Lobachevskii Journal of Mathematics http://ljm.ksu.ru ISSN 1818-9962 Vol. 24, 2006, 73–134

Eldar Straume
A GEOMETRIC STUDY OF MANY BODY SYSTEMS
(submitted by V. V. Lychagin)

Abstract. An $n$ -body system is a labelled collection of n point masses in a Euclidean space, and their congruence and internal symmetry properties involve a rich mathematical structure which is investigated in the framework of equivariant Riemannian geometry. Some basic concepts are $n$ -conﬁguration, conﬁguration space, internal space, shape space, Jacobi transformation and weighted root system. The latter is a generalization of the root system of SU(n), which provides a bookkeeping for expressing the mutual distances of the point masses in terms of the Jacobi vectors. Moreover, its application to the study of collinear central $n$ -conﬁgurations yields a simple proof of Moulton’s enumeration formula. A major topic is the study of matrix spaces representing the shape space of $n$ -body conﬁgurations in Euclidean k-space, the structure of the m-universal shape space and its O(m)-equivariant linear model. This also leads to those “orbital ﬁbrations” where SO(m) or O(m) act on a sphere with a sphere as orbit space. A few of these examples are encountered in the literature, e.g. the special case $S^{5} ∕ O (2) \approx S^{4}$ was analyzed independently by Arnold, Kuiper and Massey in the 1970’s.

Contents
1.  Introduction
   1.1.  A brief overview
   1.2.

n

-conﬁguration space, internal space and shape space
1.3. Geometrization, symmetry and reduction
1.4. Congruence and internal symmetry in

n

-body spaces
   1.5.  Jacobi vectors and the centered conﬁguration space
2.  Jacobi transformations from a categorical viewpoint
   2.1.  Transformations of

n

-body systems and Jacobi’s approach
   2.2.  Equivalent characterizations of Jacobi transformations
   2.3.  Construction of Jacobi vectors
3.   Orthogonal transformation groups on matrix spaces
   3.1.   Compact transformation groups and orbital decomposition
   3.2.  Transformations of the space of symmetric matrices
   3.3.  Algebraic realization of orbit spaces and orbital stratiﬁcation
   3.4.  Topology of the shape space and related spaces
4.  Geometric invariants of

n

-body systems
5. The weighted root system of an

n

-body system
5.1. Distance functions and the

Ψ

-root system
   5.2.  The standard weighted root system
   5.3.  Weighted root systems and their metric invariants
   5.4.  On the role of the mass distribution
6.  Collinear central conﬁgurations revisited
References

1. Introduction

An $n$ -body system is deﬁned to be a labelled collection of $n$ point masses (or particles) $P_{i}$ of mass $m_{i}$ in Euclidean 3-space $ℝ^{3}$ , and it is of general importance to ﬁnd appropriate mathematical models to describe and analyze such a system. We have in mind few-body systems as well as many-body systems, ranging from diﬀerent areas such as celestial and quantum mechanics or quantum chemistry ( $n$ -body problem, n-atomic molecules etc.). Despite the diversity of the applications they all share a fundamental underlying mathematical structure, in terms of kinematic concepts and internal space geometry, and the role of the mass distribution. Here we shall focus attention on these basic structures, in a modern geometric and topological setting with orthogonal transformation groups in the forefront. This approach also establishes similar results for point masses in any Euclidean spaces $ℝ^{d}, d \geq 2$ .

In this introductory section we ﬁrst give an overview of the paper, which take up several diﬀerent topics. Then, in the following subsections we introduce some basic concepts and constructions which we shall return to later. Here the presentation is rather informal or expository, with some comments on the history.

1.1. A brief overview. Our approach is to combine symmetry and kinematic geometric principles in the framework of equivariant geometry, exhibiting the importance of the classical orthogonal transformation groups and the associated orbit space constructions. This enables us to investigate in a unifying way the notions of congruence, internal conﬁguration space and internal symmetry group. Internal symmetries of $n$ -body systems are investigated in Section 2 from a categorical viewpoint based on pure geometrical principles. As we shall explain, for various reasons they should be referred to as Jacobi transformations, but in the physics literature they are also encountered as kinematic rotations or democracy transformations, cf. [17].

Section 3 is largely devoted to the study of the topology of shape spaces in general, and here it is natural to consider $n$ -body systems in higher dimensional Euclidean spaces $ℝ^{d}$ as well. We shall exploit the fact that congruence and internal symmetry for point masses in $ℝ^{d}$ combine together to a tensor product representation of some $O (d) \times O (m)$ , acting on the matrix space $M (d, m) ≃ ℝ^{d m}$ by matrix multiplication. In short, a typical shape space is the orbit space $M^{*} = S^{p} ∕ O (d)$ , where $S^{p}$ is the unit sphere of $M (d, m)$ . Now $M^{*}$ inherits the action of $O (m)$ as its symmetry group, but on the other hand, $M^{*}$ also naturally embeds as an $O (m)$ -invariant subset of the linear space $S y m^{0} (m)$ of symmetric matrices with zero trace. Of particular interest are the cases where $S^{p} ∕ H \approx S^{q}$ is also a sphere, $H = S O (m)$ or $O (m)$ . They yield an inﬁnite family of ”orbital ﬁbrations” $S^{p} \to S^{q}$ , which for $q = 1, 2, 4$ are the Hopf ﬁbrations, cf. Section 3.4.2.

In the study of many body systems, we apply the orbit space reduction to the conﬁguration manifold rather than to its cotangent bundle, as in the reduction method of Marsden-Weinstein which, for example, Iwai [13] applies to the Hamiltonian system describing classical molecular dynamics. On the other hand, R. Littlejohn and his collaborators (cf. e.g. [16], [17], [18]) have investigated the gauge ﬁelds that arise on the reduced (i.e. internal) conﬁguration space, and our work is directly related to the geometric framework of their investigations.

In Section 4 we determine the geometric invariants of $n$ -body systems, namely polynomial functions on the conﬁguration space which are both congruence invariants and internal symmetry (or democracy) invariants. The topic is certainly well understood in classical invariant theory, but we are also seeking symmetrized expressions for centered $n$ -body conﬁgurations, that is, their center of mass is ﬁxed at the origin.

In Section 5 we introduce the notion of the weighted root system of an $n$ -body system with a given mass distribution. This is a geometric invariant which generalizes the notion of a root system of Cartan type $A_{n - 1}$ in classical Lie theory, and its underlying structure is actually inherent in various contexts. For example, it encodes the data of the relative positions of the binary collision varieties in the conﬁguration space or shape space. We shall introduce it as a bookkeeping device for expressing the mutual distances between the n bodies in terms of the Jacobi vectors.

In Section 6 we present, as a nice application of the weighted root system, a simple proof of Moulton’s classiﬁcation of collinear central $n$ -conﬁgurations (cf. [23]) which is also well adapted for numerical computations. Recall that these are the conﬁgurations characterizing n collinear masses capable of a rigid uniform rotation under the mutual gravitational forces.

The present paper is essentially a preprint (with the same title) of the author from April 2002, prompted by the paper Hsiang [9] and the succeeding joint work [11]. Our Sections 5-6 recall the contents of [11], but in the present paper the topology of general shape spaces, the universal shape space and its linear model, is a major topic. It turns out that some few-body shape spaces have an interesting history in the literature. For example, it was surprising to ﬁnd that the quotient space of complex projective plane $ℂ P^{2}$ modulo complex conjugation is topologically a 4-sphere, namely $S^{5} ∕ O (2) \approx S^{4} .$ Several independent and diﬀerent proofs of this fact had already been published (cf. Arnold [2], [3], Kuiper [15], Massey [21]). Now we also ﬁnd that our Section 3 has some overlapping with the more recent paper Atiyah-Berndt [1].

1.2. $n$ -conﬁguration space, internal space and shape space. The location of an $n$ -body system is conveniently represented by its $n$ -conﬁguration $X = (a_{1}$ , $a_{2}$ , …, $a_{n})$ where $a_{i}$ is the position vector of $P_{i}$ . The n-tuple $X$ is regarded as a vector in the free $n$ -conﬁguration space

{\hat{M}}_{n} = ℝ^{3} \times ℝ^{3} \times . . . . \times ℝ^{3} ≃ ℝ^{3 n},

(1)

namely a Euclidean 3n-space with an orthogonal splitting reﬂecting the individual positions of the n point masses, which may possibly coincide. The mass distribution $(m_{1}, m_{2}, . ., m_{n})$ is tacitly assumed to be ﬁxed unless otherwise stated, and in general we assume $m_{i} > 0$ . The centered $n$ -conﬁguration space is the subspace of dimension $3 n - 3$

M_{n} \subset {\hat{M}}_{n} : \sum m_{i} a_{i} = 0,

(2)

consisting of those $n$ -conﬁgurations with its center of mass at the origin. ${\hat{M}}_{n}$ has the following mass dependent (Jacobi) kinematic metric, namely the inner product of $X$ and $Y = (b_{1}, b_{2}, . ., b_{n})$ is

〈X, Y〉 = \sum m_{i} a_{i} \cdot b_{i}

(3)

The isometry group of ${\hat{M}}_{n}$ is the associated Euclidean group

E ({\hat{M}}_{n}) = {\hat{M}}_{n} \hat{\times} O ({\hat{M}}_{n}) ≃ ℝ^{3 n} \hat{\times} O (3 n) = E (3 n)

(4)

which we have expressed in the usual way as a semidirect product of the subgroups of translations and orthogonal transformations respectively.

Two $n$ -conﬁgurations $X$ and $Y$ are regarded as congruent if their i-th components $a_{i}$ and $b_{i}$ diﬀer by the same Euclidean motion in $ℝ^{3}$ for each i. Thus the congruence relation is deﬁned by the natural (diagonal) $E (3)$ -action on ${\hat{M}}_{n}$ , by which $E (3)$ (resp. $O (3)$ ) is embedded as a subgroup of $E (3 n)$ (resp. $O (3 n)$ ). The space ${\bar{M}}_{n}$ of congruence classes is the (congruence) moduli space, also referred to as the internal space. It may be viewed as the result of the two-step orbit space construction

{\hat{M}}_{n} \to \frac{{\hat{M}}_{n}}{ℝ^{3}} ≃ M_{n} \to \frac{M_{n}}{O (3)} = \frac{{\hat{M}}_{n}}{E (3)} = {\bar{M}}_{n}

(5)

where in the ﬁrst step the translation-reduced space is identiﬁed with $M_{n}$ . The second step is the orbit space construction of the transformation group ( $O (3), M_{n}$ ), from which it follows that ${\bar{M}}_{n}$ is a stratiﬁed manifold of dimension $3 n - 6$ (cf. Section 3).

Next, let us divide ”congruence” into ”shape” and ”size” and use the squared norm function $I$ (cf. (7)), namely the polar moment of inertia, as a measure of the size of an $n$ -conﬁguration (or of its congruence class). Then ${\bar{M}}_{n}$ has the structure of a cone, where each ray emanates from the cone vertex (or base point) $O = (I = 0)$ and the ray represents a ﬁxed shape. The shape space is the space of rays, which we may regard as the orbit space

\frac{{\bar{M}}_{n} - \{0\}}{ℝ^{+}} = \frac{M_{n} - \{0\}}{O (3) \times ℝ^{+}} ≃ M_{n}^{*}

(6)

where $O (3) \times ℝ^{+}$ is the group of similarity transformations of Euclidean 3-space with the induced (diagonal) action on $M_{n}$ . However, each ray has a unique point where $I = 1$ , and therefore it is more convenient to identify $M_{n}^{*}$ with the subset of ${\bar{M}}_{n}$ consisting of classes of unit size $I = 1$ . Thus, the internal space ${\bar{M}}_{n}$ is naturally a cone over the shape space $M_{n}^{*}$ .

1.3. Geometrization, symmetry and reduction. The viewpoint that kinematics is a geometric discipline has a long history. For an $n$ -body motion $t \to X (t)$ , the fundamental kinematic quantities are

\begin{align} I & = {|X|}^{2} = \sum m_{i} {|a_{i}|}^{2} & T & = \frac{1}{2} {|\dot{X}|}^{2} = \frac{1}{2} \sum m_{i} {|{\dot{a}}_{i}|}^{2} & (7) \\ Ω & = X \times \dot{X} = \sum m_{i} a_{i} \times {\dot{a}}_{i} & p & = \sum m_{i} {\dot{a}}_{i} \end{align}

These are the moment of inertia, kinetic energy, angular and linear momentum, respectively. In particular, the hyperradius $ρ = \sqrt{I}$ is the norm of $X$ in the $n$ -conﬁguration space (1) with respect to the inner product (3), and using $T$ the same metric may be presented as the kinematic Riemannian metric

d s^{2} = 2 T d t^{2} = \sum m_{i} (d x_{i}^{2} + d y_{i}^{2} + d z_{i}^{2})

(8)

Furthermore, dynamics was incorporated in this geometric setting by the classical geometrization procedure which dates at least back to the early 19th century. Let us brieﬂy recall the basic idea behind this, namely Jacobi’s reformulation of Lagrange’s least action principle, which goes as follows. When dynamics is taken into account and the above $n$ -body motion is due to a force ﬁeld $\nabla U$ derived from the potential energy $- U$ , the total energy $h = T - U$ is conserved and the metric (8) should be conformally modiﬁed to the following dynamical metric

d s_{h}^{2} = (U + h) d s^{2}

(9)

depending on a ﬁxed energy level $h .$ Then the trajectories of total energy $h$ can be recovered as the geodesics of this metric. This applies, for example, to physically important $n$ -body systems with potential functions (such as the Newtonian or Coulomb potential) depending on the pairwise distances $r_{i j}$ $= |a_{i} - a_{j}|$ . Therefore, they are invariant under the Euclidean group $E (3)$ , and consequently both linear and angular momentum of the motion are conserved.

Since the early days it has been an important issue how to fully utilize conservation laws. Sometimes these are ﬁrst integrals associated with symmetry groups, and the reduction of integration problems using continuous (or inﬁnitesimal) symmetries dates back to Sophus Lie’s work in the 1870’s.

Clearly, the most eﬀective usage of the invariance of linear momentum $p$ is to choose an inertial frame at the center of mass, thereby reducing the conﬁguration space ${\hat{M}}_{n}$ to its subspace $M_{n}$ (2) and hence the associated linear momentum vanishes. A much harder problem is to further use the invariance of $Ω$ , which is related to the congruence action of $O (3)$ (or $S O (3)$ ) on $M_{n}$ and its cotangent bundle. Namely, if $T^{*}$ denotes the cotangent bundle construction, consider the two ways of reducing the phase space

T^{*} (\frac{M_{n}}{O (3)}) = T^{*} {\bar{M}}_{n}, \frac{T^{*} (M_{n})}{O (3)}

to dimension $6 n - 12$ and $6 n - 9$ , respectively. This suggests that the orbit space reduction is most eﬀective at the level of the conﬁguration space rather than the phase space.

In this paper we shall focus attention on two main issues. Firstly, it is important to construct appropriate coordinates for $M_{n}$ , which is actually a problem with no canonical or generally ”best” solution. However, from the representation theory of $O (3)$ and natural guidelines suggested by the splitting or invariance of the kinematic quantities, we are led to a natural approach whose origin may, in fact, be ascribed to Jacobi. Secondly, we inquire about the structure of the internal space ${\bar{M}}_{n}$ , as an orbit space of $M_{n}$ modulo $O (3) .$ It has the natural kinematic metric

d {\bar{s}}^{2} = 2 \bar{T} d t^{2}

(10)

where $\bar{T} = T - T^{ω}$ and $T^{ω}$ is the kinetic energy due to purely rotational motion. Indeed, this metric is Riemannian and coincides with the induced orbital distance metric (see e.g. [30]), and the reduction map $M_{n} \to {\bar{M}}_{n}$ is a (stratiﬁed) Riemannian submersion. As a consequence, the geodesics in ${\bar{M}}_{n}$ are the image curves of those geodesics (i.e., linear motions of constant speed) in $(M_{n}, d s^{2})$ with vanishing angular momentum.

On the other hand, the appropriate reduction of the dynamical equations in $M_{n}$ to the level of ${\bar{M}}_{n}$ is generally a hard problem even today. In $M_{n}$ the geometrization procedure yields the conformal modiﬁcation of the kinematic metric (8) leading to the dynamical metric (9). Similarly, one may search for a similar geometrization procedure at the internal space level, which should yield a dynamical Riemannian metric on ${\bar{M}}_{n}$ of type

d {\bar{s}}_{h, Ω}^{2} = F (U, h, Ω) d {\bar{s}}^{2}

(11)

in analogy with (9). However, this only works in special cases, for example, for the (classical) planar $n$ -body problem, but we leave this topic here.

Actually, in his study of celestial mechanics Jacobi himself abandoned the above Riemannian geometric approach in favor of the increasingly successful Hamiltonian formalism where, at the phase space level (symplectic geometry), ideas involving symmetry, conservation laws and integrability questions have been continuously developed up to present time. On the other hand, with the modern techniques of equivariant geometry, Lie transformation groups and related reduction theory and quotient constructions, the framework of Riemannian geometry and modern diﬀerential geometric techniques are now more readily applicable for the study of $n$ -body problems. We also refer to Littlejohn-Reinsch [17] for a general discussion with many references to the gauge kinematic and dynamics of many particle systems. For real historical background information on the geometrization of physics we also propose the survey article L $\ddot{u}$ tzen [19] on the 19th century interactions between mechanics and diﬀerential geometry.

1.4. Congruence and internal symmetry in $n$ -body spaces. Let us explain the interaction of the notions of congruence and internal symmetry for $n$ -body systems, where by congruence we mean $O (3)$ -congruence and assume the translational degrees of freedom have been eliminated. For this purpose we introduce a notion somewhat more general than a free m-conﬁguration space (1), namely an m-body space

M ≃ ℝ^{3 m} : (O (3), m ρ_{3})

is a Euclidean 3m-dimensional space with an orthogonal representation of type $(O (3), m ρ_{3})$ , that is, $m$ copies of the standard representation $ρ_{3}$ of $O (3)$ . However, particles and mass distributions are not mentioned, and there is no speciﬁc decomposition into 3-dimensional invariant subspaces $ℝ^{3}$ as in (1). As an example, we have in mind the centered $n$ -conﬁguration space $M_{n}$ (2), where $m = n - 1$ and the original mass distribution is disguised in the Euclidean metric (3).

On the other hand, associated with $ℝ^{3 m}$ is the totality $𝒟 (ℝ^{3 m})$ of all orthogonal, $O (3)$ -invariant and ordered decompositions

ℝ^{3 m} = V_{1} \oplus V_{2} \oplus . . . \oplus V_{m}, V_{i} ≃ ℝ^{3} .

(12)

Then, for a speciﬁc choice of decomposition we can identify $ℝ^{3 m}$ with the free m-conﬁguration space ${\hat{M}}_{m}$ as in (1) and associate a particle $P_{i}$ with position vector $a_{i}$ to each summand $V_{i}$ . Moreover, the metric on $ℝ^{3 m}$ viewed as a metric on ${\hat{M}}_{m}$ should have the form (3), so we also need a mass distribution. However, by suitably scaling of the vectors $a_{i}$ we are actually free to choose any mass distribution since a vector in $ℝ^{3 m}$ written as an m-conﬁguration $(b_{1} {, b}_{2} {, . ., b}_{m})$ determines position vector $a_{i}$ and mass $m_{i}$ modulo the constraint $b_{i} = \sqrt{m_{i}} a_{i}$ for each $i$ (see Section 2.1).

In any case, $ℝ^{3 m}$ has the given congruence group $O (3) \subset O (3 m)$ , and we deﬁne the internal (or inner) symmetries to be those transformations $ϕ \in O (3 m)$ which commute with $O (3)$ , that is, $ϕ ψ =$ $ψ ϕ$ for each $ψ \in O (3) .$ By standard representation theory (cf. Schur’s lemma) these $ϕ$ constitute a subgroup isomorphic to $O (m),$ which we shall refer to as the (internal) symmetry group. The two subgroups of $O (3 m)$ intersect at $ℤ_{2} = \{\pm I d\}$ and hence combine to a subgroup

\frac{O (3) \times O (m)}{ℤ_{2}} \to O (3 m)

which is also described mathematically by the tensor product $ρ_{3} \otimes ρ_{m}$ of their standard representations.

