [Chenciner2013a]; [Chenciner and Jiménez-Pérez2013]; [Heckman and Zhao2015]) a convex polytope ${\cal P}$ contained in the $(p-1)$ simplex

where we recall that the $r_{ij}=||\overrightarrow{r}_{i}-\overrightarrow{r}_{j}||$ are the mutual distances between the bodies. This polytope is the set of ordered $p$-tuples $(\nu_{1}\geq\cdots\geq\nu_{p})$ of positive real numbers such that $\{\pm i\nu_{1},\ldots,\pm i\nu_{p}\}$ is the spectrum of the $J$-skew-hermitian matrix $SJ+JS$ representing the angular momentum of the relative equilibrium motion of $x_{0}$ defined by some $J$. Once chosen an orthonormal basis of $E\equiv\mathbb{R}^{2p}$, $\cal P$ can be described as the image of the frequency map

which, to a complex structure $J\in U(p)/SO(2p)$, that is to an identification of $E$ with $\mathbb{C}^{p}$ such that the multiplication by $i$ is an isometry, associates the ordered spectrum of the $J$-hermitian matrix $J^{-1}S_{0}J+S_{0}$, where $S_{0}$ is the inertia matrix of the chosen configuration. Up to some zeroes coming from the difference in dimensions, the inertia $S_{0}=X_{0}X_{0}^{tr}$ and the intrinsic inertia $B_{0}=X_{0}^{tr}X_{0}$ have the same spectrum. In particular, recall that their common trace is the moment of inertia with respect to the center of mass; by a formula of Leibniz, it is equal to

As it depends only on the inertia matrix $S_{0}$ of the configuration, the polytope ${\cal P}$ is defined as well for any solid body,

It was proved in [Chenciner2013a] and [Chenciner and Jiménez-Pérez2013] that ${\cal P}$ is a Horn polytope, more precisely that if $\{\sigma_{1}\geq\cdots\geq\sigma_{2p}\}$ is the ordered spectrum of $S_{0}$, ${\cal P}$ is the set of ordered spectra of real symmetric $p\times p$ matrices of the form $c=a+b$, where

Choosing other partitions $\Pi$ of the spectrum of $S_{0}$ into two subsets with $p$ elements, one defines in the same way Horn polytopes ${\cal P}_{\Pi}$ which turn out to be subpolytopes of ${\cal P}$ (this is a non trivial fact⁷⁷Intuitively, each piece of the partition defining ${\mathcal{P}}$ is as “separated” as possible; in contrast, in Corollary 14 the polytope associated to the partition $\{\nu_{1},\nu_{2},\ldots,\nu_{d}\}\sqcup\{0,0,\ldots,0\}$ is reduced to a point. which is proved in [Fomin et al.2005]). It was noticed in [Chenciner2013a] that, given some (periodic) relative equilibrium of a central configuration, a bifurcation to a family of (quasi-periodic) relative equilibria of balanced configurations can occur only if the corresponding point in the frequency polytope ${\cal P}$ lies in some face of one of these subpolytopes ${\cal P}_{\Pi}$. We are interested in identifying the faces which actually correspond to such bifurcations.

From Proposition 7 we deduce

The identification of the frequencies $\nu_{i}=\sigma_{i}+0$ of the angular momentum is an immediate consequence of the nature of $J$ in the proposition. It remains to prove that the corresponding vertex is a boundary vertex and not an interior one. It is enough to prove that it is a vertex of the polytope $\mathcal{P}$ associated to the partition $\sigma_{-}\cup\sigma_{+}$ of the spectrum of the inertia matrix $S_{0}$ with $\sigma_{-}=\{\sigma_{1},\sigma_{3},\ldots,0\ldots,0\}$ and $\sigma_{+}=\{\sigma_{2},\sigma_{4},\ldots,0,\ldots,0\}$, where $\sigma_{1}\geq\cdots\sigma_{d}$ (see [Chenciner2013a]; [Chenciner and Jiménez-Pérez2013]). This comes from the fact that whatever be $d$, odd or even, the number of non zero terms in $\sigma_{-}$ (resp. $\sigma_{+}$) is the same as the number of zeroes in $\sigma_{+}$ (resp. $\sigma_{-}$); hence there is a vertex of $\mathcal{P}$ which corresponds to a permutation coupling each $\sigma_{i}$ with a 0.