We also point out that $O (m)$ acts naturally on the set of decompositions (12). This action is, in fact, transitive and has isotropy group of type $O {(1)}^{m}$ , and hence establishes a 1-1-correspondence

𝒟 (ℝ^{3 m}) ≃ \frac{O (m)}{O {(1)}^{m}}

To be more explicit, let us ﬁx some decomposition $(12)$ and choose orthonormal bases to identify each $V_{i}$ with $ℝ^{3}$ , and hence identify $ℝ^{3 m}$ in (12) with the space $M (3, m)$ of real $3 \times m$ - matrices

X = (x_{1} {, x}_{2} {, . ., x}_{m})

(13)

with the standard Euclidean norm square

{|X|}^{2} = t r a c e (X^{t} X) = \sum_{i = 1}^{m} {|x_{i}|}^{2},

(14)

where $x_{i} \in$ $V_{i}$ is the i-th column vector of $X .$ Then the action of $ψ \in O (3)$ and $ϕ \in O (m)$ is just matrix multiplication on the left and right side respectively, inducing a joint left action on matrices by

(ψ, ϕ) X = ψ X ϕ^{- 1} (matrix multiplication)

(15)

Thus congruence and symmetry combined together is the following tensor product representation

(O (3) \times O (m), ρ_{3} \otimes ρ_{m}, M (3, m)),

(16)

whose orbit structure will be analyzed by combining two consecutive orbit space constructions

M (3, m) \to \frac{M (3, m)}{O (3)} = \bar{M} (3, m) \to \frac{\bar{M} (3, m)}{O (m)} = \frac{M (3, m)}{O (3) \times O (m)} ≃ C (Δ)

(17)

Of primary interest is the ﬁrst orbit space, namely the internal space $\bar{M} = \bar{M} (3, m)$ with its induced metric structure. The symmetry group $O (m)$ descends faithfully to an induced transformation group on $\bar{M}$ which is, in fact, its isometry group. Thus, the last step in (17) yields the ﬁnal orbit space which is geometrically the Riemannian cone over a spherical triangle $Δ$ (resp. a circular arc if $m = 2$ ):

C (Δ) = \{\begin{matrix} ℝ^{3} ∕ B_{3} & if m \geq 3 \\ ℝ^{2} ∕ B_{2} & if m = 2 \end{matrix}

(18)

Of course, this is a Euclidean cone. However, as indicated it is also the fundamental domain of the Weyl group $B_{3}$ (resp. $B_{2}$ if $m = 2$ ) in classical Lie theory, and this tells us that $C (Δ)$ embeds isometrically into $M (3, m)$ as a cross section of the transformation group (16). Namely, $C (Δ)$ hits every orbit of that group at a unique point and, moreover, it is perpendicular to every orbit.

We refer to Section 3 for further analysis of the above orbit spaces, in a more general setting involving all matrix spaces $M (d, m)$ with the transformation groups $O (d) \times O (m),$ for any $d \geq 1$ .

1.5. Jacobi vectors and the centered conﬁguration space. In dynamics it is the conservation of linear momentum $p$ that enables one to reduce the $n$ -body problem to an (n-1)-body problem plus a trivial 1-body problem for the motion of the center of mass. To explain this, consider the canonical orthogonal and $O (3)$ -invariant decomposition of the free $n$ -conﬁguration space

{\hat{M}}_{n} = M_{n} \oplus Δ ℝ^{3}, cf. (1)

(19)

where the subspace $M_{n} ≃ ℝ^{3 n - 3}$ is the centered $n$ -conﬁguration space (2) and the ”diagonal” $Δ ℝ^{3} = \{(a, a, . ., a)\}$ is its orthogonal complement. For a motion $X (t)$ in ${\hat{M}}_{n}$ the vector $a = a (t)$ is, indeed, the center of mass of $X (t)$ . Now, conservation of $p$ means our inertial frame of reference will remain inertial if we translate the frame so that $a$ becomes the new origin. Therefore, with respect to the new frame, the motion will take place in the summand $M_{n}$ in (19). This simple reduction is a key step in the integration of the classical Kepler problem, where $n = 2$ and $M_{2} ≃ ℝ^{3} ≃ {\hat{M}}_{1}$ is the conﬁguration space of a ﬁctitious 1-body system.

However, for $n > 2$ $M_{n}$ is just an (n-1)-body space and there is no canonical way of further decomposing $M_{n}$ to become the conﬁguration space of $n - 1$ ﬁctitious particles with appropriate position vectors and mass distribution, cf. (12). This fact is clearly reﬂected by the variety of types of coordinates for $M_{n}$ which can be found in the literature. Recall, for example, the eﬀorts of Lagrange, Jacobi and Delaunay who constructed their own ”good” coordinates to study the classical $3$ -body problem (cf. e.g. Marchal [20]).

Of particular interest to us is Jacobi’s approach, which by repeated applications generalizes to $n$ bodies, but again there is no canonical way of doing so. Anyhow, our interpretation of his basic idea is that a solution of the above splitting problem amounts to the construction of a transformation

\begin{align} Ψ & : M_{n} \to {\hat{M}}_{n - 1} & (20) \\ : (a_{1}, a_{2}, . ., a_{n}) \to (x_{1} {, x}_{2} {, . ., x}_{n - 1}) \end{align}

with the ”appropriate” properties (see below), connecting the (n-1)-body space $M_{n}$ (with the Jacobi metric (3)) to a standard model, namely the free conﬁguration space ${\hat{M}}_{n - 1}$ with all masses equal to 1.

By viewing the vectors $x_{i} \in ℝ^{3}$ as the columns of a matrix (13) we may identify ${\hat{M}}_{n - 1}$ with the matrix space

M (3, n - 1) = ℝ_{}^{3} \otimes ℝ^{n - 1} = \sum_{i = 1}^{n - 1} \oplus ℝ_{i}^{3} = \sum_{j = 1}^{3} \oplus ℝ_{j}^{n - 1}

(21)

with the norm as in (14), congruence group $O (3)$ , symmetry group $O (n - 1)$ and their joint tensor product representation of $O (3) \times O (n - 1)$ , see (15), (16). In (21) we have also indicated that the column and row vectors of a matrix belong to two diﬀerent Euclidean spaces, having the standard action of $O (3)$ and $O (n - 1)$ respectively.

Now, what are those ”appropriate” properties Jacobi transformations such as $Ψ$ should satisfy? In short, the answer is that $Ψ$ is just an $O (3)$ -equivariant isometry, that is,

i) Ψ (g X) = g Ψ (X), g \in O (3), ii) |X| = |Ψ (X)|

(22)

Let $Ψ$ in (20) be a given transformation of this type. It associates to a centered $n$ -conﬁguration $X = (a_{1}, a_{2}, . ., a_{n})$ its Jacobi vector matrix

X = Ψ (X) = (x_{1}, x_{2}, . ., x_{n - 1}) \in M (3, n - 1)

whose column vectors $x_{i}$ will be referred to as the corresponding Jacobi vectors. On the other hand, if $Ψ^{'}$ is another Jacobi transformation, then the composition $Ψ^{'} \circ Ψ^{- 1} = ϕ$ is still an $O (3)$ -equivariant isometry. Consequently, we can write $Ψ^{'} = ϕ \circ Ψ$ where

ϕ : M (3, n - 1) \to M (3, n - 1)

(23)

is a Jacobi transformation of the standard model (21), namely an orthogonal transformation commuting with $O (3) .$ In other words, $ϕ$ belongs to the symmetry group $O (n - 1)$ (acting on the matrices $X$ by right multiplication). This explains the non-uniqueness of $Ψ$ ! Brieﬂy, by knowing only one of the $Ψ^{'} s$ we obtain all of them by composing $Ψ$ with any $ϕ \in O (n - 1)$ .

So far, however, we have not constructed a single $Ψ$ in (20), but we have just seen that this is all we need to do. We refer to Section 2.2 for the explicit construction of our stan $dard$ choice $Ψ = Ψ_{0}$ of Jacobi transformation and hence also our preferred choice of Jacobi vectors $x_{i}$ as distinguished linear combinations of the vectors $a_{j}$ .

Remark 1.1. Quantities which are invariant under the transformation group $(O (n - 1), M_{n})$ are also insensitive to the diﬀerent orderings of the $n$ bodies. Therefore, this group is sometimes referred to as the democracy group. Elements $ϕ \in S O (n - 1)$ are also named kinematic rotations. See e.g. [17].

2. Jacobi transformations from a categorical viewpoint

Loosely speaking, there is the category of m-body spaces whose objects are the Euclidean spaces $M ≃ ℝ^{3 m}$ with an orthogonal transformation group of type $(O (3), m ρ_{3})$ , and the morphisms will be called Jacobi transformations, namely they are the $O (3)$ -equivariant isometries $M \to M^{'}$ between the spaces. Here the matrix space $M (3, m) = {\hat{M}}_{m}$ is a distinguished object which also serves as our standard model (21). Therefore, for a given m-body space $M$ a ”good” coordinate system for $M$ amounts to the choice of an appropriate Jacobi transformation

Ψ : M \to M (3, m)

and the columns of the matrix $X \in M (3, m)$ will be the Jacobi vectors with respect to $Ψ$ . We are primarily interested in the centered $n$ -conﬁguration space $M = M_{n}$ (where $m = n - 1$ ).

In the following two subsections we shall characterize Jacobi transformations in several equivalent ways, and we start with the classical Jacobi construction which also justiﬁes our usage of terminology.

2.1. Transformations of $n$ -body systems and Jacobi’s approach. In the Introduction we actually started (less formally) with the category of $n$ -body systems, whose objects are n-tuples $(P_{1}, P_{2}, . . ., P_{n})$ of point masses $P_{i} = (a_{i}, m_{i})$ with $a_{i}$ and $m_{i}$ as position vector and mass respectively. In this setting, what should be the appropriate morphisms

Φ : (P_{1}, . . ., P_{n}) \to (P_{1}^{'}, . . ., P_{n}^{'}) ?

(24)

After all, the usefulness of such transformations with the desired properties, to simplify the further analysis, is well documented in both classical and quantum mechanics. First of all, we propose to consider transformations

Φ : \begin{matrix} (a_{1}, . . . ., a_{n}) \to (a_{1}^{'}, . . . ., a_{n}^{'}) \\ (m_{1}, . . ., m_{n}) \to (m_{1}^{'}, . . ., m_{n}^{'}) \end{matrix}

(25)

where the ﬁrst map in (25) is an invertible linear transformation on ${\hat{M}}_{n} = ℝ^{3 n}$ which may possibly depend on the masses $m_{i}$ , with the choice of ”new” masses $m_{i}^{'}$ constrained in some way. But $Φ$ also transforms the motions of $n$ -body system and hence also the ﬁrst three basic kinematic quantities in (7), so let us demand that $Φ$ preserves them, namely

\begin{align} I & = \sum m_{i} {|a_{i}|}^{2} = \sum m_{j}^{'} {|a_{j}^{'}|}^{2} \\ 2 T & = \sum m_{i} {|{\dot{a}}_{i}|}^{2} = \sum m_{j}^{'} {|{\dot{a}}_{j}^{'}|}^{2} & (26) \\ Ω & = \sum m_{i} a_{i} \times {\dot{a}}_{i} = \sum m_{j}^{'} a_{j}^{'} \times {\dot{a}}_{j}^{'} \end{align}

Motivated by an idea of Jacobi, we deﬁne the (i,j)-basic Jacobi transformation for $i \neq j$

ρ_{i j} : (P_{1}, . . ., P_{n}) \to (P_{1}^{'}, . . ., P_{n}^{'})

(27)

by demanding $P_{k}^{'} = P_{k}, m_{k}^{'} = m_{k}$ for $k \neq i, j$ , and

\begin{align} P_{i}^{'} & = (a_{i}^{'}, m_{i}^{'}) = (a_{i} - a_{j}, \frac{m_{i} m_{j}}{m_{i} + m_{j}}) & (28) \\ P_{j}^{'} & = (a_{j}^{'}, m_{j}^{'}) = (\frac{m_{i} a_{i} + m_{j} a_{j}}{m_{i} + m_{j}}, m_{i} + m_{j}) \end{align}

In particular, $P_{j}^{'}$ is the center of mass of $P_{i}$ and $P_{j}$ , and $m_{i}^{'}$ is their reduced mass

μ_{i j} = \frac{m_{i} m_{j}}{m_{i} + m_{j}}

(29)

Among the basic Jacobi transformations let us also include the following mass normalizing transformation

ρ_{0} : (a_{i}, m_{i}) \to (\sqrt{m_{i}} a_{i}, 1), \forall i

(30)

and the permutation transformations

\bar{σ} : (P_{1}, . . ., P_{n}) \to (P_{σ (1)}, . . ., P_{σ (n)})

(31)

which permute the vectors $a_{i}$ and masses $m_{i}$ covariantly, namely there is one for each permutation $σ$ of ${1, 2, . ., n}$ . It is straightforward to check that the above transformations, and hence all their compositions, have the invariance property (26).

Next, we inquire whether the above special transformations already generate all those $Φ$ with the property (26). This is aﬃrmatively settled in the following subsection, but ﬁrst we shall make some clarifying observations. Note that velocity vectors ${\dot{a}}_{i}$ of a motion are transformed in the same way as the position vectors $a_{i}$ and may well be regarded as vectors $b_{i}$ independent of the $a_{i}$ ’s. In particular, invariance of $I$ means the same as invariance of $T,$ namely that $Φ$ is an isometry. Therefore, the remaining issue is the nature of the invariance of $Ω .$

Consider the $3 n \times 3 n$ -matrix of a linear transformation $Φ,$ regarded as an $n \times n$ -matrix

[Φ] = (\begin{matrix} B_{11} & B_{12} & . . . . . \\ B_{21} & B_{22} & . . . . . \\ . . . . & . . . & . . . . \end{matrix})

(32)

where each block $B_{i j}$ is a $3 \times 3$ -matrix. Such an $Φ$ expresses each $a_{j}^{'}$ as a linear combination of the old ones, that is,

a_{i}^{'} = \sum_{j = 1}^{n} β_{i j} a_{j}, i = 1, . ., n,

(33)

if and only if each 3-block is a scaling matrix $B_{i j} = β_{i j} I d,$ where $I d$ denotes the identity matrix. Equivalently, $Φ$ commutes with the (diagonal) action of $G L (3)$ on $ℝ^{3 n}$ and hence (by representation theory) belongs to the subgroup $G L (n)$ embedded into $G L (3 n)$ by the tensor product action of $G L (3) \times G L (n) .$

On the other hand, assuming (33) holds $Φ$ will be an isometry (i.e. $I$ is preserved) if and only if the $n \times n$ -matrix $(β_{i j})$ is ”orthogonal” in the sense that

(β_{i j}) \in D^{' - 1} O (n) D \subset G L (n)

(34)

where $D = d i a g (\sqrt{m_{1}}, . . ., \sqrt{m_{n}}), D^{'} = d i a g (\sqrt{m_{1}^{'}}, . ., \sqrt{m_{n}^{'}}) .$ Finally, assuming (33) and (34) it is easy to verify that $Ω$ is also preserved.

2.2. Equivalent characterizations of Jacobi transformations. It turns out that the invariance properties (26) are satisﬁed solely by demanding invariance of the vector $Ω .$ This is elucidated by the following proposition where the angular momentum construction is analyzed from a purely algebraic viewpoint.

Proposition 2.1. Consider two n-tuples of vectors in 3-space

X = (x_{1}, x_{2}, . ., x_{n}), Y = (y_{1}, y_{2}, . ., y_{n})

and deﬁne their angular momentum vector by

Ω = Ω (X, Y) = \sum_{i = 1}^{n} x_{i} {\times y}_{i}

Regard $X, Y$ as vectors in $ℝ^{3 n}$ $= ℝ^{3} \times . . . . \times ℝ^{3}$ and consider general linear transformations $Φ : ℝ^{3 n} \to ℝ^{3 n},$

X \to X^{'} = (x_{1}^{'}, x_{2}^{'}, . ., x_{n}^{'}), Y \to Y^{'} = (y_{1}^{'}, y_{2}^{'}, . ., y_{n}^{'})

Then $Φ$ preserves angular momentum, that is, for all $X, Y$

Ω (X, Y) = Ω (X^{'}, Y^{'})

if and only if $Φ$ is an $O (3)$ -equivariant isometry.

Proof. The $3 \times 3$ -matrices

S_{k} = (s_{i j}^{(k)}), k = 1, 2, 3,

whose only non-zero entries are $s_{i j}^{(k)} = - s_{j i}^{(k)} = 1$ whenever $(i, j, k)$ is an even permutation of $(1, 2, 3),$ constitute a basis for the Lie algebra $s o (3)$ of all skew-symmetric matrices $S$ . The action of $S O (3)$ on $ℝ^{3 n}$ embeds each $S$ as a ”diagonal block” matrix

Δ (S) \in s o (3 n)

with $n$ copies of $S$ along the diagonal. Observe that the matrices $Δ (S_{k})$ actually represent the three components $Ω_{k}$ of the vector $Ω$ , as skew-symmetric bilinear forms on $ℝ^{3 n}$ expressed as matrix products, namely

Ω_{k} (X, Y) = X^{t} Δ (S_{k}) Y

where $X, Y$ are regarded as column vectors with 3n components.

Let us identify a transformation $Φ$ with its matrix in $G L (3 n) .$ Those transformations leaving $Ω_{k}$ invariant constitute a matrix group $G_{k}$ deﬁned by the constraint

Φ^{t} Δ (S_{k}) Φ = Δ (S_{k})

and hence the group leaving $Ω$ invariant is the intersection $G = G_{1} \cap G_{2} \cap G_{3}$ deﬁned by the constraint

Φ^{t} Δ (S) Φ = Δ (S),  for all S \in s o (3)

(35)

The proof amounts to show $G = O (n) \subset O (3 n)$ , namely that $G$ is the group of isometries commuting with $O (3)$ (or equivalently with $S O (3)$ ).

If we knew $Φ$ is orthogonal, then we could have applied the exponential function to (35) to conclude that $Φ$ commutes with $S O (3)$ and hence belongs to $O (n) .$ We claim, however, the isometry condition is itself a consequence of (35). To see this, write a typical matrix of the Lie algebra $L (G)$ as an $n \times n$ -matrix $(A_{i j})$ with blocks $A_{i j}$ , see (32). Then the inﬁnitesimal version of (35) simply reads

A_{i j}^{t} S + S A_{i j} = 0, for all i,j and all S \in s o (3),

or equivalently

A_{i j} = - A_{j i}, A_{i j} = α_{i j} I d (scaling matrix)

However, these are precisely the conditions deﬁning the Lie subalgebra $s o (n)$ and consequently $G$ and $O (n)$ have the same connected component $S O (n) .$

Finally, to show $G = O (n)$ we use the fact that $G$ lies in the normalizer of $S O (n)$ in $G L (3 n)$ , namely in the image of $G L (3) \times O (n)$ by the tensor product representation. Therefore, we may assume $Φ = Δ (B)$ where $B \in G L (3)$ and hence by (35)

B^{t} S B = S for all S \in s o (3)

This condition says $B$ leaves invariant all skew-symmetric bilinear forms in $ℝ^{3}$ , and consequently $B = \pm I d .$ This completes the proof. _

The basic Jacobi transformations (27), (30), (31) are maps between n-conﬁguration spaces with possibly diﬀerent mass distribution. However, using normalizing transformations $ρ_{0}$ , as in (30), any composition may be ”pulled back” to a transformation $M (3, n) \to M (3, n)$ and with all masses $m_{i} = 1$ . In this interpretation, the (i,j)-basic Jacobi transformation $ρ_{i j}$ will be a (2-dimensional) rotation ${\hat{ρ}}_{i j} \in O (n)$ and it is easy to check that ${\hat{ρ}}_{12}, {\hat{ρ}}_{23}, {\hat{ρ}}_{13}$ generate a dense subgroup of $S O (3) \subset S O (3) \times S O (n - 3) \subset O (n) .$ Therefore, all rotations ${\hat{ρ}}_{i j}$ together with the permutations (31) generate a dense subgroup of $O (n) .$

As a consequence of the above observations we now state the following result on the characterization of Jacobi transformations, a terminology justiﬁed by (iii) below. In view of Section 1.3 we shall not attempt to deﬁne the angular momentum vector $Ω$ in the broad category of $n$ -body spaces, but in the following theorem we refer to $n$ -conﬁguration spaces as deﬁned in Section 1.1.

Theorem 2.2. The following four classes of linear transformations $Φ$ between $n$ -conﬁguration spaces are identical :

(i) $Φ$ is an isometry which preserves the angular momentum vector $Ω$ .

(ii) $Φ$ preserves $Ω$ .

(iii) $Φ$ is in the closure of the set of transformations generated by the basic Jacobi transformations.

(iv) $Φ$ is an $O (3)$ -equivariant isometry.

The Jacobi transformations on a ﬁxed ( $n$ -body or $n$ -conﬁguration) space are also its symmetries (cf. Section 1.3), and they constitute a group isomorphic to $O (n) .$ In particular, the symmetry group for our standard model, namely the matrix space $M (3, n)$ , is the group $O (n)$ acting by matrix multiplication on the right side.

Transformations which preserve angular momentum and kinetic energy are certainly useful in quantum mechanics. Here one tries to keep operators ”separable”, that is, with no cross terms (cf. e.g. [6], $§$ 10.1). As a consequence of the above results Jacobi transformations provide all possible linear combinations $a_{j}^{'}$ of the vectors $a_{i}$ which preserve the kinetic energy operator, in the sense that

T = - \frac{ℏ^{2}}{2} [\frac{1}{m_{1}} \nabla_{a_{1}}^{2} + . . . + \frac{1}{m_{n}} \nabla_{a_{n}}^{2}] = - \frac{ℏ^{2}}{2} [\frac{1}{m_{1}^{'}} \nabla_{a_{1}^{'}}^{2} + . . . + \frac{1}{m_{n}^{'}} \nabla_{a_{n}^{'}}^{2}]

(36)

A reduction of variables is achieved by choosing the transformation $Φ$ so that $a_{n}^{'}$ becomes the center of mass vector and hence vanishes relative to a center of mass coordinate system. Then, for example, the (time dependent) Schr $\ddot{o}$ dinger equation with potential energy $V$ reduces to

i ℏ \frac{\partial}{\partial t} Ψ = [- \frac{ℏ^{2}}{2} (\sum_{i = 1}^{n - 1} \frac{1}{m_{i}^{'}} \nabla_{a_{i}^{'}}^{2}) + V (a_{1}^{'}, . . ., a_{n - 1}^{'})] Ψ

where $Ψ = Ψ (a_{1}^{'}, . . ., a_{n - 1}^{'}, t)$ . In the following subsection an explicit construction of such coordinates is given, and by usage of the symmetry group $O (n)$ they can be modiﬁed to satisfy additional properties for a speciﬁc purpose.

On the other hand, in the study of atomic structures one of the masses, say $m_{n}$ (the nucleus), may be relatively large and then non-Jacobian coordinates may also turn out to be useful. However, in that case mass polarization terms cannot be avoided, namely mixed terms $\nabla_{a_{i}^{'}} \cdot \nabla_{a_{j}^{'}}$ will appear in the kinetic energy operator (see e.g. Appendix 8 in [5]).

2.3. Construction of Jacobi vectors. Here we shall construct an explicit Jacobi transformation (20), or rather its inverse

Φ = Ψ^{- 1} : X = (x_{1}, x_{2}, . . ., x_{n - 1}) \to (a_{1}, a_{2}, . ., a_{n - 1}, - \frac{1}{m_{n}} \sum_{i = 1}^{n - 1} m_{i} a_{i})

(37)

from $M (3, n - 1)$ to the centered conﬁgurations space $M_{n}$ . By deﬁnition, $Φ$ must be an $O (3)$ -equivariant isometry, and hence in analogy with (22)

\begin{align} i) Φ (g X) & = g Φ (X) g \in O (3) & (38) \\ ii) |Φ (X)| & = |X|, X \in M (3, n - 1) \end{align}

and let us also add the following ”normalizing” condition :

(x_{1}, x_{2}, . ., x_{k}, 0, . . ., 0) \to (a_{1}, a_{2}, . ., a_{k}, b^{(k)}, . ., b^{(k)}),  for k < n - 1

(39)

Namely, the Jacobi vectors $x_{i}$ should vanish for $i > k$ if and only if the particles $P_{i}$ ”collide” for $i > k$ , that is, they occupy the same position $b^{(k)}$ . Equivalently, the Jacobi vectors deﬁne an orthogonal decomposition

M_{n} = ℝ_{1}^{3} \oplus ℝ_{2}^{3} \oplus . . . \oplus ℝ_{n - 1}^{3}

where $ℝ_{k}^{3}$ $≃ ℝ^{3}$ is the image of the imbedding

(0, . ., 0, x_{k}, 0, . ., 0) \to (0, . ., 0, a_{k} - b^{(k - 1)}, b_{(k)}, \dots, b_{(k)})

(40)

which actually represents a collinear $3$ -body system in the sense that $P_{1}, P_{2}, . . ., P_{k - 1}$ are located at the origin and $P_{k + 1,} . . . ., P_{n}$ at the common position vector

b_{(k)} = b^{(k)} - b^{(k - 1)} .

The condition that $X = (a_{1}, . ., a_{k}, b^{(k)}, . . ., b^{(k)})$ in (39) belongs to $M_{n}$ yields the formula

b^{(k)} = \frac{- 1}{m^{(k)}} \sum_{i = 1}^{k} m_{i} a_{i}, b^{(0)} = 0,

(41)

where by deﬁnition

m^{(k)} = m_{k + 1} + . . . + m_{n}, m^{(0)} = \sum m_{i} = \bar{m}

As a consequence of (39) we also note that the matrix $[Φ]$ formally representing $Φ$ by matrix multiplication

[Φ] (\begin{matrix} x_{1} \\ x_{2} \\ : \\ x_{n - 1} \end{matrix}) = (\begin{matrix} a_{1} \\ a_{2} \\ : \\ a_{n - 1} \end{matrix})

(42)

is lower triangular.

By combining (40) with the $O (3)$ -equivariance of $Φ$ we obtain the following Jacobi vectors

x_{k} = ζ_{k} (a_{k} - b^{(k - 1)}) = ζ_{k} (a_{k} + \frac{1}{m^{(k - 1)}} \sum_{i = 1}^{k - 1} m_{i} a_{i})

(43)

for suitable constants $ζ_{k} \neq 0,$ whose square is determined by the isometry condition ii) of (38), namely

ζ_{k}^{2} = \frac{m_{k} m^{(k - 1)}}{m^{(k)}}, 1 \leq k \leq n - 1

(44)

For example, for $k = 1$ we have $x_{1} = ζ_{1} a_{1}$ with

{|x_{1}|}^{2} = ζ_{1}^{2} {|a_{1}|}^{2} = {|(a_{1}, b^{(1)}, . . ., b^{(1)})|}^{2} = {|a_{1}|}^{2} (m_{1} + \frac{m_{1}^{2}}{(\bar{m} - m_{1})})

and hence $ζ_{1}^{2} = m_{1} \bar{m} ∕ (\bar{m} - m_{1}) .$

Now it is easy to describe all $Φ$ satisfying the condition (39), namely $Φ$ is uniquely determined by the choice of sign of the numbers $ζ_{k}$ , $1 \leq k \leq n - 1 .$ There are $2^{n - 1}$ such choices, and a natural choice is to take all $ζ_{k} > 0$ , namely our distinguished $Φ = Φ_{0}$ is deﬁned by taking

ζ_{k} = \sqrt{\frac{m_{k} m^{(k - 1)}}{m^{(k)}}}, 1 \leq k \leq n - 1

(45)

Its inverse $Ψ_{0} = Φ_{0}^{- 1}$ will be regarded as our standard Jacobi transformation, and hence the standard Jacobi vectors are the columns of the matrix $X = Ψ_{0} (X)$ . Then the corresponding lower triangular matrix $[Ψ_{0}] = {[Φ_{0}]}^{- 1}$ (see (42)), has entries

{[Ψ_{0}]}_{i k} = ζ_{i} . \{\begin{matrix} 1 & i = k \\ \frac{m_{k}}{m^{(i - 1)}}, & 1 \leq k < i \leq n - 1 \\ 0 & 1 \leq i < k \leq n - 1 \end{matrix}

(46)

derived from the relation (43). On the other hand, the entries of $[Φ_{0}]$ $= (l_{i k})$ are

{[Φ_{0}]}_{i k} = l_{i k} = ζ_{k}^{- 1} . \{\begin{matrix} 1 & i = k \\ - \frac{m_{k}}{m^{(k)}}, & 1 \leq k < i \leq n - 1 \\ 0 & 1 \leq i < k \leq n - 1 \end{matrix}

(47)

where in each column the entries below the diagonal are identical. Moreover, the square sum for each row is given by the simple formula

l_{k 1}^{2} + . . . + l_{k k}^{2} = \frac{1}{\bar{m}} \frac{\bar{m} - m_{k}}{m_{k}}, 1 \leq k \leq n - 1

Theorem 2.3. The following Jacobi transformation

{\tilde{Ψ}}_{0} = ρ_{0} \circ ρ_{1 n} \circ ρ_{2 n} \circ . . . \circ ρ_{n - 1, n} : {\hat{M}}_{n} \to M (3, n)

maps $(a_{1}, . . ., a_{n})$ to $(x_{1}, . . ., x_{n})$ , where $x_{n}$ $= {\bar{m}}^{- 1} \sum m_{i} a_{i}$ is the center of mass and the remaining $x_{i}$ are given by the expressions (43), (45). In particular, the restriction of ${\tilde{Ψ}}_{0}$ to the centered conﬁguration space yields $x_{n} = 0$ and it coincides with the standard Jacobi transformation

Ψ_{0} : M_{n} \to M (3, n - 1) : (a_{1}, . . ., a_{n}) \to (x_{1}, . . ., x_{n - 1})

The proof follows by an explicit calculation of $x_{k}$ as a linear combination of $a_{1}, . . ., a_{k},$ which shows that $x_{k}$ coincides with the vector in (43) for $k < n$ , and furthermore, $x_{n}$ is the center of mass of the $n$ -conﬁguration $X$ $= (a_{1}, . . ., a_{n}) .$

The above transformation $Ψ_{0} : M_{n} \to M (3, n - 1)$ turns out, in fact, to be identical with the coordinate transformation described in Theorem 1 of Hsiang [9], derived by a similar but diﬀerent ”naturality” principle. For convenience, explicit formulae (43) for the standard Jacobi vectors when $n = 3, 4$ are listed below :

\begin{align} n = 3 : & x_{1} = \sqrt{\frac{m_{1} \bar{m}}{m_{2} + m_{3}}} a_{1}, x_{2} = \sqrt{\frac{m_{2} (m_{2} + m_{3})}{m_{3}}} (a_{2} + \frac{m_{1}}{m_{2} + m_{3}} a_{1}) \\ n = 4 : & x_{1} = \sqrt{\frac{m_{1} \bar{m}}{m_{2} + m_{3} + m_{4}}} a_{1} & (48) \\ x_{2} = \sqrt{\frac{m_{2} (m_{2} + m_{3} + m_{4})}{m_{3} + m_{4}}} (a_{2} + \frac{m_{1}}{m_{2} + m_{3} + m_{4}} a_{1}) \\ x_{3} = \sqrt{\frac{m_{3} (m_{3} + m_{4})}{m_{4}}} [a_{3} + \frac{1}{m_{3} + m_{4}} (m_{1} a_{1} + m_{2} a_{2})] \end{align}

3. Orthogonal transformation groups on matrix spaces

By applying a ﬁxed Jacobi transformation $Ψ : M_{n} \to M (3, n - 1)$ , quantities and constructions in the centered conﬁguration space $M_{n}$ are transported to its matrix model $M (3, m)$ with $m = n - 1$ , as explained in Section 1.4. In this setup we may and shall, however, discuss more generally the matrix space

M = M (d, m) ≃ ℝ^{d} \otimes ℝ^{m}

with its Euclidean norm (14) and orthogonal transformation group

G = G_{1} \times G_{2} = O (d) \times O (m)

acting by the tensor product $ρ_{d} \otimes ρ_{m}$ , namely $(ψ, ϕ) \in G$ acts on matrices $X$ by

((ψ, ϕ), X) \to ψ X ϕ^{- 1} (matrix multiplication)

(49)

Physically, we have in mind $M$ as the conﬁguration space for $m$ free bodies (or $m + 1$ bodies with ﬁxed center of mass) in $d$ -dimensional Euclidean space, and the orbit space $\bar{M} = M ∕ G_{1}$ is the internal conﬁguration space with the induced action of $G_{2}$ as the internal symmetries. Therefore, for further investigation of the topology of $\bar{M}$ it is natural to analyze its induced $G_{2}$ -orbital structure. Thus we shall investigate two successive equivariant systems $(G_{1}, M)$ and $(G_{2}, \bar{M})$ , where by an equivariant system $(K, Y)$ we mean in general a (compact) Lie group $K$ acting on a space $Y$ . We also say $(K, Y)$ is a transformation group or we simply say $Y$ is a $K$ -space, and it is convenient to refer to the orbit space construction and the orbit map

π : Y \to Y ∕ K

(50)

as a K-orbital ﬁbration. In (50) the ”ﬁbers” are the orbits, and therefore $π$ may not be the projection of a ﬁber bundle (or ﬁbration) in the usual sense since in general the orbit type is not unique (see below).

3.1. Compact transformation groups and orbital decomposition. First we shall recall some basic facts concerning equivariant systems $(K, ℳ)$ where $K$ is a compact Lie group acting smoothly on a manifold $ℳ$ , that is, the action map $K \times ℳ \to ℳ$ is $C^{\infty}$ -smooth. As a good reference on this topic we propose, for example, the book by Bredon [7]. Then we specialize to the compact linear groups, and of particular interest are the natural transformation groups on the matrix spaces.

3.1.1. Basic deﬁnitions and constructions. The space $ℳ$ splits into a disjoint union of $K$ -orbits; these are minimal (or homogeneous) $K$ -spaces in the obvious sense. The isotropy (or stability) groups $K_{p}$ at points $p$ along the same orbit constitute a single conjugacy class $(K_{α})$ of subgroups representing the type of the orbit. For simplicity we say the orbit is of type $K_{α}$ , and as a $K$ -space such an orbit is naturally equivalent to the coset space $K ∕ K_{α} .$ By grouping together orbits of the same type, $ℳ$ decomposes as a union of orbit strata which are smooth submanifolds $ℳ_{α}$ , all of which have dimension less than $ℳ$ except the principal stratum $ℳ_{ω}$ , which is an open and dense submanifold. The corresponding principal type $K_{ω}$ is characterized as the unique smallest in the sense that (up to conjugation) $K_{ω} \subset K_{α}$ for each $α$ .

The orbit map

π : ℳ \to \bar{ℳ} = ℳ ∕ K

(51)

induces a smooth functional structure on the orbit space, in the sense that a function $f$ on $\bar{ℳ}$ is called smooth if the composition $f \circ π$ is a smooth function on $ℳ$ . Then $\bar{ℳ}$ is a diﬀerential space, but it may not be even locally Euclidean and hence it is not a manifold in general. However, still it has the nice and rich structure of a stratiﬁed smooth manifold with the image sets ${\bar{ℳ}}_{α} = π (ℳ_{α})$ as strata, and they are actually smooth manifolds. In this way we may regard $ℳ$ as a ﬁnite union of smooth orbit bundles denoted by

K ∕ K_{α} ↪ ℳ_{α} \to^{π_{α}} {\bar{ℳ}}_{α}

(52)

whose ﬁbers are those orbits of a ﬁxed type $K_{α}$ and $π_{α}$ is, in fact, the projection of an actual ﬁber bundle.

For convenience, let us recall the general notion of a (locally trivial) ﬁber bundle, typically illustrated by a sequence like (52)

F ↪ E \to^{π} B

(53)

where $π$ is the projection map, $E$ (resp. $B$ ) is the total (resp. base) space, and $F$ denotes the typical ﬁber, that is, the ﬁbers $E_{b} = π^{- 1} (b)$ are homeomorphic to $F$ for each $b \in B$ . The simplest example is the product bundle, where $E = B \times F$ and $π$ is the obvious projection. An isomorphism between two bundles over $B$ is given by a ﬁber preserving homeomorphism $ϕ : E_{1} \to E_{2}$ between their total spaces, and a bundle is trivial if it is isomorphic to a product bundle. The local triviality property (which is part of the deﬁnition) means that $B$ is covered by open sets $U$ such that the portion $π^{- 1} (U)$ over $U$ is a trivial bundle, that is, $π^{- 1} (U) ≃ U \times F$ . Finally, the bundle is smooth if all the above spaces are smooth manifolds and the maps involved are smooth.

In particular, a $K$ -orbit bundle like (52) is deﬁned by an equivariant system $(K, E)$ with a single orbit type, and moreover, $B$ is the orbit space $E ∕ K$ and hence the ﬁbers are the $K$ -orbits. A principal bundle (or principal ﬁbration) is the special case

K ↪ E \to E ∕ K = B

(54)

of (52) where $K$ acts freely (i.e. all isotropy groups are trivial).

When $K$ acts by isometries on a Riemannian manifold $ℳ$ , there is an induced Riemannian metric on ${\bar{ℳ}}_{α}$ so that the projection map $π_{α}$ in (52) is a so-called Riemannian submersion. Namely, the tangent map $d π_{α}$ at any point $p \in ℳ_{α}$ restricts to an isometry between the horizontal space $ℋ_{p}$ (i.e. vectors perpendicular to the orbit $\bar{p} = K (p)$ and the tangent space of ${\bar{ℳ}}_{α}$ at the point $\bar{p}$ . In this way the total orbit space $\bar{ℳ}$ in (51) becomes a stratiﬁed Riemannian space; in particular, the strata are Riemannian manifolds. Moreover, $\bar{ℳ}$ has also a global orbital distance metric which measures the distance between orbits in $ℳ$ (c.f. e.g. [9], [30]). This metric is certainly determined by its restriction to the principal stratum ${\bar{ℳ}}_{ω}$ , which in turn is derived from the Riemannian metric on this stratum. Brieﬂy, the Riemannian manifold ${\bar{ℳ}}_{ω}$ determines the Riemannian structure on each ${\bar{ℳ}}_{α}$ .

Next, we turn to the so-called slice theorem which reduces the local study of $(K, ℳ)$ to linear representation theory, as follows. The local representation of the isotropy group $H = K_{p}$ at a given point $p$ , denoted by ${(l o c a l)}_{p}$ in (55), is the induced linear action (via diﬀerentiation) on the tangent space $T_{p} ℳ$ of $ℳ$ at $p$ . This splits into the isotropy and slice representation and we write

T_{p} ℳ = T_{p} K (p) \oplus ℋ_{p}, {(l o c a l)}_{p} = {(I s o)}_{p} + {(s l i c e)}_{p},

(55)

to indicate that the isotropy representation is the $H$ -action on the tangent space $T_{p} K (p)$ of the orbit $K (p)$ , and the slice representation is the induced action on some $H$ -invariant complementary subspace $ℋ_{p}$ (which exists since $H$ is compact). For example, when $ℳ$ is Riemannian we take $ℋ_{p}$ to be the horizontal space, that is, the orthogonal complement of $T_{p} K (p)$ .

Now, the slice theorem says the orbit $K (p)$ has a $K$ -invariant tubular neighborhood which has the structure of a twisted product $K \times_{H} ℋ_{p}$ , consisting of equivalence classes $[k, v]$ in $K \times ℋ_{p}$ modulo the relation $[k, v] = [k h^{- 1}, h v]$ , for $k \in K, h \in H, v \in$ $ℋ_{p}$ . The ”tube” $K \times_{H} ℋ_{p}$ has the structure of a vector bundle over $K (p) ≃ K ∕ H$ with ﬁber $ℋ_{p}$ , and moreover, as a $K$ -space it has the following left action

k^{'} [k, v] = [k^{'} k, v], k^{'} \in K

and consequently the isotropy groups at points in the tubular neighborhood of $K (p)$ looks like

K_{[k, v]} = k H_{v} k^{- 1}

In particular, by calculating the slice representation $(K_{p}, ℋ_{p})$ as the diﬀerence

{(s l i c e)}_{p} = {(l o c a l)}_{p} - {(I s o)}_{p}

(56)

one can determine the orbit types that occur in the neighborhood. Moreover, there is the following isomorphism of orbit spaces

(K \times_{K_{p}} ℋ_{p}) ∕ K ≃ ℋ_{p} ∕ K_{p}

(57)

which enables one to analyze the local smooth structure of orbit spaces by successive application of the slice theorem.

Finally, we recall the notion of a fundamental domain for $(K, ℳ)$ , or equivalently a cross-section for the orbit map (51). This is a closed subset $Σ$ $\subset$ $ℳ$ which intersects each orbit in a unique point, and consequently there is an identiﬁcation

Σ ≃ \bar{ℳ}

(58)

In the special case that $ℳ$ is Riemannian and $Σ$ is an orthogonal cross-section, and hence the submanifold $Σ$ is also perpendicular to each orbit, the identiﬁcation (58) is in fact an isometry. We remark, however, that fundamental domains in the above strict sense exist only in special cases – but fortunately including those orthogonal transformation groups that we shall investigate below.

3.1.2. Compact linear groups on Euclidean spaces . Now, let us consider pairs $(K, V)$ and the corresponding orbit map

π : V \to \bar{V} = V ∕ K

(59)

where $K$ is a compact Lie group acting orthogonally on a Euclidean space $V$ . Since the orbits are compact it is a well known fact that they can be separated by $K$ -invariant polynomial functions $𝔭_{i} .$ Namely, we can choose a Hilbert basis or ”suﬃciently many” separating and invariant polynomial functions $𝔭_{i}$ as the components of a map

𝔭 = (𝔭_{1}, 𝔭_{2}, . . ., 𝔭_{N}) : V \to ℝ^{N}

which induces an embedding

\bar{𝔭} : \bar{V} \to 𝔭 (V) \subset ℝ^{N}

(60)

and hence identiﬁes the orbit space with its image $𝔭 (V)$ . The latter is a semi-algebraic subset, that is, it is deﬁned by polynomial identities and inequalities in $ℝ^{N}$ .

Concerning smoothness, we recall that $\bar{V}$ has, on the one hand, the structure of a diﬀerential space such that a function $f$ on $\bar{V}$ is ”smooth” if and only if the composition $f \circ π$ is a smooth function on $V$ . On the other hand, there is another approach deﬁning a function on $𝔭 (V)$ to be ”smooth” if it is the restriction of a smooth function on $ℝ^{N} .$ Fortunately, the two notions of smoothness are identical, and therefore we may regard the bijective correspondence $\bar{𝔭}$ in (60) to be a diﬀeomorphism, cf. Schwarz [24]. Consequently, $f$ is ”smooth” if and only if $f \circ π = \bar{f} \circ 𝔭$ for some smooth function $\bar{f}$ on $ℝ^{N} .$ Note, however, the orbital distance metric on $\bar{V}$ $≃ 𝔭 (V)$ is certainly not induced from the Euclidean space in (60).