In the equal mass three-body problem, the bifurcation from an equilateral periodic relative equilibrium of a family of isosceles quasi-periodic relative equilibria in $\mathbb{R}^{4}$ with two frequencies cannot originate from the planar Lagrange solution but only from an equilateral relative equilibrium whose angular momentum is equivalent to the complex structure $J_{0}$ (Fig. 4).

Figure 5 depicts the frequency polytope of the regular tetrahedron configuration in $\mathbb{R}^{6}$. Generically, only two possibilities exist for the inertia eigenvalues $\sigma_{1},\sigma_{2},\sigma_{3}$:

The first case is what becomes the example depicted in [Chenciner2013a] under the assumption that $\sigma_{4}=\sigma_{5}=\sigma_{6}=0$. An example is the regular tetrahedron with one of the masses much smaller than the three others. Another one is the regular tetrahedron with masses $m_{1}=m_{3}>>m_{2}=m_{4}$. An example of the second one is the regular tetrahedron with almost equal masses.

Figure 6 indicates the angular momentum frequencies corresponding to the sizes and vertices when the frequency of the corresponding relative equilibrium equals 1 (if not, all the frequencies should be multiplied by this frequency $\omega$).

There are four distinct partitions $\Pi$ of the spectrum $\{\sigma_{1},\sigma_{2},\sigma_{3},0,0,0\}$:

1. (1)

   $\Pi_{0}=\{\sigma_{1},\sigma_{2},\sigma_{3}\}\cup\{0,0,0\}$; the corresponding Horn polytope is reduced to one point, the “generic bifurcation vertex” $A$, which is the only place where a periodic relative equilibrium in $\mathbb{R}^{6}$ of the given central configuration could bifurcate into a family of quasi-periodic relative equilibria in $\mathbb{R}^{6}$ with three frequencies of balanced configurations with $\lambda_{1},\lambda_{2},\lambda_{3}$ all distinct⁸⁸The notations $A,B,C,D$ for the vertices have, of course, no relation with the matrices $A,B,C$..

2. (2)

   $\Pi_{i}=\{\sigma_{i},0,0\}\cup\{\sigma_{j},\sigma_{k},0\},\,i=1,2,3,$ the frequency polytope, which contains all the others, corresponding to $i=2$.

The vertex $B$ corresponds to a periodic relative equilibrium motion in $\mathbb{R}^{4}$ which could bifurcate into a family of quasi-periodic relative equilibria with two frequencies in $\mathbb{R}^{4}$ of balanced configurations with $\lambda_{1}=\lambda_{2}\not=\lambda_{3}$. The eigenplanes of the instantaneous rotation $\Omega$ would tend respectively to $\{\rho_{1},\rho_{2}\}$ and $\{\rho_{3},v_{1}\}$, where in agreement with the notations in Sect. 1.3, $v_{1}$ is any non-zero vector orthogonal to the image of the balanced configuration in question. It follows that the bifurcation happens at the vertex $B$, which corresponds to a relative equilibrium directed by the complex structure having the planes $\{\rho_{1},\rho_{2}\}$ and $\{\rho_{3},v_{1}\}$ as complex lines.

Analogous descriptions hold for the vertex $C$ ($\lambda_{1}\not=\lambda_{2}=\lambda_{3}$) and the vertex $D$ ($\lambda_{1}=\lambda_{3}\not=\lambda_{2}$).