In the Euclidean space $V$ size and shape are quantities preserved by orthogonal transformations. The norm $ρ = |X|$ is a size function which measures the distance from the origin $O$ , namely the arc-length along rays emanating from $O$ . A nonzero vector can be uniquely scaled to unit size by a homothety transformation $X \to c X$ $(c > 0)$ , and by expressing the Euclidean metric $d s^{2}$ of $V$ in polar coordinates we actually describe $V$ as a Riemannian cone over its unit sphere $V^{1} :$

(V^{1}, d θ^{2}) \subset (V, d s^{2}), d s^{2} = d ρ^{2} + ρ^{2} d θ^{2}

(61)

where $d θ^{2}$ denotes the induced spherical metric on $V^{1}$ .

On the other hand, the $K$ -orbit space $\bar{V}$ and its subspace $V^{*} = V^{1} ∕ K$ are stratiﬁed Riemannian spaces as explained above. Moreover, since the norm function $ρ$ is $K$ -invariant it is also a size function on $\bar{V}$ and then $V^{*} = (ρ = 1)$ becomes the unit distance ”sphere”. Furthermore, in analogy with (61) $\bar{V}$ inherits the structure of a stratiﬁed Riemannian cone over its ”sphere” $V^{*}$ :

(V^{*}, d σ^{2}) \subset (\bar{V}, d {\bar{s}}^{2}), d {\bar{s}}^{2} = d ρ^{2} + ρ^{2} d σ^{2}

(62)

where $d σ^{2}$ denotes the orbital distance metric induced from $d θ^{2}$ . In this cone the rays emanating from the cone vertex (or base point), still denoted by $O$ , are also geodesics for the metric $d {\bar{s}}^{2}$ , and homothety transformations move points $\neq O$ along these rays.

3.1.3. The conﬁguration space for m-body systems in d-space. Henceforth, we take $V$ to be the matrix space $M = M (d, m)$ with the orthogonal action of $G = O (d) \times O (m)$ as in (49). By regarding $M$ as the conﬁguration space for an m-body system in $ℝ^{d}$ , in analogy with the ”physical world” case $d = 3$ described in Section 1, $G_{1} = O (d)$ and $G_{2} = O (m)$ play the roles of congruence and internal symmetry groups respectively. Then there is the following two-step orbital decomposition of stratiﬁed Riemannian spaces

M \to \bar{M} = \frac{M}{G_{1}} \to \frac{\bar{M}}{G_{2}} = \frac{M}{G}

(63)

where the internal conﬁguration space $\bar{M} = \bar{M} (d, m)$ and its symmetry group constitute a natural equivariant system $(G_{2}, \bar{M})$ . Again, the metric (62) on $\bar{M}$ will be referred to as the kinematic metric, since in the physical case $d \leq 3$ it is, in fact, representing the internal kinetic energy

\bar{T} = \frac{1}{2} d {\bar{s}}^{2} = T - T^{ω} = \frac{1}{2} d s^{2} - T^{ω},

namely the total kinetic energy $T$ minus the purely rotational kinetic energy $T^{ω}$ of the $n$ -body system, see e.g. [30].

The internal symmetry orbit space $\bar{M} ∕ G_{2}$ , which is a cone in $ℝ^{μ}$ , $μ = min \{d, m\}$ , is actually a Weyl chamber of Cartan type $B_{μ}$ . Indeed, the $G$ -representation on $M$ is the isotropy representation of the symmetric space $O (d + m) ∕ O (d) \times O (m)$ , which is of type $B_{μ}$ , and as such it is well understood in terms of the classical invariant theory of matrices and canonical forms modulo orthogonal transformation groups.

Let $M^{1}$ be the unit sphere of $M .$ In view of the cone structure (61), (62), we may faithfully replace (63) by its unit distance ”sphere” version

M^{1} \to M^{*} = \frac{M^{1}}{G_{1}} \to \frac{M^{*}}{G_{2}} = \frac{M^{1}}{G} = M^{* *}

(64)

where $M^{*} = M^{*} (d, m)$ will be referred to as the shape space. Its $G_{2}$ -orbit space $M^{* *}$ is a spherical simplex and it is explicitly described in (85) below.

The orbit types of the above transformation groups will be described in detail. In principle, the topological structure of the shape space $M^{*}$ can be reconstructed and analyzed as the union of well understood orbit bundles, but in this paper we shall only describe the global topology of $M^{*}$ in those few cases where it is a manifold. In particular, the cases $d = m - 1$ or $d = m$ are rather simple since $(G_{2}, M^{*})$ is essentially isomorphic to a linear model, that is, a sphere $S^{N}$ or disk $D^{N + 1}$ with an orthogonal action of $G_{2}$ , see Section 3.4.3. Before turning to these topics, let us have a closer look at some important classical matrix spaces and their natural transformation groups.

3.2. Transformations of the space of symmetric matrices. For $d \leq m$ , consider the following sets of $m \times m$ -matrices

\begin{matrix} S y m^{+} {(m)}_{\leq d} & \subset & S y m^{+} (m) & \subset & S y m (m) \subset M (m, m) \\ \cup & \cup & \cup \\ B^{+} (d) ≃ B^{+} {(m)}_{\leq d} & \subset & B^{+} (m) & \subset & B (m) \subset D i a g (m) ≃ ℝ^{m} \end{matrix}

(65)

where in the ﬁrst row $S y m (m)$ is the space of symmetric matrices, $S y m^{+} (m)$ is the convex cone of positive semideﬁnite matrices, and $S y m^{+} {(m)}_{\leq d}$ is the subcone of matrices of rank at most $d .$ In the second row $D i a g (m)$ is the space of diagonal matrices $d i a g (λ_{1}, . . ., λ_{m})$ , containing the cone

B (m) = \{Y = d i a g (λ_{1}, λ_{2}, . . ., λ_{m}); λ_{1} \geq λ_{2} \geq . .. \geq λ_{m}\},

(66)

and the subcones $B^{+} (m)$ and $B^{+} {(m)}_{\leq d}$ are given by the additional constraint $λ_{m}$ $\geq 0$ , and $λ_{i} = 0$ for all $i > d$ , respectively.

With the usual inner product on $M (m, m)$

Y_{1} \cdot Y_{2} = t r a c e (Y_{1}^{t} Y_{2}),     cf. (14)

where $Y^{t}$ means the transpose of $Y$ , $O (m)$ acts orthogonally by conjugation

(ϕ, Y) \to ϕ Y ϕ^{- 1}, ϕ \in O (m), Y \in M (m, m)

(67)

and the subsets in the ﬁrst row of (65) are clearly invariant. Thus the transformation group $(O (m), S y m (m))$ is the second symmetric tensor product $S^{2} ρ_{m}$ of the standard representation $ρ_{m} =$ $(O (m), ℝ^{m})$ , and let us recall the classical result about diagonalization of symmetric matrices. Namely, every symmetric matrix is conjugate to a unique matrix - its canonical form - in the cone $B (m)$ , which is therefore a fundamental domain (or cross section) of the orbit map

S y m (m) \to \frac{S y m (m)}{O (m)} ≃ \frac{ℝ^{m}}{S_{m}} ≃ B (m) \subset S y m (m)

(68)

As indicated, $B (m)$ is also a fundamental domain (or cross section) for $(S_{m}, ℝ^{m})$ , where $S_{m}$ is the symmetric group acting on $ℝ^{m}$ by permuting the coordinates $λ_{i} .$

Fundamental domains for the $O (m)$ -action restricted to invariant subspaces of $S y m (m)$ are the corresponding subsets of $B (m)$ in the second row of (65), consequently

B^{+} (m) ≃ \frac{ℝ^{m}}{B_{m}} ≃ \frac{S y m^{+} (m)}{O (m)} \supset \frac{S y m^{+} {(m)}_{\leq d}}{O (m)} ≃ \frac{ℝ^{d}}{B_{d}} ≃ B^{+} (d)

(69)

As indicated, we recognize $B^{+} (k)$ as a Weyl chamber for the compact connected Lie group $S O (2 k + 1)$ , namely a fundamental domain of the canonical representation of the Weyl group $B_{k} \supset S_{k}$ acting on $ℝ^{k}$ as a group generated by reﬂections.

Note that $S y m (m)$ is not an irreducible $O (m)$ -space. In fact, there is the orthogonal and invariant decomposition

S y m (m) = S y m^{0} (m) \oplus ℝ^{1}

(70)

where the trivial summand $ℝ^{1}$ is spanned by the identity matrix, and the ﬁrst summand, which consists of matrices of trace zero, gives the irreducible representation $S^{2} ρ_{m} - 1$ with fundamental domain $B^{0} (m)$ $\subset B (m)$ deﬁned by $\sum λ_{i} = 0 :$

B^{0} (m) ≃ \frac{ℝ^{m - 1}}{S_{m}} ≃ \frac{S y m^{0} (m)}{O (m)}

(71)

Here we have also recognized $B^{0} (m)$ as a fundamental domain of the canonical representation of the Weyl group $A_{m - 1} = S_{m}$ of $S U (m) .$

In addition to its Euclidean norm where $|Y| = \sqrt{t r a c e (Y^{2})}$ , the positive cone $S y m^{+} (m)$ has another $O (m)$ -invariant size function $ρ^{*} (Y) = t r a c e (Y)$ , with corresponding ”unit sphere”

S y m^{*} (m) = \{Y \in S y m^{+} (m); t r a c e (Y) = 1\}

(72)

For later reference, we note that its intersection with the Euclidean unit sphere in $S y m (m)$ is the subset

S y m^{*} (m) \cap S y m {(m)}^{1} = S y m^{*} {(m)}_{1} = \{Y \in S y m^{*} (m); r k (Y) = 1\}

(73)

Let $Y \in B (m) .$ The tangent space at $Y$ of the $O (m)$ -orbit through $Y$ in $S y m (m)$ is the subspace

[s o (m), Y] = \{S Y - Y S; S is skew-symmetric\} \subset S y m (m),

and it is perpendicular to $B (m)$ since the inner product with any $Y^{^{'}} \in B (m)$ is

(S Y - Y S) \cdot Y^{'} = t r a c e ((S Y - Y S) Y^{'}) = 0

A direct consequence of this observation is as follows :

Remark 3.1. The fundamental domain $B (m)$ of $(O (m), S y m (m))$ is an orthogonal cross section, in the sense that it is perpendicular to the $O (m)$ -orbits. Therefore, the orbit space of $S y m (m)$ (with the orbital distance metric) is isometric to $B (m) .$ Consequently, orbit spaces of various $O (m)$ - invariant subspaces of $S y m (m)$ in (65) are isometric to the corresponding subsets of $B (m) .$

Finally, let us determine the orbit types of the $O (m)$ -action on symmetric matrices. To this end, consider a matrix $Y \in B (m)$ and let the ”strings” of equal entries $λ_{i}$ in (66) have length $m_{1}, m_{2}, . . .,$ so that

m = m_{1} + m_{2} + . . . + m_{p} (m_{i} > 0, p > 0)

is a partition of $m$ and $p$ is the number of diﬀerent $λ_{i}^{'} s$ . It is easy to see that the isotropy group at $Y$ is the ”block” orthogonal matrix group

O (m_{1}, m_{2}, . . ., m_{p}) = O (m_{1}) \times O (m_{2}) \times . . . \times O (m_{p}) \subset O (m)

(74)

and therefore the partition function $π (m)$ of $m$ enumerates the diﬀerent orbit types. The orbits are in fact connected and therefore they are also $S O (m)$ -orbits, that is,

\frac{S y m (m)}{S O (m)} = \frac{S y m (m)}{O (m)} ≃ B (m)

3.3. Algebraic realization of orbit spaces and orbital stratiﬁcation. With the results from the previous subsection as a basis, the construction of the orbit spaces in (63) or (64) as semialgebraic subsets is based upon the following two simple but fundamental properties of matrices :

The polynomial map
$𝔭 : M (d, m) \to S y m (m), X \to Y = X^{t} X$ (75)

preserves the matrix rank, and its image consists of all positive semideﬁnite symmetric matrices $Y$ of rank $\leq min \{d, m\} .$ Moreover, for $ψ \in O (d), ϕ \in O (m),$
$𝔭 (ψ X ϕ^{- 1}) = ϕ 𝔭 (X) ϕ^{- 1} = ϕ Y ϕ^{- 1}$ (76)

and the entries of $Y$ are the inner products $x_{i} \cdot x_{j}$ of the column vectors $x_{i}$ of $X$ .
Consider a ﬁnite collection of vectors $x_{j}$ in a Euclidean space $ℝ^{d} .$ Then the collection is uniquely determined, up to $O (d)$ -congruence, by the inner products $x_{i} \cdot x_{j}$ . For a proof we refer to Weyl [31], page 52.

The map $𝔭$ in (75) is clearly constant on each $O (d)$ -orbit in $M = M (d, m)$ and induces a surjective map

\bar{𝔭} : \bar{M} (d, m) \to S y m^{+} {(m)}_{\leq d} \subset S y m (m)

(77)

which is, moreover, injective by the above mentioned result of Weyl. In other words, the components of $𝔭$ , namely the collection of inner products $x_{i} \cdot x_{j}$ , constitute a Hilbert basis in the sense of Section 3.1.1. Consequently, $\bar{𝔭}$ is a diﬀeomorphism between diﬀerential spaces which identiﬁes $\bar{M} = M ∕ O (m)$ with the set of positive semideﬁnite symmetric $m \times m$ -matrices of rank at most $d .$ In particular,

\bar{M} (d, m) = \bar{M} (m, m) ≃ S y m^{+} (m) if d \geq m,

(78)

so we will henceforth assume without loss of generality that $d \leq m .$ Thus we have explained why all orbit spaces involved are realizable as appropriate spaces of symmetric matrices, amenable to the setting in Section 3.2.

3.3.1. Cross sections and canonical forms. In general, the inclusions $M (k, m) \subset M (k + 1, m)$ , deﬁned by taking the last row to be zero, induce the increasing ﬁltration

\bar{M} (1, m) \subset \bar{M} (2, m) \subset . . . \subset \bar{M} (m, m) ≃ S y m^{+} (m)

(79)

and this coincides with the matrix rank ﬁltration of the positive cone $S y m^{+} (m)$ , namely for $k + i \leq m$

\bar{M} (k, m) = \bar{M} {(k + i, m)}_{\leq k} ≃ S y m^{+} {(m)}_{\leq k}

On the other hand, $G_{2} = O (m)$ acts on the spaces in (77). The action on $\bar{M} = \bar{M} (d, m)$ is induced from the action (49), and $G_{2}$ acts on $S y m (m)$ by conjugation (67). From the equivariance property (76) it follows that these actions commute with the map $\bar{𝔭}$ , that is, $\bar{𝔭}$ is an isomorphism of $G_{2}$ -spaces and hence induces a diﬀeomorphism between $G_{2}$ -orbit spaces

\tilde{𝔭} : \frac{M}{G} = \frac{\bar{M}}{G_{2}} \to \frac{S y m^{+} {(m)}_{\leq d}}{G_{2}} ≃ B^{+} (d) \subset S y m^{+} {(m)}_{\leq d} ≃ \bar{M}

(80)

The cross section $B^{+} (d) \subset$ $\bar{M}$ in (80), being transversal to the $G_{2}$ -orbits in $\bar{M}$ , further lifts to a cross section $M^{+} (d)$ of the composite orbit map $M \to M ∕ G,$ as follows :

B^{+} (d) ≃ M^{+} (d) \subset M : \{\begin{matrix} X = d i a g (r_{1}, r_{2}, . . ., r_{d}) \\ r_{1} \geq r_{2} \geq . . . \geq r_{d} \geq 0 \end{matrix}

(81)

where $X = (x_{i j})$ has $x_{i i} =$ $r_{i}$ and zero entries otherwise. Thus we have also established the diagonalization procedure saying that every matrix in $M = M (d, m)$ can be transformed by the action (49) to a unique matrix - its canonical form - in the subset $M^{+} (d)$ . As a (simplicial) Euclidean cone in the $\{r_{i}\}$ -coordinate space $ℝ^{d}$ , $M^{+} (d)$ is also the Weyl chamber of the Weyl group $B_{d}$ , cf. (69).

3.3.2. Orthogonality of cross sections. In view of Remark 3.1, the cone $M^{+} (d)$ in (81) is orthogonal to the $G$ -orbits and is therefore an orthogonal cross section for $(G, M)$ . Hence, we have an isometry $M ∕ G ≃ M^{+} (d)$ of Euclidean cones with the metric

d s^{2} = \sum_{i = 1}^{d} d r_{i}^{2} = \frac{1}{4} \sum_{i = 1}^{d} \frac{1}{λ_{i}} d λ_{i}^{2}

(82)

Here $\{r_{i}\}$ from (81) and $\{λ_{i}\}$ from (66) are coordinate systems for the cone and are related by the polynomial map $𝔭$ in (75), which restricts to a diﬀeomorphism

𝔭 : M^{+} (d) \to B^{+} (d) \subset S y m^{+} (m), r_{i} \to r_{i}^{2} = λ_{i}

(83)

Thus, the $G$ -orbit of a given matrix $X \in M = M (d, m)$ has coordinates $λ_{i}$ interpreted as the eigenvalues of $X^{t} X$ . In the special case that $d = m$ and $X$ is symmetric, the numbers $r_{i}$ are the absolute values of the eigenvalues of $X .$

Remark 3.2. At this point, observe that the internal conﬁguration space $\bar{M} = \bar{M} (d, m)$ has two metrics, namely the kinematic metric (i.e. the $O (d)$ -orbital distance metric) and the induced ”fake” metric as a subset $S y m^{+} {(m)}_{\leq d}$ of the Euclidean space $S y m (m)$ , cf. (77). For both metrics the symmetry group $O (m)$ is actually an isometric transformation group with $B^{+} (d)$ as a fundamental domain (i.e. cross section). In the ”fake” metric $B^{+} (d)$ inherits the Euclidean metric $\sum d λ_{i}^{2}$ and is by Remark 3.1 an orthogonal cross section, whereas in the kinematic (but still Euclidean) metric (82) of $\bar{M}$ the orthogonality property of $B^{+} (d)$ fails. See also Remark 3.4 below.

3.3.3. Rank and subrank stratiﬁcation of matrix spaces. The subrank stratiﬁcation of the matrix space $M = M (d, m)$ is a natural reﬁnement of the usual rank stratiﬁcation. The latter coincides with the orbit type stratiﬁcation of the action of $G_{1}$ (or $G_{2}$ ) on $M$ and the mentioned reﬁnement coincides with the orbit type stratiﬁcation of the action of the full group

G = G_{1} \times G_{2} = O (d) \times O (m)

In order to describe the combinatorial structure involved we turn to the two-step orbital decomposition (64), where the ﬁrst orbit space is the shape space

M^{*} = M^{*} (d, m) ≃ S y m^{*} {(m)}_{\leq d}, cf. (72)

(84)

and the ﬁnal $G$ -orbit space is

M^{* *} = M^{* *} (d, m) = M^{*} ∕ G_{2} = M^{1} ∕ G

which we may identify with the following fundamental domain in the sphere $M^{1}$ :

Δ^{d - 1} = M^{+} (d) \cap M^{1} : r_{1} \geq r_{2} \geq . . . \geq r_{d} \geq 0, \sum r_{i}^{2} = 1

(85)

This is a spherical simplex whose structure, indeed, identiﬁes it with the spherical Weyl chamber of type $B_{d}$ (see (69)); in particular, it is homeomorphic to the closed disk $D^{d - 1}$ .

Thus we also conclude that the above simplex (85) is a cross section for the transformation group $(G, M^{1})$ , and hence any matrix in $M^{1}$ is $G$ -equivalent to a unique matrix - its canonical form - in the simplex. On the other hand, a typical matrix of rank $k$ in (85) may be characterized by the following numerical data

X : \{\begin{matrix} r_{1} = r_{k_{1}} > r_{k_{1} + 1} = r_{k_{1} + k_{2}} > r_{k_{1} + k_{2} + 1} = . . . . = r_{k} > 0 = r_{k + 1} \\ 1 \leq \sum_{i = 1}^{p} k_{i} = k \leq d, k_{i} > 0, p > 0, \end{matrix}

(86)

and then its subrank is deﬁned to be the corresponding (unordered) partition of $k$ into $p$ positive integers

κ = (k_{1}, k_{2}, . . ., k_{p}), |κ| = k_{1} + . . . + k_{p} = k

(87)

which records the length of the strings of equal numbers in (86).

We label the rank (resp. subrank) stratum by the subscript $k$ (resp. $κ$ ) of the corresponding sets. Thus $X$ belongs to $M_{k}^{1}$ if and only if its $G_{1}$ -orbit has the type $O (d - k)$ (cf. also (100) below). Moreover, by (85) the image of $M_{k}^{1}$ in $M^{* *}$ is the ”semi-open” simplicial disk

M_{k}^{* *} ≃ Δ_{k}^{k - 1} \subset Δ_{k}^{d - 1} : r_{1} \geq r_{2} \geq . . . \geq r_{k} > r_{k + 1} = 0, \sum r_{i}^{2} = 1,

which is subdivided into its various subrank strata

M_{κ}^{* *} ≃ Δ_{κ}^{k - 1} \subset Δ_{k}^{k - 1} \subset Δ^{d - 1} ≃ M^{* *}

(88)

In particular, the principal stratum has subrank $κ_{0} = (1, 1, . . ., 1)$ with $|κ_{0}| = d$ , and $M_{κ_{0}}^{* *} ≃ Δ_{κ_{0}}^{d - 1}$ is the interior of the spherical simplex $Δ^{d - 1}$ and is therefore homeomorphic to $ℝ^{d - 1}$ .

To see why the subrank stratiﬁcation actually coincides with the $G$ -orbit type stratiﬁcation, consider the isotropy group $G_{X}$ of the matrix in (86), using the following notation for groups :

\begin{align} Δ H & = d i a g (H \times H) ≃ H (diagonal embedding) \\ O (k_{1}, . ., k_{p}) & = O (k_{1}) \times . . . \times O (k_{p}) \subset O (k) & (89) \\ G (k_{1}, . ., k_{p}) & = O (d - k) \times Δ O (k_{1}, . ., k_{p}) \times O (m - k) \subset G \end{align}

Then it is not diﬃcult to show by direct calculations with matrices that

G_{X} = \{(g_{1}, g_{2}) \in O (d) \times O (m); g_{1} X g_{2}^{- 1} = X\} = G (k_{1}, . ., k_{p})

(90)

(cf. Table 1, #10, in [27]), and hence the isotropy types uniquely characterize the subrank strata, as claimed.

We are particularly interested in the stratiﬁcation of $M^{*}$ , but now there are two natural options, namely the induced subrank stratiﬁcation and the $G_{2}$ -orbit type stratiﬁcation. Our next claim, however, is that they are identical, and hence one can describe $M^{*}$ is a union of $G_{2}$ -orbit bundles lying over the various strata (88).

To calculate the $G_{2}$ -orbit types and how they correspond to the subrank strata we proceed as follows. Let $X^{*} \in M^{*}$ be the image of a matrix $X$ of rank $k$ and consider the ”large” orbit $G (X) \subset M$ . As a $G_{1}$ -space all the orbits of $G (X)$ have the same type as the ”small” orbit $G_{1} (X)$ . Moreover, the orbit space $G (X) ∕ G_{1} \subset M^{*}$ coincides with the $G_{2}$ -orbit through $X^{*}$ , namely the set $G_{2} (X^{*})$ . Thus, the ”large” $G$ -orbit through $X$ has the structure of a $G_{1}$ -orbit bundle, and in terms of homogeneous spaces (i.e. coset spaces of groups) this ﬁber bundle can be described as follows :

\begin{matrix} G_{1} (X) & \to & G (X) & \to & G_{2} (X^{*}) \\  &  &  \\ \frac{O (d)}{O (d - k)} & \to & \frac{G}{G (k_{1}, k_{2}, . ., k_{p})} & \to & \frac{O (m)}{O (k_{1}, k_{2}, . ., k_{p}) \times O (m - k)} \end{matrix}

(91)

The base space $G_{2} (X^{*})$ of the ﬁber bundle is in turn a ﬁber of another bundle, namely the multistratum $M_{κ}^{*}$ which as a $G_{2}$ -orbit bundle ﬁbers as follows

\frac{O (m)}{O (k_{1}, k_{2}, . ., k_{p}) \times O (m - k)} \to M_{κ}^{*} \to M_{κ}^{* *} ≃ Δ_{κ}^{k - 1}

(92)

In this way the subrank stratiﬁcation of the shape space

M^{*} (d, m) = ⋃_{k = 1}^{d} M_{k}^{*} = ⋃_{k = 1}^{d} ⋃_{|κ| = k} M_{κ}^{*}

has strata which are ﬁber bundles of type (92), and there are altogether

π (1) + π (2) + . . . + π (d)

(93)

such bundles (or strata), where $π (k)$ is the partition function of $k .$ The principal stratum $M_{κ_{0}}^{*}$ constitutes an open and dense subset of $M^{*}$ , and the principal orbit bundle

\frac{O (m)}{O {(1)}^{d} \times O (m - d)} \to M_{κ_{0}}^{*} \to M_{κ_{0}}^{* *} \approx ℝ^{d - 1}

(94)

is actually trivial (e.g. since the base space is contractible). The orbit type, either for the $G_{2}$ -action on $M^{*}$ or the $G$ -action on $M^{1}$ , for a stratum of subrank $κ = (k_{1}, k_{2}, . . ., k_{p})$ can be read oﬀ from (91).

Example 3.3. $d = 3 :$ $M (3, n - 1) ≃ M_{n}$ is the centered conﬁguration space for $n$ -body systems in 3-space. See Section 3.4.4 for a brief description of the shape spaces $M_{n}^{*}$ when $n$ is small. In general, for $n > 3$ their $G_{2}$ -orbit space

\frac{M_{n}^{*}}{G_{2}} = M^{* *} (3, n - 1) ≃ Δ^{2}

is the spherical triangle on the sphere $r_{1}^{2} + r_{2}^{2} + r_{3}^{2} = 1$ with vertices

A = (1, 0, 0), B = (1 ∕ \sqrt{2}, 1 ∕ \sqrt{2}, 0), C = (1 ∕ \sqrt{3}, 1 ∕ \sqrt{3}, 1 ∕ \sqrt{3})

On the circle $r_{3} = 0$ lies the closed circular arc of length $π ∕ 4$

[A, B] : r_{1} \geq r_{2} \geq 0

which consists of those points of rank $\leq 2$ . It has three strata, namely the vertex $A$ (of rank 1), the vertex $B$ and the open arc $(A, B)$ (both of rank 2). However, the open arcs

(A, C) : r_{1} > r_{2} = r_{3} > 0, (B, C) : r_{1} = r_{2} > r_{3} > 0

constitute the same stratum of subrank $κ = (2, 1)$ . Thus, although the triangle decomposes into $7 = 3 + 3 + 1$ strata components, namely vertices, open edges and the interior of $Δ^{2}$ , there are only 6 multistrata (since one of them has $2$ components), in accordance with the enumeration formula (93).

Remark 3.4. In general, the kinematic metric on the shape space $M^{*}$ is uniquely determined by its restriction to the principal stratum, which is a Riemannian manifold $(M_{κ_{0}}^{*}, d σ^{2})$ . There is a general procedure, in the setting of equivariant diﬀerential geometry, for the calculation (or description) of the Riemannian connection on such a principal orbit bundle, see e.g. [4]. The simplest case $d = 3$ , as in the above example, is analyzed in Hsiang [9], but similar calculations can also be done for $d > 3$ . We shall, however, leave the geometric issues here and concentrate on the topological structures in the sequel.

3.4. Topology of the shape space and related spaces. For ﬁxed $m$ the largest shape space (with $d = m$ ) is

M^{*} (m, m) ≃ S y m^{*} (m)

(95)

and by (84) it contains any other shape space

M^{*} (d, m) = M^{*} {(m, m)}_{\leq d} ≃ S y m^{*} {(m)}_{\leq d}

as the union of those strata of matrix rank at most $d$ . Therefore, we shall refer to the space (95) as the m-universal shape space. Thus there is the increasing and $O (m)$ -invariant rank ﬁltration

M^{*} (1, m) \subset M^{*} (2, m) \subset . . . \subset M^{*} (m, m)

(96)

and corresponding $O (m)$ -orbit spaces $\approx Δ^{d - 1},$ $1 \leq d \leq m,$

M^{* *} (1, m) \subset M^{* *} (2, m) \subset . . . \subset M^{* *} (m, m) = Δ^{m - 1}

(97)

where the ﬁrst three spaces in the chain (97) are, indeed, the inclusions of simplices

\{A\} = Δ^{0} \subset [A, B] = Δ^{1} \subset [A, B, C] = Δ^{2}

from Example 3.3.

By inspection the chain in (96) starts with

\begin{align} M^{*} (1, m) & = S^{m - 1} ∕ O (1) = ℝ P^{m - 1} \\ M^{*} (2, m) & = S^{2 m - 1} ∕ O (2) = ℂ P^{m - 1} ∕ ℤ_{2} & (98) \end{align}

where

ℝ P^{m - 1} \subset ℂ P^{m - 1} = S^{2 m - 1} ∕ S O (2)

are the real and complex the projective (m-1)-space, and $ℤ_{2} = O (2) ∕ S O (2)$ acts on $ℂ P^{m - 1}$ by complex conjugation with $ℝ P^{m - 1}$ as ﬁxed point set. In particular, the vertex $Δ^{0} = M^{* *} (1, m)$ of the simplex $Δ^{m - 1}$ in (97) is the single $O (m)$ -orbit $M^{*} (1, m)$ , with the topology of a real projective space as noted above.

The topology of the next shape space $M^{*} (2, m)$ is more diﬃcult to describe. One approach is to utilize the fact that it is a $O (m)$ -space of cohomogeneity one, that is, with one-dimensional orbit space. This holds since

\frac{M^{*} (2, m)}{O (m)} = M^{* *} (2, m) ≃ Δ^{1} = [A, B]

is an interval. We also know the orbit types $(K_{1}), (K_{2}), (H)$ corresponding to the three strata $A, B$ and $(A, B)$ respectively. Therefore, the space can be described topologically in terms of its orbit types by the construction (cf. e.g. [29], Vol.1, Chap. IV) known as the equivariant union

M^{*} (2, m) = M (π_{1}) \cup M (π_{2})

of the mapping cylinders $M (π_{i})$ of the canonical projections $π_{i} : O (m) ∕ H \to O (m) ∕ K_{i}$ , where in our case

\begin{align} K_{1} & = O (1) \times O (m - 1), K_{2} = O (2) \times O (m - 2), & (99) \\ H & = K_{1} \cap K_{2} = O {(1)}^{2} \times O (m - 2) \end{align}

The space $M^{*} (2, 2)$ is a 2-disk and $M^{*} (2, 3)$ is actually homeomorphic to $S^{4}$ , see (104) and Section 3.4.4). For $m > 3$ $M^{*} (2, m)$ fails to be a manifold in a neighborhood of the singular $O (m)$ -orbit

A = O (m) ∕ K_{1} = M^{*} (1, m) ≃ ℝ P^{m - 1},

but two copies of $M^{*} (2, m)$ glued together along $A$ yields, indeed, the diﬀerentiable manifold $S^{2 m - 1} ∕ S O (2) = ℂ P^{m - 1}$ . Below we will return to this construction and further investigate the topology of the spaces $M^{*} (d, m)$ for $2 < d \leq m .$

3.4.1. Local and global topology. By referring to Section 3.1 concerning compact transformation groups and the slice theorem, let us ﬁrst recall a nice property of regular representations of classical groups such as $O (d)$ . Namely, we consider the linear action of $O (d)$ on $ℝ^{N}$ by some representation of type $Φ = m ρ_{d} + τ_{q}$ , $N = m d + q$ , which is $m$ copies of the standard representation $ρ_{d}$ plus a $q$ -dimensional trivial summand $τ_{q}$ . Then the orbit types constitute the following ”connected” string of regular subgroups

O (k); d - m \leq k \leq d

(100)

where $O (k) = 1$ if $k \leq 0$ . To calculate the slice representation $Φ_{k}$ at a point with isotropy group $O (k)$ we calculate the diﬀerence as in (56), with the local representation equal to $Φ$ restricted to $O (k)$ , that is, $m ρ_{k} + (t r i v i a l)$ . Moreover, the isotropy representation $I s o (d, k)$ of $O (k)$ is the ”linearized” action on the Stiefel manifold $O (d) ∕ O (k) = S O (d) ∕ S O (k)$ at the base point, and a simple calculation (involving the adjoint representation of $O (d)$ ) yields the representation $(d - k) ρ_{k} + (t r i v i a l)$ , consequently

Φ_{k} = Φ ∣_{O (k)} - I s o (d, k) \equiv (m - d + k) ρ_{k} (mod trivial )

(101)

From this we conclude that each slice representation inherits the regularity property of the original action $Φ$ .

Now we turn to the ”left side” action of $O (d)$ on the matrix space $M = M (d, m) ≃ ℝ^{m d}$ , which is just the regular representation $Φ = m ρ_{d}$ . However, we shall rather consider the restricted action on the unit sphere $M^{1}$ , noting that the only change in the above calculations is that the local representations lose a trivial summand $τ_{1}$ . The orbit types still constitute the string (100) and also (101) holds, except that $k < d$ in (100) since $O (d)$ has no ﬁxed point on the sphere $M^{1}$ .

Lemma 3.5. (i) The m-universal shape space $M^{*} (m, m)$ $≃ S y m^{*} (m)$ is a manifold with boundary

\partial M^{*} (m, m) = M^{*} (m - 1, m) ≃ S y m^{*} {(m)}_{\leq m - 1}

(ii) For $p > 0,$ $M^{*} (d, d + p)$ is a manifold (and with no boundary) if and only if $p = 1$ or $d = 1$ .

Proof. By (57), the local topology of $M^{*} (m, m)$ around an $O (m)$ -orbit of type $O (k)$ is the topology of the orbit space of the slice representation $(O (k), Φ_{k})$ in $M^{1}$ , with $m = d$ in (101). Hence, part (i) of the lemma follows by induction on $m \geq 1$ , since at the initial step $m = 1$ the orbit space of $(O (1), ρ_{1})$ is the half-line $ℝ_{+}^{1} = [0, \infty) .$

For the proof of (ii) we may assume $d > 1$ , and the slice representation (101) of $O (k)$ is $Φ_{k} = (k + p) ρ_{k}$ (mod trivial). By induction the proof reduces to the crucial singular case $(O (1), (1 + p) ρ_{1})$ , where $O (1)$ acts by inversion $v \to - v,$ and here the orbit space is the cone $\frac{ℝ^{p + 1}}{O (1)} = C (ℝ P^{p})$ over the real projective p-dimensional space.This is a manifold if and only if $p = 1$ , in which case the orbit space is homeomorphic to the Euclidean plane

\frac{ℝ^{2}}{O (1)} = C (ℝ P^{1}) \approx C (S^{1}) = ℝ^{2}

(102)

Next, let us actually determine the topological type of the compact manifold $M^{*} (m, m) ≃ S y m^{*} (m)$ and its boundary. By the $O (m)$ -action on $M (m, m)$ every $X = (x_{i j})$ can be mapped to the subset

T^{+} (m) \subset M (m, m)

of upper triangular matrices with diagonal entries $x_{i i} \geq 0,$ and therefore the restriction of the polynomial map in (75)

𝔭 : T^{+} (m) \to S y m^{+} (m)

(103)

is still surjective. The subset $T^{+} {(m)}_{m}$ of matrices $X$ with all $x_{i i} > 0$ , i.e. of maximal rank $m$ , is clearly diﬀeomorphic to the Euclidean space $ℝ^{m (m + 1) ∕ 2}$ , and it is easy to verify that diﬀerent matrices $X$ lie on diﬀerent $O (m)$ -orbits. Therefore the following lemma must hold.

Lemma 3.6. The polynomial map in (103) further restricts to a diﬀeomorphism

𝔭 : T^{+} {(m)}_{m} ≃ S y m^{+} {(m)}_{m} ≃ ℝ^{m (m + 1) ∕ 2}

with the set of positive deﬁnite symmetric matrices. Hence, the interior of $S y m^{*} (m)$ is the open ”sphere-octant”

S y m^{*} {(m)}_{m} ≃ T^{+} {(m)}_{m} \cap M^{1} (m, m) : \sum x_{i j}^{2} = 1, x_{i i} > 0

which is diﬀeomorphic to $ℝ^{m (m + 1) ∕ 2 - 1}$ .

We know from the previous two lemmas that $S y m^{*} (m)$ is a compact manifold with boundary and its interior is an open disk. Using some manifold theory this information actually suﬃces to conclude that $S y m^{*} (m)$ is a closed disk. For example, one may apply the so-called collar neighborhood theorem and the h-cobordism theorem, cf. Milnor [22]). Anyhow, we have the topological types

\begin{align} M^{*} (m, m) & ≃ S y m^{*} (m) \approx D^{m (m + 1) ∕ 2 - 1} & (104) \\ M^{*} (m - 1, m) & ≃ S y m^{*} {(m)}_{\leq m - 1} \approx S^{m (m + 1) ∕ 2 - 2} \end{align}

and taking the cone over these spaces yields the topological types

\begin{array}{l} \bar{M} (m, m) & \approx ℝ^{m (m + 1) ∕ 2 - 1} \times [0, \infty) (half-space) \\ \bar{M} (m - 1, m) & ≃ \partial \bar{M} (m, m) \approx ℝ^{m (m + 1) ∕ 2 - 1} (Euclidean space) \end{array}

Remark 3.7. The linear model construction in Section 3.4.3 provides another proof of the homeomorphisms (104).

3.4.2. Branched coverings and generalized Hopf ﬁbrations. The usage of $S O (d)$ rather than $O (d)$ as the congruence group acting on $M (d, m)$ may lead to a diﬀerent ”shape space”, and we shall explain this distinction below. First, observe that for $k \geq 1$ we have connected and hence equal orbits

O (d) ∕ O (k) ≃ S O (d) ∕ S O (k)

and consequently the orbit spaces of $S O (d)$ and $O (d)$ will be identical as long as $d > m,$ see (100). However, due to (78) we have been assuming $d \leq m$ and therefore $k = 0$ also occurs in the string (100), so the two orbit spaces cannot be identical. In order to explain the diﬀerence the following discussion will be helpful.

Consider a space $\tilde{Q}$ with a given involution $σ$ (i.e. a transformation of order two) and hence we shall regard $ℤ_{2} = \{I d, σ\}$ as a transformation group on $\tilde{Q}$ . We denote the orbit space of $ℤ_{2}$ by $Q$ and the ﬁxed point set by $Σ$ . We also assume $Σ \neq \emptyset$ and regard it as a subset of both $\tilde{Q}$ and $Q$ . Then the orbit map

π : \tilde{Q} \to \frac{\tilde{Q}}{ℤ_{2}} = Q

(105)

is an example of a ramiﬁed double covering which is ramiﬁed along $Σ$ . Conversely, starting from $Q$ and $Σ$ we may reconstruct $\tilde{Q}$ as the double of $Q$

2 Q = Q \cup_{Σ} Q ≃ \tilde{Q}

by taking two copies of $Q$ and identify (or glue together) their ”singular” set $Σ$ . However, the pair $(Q, Σ)$ does not always lead to a unique double space, so we shall rather have in mind a speciﬁed ramiﬁed covering as in (105). Thus, we deﬁne for $d \leq m$

2 M^{*} (d, m) = M^{*} (d, m) \cup_{M^{*} (d - 1, m)} M^{*} (d, m)

(106)

and refer to the orbit map

2 M^{*} (d, m) ≃ \frac{M^{1} {(d, m)}_{}}{S O (d)} \to \frac{M^{1} {(d, m)}_{}}{O (d)} = M^{*} (d, m)

(107)

of the action of $O (d) ∕ S O (d) =$ $ℤ_{2}$ , which is, indeed, a double covering ramiﬁed along the ﬁxed point set $M^{*} (d - 1, m)$ of $ℤ_{2}$ .

Lemma 3.8. For $p \geq 0$ and $d > 1$ , the space $2 M^{*} (d, d + p)$ is a manifold (and with no boundary) if and only if $p = 0$ or $d = 2$ .

Proof. For $d = 2$ the $S O (2)$ -orbit space is the complex projective space $ℂ P^{p + 1}$ , so let us assume $d > 2$ . Using again the slice theorem and an inductive argument, the crucial case will be the topology of the orbit space of $(S O (2), (p + 2) ρ_{2})$ , namely the cone

\frac{ℝ^{2 (2 + p)}}{S O (2)} = C (\frac{S^{2 (2 + p) - 1}}{S O (2)}) = C (ℂ P^{p + 1})

over the complex projective space $ℂ P^{p + 1}$ . It is well known that this is a manifold (i.e. locally Euclidean) if and only if $p = 0$ , in which case it is a cone homeomorphic to Euclidean 3-space

C (ℂ P^{1}) \approx C (S^{2}) = ℝ^{3}

(108)

Lemma 3.9. The induced smooth functional structure on the cone (102) (resp. (108)) is a reﬁnement of the standard diﬀerentiable structure of $ℝ^{2}$ (resp. $ℝ^{3}$ ). The structures are identical away from the origin (cone vertex), where the cone fails to be a smooth manifold.

Proof. Consider the case (108), namely the orbit space of $(S O (2), 2 ρ_{2})$ acting on $ℝ^{2} \times ℝ^{2}$ . To describe it algebraically, let $u, v$ denote vectors in $ℝ^{2}$ and deﬁne the polynomial map

𝔭 = (X, Y, Z, W) : ℝ^{2} \times ℝ^{2} \to ℝ^{4}, see (60)

where $X = {|u|}^{2} - {|v|}^{2}$ , $Y = 2 u \cdot v$ , $Z = 2 (u \times v) \cdot k$ , $W = {|u|}^{2} + {|v|}^{2}$ , which identiﬁes the orbit space with the cone in 4-space

X^{2} + Y^{2} + Z^{2} = W^{2}, W \geq 0

By projecting the cone onto the coordinate $(X, Y, Z)$ -space we identify it with $ℝ^{3}$ as in (108), with (standard) coordinate functions $X, Y, Z$ which are certainly smooth in the induced orbital functional structure. However, at the origin the function $f = \sqrt{X^{2} + Y^{2} + Z^{2}}$ is not smooth with respect to the standard structure, although its composition with the orbit map is by our construction the smooth function $W .$ The case (102) is similar but simpler, using $𝔭 = (X, Y, W)$ and $(u, v) \in ℝ^{1} \times ℝ^{1} .$ _

According to (104) and (107) we have constructions of orbital ﬁbrations of spheres over spheres involving two types of orthogonal transformation groups, as follows :

:

Case 1 : Consider the regular representation

(O (m - 1), m ρ_{m - 1})

, acting on the unit sphere

M^{1} (m - 1, m) = S^{(m - 1) m - 1}

with orbit space as in the second line of (104),

S^{(m - 1) m - 1} \to \frac{S^{(m - 1) m - 1}}{O (m - 1)} \approx S^{m (m + 1) ∕ 2 - 2}, m = 2, 3, 4, . . .