The broken edge $AB$ ($\nu_{3}=\sigma_{3}$ reflected in $\nu_{2}=\sigma_{3}$) corresponds to periodic relative equilibria in $\mathbb{R}^{6}$ which could bifurcate into a family of quasi-periodic relative equilibria with two frequencies in $\mathbb{R}^{6}$ of balanced configurations with $\lambda_{1}=\lambda_{2}\not=\lambda_{3}$ and whose instantaneous rotation $\Omega$ would have a four-dimensional eigenspace $\{\rho_{1},\rho_{2},v_{1},v_{2}\}$ and a two-dimensional eigenplane $\{\rho_{3},v_{3}\}$. The edge is parametrized by the choice of a complex structure in the four-dimensional eigenspace.

In the same way, the (possibly broken) edge $AC$ ($\nu_{1}=\sigma_{1}$) corresponds to periodic relative equilibria in $\mathbb{R}^{6}$ which could bifurcate into a family of quasi-periodic relative equilibria with two frequencies in $\mathbb{R}^{6}$ of balanced configurations with $\lambda_{1}\not=\lambda_{2}=\lambda_{3}$ and whose instantaneous rotation $\Omega$ would have a four-dimensional eigenspace $\{\rho_{2},\rho_{3},v_{2},v_{3}\}$ and a two-dimensional eigenplane $\{\rho_{1},v_{1}\}$. The edge is, as above, parametrized by the choice of a complex structure in the four-dimensional eigenspace. Finally, the same description holds for the interior side $AD$.

On the contrary, apart from the vertices $B,D,C$, the edge $BC$ ($\nu_{3}=0$) does not correspond to possible bifurcations. This is because it is the interior of the frequency polytope when the dimension of $E$ goes down to 4. This remark indicates in more general situations what faces of the frequency polytope (and subpolytopes) are bifurcation values.

According to Corollary 13, any balanced configuration close enough to the regular tetrahedron with only two different masses, say $m_{1}=m_{3}\not=m_{2}=m_{4}$, is a rhombus configuration:

In such a simple case, it is possible to give explicit formulæ for the bifurcating families. Given real numbers $(\alpha,\beta,\gamma_{1},\gamma_{2})$, which we may suppose all positive, such that $m_{1}\gamma_{1}=m_{2}\gamma_{2}$, we define a configuration $x_{0}$ of four bodies in $\mathbb{R}^{3}$ with center of mass at the origin by

The mutual distances are

In the $\mu^{-1}$-orthonormal basis $\{u_{1},u_{2},u_{3}\}$ of $\mathcal{D}^{*}$ formed by the vectors

$x_{0}$ is represented by the $3\times 3$ matrix whose columns are Jacobi vectors

The corresponding inertia matrices are

with

while the Wintner–Conley endomorphism $A:\mathcal{D}^{*}\to\mathcal{D}^{*}$ is

This is the case when $\gamma_{1}=\gamma_{2}=0$, that is $4b-a-f=0$. We check property (H) (see Sect. 1.5.2) for the symmetric ($m_{1}=m_{3},\;b^{\prime}=b^{\prime\prime}=b,\;d^{\prime}=d^{\prime\prime}=d$, see Sect. 3.5) balanced configurations in the neighborhood of the planar rhombus central configuration $x_{0}$. As $ImB_{0}$ is the plane $x=0$, $x_{0}$ is characterized by the equality of the last two eigenvalues of $A_{0}$, that is

Finally, the ellipsoid of inertia $B_{0}$ of a rhombus planar configuration is degenerate (i.e. round) if and only if $m_{1}a=m_{2}f$. Taking the Newtonian value $\varphi(s)=-\frac{1}{2}s^{-\frac{3}{2}}$ and supposing that $x_{0}$ is central, the condition of degeneracy becomes

or, normalizing the masses by setting $m_{2}=1$,

which defines three values $\gamma,1,1/\gamma$, with $\gamma\simeq 0.575$. Hence,