(109)

Here the initial case $m = 2$ is the trivial Hopf ﬁbration or double covering

O (1) ↪ S^{1} \to^{π} ℝ P^{1} = \frac{S^{1}}{O (1)} ≃ S^{1}

(110)

:

Case 2 : Consider the regular representation

(S O (m), m ρ_{m})

, acting on the unit sphere

M^{1} (m, m) = S^{m^{2} - 1}

. The orbit space coincides with the ”double” construction (106) applied to the ﬁrst line of (104)

\frac{M^{1} (m, m)}{S O (m)} = 2 M^{*} (m, m) \approx 2 D^{m (m + 1) ∕ 2 - 1} = S^{m (m + 1) ∕ 2 - 1}

and there is the orbit map

S^{m^{2} - 1} \to \frac{S^{m^{2} - 1}}{S O (m)} \approx S^{m (m + 1) ∕ 2 - 1}, m = 2, 3, 4, . . . .

(111)

which for $m = 2$ is the well known Hopf ﬁbration

S O (2) ↪ S^{3} \to^{π} ℂ P^{1} = \frac{S^{3}}{S O (2)} ≃ S^{2}

(112)

Remark 3.10. With the induced diﬀerential structure the above quotient spheres are, of course, stratiﬁed diﬀerentiable manifolds (cf. Section 3.1) which are also locally Euclidean spaces since they are spheres in the topological sense. In fact, for $m = 2$ they are the standard sphere (i.e. $S^{1}$ or $S^{2}$ ) in the diﬀerentiable sense as well. However, for $m > 2$ there are more than one orbit type, and with the induced diﬀerential structure they are actually not diﬀerentiable manifolds. However, this artiﬁcial situation can be remedied by ”relaxing” the diﬀerential structure so that the quotient sphere becomes the standard sphere while the map $π$ is still diﬀerentiable, see Section 3.4.3.

Remark 3.11. The above orbital ﬁbrations are natural maps between spheres with speciﬁc properties, with possible future applications in physics. Indeed, the Hopf ﬁbration (112) is a simple special case with several well known applications. For example, it describes the geometry of a magnetic monopole, and Dirac made the major discovery that the ﬁbration could explain (in a modern language) the quantization of electric charge. Here $S^{3}$ is the unit sphere of $ℝ^{4} = ℂ^{2}$ , the action of $S O (2) ≃ U (1)$ extends to $ℂ^{2}$ by complex scalar multiplication, and the point magnetic source is at the origin of $ℂ^{2} ∕ U (1) \approx ℝ^{3}$ . The 4-space $ℝ^{4}$ in this setting is also referred to as the Kaluza-Klein model of the Dirac monopole.

Arnold [3] has considered the three special cases $m = 2, 3, 5$ of (109) from a diﬀerent viewpoint. Namely, they ﬁt into the following unifying pattern with projective spaces over $ℝ, ℂ$ or $ℍ$ as intermediate quotient spaces :

\begin{align} i) & S^{1} \to ℝ P^{1} ≃ S^{1} \\ ii) & S^{5} \to ℂ P^{2} \to ℂ P^{2} ∕ O (1) \approx S^{4} & (113) \\ iii) & S^{19} \to ℍ P^{4} \to ℍ P^{4} ∕ 〈A u t ℍ, ℤ_{2}〉 \approx S^{13} \end{align}

where $O (1) ≃ ℤ_{2}$ acts by (complex) conjugation and $A u t ℍ ≃ S O (3)$ is the automorphism group of $ℍ$ (quaternions).The last two cases are two-step orbital ﬁbrations deﬁned by groups $H \to K \to K ∕ H$ , that is, $K$ acts on the sphere on the left side in (113), and the intermediate (projective) space is the orbit space of the normal subgroup $H$ , with the induced action of $K ∕ H$ . The groups corresponding to the last two cases of (113) are as follows

ii) S O (2) \to O (2) \to O (1) iii) S p (1) \to O (4) \to O (3)

where $O (3) = S O (3) \times ℤ_{2}$ $≃ 〈A u t ℍ, ℤ_{2}〉$ . However, from this viewpoint it is also natural to include the cases $m = 2, 4$ of (111), where there is no group $ℤ_{2}$ . Case $m = 2$ is the Hopf ﬁbration (112) and the new case $m = 4$ reads

S^{15} \to ℍ P^{3} \to ℍ P^{3} ∕ A u t ℍ \approx S^{9}

(114)

with the corresponding groups $S p (1) \to S O (4) \to S O (3)$ .

3.4.3. The m-universal linear model. We shall describe another property of the m-universal shape space $M^{*} (m, m) ≃ S y m^{*} (m),$ which by (104) is known to be a disk whose boundary sphere is $M^{*} (m - 1, m)$ . In fact, what is remarkable is that the equivariant system $(O (m), S y m^{*} (m))$ has a linear model, namely it ”resembles” closely a Euclidean disk with $O (m)$ acting as an orthogonal transformation group. To make this correspondence precise, let us ﬁrst inquire what is the appropriate linear model.

Recall from Section 3.2 that $O (m)$ acts orthogonally on the space $S y m (m)$ of symmetric matrices of dimension $m$ , namely by the symmetric tensor product representation $S^{2} ρ_{m}$ . Let $S y m^{0} (m)$ be the subspace of matrices of trace zero, where $O (m)$ acts by the irreducible representation $S^{2} ρ_{m} - 1$ , and let

D^{0} (m) \subset S y m^{0} (m) cf. (70)

be the unit disk centered at the origin. Then we make the following deﬁnition which will be justiﬁed below.

Deﬁnition 3.12. The above equivariant system $(O (m), D^{0} (m))$ is the linear model of $(O (m), M^{*} (m, m))$ , and the restriction to the boundary sphere $\partial D^{0} (m)$ is the linear model of $\partial M^{*} (m, m) = M^{*} (m - 1, m)$ .

Now, we shall construct a 1-1 correspondence between $S y m^{*} (m)$ and the disk $D^{0} (m)$ which is smooth and $O (m)$ -equivariant, that is, the map commutes with the action of $O (m)$ . To each matrix $Y \in S y m^{*} (m)$ , namely a positive semideﬁnite matrix with trace 1, we associate the matrix

Y_{0} = \frac{1}{r_{m}} (Y - \frac{1}{m} I d) \in S y m^{0} (m),   where r_{m} = \sqrt{1 - \frac{1}{m}},

and note that $Y_{0}$ is perpendicular to the identity $I d$ and

{|Y_{0}|}^{2} = \frac{1}{r_{m}^{2}} ({|Y|}^{2} - \frac{1}{m}) = \frac{1}{r_{m}^{2}} (t r a c e (Y^{2}) - \frac{1}{m}) \leq \frac{1}{r_{m}^{2}} (t r a c e {(Y)}^{2} - \frac{1}{m}) = 1

Therefore, the $O (m)$ -equivariant aﬃne transformation

S y m (m) \to S y m (m) : Y \to \frac{1}{r_{m}} (Y - \frac{1}{m} I d) = Y_{0}

restricts to an embedding

S y m^{*} (m) \to D^{0} (m), Y \to Y_{0}

(115)

between disks of the same dimension, which maps the geometric center $\frac{1}{m} I d$ to the origin, i.e., the center of $D^{0} (m)$ .

Thus, by a translation and homothety inside the Euclidean space $S y m (m)$ the convex subset $S y m^{*} (m)$ becomes, somehow, an ”inward” equivariant deformation of $D^{0} (m)$ , which by (73) ﬁxes the subset $S y m^{*} {(m)}_{1} ≃ ℝ P^{m - 1}$ lying on the boundary sphere $\partial D^{0} (m)$ . In the simplest case $m = 2$ we have more explicitly

S y m^{*} (2) = \{(\begin{matrix} u & v \\ v & 1 - u \end{matrix}); {(u - \frac{1}{2})}^{2} + v^{2} \leq \frac{1}{4}\} ≃ D^{2}

(116)

and the embedding (115) is actually a diﬀeomorphism.

However, the embedding (115) is not surjective for $m > 2$ , so let us explain how to further deform equivariantly to make the embedding ﬁll the whole unit disk. First of all, by the convexity of $S y m^{*} (m)$ it follows that each ray in $D^{0} (m)$ from the origin passes through a unique point of the embedded sphere $\partial S y m^{*} (m)$ . In particular, by an additional scaling we obtain a 1-1 correspondence between the boundary spheres of the two disks, and the ﬁnal composition

ψ_{m} : \partial S y m^{*} (m) \to \partial D^{0} (m) = S^{m (m + 1) ∕ 2 - 2}, Y \to Y_{0} \to \frac{1}{|Y_{0}|} Y_{0}

(117)

is certainly an $O (m)$ -equivariant and smooth homeomorphism.

On the other hand, the above map may be extended to the whole disk, as follows. First of all, each ray from the origin intersects the embedded disk $S y m^{*} (m)$ in a segment. So, let us stretch the segment outward along the ray until it has unit length, but with no stretching in a neighborhood of the origin. Moreover, the stretching must be speciﬁed by a function on the orbit space since $O (m)$ -equivalent segments must be stretched in the same way to make $ψ_{m}$ equivariant. Following this procedure we certainly obtain an $O (m)$ -equivariant and smooth homeomorphism

ψ_{m} : S y m^{*} (m) \to^{\approx} D^{0} (m)

(118)

which extends the map in (117). Such an equivariant and smooth homeomorphism is not unique, of course. The ambiguity lies in the group $D i f f_{O (m)} (D^{0} (m))$ consisting of all equivariant diﬀeomorphisms of the linear model. Yet, another construction of equivariant homeomorphisms like $ψ_{m}$ is described brieﬂy at the end of this subsection.

In summary, we arrive at the following result :

Theorem 3.13. There is an $O (m)$ -equivariant and diﬀerentiable homeomorphism $ψ_{m}$ from the m-universal shape space $M^{*} (m, m) ≃ S y m^{*} (m)$ to its linear model $D^{0} (m) ≃ D^{m (m + 1) ∕ 2 - 1}$ . For $m = 2$ this map is a diﬀeomorphism.

By referring to the following diagram

\begin{matrix} M^{1} (m, m) \\ ↓ π \\ ℝ & \leftarrow^{g} & S y m^{*} (m) & \to^{ψ_{m}} & D^{0} (m) \end{matrix}

(119)

let us explain the diﬀerential structure of $S y m^{*} (m)$ induced via the orbit map $π$ . Namely, the function $g$ is said to be smooth if the composed map $g \circ π$ is smooth. Moreover, smoothness of $ψ_{m}$ means the composed map $ψ_{m} \circ π$ is smooth, and therefore we actually know (by our construction) that $ψ_{m}$ is smooth.

On the other hand, for $m > 2$ there also exists a smooth map $g$ such that $g \circ ψ_{m}^{- 1}$ is not smooth, as a function on the Euclidean disk $D^{0} (m)$ , and consequently $ψ_{m}^{- 1}$ cannot be smooth. Here we also refer to the discussion at the end of Section 3.4.4, together with Lemma 3.9 and Remark 3.10. Brieﬂy, the diﬀerential structure on $S y m^{*} (m)$ induced via $π$ is a strict ”reﬁnement” of the standard structure which $S y m^{*} (m)$ inherits from the Euclidean disk (via $ψ_{m}$ ).

Remark 3.14. We have seen that the m-universal shape space $S y m^{*} (m)$ has two naturally induced smooth structures, namely induced via $π$ and $ψ_{m}^{- 1}$ respectively, and they are diﬀerent when $m > 2$ . In fact, $S y m^{*} (m)$ is not a smooth manifold in the ﬁrst case. But in both cases $π$ and $ψ_{m}$ are smooth maps, but the standard structure (induced via $ψ_{m}^{- 1}$ ) is the only one that makes $ψ_{m}$ an $O (m)$ -equivariant diﬀeomorphism.

Finally, let us turn to the natural simplicial structures of the orbit spaces of $S y m^{*} (m)$ and the disk $D^{0} (m)$ , namely the following two spherical simplices

\begin{align} \frac{S y m^{*} (m)}{O (m)} & ≃ Δ^{m - 1} = \frac{S^{m - 1}}{B_{m}} : r_{1} \geq r_{2} \geq . . . \geq r_{m} \geq 0, \sum r_{i}^{2} = 1 & (120) \\ \frac{D^{0} (m)}{O (m)} & ≃ {\bar{Δ}}^{m - 1} = \frac{D^{m - 1}}{S_{m}} : λ_{1} \geq λ_{2} \geq . . . \geq λ_{m}, \sum λ_{i} = 0, \sum λ_{i}^{2} \leq 1 \end{align}

Despite the above theorem, which implies that $ψ_{m}$ induces an $O (m)$ -orbit type strata preserving homeomorphism ${\bar{ψ}}_{m} : Δ^{m - 1} \to {\bar{Δ}}^{m - 1}$ , the two structures in (120) are conspicuously diﬀerent. The reason is that the simplicial structure in the ﬁrst case of reﬂects the induced subrank stratiﬁcation of $S y m^{*} (m)$ (cf. Section 3.3.3), namely the common reﬁnement of the rank and $O (m)$ -orbit type stratiﬁcation, whereas the simplicial structure in the second case is merely reﬂecting the pure $O (m)$ -orbital stratiﬁcation of the orthogonal transformation group $O (m)$ on $D^{0} (m)$ . Thus, by passing from $S y m^{*} (m)$ to its linear model $D^{0} (m)$ the notion of rank is seemingly lost. Therefore, let us also investigate how the rank strata can be recognized in the linear model itself.

Recall that $M^{*} (k, m) \subset S y m^{*} (m)$ is the subspace lying above $Δ^{k - 1} : r_{k + 1} = . . . = r_{m} = 0$ , and the rank $k$ stratum is deﬁned by the subset $Δ_{k}^{k - 1} \subset Δ^{k - 1} : r_{k} > 0$ . This stratum has all the $O (m)$ -orbit types

O (k_{1}, k_{2}, . ., k_{p}) \times O (m - k), \sum k_{i} = k, cf. (89)

(121)

labelled by the various subranks $κ = (k_{1}, . ., k_{p})$ which record the strings of equalities among the numbers $r_{i} > 0$ , see (86), (87).

On the other hand, in the linear disk model any pattern of equality strings among the $λ_{i}$ ’s corresponds in the same way to a tuple $κ^{'} = (k_{1}, . ., k_{p}, k_{p + 1})$ of positive integers, where $k_{1}$ is the number of $λ_{i}$ ’s in (120) equal to $λ_{1}$ , $k_{2}$ is the number of $λ_{i}$ ’s equal to $λ_{k_{1} + 1}$ etc., and the factor $O (m - k)$ in (121) is replaced by $O (k_{p + 1})$ . In any case, the last factor $O (m - k)$ (resp. $O (k_{p + 1})$ ) of the isotropy group (121) is no more special than the other factors $O (k_{i})$ , and this clearly explains why the rank is not determined by the $O (m)$ -orbit type.

We claim, however, the rank $k$ is determined in the linear model by the identity $k_{p + 1} = m - k$ provided the obvious condition $\sum λ_{i}^{2} = 1$ holds. The latter condition merely says $M^{*} (k, m)$ embeds as a subset of the sphere $\partial D^{0} (m)$ and hence away from the interior $D^{0} (m)$ of the disk. The interior is, of course, given by $\sum λ_{i}^{2} < 1$ and here the rank is $k = m$ . Moreover, by removing $k_{p + 1}$ from the tuple $κ^{'}$ we are left with the correct subrank tuple $κ$ . The above claim about $k$ will be settled below.

Any $O (m)$ -equivariant homeomorphism

ψ : S y m^{*} (m) \to^{\approx} D^{0} (m)

such as $ψ_{m},$ for example, induces an $O (m)$ -orbital strata preserving homeomorphism $\bar{ψ} : Δ^{m - 1} \to {\bar{Δ}}^{m - 1}$ . Conversely, let us see how to start from $\bar{ψ}$ and construct an appropriate lifting $ψ$ as above. The idea is to construct $\bar{ψ}$ as a map between fundamental domains with the $O (m)$ -isovariant property, that is, a point and its image point have the same isotropy group. Then $ψ$ will be the unique equivariant extension to all of $S y m^{*} (m)$ . To choose appropriate fundamental domains, ﬁrst observe that ${\bar{Δ}}^{m - 1} \subset D^{0} (m)$ is a fundamental domain if the numbers $λ_{i}$ in (120) are regarded as the entries of a diagonal matrix. Similarly, the following subset of diagonal matrices

{\tilde{Δ}}^{m - 1} \subset S y m^{*} (m) : λ_{1} \geq λ_{2} \geq . . . \geq λ_{m} \geq 0, \sum λ_{i} = 1

is a fundamental domain in $S y m^{*} (m)$ . Then the $O (m)$ -orbit map projection ${\tilde{Δ}}^{m - 1} \to$ $Δ^{m - 1}$ is just $λ_{i} \to \sqrt{λ_{i}} = r_{i}$ , see (83), and we note that $\partial {\tilde{Δ}}^{m - 1}$ $≃ {\tilde{Δ}}^{m - 2}$ is the subset with $λ_{m} = 0$ .

For example, the above equivariant homeomorphism $(118)$ between the boundary spheres corresponds to the $O (m)$ -isovariant map

\frac{S^{m - 2}}{B_{m - 1}} ≃ {\tilde{Δ}}^{m - 2} = \partial {\tilde{Δ}}^{m - 1} \to^{{\bar{ψ}}_{m}} \partial {\bar{Δ}}^{m - 1} ≃ \frac{S^{m - 2}}{S_{m}}

(122)

which sends $Y = d i a g (λ_{1}, . ., λ_{m - 1}, 0)$ to the matrix

\frac{1}{|Y_{0}|} Y_{0} = \frac{1}{λ} d i a g (λ_{1} - 1 ∕ m, . ., λ_{m - 1} - 1 ∕ m, - 1 ∕ m)

(123)

where $λ = \sqrt{λ_{1}^{2} + . . . + λ_{m - 1}^{2} - 1 ∕ m}$ .

The map (122) is isovariant since the inequality pattern among the entries of the matrix $Y$ and its image is preserved. From this it is also clear that $λ_{k + 1} = 0$ if and only if the $m - k$ last entries in (123) are identical. This also settles the above claim concerning the rank $k$ recognition in the linear model.

3.4.4. The simplest shape spaces. We return to the shape spaces $M^{*} (3, m)$ together with their $S O (3)$ -version for $m = 2, 3, 4$ . Recall from Section 1 that $M^{*} (3, n - 1) =$ $M_{n}^{*}$ is the shape space for the $n$ -body problem, and for $n = 3, 4, 5$ the topological classiﬁcation of these spaces are as follows :

\begin{align} n & = 3 : M_{3}^{*} ≃ \frac{M^{1} (3, 2)}{O (3)} = \frac{M^{1} (3, 2)}{S O (3)} = \frac{M^{1} (2, 2)}{O (2)} = D^{2} \\ n & = 4 : M_{4}^{*} ≃ \frac{M^{1} (3, 3)}{O (3)} \approx D^{5}, \frac{M^{1} (3, 3)}{S O (3)} \approx 2 D^{5} = S^{5} & (124) \\ n & = 5 : M_{5}^{*} ≃ \frac{M^{1} (3, 4)}{O (3)} \approx S^{8}, \frac{M^{1} (3, 4)}{S O (3)} \approx 2 S^{8} = S^{8} \cup_{P} S^{8} (P = ℂ P^{3} ∕ ℤ_{2}) \end{align}

We shall make some further comments on the cases $n = 3, 4 .$

Case $n = 3$ . With the induced metric the disk $D^{2}$ is a hemisphere of the base space

2 D^{2} = \frac{M^{1} (2, 2)}{S O (2)} = ℂ P^{1} = S^{2} (1 ∕ 2)

(125)

of the Hopf ﬁbration (112), which is a round sphere of radius $1 ∕ 2$ . Its equator circle $M^{*} (1, 2) = ℝ P^{1} = S^{1} (1 ∕ 2)$ represents the shapes of degenerate (i.e. collinear) 3-conﬁgurations. But in the study of the 3-body problem it is, in fact, natural to use the whole sphere as the shape space. The reason is that a non-degenerate 3-conﬁguration in 3-space is geometrically a triangle which can be oriented in two diﬀerent ways, by the ordering of the vertices. Then the two hemispheres in (125) represent triangles with opposite orientation, cf. [10], [12]. In this way a 3-body motion corresponds to a continuous motion of a mass triangle whose orientation changes when the motion passes through an eclipse, that is, when the shape curve crosses the equator circle.

Case $n = 4$ . The boundary of the disk $D^{5}$ in (124) is the shape space of coplanar 4-conﬁgurations, that is, the sphere

M^{*} (2, 3) = ℂ P^{2} ∕ ℤ_{2} \approx S^{4} ((cf. (109) with m = 3)

(126)

and the action of its isometry group $O (3)$ is equivalent (by Theorem 3.13) to that of its linear model $(O (3), S^{4}, S^{2} ρ_{3} - 1)$ . Namely, the linear model is the space of symmetric 3 $\times$ 3-matrices with zero trace and unit norm, with the natural action of $O (3)$ by conjugation.

The fact that the quotient space $ℂ P^{2} ∕ ℤ_{2}$ is homeomorphic to $S^{4}$ was already known to L.S. Pontryagin in the 1930’s, according to Arnold [3], and we refer to [2], [15], [21] for diﬀerent proofs of this speciﬁc result. Massey also observed that the induced $S O (3)$ -action on $ℂ P^{2} ∕ ℤ_{2}$ has the same orbit structure as that of the above linear model $(S O (3), S^{4})$ , and the existence and construction of an equivariant homeomorphism (such as (117)) was, in fact, a problem stated by Massey. Moreover, Arnold [2] has constructions which are very close to our linear model construction.

Arnold also discusses the diﬀerentiable structure of $S^{4}$ , as a quotient space $ℂ P^{2} ∕ ℤ_{2}$ , and it is stated that $ℂ P^{2} ∕ ℤ_{2}$ is diﬀeomorphic to $S^{4}$ (cf. $§$ 1 in [2]). However, what is shown is that the composite map

M^{1} (2, 3) = S^{5} \to ℂ P^{2} \to ℂ P^{2} ∕ ℤ_{2} \to^{\approx} S^{4}

(127)

from $S^{5}$ to $S^{4}$ is diﬀerentiable (in the usual sense). Both $ℂ P^{2}$ and $ℂ P^{2} ∕ ℤ_{2}$ have the induced smooth functional structure as quotient spaces of $S^{5}$ , but only $ℂ P^{2}$ becomes a diﬀerentiable manifold in this way. So, Arnold ”relaxes” the diﬀerential structure on $ℂ P^{2} ∕ ℤ_{2}$ so that the last map in (127) becomes a diﬀeomorphism, that is, he deﬁnes the diﬀerential structure to be the standard structure mentioned in Remark 3.14. This makes the last map in (127) a diﬀeomorphism while the composed map $S^{5} \to S^{4}$ in (127) is still a smooth map. The same applies, of course, to all the constructions $S^{p} \to S^{p} ∕ K \to^{\approx} S^{q}$ in Section 3.4.2, where none of the ”quotient spheres” $S^{p} ∕ K$ is really a smooth manifold when $q \geq 4$ .

Finally, let us explain why the ”quotient spheres” are not smooth manifolds. Since all cases are analogous we consider again the simplest case

M^{*} (2, 3) = S^{5} ∕ O (2) = ℂ P^{2} ∕ ℤ_{2} \approx S^{4}

(128)

viewed as an $O (2)$ -orbit space with the induced smooth structure. Then our claim is that the orbit space in (128) is not a diﬀerentiable manifold in a neighborhood of the subset $A = M^{*} (1, 3) ≃ ℝ P^{2}$ . To see this we shall apply the slice theorem (cf. Section 3.1.1), according to which each $O (2)$ -orbit in $S^{5}$ belongs to a tubular neighborhood $U$ whose image $\bar{U} = U ∕ O (2)$ in the orbit space is an open set of type (57), namely diﬀeomorphic to the orbit space of the slice representation. The set $A$ represents those orbits of type $O (1)$ , and the slice representation of $O (1)$ acts on $ℝ^{4}$ with the eigenvalues $(- 1, - 1, 1, 1)$ , consequently any point on $A$ has an open neighborhood diﬀeomorphic to

\bar{U} = \frac{O (2) \times_{O (1)} (ℝ^{2} \oplus ℝ^{2})}{O (2)} ≃ \frac{ℝ^{2} \oplus ℝ^{2}}{O (1)} ≃ \frac{ℝ^{2}}{O (1)} \times ℝ^{2}

(129)

where the ﬁrst factor is of type (102) and is transversal to the set $A$ and the second is $≃ \bar{U} \cap A$ . However, the product space in (129) is homeomorphic to $ℝ^{4}$ , but in view of Lemma 3.9 it is not a diﬀerentiable manifold.

4. Geometric invariants of $n$ -body systems

By a geometric invariant on the centered conﬁguration space

M_{n} : \sum m_{i} a_{i} = 0

we mean a polynomial function $F (a_{1}, . . ., a_{n})$ which is invariant under congruence and internal symmetries. We will describe the ring of all these invariants by calculating the ring of invariants for matrix spaces $M (d, m)$ in general. Denote a typical matrix by

X = (x_{1}, . . ., x_{m}) = [x_{1}^{*}, . . ., x_{d}^{*}]

where the $x_{i}$ and $x_{j}^{*}$ are the column and row vectors respectively. The product group $G L (d) \times G L (m)$ acts on $M (d, m)$ by matrix multiplication

X \to ψ X ϕ^{t}, (ψ, ϕ) \in G L (d) \times G L (m),

and $O (d) \times O (m)$ is the subgroup leaving invariant the standard metric form

I = {|X|}^{2} = t r a c e (X^{t} X) = \sum_{i = 1}^{m} {|x_{i}|}^{2} = \sum_{j = 1}^{d} {|x_{j}^{*}|}^{2},

(130)

On the other hand, for ﬁxed mass distribution $μ = (m_{1}, . ., m_{m}),$ $m_{i} > 0,$ let us also consider the mass dependent metric deﬁned by

I^{(μ)} = t r a c e (D X^{t} X D) = \sum_{i = 1}^{m} m_{i} {|x_{i}|}^{2},

(131)

where $D = d i a g (\sqrt{m_{1}}, \sqrt{m_{2}}, . ., \sqrt{m_{m}})$ and the corresponding isometry subgroup is $O (d) \times O^{*} (m) \subset$ $G L (d) \times G L (m)$ , where

O^{*} (m) = D O (m) D^{- 1} \subset G L (m)

(132)

is the subgroup leaving invariant the metric form $\sum m_{i} x_{i}^{2}$ on $ℝ^{m}$ (space of row vectors with right side action of $G L (m)$ ).

Lemma 4.1. The invariant ring on $M = M (d, m)$ , $d \leq m,$ under the action of $O (d) \times O^{*} (m)$ , is the polynomial ring with generators

I_{k} = \sum_{i_{1} < i_{2} . . . . < i_{k}} m_{i_{1}} m_{i_{2}} . . . m_{i_{k}} . {|x_{i_{1}} \land x_{i_{2}} \land . . . \land x_{i_{k}}|}^{2}, 1 \leq k \leq d,

(133)

where $I_{1} = I^{(μ)}$ and the exterior product space $\land^{k} ℝ^{m}$ has the standard norm.

Proof. First assume $m_{i} = 1$ for all i. It is easy to verify the identity

I_{k} = \sum_{i_{1} < i_{2} . . . . < i_{k}} {|x_{i_{1}} \land x_{i_{2}} \land . . . \land x_{i_{k}}|}^{2} = \sum_{i_{1} < i_{2} . . . . < i_{k}} {|x_{i_{1}}^{*} \land x_{i_{2}}^{*} \land . . . \land x_{i_{k}}^{*}|}^{2}

(134)

for each $k = 1, . ., d .$ Then, from the column vector version it is clear that $I_{k}$ is invariant under the action of $O (d),$ and similarly the row vector expression is invariant under the action of $O (m)$ . Hence, $I_{k}$ is an invariant of $G .$

Let us apply the reduction principle for orthogonal transformation groups (cf. e.g. [28]). Namely, we ﬁrst calculate the reduced group of the action of $G = O (d) \times O (m)$ on $M$ , which is the quotient group

\bar{G} = N_{G} (H) ∕ H ≃ B_{d}

and is ﬁnite in our case, where

H = Δ O {(1)}^{d} \times O (m - d) \subset G, cf. (89)

is the principal isotropy group and $N_{G} (H)$ is its normalizer in $G$ . Next we determine the ﬁxed point set of $H$

ℝ^{d} = F (H, M) : X = d i a g (r_{1}, r_{2}, . ., r_{d}),   cf. (81)

which contains the fundamental domain

M^{+} (d) \subset ℝ^{d} \subset M : r_{1} \geq r_{2} \geq . . . . \geq r_{d} \geq 0

of the $G$ -action, identiﬁed with the orbit space of $G$ in Section 3.3.1. Moreover, for the $G$ -orbit through $X$ , $r_{i}^{2} =$ $λ_{i}$ are the eigenvalues of $X^{t} X$ .

Finally, we consider the induced action of $\bar{G}$ on $ℝ^{d}$ , which is a group generated by reﬂections and as such it is the Weyl group $B_{d}$ , having the above cone $M^{+} (d)$ as a fundamental domain. Now, the reduction principle says that the restriction of polynomials induces an isomorphism between the invariant rings of $(G, M)$ and $(\bar{G}, ℝ^{d})$ . But the latter ring is certainly generated by the elementary symmetric functions

I_{1} = \sum λ_{i}, I_{2} = \sum_{i < j} λ_{i} λ_{j}, I_{3} = \sum_{i < j < k} λ_{i} λ_{j} λ_{k}, . . . .

which are also the restrictions of the functions in (134), and this proves the lemma for masses $m_{i} = 1 .$

Next, let $M {(d, m)}^{(μ)}$ be the matrix space with the mass dependent metric (131) and consider the transformation

M {(d, m)}^{(μ)} \to M (d, m), X \to X D = (\sqrt{m_{1}} x_{1}, \sqrt{m_{2}} x_{2}, . . ., \sqrt{m_{m}} x_{m})

where the left and right hand space have metrics (131) and (130) respectively. The transformation is a $G$ -equivariant isometry, where $O (m)$ acts on $M {(d, m)}^{(μ)}$ by ﬁrst applying the isomorphism

O (m) \to D O (m) D^{- 1} = O^{*} (m), cf. (132)

Hence, the transformation also induces an isomorphism of the corresponding invariant rings. In eﬀect, the invariant ring of $M {(d, m)}^{(μ)}$ is obtained by replacing each vector $x_{i}$ in (134) by $\sqrt{m_{i}} x_{i} .$ _

In particular, the centered conﬁguration space $M_{n} ≃ M (3, n - 1)$ has the following basic geometric invariants, in terms of the Jacobi vectors and classical vector operations :

I_{1} = \sum_{} {|x_{i}|}^{2}, I_{2} = \sum_{i < j} {|x_{i} \times x_{j}|}^{2}, I_{3} = \sum_{i < j < k} {|(x_{i} \times x_{j}) \cdot x_{k}|}^{2}

(135)

for all $n \geq 4 .$ (For $n = 3$ the generators are $I_{1}$ and $I_{2}$ ). The expressions for $I_{k}$ hold for any choice of Jacobi vectors, but still it remains to express them as functions of $a_{1}, . . ., a_{n} .$ To this end, consider the free $n$ -conﬁguration space or matrix space

{\hat{M}}_{n} = M {(3, n)}^{(μ)} = \{X = (a_{1}, a_{2}, . . ., a_{n})\}

with the metric form $I^{(μ)} = \sum m_{i} {|a_{i}|}^{2}$ as in (131. By the above lemma its invariant ring with respect to the group $O (3) \times O^{*} (n)$ has the basic generators

\begin{align} I_{1} & = \sum_{} m_{i} {|a_{i}|}^{2}, I_{2} = \sum_{i < j} m_{i} m_{j} {|a_{i} \times a_{j}|}^{2}, & (136) \\ I_{3} & = \sum_{i < j < k} m_{i} m_{j} m_{k} {|(a_{i} \times a_{j}) \cdot a_{k}|}^{2} \end{align}

and these are still complete and independent as invariants of the subgroup $O (3) \times O^{*} (n - 1)$ acting on the subspace $M_{n} \subset {\hat{M}}_{n}$ , as long as $3 \leq n - 1 .$ Here $O^{*} (n - 1)$ $\subset O^{*} (n)$ is the internal symmetry group of $M_{n}$ and is the subgroup leaving $M_{n}$ invariant, or equivalently, the subgroup of $O^{*} (n)$ which under the action on $ℝ^{n}$ (row vectors) ﬁxes the vector $(m_{1}, m_{2}, . . ., m_{n}) .$ This proves the following result.

Theorem 4.2. The ring of geometric invariants of the centered $n$ -conﬁguration space $M_{n}$ is the polynomial ring generated by $I_{1}, . ., I_{q}$ in (136), $q = min {3, n - 1}$ .

Remark 4.3. Starting from the Jacobi vector expressions (135) of the invariants, a Jacobi transformation may transform them to expressions involving the vectors $a_{i}$ . In this way, however, we may arrive at a non-symmetric expression since $\sum m_{i} a_{i} = 0$ . Below we shall give examples to illustrate the non-uniqueness of the symmetrization procedure and also give a geometric interpretation of $I_{n - 1}$ for $n = 3$ or $4 .$

Case $n = 3$ . Let $𝒜$ be the area of the triangle spanned by the vector triple $(a_{1}, a_{2}, a_{3})$ with $\sum m_{i} a_{i} = 0 .$ By simple trigonometry

2 𝒜 = \frac{\bar{m}}{m_{3}} |a_{1} \times a_{2}| = \frac{\bar{m}}{m_{1}} |a_{2} \times a_{3}| = \frac{\bar{m}}{m_{2}} |a_{3} \times a_{1}|

On the other hand, from the Jacobi vector formula (43),

x_{1} \times x_{2} = ζ_{1} ζ_{2} a_{1} \times a_{2}

and consequently

|x_{1} \times x_{2}| = ζ_{1} ζ_{2} |a_{1} \times a_{2}| = 2 \sqrt{\frac{m_{1} m_{2} m_{3}}{\bar{m}}} 𝒜

and

\begin{array}{l} I_{2} & = {|x_{1} \times x_{2}|}^{2} = \bar{m} \frac{m_{i} m_{j}}{m_{k}} {|a_{i} \times a_{j}|}^{2} \\ = \bar{m} \frac{m_{1} m_{2}}{3 m_{3}} {|a_{1} \times a_{2}|}^{2} + \bar{m} \frac{m_{2} m_{3}}{3 m_{1}} {|a_{2} \times a_{3}|}^{2} + \bar{m} \frac{m_{3} m_{1}}{3 m_{2}} {|a_{3} \times a_{1}|}^{2} \end{array}

where $\{i, j, k\} = \{1, 2, 3\}$ and the last sum is a symmetrization with all three terms equal, in fact. However, the above expression for $I_{2}$ and the symmetric expression in (136) are identical as a function on $M_{3} .$

Case $n = 4$ . Let $𝒱$ be the volume of the tetrahedron spanned by the vector quadruple $(a_{1}, a_{2}, a_{3}, a_{4})$ satisfying $\sum m_{i} a_{i} = 0 .$ Then $\pm 6 𝒱$ equals the triple product of $a_{i} - a_{1}$ , $i = 2, 3, 4,$ and

6 𝒱 = \frac{\bar{m}}{m_{4}} |(a_{1} \times a_{2}) \cdot a_{3}| = \frac{\bar{m}}{m_{1}} |(a_{2} \times a_{3}) \cdot a_{4}| etc.

Again, by expressing the Jacobi vectors $x_{i}$ in terms of the $a_{i}$ it follows

|(x_{1} \times x_{2}) \cdot x_{3}| = ζ_{1} ζ_{2} ζ_{3} |(a_{1} \times a_{2}) \cdot a_{3}| = 3! \sqrt{\frac{m_{1} m_{2} m_{3} m_{4}}{\bar{m}}} 𝒱

and consequently

\begin{array}{l} I_{3} & = {|(x_{1} \times x_{2}) \cdot x_{3}|}^{2} = \bar{m} \frac{m_{i} m_{j} m_{k}}{m_{l}} {|(a_{i} \times a_{j}) \cdot a_{k}|}^{2} \\ = \bar{m} \frac{m_{1} m_{2} m_{3}}{4 m_{4}} {|(a_{1} \times a_{2}) \cdot a_{3}|}^{2} + \bar{m} \frac{m_{2} m_{3} m_{4}}{4 m_{1}} {|(a_{2} \times a_{3}) \cdot a_{4}|}^{2} + etc. \end{array}

where $\{i, j, k, l\} = \{1, 2, 3, 4\}$ and the four terms in the last sum are equal. Again, $I_{3}$ in (136) is an another symmetric expression for the same function on $M_{4} .$

Remark 4.4. The cases $n = 3$ or $4$ are special since, for example, their shape space $M_{n}^{*}$ is topologically a disk $D^{2}$ or $D^{5}$ respectively. Moreover, in these cases $M_{n}^{*}$ has a unique geometric center, and this may be characterized in several ways. It is the unique ﬁxed point of the symmetry group $O (n - 1)$ , and the point has the same distance to each of the binary collision varieties. Furthermore, it is also the maximum point of $I_{2}$ or $I_{3}$ respectively. That is, for a ﬁxed size $I_{1} = 1$ the triangle (resp. tetrahedron) with the largest area (resp. volume) is at the geometric center. In terms of Jacobi vectors the condition is that $x_{i} \cdot x_{j} = 0$ and $|x_{i}| = |x_{j}|$ for $i \neq j .$

5. The weighted root system of an $n$ -body system

A typical potential function $U (a_{1}, a_{2}, . . ., a_{n})$ of an $n$ -body problem depends only on the pairwise distances

r_{i j} = |a_{i} - a_{j}|, 1 \leq i < j \leq n,

and for certain applications one would like to express them in terms of Jacobi vectors $x_{i}$ , say

X = (x_{1}, . ., x_{n - 1}) \in M (3, n - 1)

is the matrix associated with a given Jacobi transformation $Ψ : M_{n} ≃ M (3, n - 1)$ . For this purpose we shall deﬁne and investigate the weighted root system, depending on the mass distribution, which is primarily a book-keeping for expanding the vectors $a_{i} - a_{j}$ as a linear combination of Jacobi vectors. It is naturally a ”weighted” version of the root system of type $A_{n - 1}$ in classical Lie theory, and they are identical when the mass distribution is uniform.

5.1. Distance functions and the $Ψ$ -root system. The mutual distances $r_{i j}$ have a nice geometric interpretation as functions on $M_{n}$ which measure the distance from speciﬁc subvarieties. To explain this, consider the linear transformation

M = M (3, n - 1) \to ℝ^{3}

deﬁned by

X \to [u, X] = \sum_{i = 1}^{n - 1} c_{i} x_{i} = X u^{t}

(137)

where $u = (c_{1}, c_{2}, . . ., c_{n - 1}) \in ℝ^{n - 1}$ is a given row vector of unit length and $u^{t}$ is the column vector. Deﬁne the codimension $3$ subspace $B = B_{u}$ $\subset$ $M$ to be the kernel

B : [u, X] = 0 .

It is easy to verify that the distance in $M$ from a point $X$ to the subspace $B$ is given by the function

δ_{B} (X) = d i s t (X, B) = |[u, X]| = {|\sum_{k = 1}^{3} {({u \cdot x}_{k}^{*})}^{2}|}^{\frac{1}{2}}

where the $x_{k}^{*}$ are the row vectors of $X$ and the inner product on $ℝ^{n - 1}$ is the standard one.

In particular, the binary collision varieties

B_{i j} \subset M_{n} : a_{i} = a_{j}, i \neq j

(138)

have associated distance functions $δ_{i j}$ of this kind, deﬁned on $M$ via a chosen Jacobi transformation $Ψ : M_{n} \to M$ . Consequently there is a unit vector (unique up to sign)

u_{i j} = (c_{i j}^{1}, c_{i j}^{2}, . . ., c_{i j}^{n - 1}) \in ℝ^{n - 1}

(139)

depending on $Ψ$ and the mass distribution, so that

δ_{i j} (X) = {|\sum_{k = 1}^{3} {(u_{i j} {\cdot x}_{k}^{*})}^{2}|}^{\frac{1}{2}}

(140)

Let us express $δ_{i j}$ in terms of the $n$ -conﬁguration $X = (a_{1}, a_{2}, . . ., a_{n})$ . For $X$ ﬁxed, let $Y_{0} = (b_{1}, b_{2}, . . . . b_{n})$ be the critical point of the distance function $δ (Y) = |Y - X|$ deﬁned on $B_{i j}$ , where by deﬁnition $b_{i} = b_{j}$ . The condition $\nabla δ = 0$ implies

b_{i} = b_{j} = \frac{m_{i} a_{i} + m_{j} a_{j}}{m_{i} + m_{j}}, b_{k} = a_{k} for k \neq i, j

and consequently in terms of reduced masses (29)

\begin{array}{l} δ_{i j}^{2} (X) & = {|Y_{0} - X|}^{2} = m_{i} {|b_{i} - a_{i}|}^{2} + m_{j} {|b_{j} - a_{j}|}^{2} \\ = \frac{m_{i} m_{j}}{m_{i} + m_{j}} {|a_{i} - a_{j}|}^{2} = μ_{i j} r_{i j}^{2} \end{array}

that is,

r_{i j} = \frac{1}{\sqrt{μ_{i j}}} δ_{i j} = \frac{1}{\sqrt{μ_{i j}}} |[u_{i j}, X]|

(141)

Therefore, we may deﬁne vectors $u_{i j} = - u_{j i}$ uniquely by the constraint

a_{i} - a_{j} = \frac{1}{\sqrt{μ_{i j}}} [u_{i j}, X], i \neq j,

(142)

and clearly they satisfy all identities of type

\frac{1}{\sqrt{μ_{i j}}} u_{i j} + \frac{1}{\sqrt{μ_{j k}}} u_{j k} = \frac{1}{\sqrt{μ_{i k}}} u_{i k} .

(143)

Deﬁnition 5.1. The $Ψ$ -root system with mass distribution $(m_{1,} . . ., m_{n})$ is the above collection of $(\binom{n}{2})$ vector pairs $\pm \frac{1}{\sqrt{μ_{i j}}} u_{i j} \in ℝ^{n - 1}, 1 \leq i < j \leq n .$ The collection $\{\pm u_{i j}\}$ of unit vectors is the normalized $Ψ$ -root system.

5.2. The standard weighted root system. It will be useful to have explicit formulas for the $Ψ_{0}$ -roots, where

Ψ_{0} : M_{n} \to M (3, n - 1)

is the standard Jacobi transformation constructed in Section 2.3. The notation used for various constants in Section 2.3 is also used below. We shall also use the notation from deﬁnition of $Ψ_{0}$ (cf. (43), (47))

a_{i} - a_{j} = \frac{ζ_{i}}{m_{i}} x_{i} - \frac{1}{ζ_{j}} x_{j} + \sum_{k = i + 1}^{j - 1} \frac{ζ_{k}}{m^{(k - 1)}} x_{k} = \frac{1}{\sqrt{μ_{i j}}} \sum_{k = 1}^{n - 1} c_{i j}^{k} x_{k}, i < j,

(where the term $\frac{1}{ζ_{j}} x_{j}$ is undeﬁned when $j = n$ ) and the vector $u_{i j} = (c_{i j}^{1}, . . ., c_{i j}^{n - 1})$ have its nonzero components $c_{i j}^{k}$ for $k$ in the range $i \leq k \leq min \{j, n - 1\} .$ In particular,

u_{n - 1, n} = (0, 0, . . ., 1)

(144)

and for $i < n - 1$

\frac{1}{\sqrt{μ_{i, i + 1}}} u_{i, i + 1} = (0, . . ., 0, \sqrt{\frac{m^{(i - 1)}}{m_{i} m^{(i)}}}, - \sqrt{\frac{m^{(i + 1)}}{m_{i + 1} m^{(i)}}}, 0, . . ., 0)

(145)

\frac{1}{\sqrt{μ_{i n}}} u_{i n} = (0, . ., 0, \sqrt{\frac{m^{(i - 1)}}{m_{i} m^{(i)}}}, \sqrt{\frac{m_{i + 1}}{m^{(i)} m^{(i + 1)}}}, . . ., \sqrt{\frac{m_{n - 1}}{m^{(i)} m^{(n - 1)}}})

(146)

whereas for $2 \leq i + 1 < j < n$

\begin{align} \frac{1}{\sqrt{μ_{i j}}} u_{i j} & = (0, . ., 0, \sqrt{\frac{m^{(i - 1)}}{m_{i} m^{(i)}}}, \sqrt{\frac{m_{i + 1}}{m^{(i)} m^{(i + 1)}}}, \dots, \sqrt{\frac{m_{k}}{m^{(k - 1)} m^{(k)}}}, \dots, & (147) \\ \sqrt{\frac{m_{j - 1}}{m^{(j - 2)} m^{(j - 1)}}}, - \sqrt{\frac{m^{(j)}}{m_{j} m^{(j - 1)}}}, 0, . ., 0) \end{align}

For a ﬁxed mass distribution, all $Ψ$ -root systems are, in fact, orthogonally equivalent to the above $Ψ_{0}$ -root system $\{\pm \frac{1}{\sqrt{μ_{i j}}} u_{i j}, i < j\}$ , see the following subsection.

Example 5.2. $n = 3$ :

\begin{array}{l} u_{12} & = (\sqrt{\frac{m_{2} \bar{m}}{(m_{2} + m_{3}) (m_{1} + m_{2})}}, - \sqrt{\frac{m_{1} m_{3}}{(m_{2} + m_{3}) (m_{1} + m_{2})}}) \\ u_{13} & = (\sqrt{\frac{m_{3} \bar{m}}{(m_{2} + m_{3}) (m_{1} + m_{2})}}, \sqrt{\frac{m_{1} m_{2}}{(m_{2} + m_{3}) (m_{1} + m_{2})}}) \end{array}

Example 5.3. $n \geq 3$ :

\begin{array}{l} n & \geq 3 : \frac{1}{\sqrt{μ_{12}}} u_{12} = (\sqrt{\frac{\bar{m}}{m_{1} (m_{2} + . . + m_{n})}}, - \sqrt{\frac{m_{3} + . . + m_{n}}{m_{2} (m_{2} + . . + m_{n})}}, 0, . . .) \\ n & \geq 4 : \frac{1}{\sqrt{μ_{23}}} u_{23} = (0, \sqrt{\frac{m_{2} + . . + m_{n}}{m_{2} (m_{3} + . . + m_{n})}}, - \sqrt{\frac{m_{4} + . . + m_{n}}{m_{3} (m_{3} + . . + m_{n})}}, 0, . . .) \\ n & \geq 5 : \frac{1}{\sqrt{μ_{34}}} u_{34} = (0, 0, \sqrt{\frac{m_{3} + . . + m_{n}}{m_{3} (m_{4} + . . + m_{n})}}, - \sqrt{\frac{m_{5} + . . + m_{n}}{m_{4} (m_{4} + . . + m_{n})}}, 0, . . .) \end{array}

5.3. Weighted root systems and their metric invariants. It follows from (143)-(147) that the (normalized) $Ψ_{0}$ -roots $u_{i k}$ and $u_{j l}$ are mutually perpendicular except when $\{i, k\} \cap \{j, l\} \neq \emptyset,$ in which case we deﬁne $α_{k}^{i, j} =$ $α_{k}^{j, i}$ to be the angle between $u_{i k}$ and $u_{k j}$ , namely

cos α_{k}^{i, j} = u_{i k} \cdot u_{k j} = - \sqrt{\frac{m_{i} m_{j}}{(m_{i} + m_{k}) (m_{k} + m_{j})}}

(148)

In particular, for equal masses the possible angles between any two non-collinear vectors are $α = \frac{1}{2} π, \frac{1}{3} π$ and $\frac{2}{3} π,$ and the $Ψ_{0}$ -roots constitute, indeed, a root system of type $A_{n - 1}$ in the usual sense. For general masses we make the following deﬁnition.

Deﬁnition 5.4. A weighted root system (of type $A_{n - 1}$ ) with mass distribution $(m_{1}, m_{2}, . . ., m_{n})$ is a collection of nonzero vectors

w_{i j} = - w_{j i}, 1 \leq i < j \leq n,

in $ℝ^{n - 1}$ such that i) $\sqrt{μ_{i j}} |w_{i j}| = 1$ , ii) $w_{i k}$ and $w_{j l}$ are perpendicular if and only if $\{i, k\} \cap \{j, l\} = \emptyset,$ and iii) the angle between $w_{i k}$ and $w_{k j}$ is $α_{k}^{i, j}$ , as deﬁned by (148).

The $Ψ_{0}$ -root system is our prototype of such a root system, and clearly the subset of n-1 vectors $\{\frac{1}{\sqrt{μ_{i, i + 1}}} u_{i, i + 1}\}$ constitutes a system of simple roots for obvious reasons, see (143), (145).

Theorem 5.5. For a given mass distribution, each $Ψ$ -root system (see Deﬁnition 5.1) is a weighted root system in the sense of Deﬁnition 5.4, and conversely, each weighted root system is the $Ψ$ -root system for a unique Jacobi transformation $Ψ : M_{n} \to M (3, n - 1)$ . Moreover, for the given mass distribution all the weighted root systems are orthogonally equivalent.

This result is a simple consequence of the fact that any $Ψ$ can be written uniquely as a composition

Ψ = ϕ \circ Ψ_{0} : M_{n} \to M (3, n - 1) \to M (3, n - 1)

where $ϕ \in O (n - 1)$ acts on $M (3, n - 1)$ by multiplication, that is, $ϕ (X) = X ϕ$ . Therefore the Jacobi vector matrices $X$ and $X^{'}$ of $Ψ_{0}$ and $Ψ$ , respectively, are related by $X^{'} = X ϕ$ . Let $\{\pm u_{i j}\}$ and $\{\pm u_{i j}^{'}\}$ be the normalized $Ψ_{0}$ -roots and $Ψ$ -roots, respectively. The deﬁning relation (142)

\sqrt{μ_{i j}} (a_{i} - a_{j}) = [u_{i j}, X] = [u_{i j}^{'}, X^{'}]

implies $u_{i j}^{'} = u_{i j} ϕ$ , and consequently the $Ψ$ -root system is the orthogonally transformed image

\{\pm \frac{1}{\sqrt{μ_{i j}}} u_{i j}^{'}\} = \{\pm \frac{1}{\sqrt{μ_{i j}}} u_{i j}\} ϕ (row vectors)

of the $Ψ_{0}$ -root system.

On the other hand, a root system is a ﬁnite subset of $ℝ^{n - 1}$ with speciﬁed inner products of the vectors, and according to a result of Weyl (see Section 3.3) these numbers determine the subset modulo orthogonal equivalence. Moreover, since the weighted root system spans all $ℝ^{n - 1}$ , any two of them are related by a unique $ϕ \in O (n - 1)$ . In particular, a weighted root system is the $Ψ$ -root system for a unique Jacobi transformation $Ψ = ϕ \circ Ψ_{0}$ .

5.4. On the role of the mass distribution. The mass distribution of a $n$ -body system manifests itself through the various mass dependent quantities constructed above, for example :

the collection $\{μ_{i j}\}$ of reduced masses (29);
the weighted and normalized root system $\{\pm u_{i j}\};$
the collection $\{α_{k}^{i, j}\}$ of angles between the vectors of a weighted root system.

The mass distribution (modulo scaling ) may, in fact, be reconstructed from any of these invariants. The reduced masses, for example, satisfy the following two conditions

\begin{array}{l} (i) μ_{i j}^{- 1} + μ_{j k}^{- 1} & > μ_{i k}^{- 1} for i, j, k diﬀerent \\ (i i) μ_{i j}^{- 1} + μ_{k l}^{- 1} & = μ_{i k}^{- 1} + μ_{j l}^{- 1} = μ_{i l}^{- 1} + μ_{j k}^{- 1} for i, j, k, l diﬀerent \end{array}

and one can deﬁne a reduced mass distribution of order $n$ to be a collection of $(\binom{n}{2})$ positive numbers $μ_{i j} = μ_{j i},$ $1 \leq i < j \leq n$ , constrained by the above two conditions. Then there is a 1-1 correspondence between the usual and the reduced mass distributions (modulo scaling) given by the formulae for $μ_{i j}$ and their inversion

m_{i} = \frac{2}{μ_{i j}^{- 1} + μ_{i k}^{- 1} - μ_{j k}^{- 1}} for i, j, k diﬀerent

On the other hand, the congruence space ${\bar{M}}_{n}$ (and hence the shape space $M_{n}^{*}$ ), with the kinematic Riemannian structure, is independent of the mass distribution. The mass distribution determines, however, the representation of congruence classes as points in ${\bar{M}}_{n}$ ; in particular, it determines the relative position of the binary collision varieties. Conversely, we can reconstruct the masses $m_{i}$ (modulo scaling) from knowledge of the position of these varieties.

To illustrate the last statement, consider the case $n = 3,$ where $M_{3}^{*}$ is a round hemisphere (or sphere, see Section 3.4.4) with a distinguished equator circle $S^{1}$ representing the shapes of the degenerate triangles. On this circle lie the three collision points $b_{12}, b_{23}, b_{13}$ , where $b_{i j}$ represents the shape of those triangles $(a_{1}, a_{2}, a_{3})$ with $a_{i} = a_{j} \neq 0$ and the center of mass at the origin. The mass distribution determines their relative position, that is, the angles (or distances) between the points $b_{i j}$ , and once their position have been ﬁxed we know how to determine the position of any shape (cf. [12]).

To describe the relative positions of the points $b_{i j}$ , consider $S^{1}$ as the circumscribed circle of the triangle $Δ = Δ (b_{12}, b_{23,} b_{31})$ in a Euclidean plane with origin at the center of $S^{1}$ . It turns out that $Δ$ is a central triangle, in the sense that the center of its circumscribed circle lies in its interior. Moreover, the three central angles

0 < β_{k} < π, \sum β_{k} = 2 π

where $β_{k}$ is opposite to the vertex $b_{i j}$ ( $i, j, k$ diﬀerent), are given in terms of the normalized mass distribution by

sin β_{k} = 2 \frac{\sqrt{m_{1} m_{2} m_{3}}}{(m_{k} + m_{i}) (m_{k} + m_{j})} (\sum m_{i} = 1)

Conversely, any central triangle can be realized in this way and the inversion formula is

m_{k} = 1 - \frac{2 sin β_{k}}{\sum_{i = 1}^{3} sin β_{i}}

(cf. [10] or [12]) where the positivity of $m_{k}$ , indeed, reﬂects the central property of the triangle $Δ$ .

The triple of angles $(β_{1}, β_{2}, β_{3})$ is in 1-1 correspondence with another triple $(α_{1}, α_{2}, α_{3})$ , where

α_{k} = π - β_{k} ∕ 2, k = 1, 2, 3

(149)

and these are the central angles of another central triangle, namely the triangle representing (up to congruence) the weighted root system of the mass distribution (cf. Section 5.3). More precisely, whereas $β_{k} = β_{i j}$ is the angular distance between $b_{i k}$ and $b_{k j}$ on the circle $S^{1}$ , $α_{k} = α_{i j}$ is the angle between the normalized root vectors $u_{i k}$ and $u_{k j}$ since

cos α_{k} = u_{i k} \cdot u_{k j} = - \sqrt{\frac{m_{i} m_{j}}{(m_{i} + m_{k}) (m_{k} + m_{j})}}, cf. (148)

Finally, we recall that the root system of a $k$ -body system is a weighted root system of type $A_{k - 1}$ . The way diﬀerent $A_{2}$ systems combine into higher rank systems $A_{k},$ $k \geq 3,$ is completely parallel to the ”standard” theory of root systems of type $A_{k}$ , and therefore the higher rank case poses no further problem.

6. Collinear central conﬁgurations revisited

Consider the classical Newtonian potential function

U = \sum_{i < j} \frac{m_{i} m_{j}}{|a_{i} - a_{j}|}

(150)

and its gradient ﬁeld $\nabla U$ with respect to the kinematic metric (8) in $M_{n}$ (or ${\hat{M}}_{n}$ ). An $n$ -conﬁguration $X = (a_{1}, . . ., a_{n})$ is called central if $\nabla U (X) = λ X$ for some constant $λ$ , namely

λ a_{i} = \frac{1}{m_{i}} \frac{\partial U}{\partial a_{i}} = \sum_{j \neq i} \frac{m_{j} (a_{j} - a_{i})}{{|a_{i} - a_{j}|}^{3}}, i = 1, . . ., n

(151)

In fact, $λ = - I^{- 1} U (X)$ where $I = {|X|}^{2}$ , since by a classical result of Euler the homogeneity of $U$ implies $\nabla U (X) \cdot X = - U (X)$ . The collection of central conﬁgurations is clearly invariant under similarity transformations. Thus we may ﬁx a scaling of the vectors, say $I = 1$ , and ask for solutions modulo $O (3)$ -congruence. Then the solutions are just the critical points of $U$ as a function restricted to the shape space $M_{n}^{*}$ . For $n = 3$ the only solution which is the shape of a non-degenerate triangle is the equilateral triangle (by Lagrange, 1772), whereas there are three degenerate triangle solutions (by Euler, 1767) and they are represented by the three Euler points on the equator circle of $M_{3}^{*}$ .

Even the enumeration of all critical points in $M_{n}^{*}$ is, in fact, still an open problem for $n \geq 4$ , and now the number also depends on the mass distribution. It is only known to be ﬁnite for $n < 5$ . However, the number of collinear solutions is known to be $n! ∕ 2$ for all $n \geq 3$ , and the ﬁrst proof was presented by Moulton [23]. More recently, Smale [26] has given a topological proof using elementary Morse theory. In this subsection we shall give a similar and quite simple proof using the weighted root system of an $n$ -body system.

As usual, we assume that $U$ in (150) is a function of n vector variables $a_{i}$ linearly related by the condition that the center of mass is at the origin. Actually this condition is a consequence of the identities (151). Anyhow, let us express $U$ as a function of the n-1 Jacobi vectors

X = (x_{1}, x_{2}, . . ., x_{n - 1}) = [x_{1}^{*}, x_{2}^{*}, x_{3}^{*}]

deﬁned by a ﬁxed Jacobi transformation

Ψ : M_{n} \to M (3, n - 1)

with associated root system $\{\pm w_{i j}\}$ . Take, for example, the standard transformation $Ψ_{0}$ and the expressions for the vectors $x_{i}$ and $w_{i j}$ in Section 2.3 and Section 5.2. The (row) vectors $w_{i j}$ and $x_{k}^{*}$ belong to the same Euclidean space $ℝ^{n - 1}$ and by deﬁnition

{|a_{i} - a_{j}|}^{2} = \sum_{k = 1}^{3} {(w_{i j} \cdot x_{k}^{*})}^{2}, w_{i j} = \frac{1}{\sqrt{μ_{i j}}} u_{i j} (cf. Section 5.1)

The gradient ﬁeld $\nabla U$ is tangential to the subvariety of collinear conﬁgurations and therefore the critical points we seek are also the critical points of $U$ restricted to the subvariety of collinear shapes, namely $M^{*} (1, n - 1) ≃ R P^{n - 2}$ . Let us represent all shapes of collinear type by $n$ -conﬁgurations $X = (a_{1}, a_{2}, . ., a_{n})$ in $M_{n}$ with position vectors $a_{i}$ along the x-axis, and therefore the Jacobi matrix $X$ has row vectors $x_{2}^{*} = x_{3}^{*} = 0$ . Hence, we shall regard $U$ as a function on the (n -1)-space

ℝ^{n - 1} : x = x_{1}^{*} = (x_{1}, . . ., x_{n - 1})

and the condition $I = 1$ means restriction to the unit sphere $S^{n - 2} : \sum x_{i}^{2} = 1$ . When antipodal points on this sphere are identiﬁed, we obtain the above projective space $R P^{n - 2}$ , see (154) below.

For convenience, we shall identify the root vector $w_{i j}$ with the linear functional

ω_{i j} : ℝ^{n - 1} \to ℝ ω_{i j} (x) = w_{i j} \cdot x,

and regard the family

Δ = \{\pm ω_{i j}\} = Δ_{+} \cup Δ_{-}

as the root system. The positive and simple roots are the subfamilies

Δ_{+} = \{ω_{i j}, i < j\} \supset Π = \{ω_{12}, ω_{23},,, ω_{n - 1, n}\},

respectively, with $ω_{i j} = - ω_{j i}$ and $ω_{i j} + ω_{j k} = ω_{i k}$ , and following the ”usual” procedure we deﬁne hyperplanes

H_{i j} = ker ω_{i j} \subset ℝ^{n - 1}

which subdivide the space into disjoint, open and connected chambers $C_{i}$ whose union is

C_{1} \cup C_{2} \cup . . . . = ℝ^{n - 1} - \cup_{i < j} H_{i j} .

Clearly, there are $n!$ chambers, and each chamber $C$ is distinguished by a speciﬁc choice of signs $ɛ_{i j} = \pm 1$ such that

x \in C \Leftrightarrow ɛ_{i j} ω_{i j} (x) > 0 for all i < j

By changing all signs one obtains the antipodal chamber. In particular, our fundamental chamber $C_{0}$ is the one with all $ɛ_{i j}$ positive, namely

C_{0} : ω_{i j} (x) > 0 for all i < j,

or equivalently in terms of the positions $a_{i} = a_{i}$ of the point masses (on the x-axis), since $ω_{i j} (x) = a_{i} - a_{j},$

a_{1} > a_{2} > . . . . > a_{n}

(152)

The above decomposition and combinatorial structure is, of course, similar to the Weyl chamber decomposition for the Weyl group $A_{n - 1} = S_{n}$ of $S U (n) .$

By (140) and (141), the expression (150) may be written

U (x) = \sum_{i < j} \frac{m_{i} m_{j}}{|ω_{i j} (x)|}

(153)

which in the fundamental chamber reads

U (x) = \sum_{i < j} \frac{m_{i} m_{j}}{ω_{i j} (x)} for x \in C_{0}

Lemma 6.1. $U (x)$ is a convex function on the chamber $C_{0}$ , and it has a unique critical point $x_{0}$ (where $U$ has a minimum) in the spherical chamber $C_{0} \cap S^{n - 2}$ .

Proof. By straightforward calculations

\begin{array}{l} \frac{\partial U}{\partial x_{k}} & = - \sum_{i < j} m_{i} m_{j} \frac{w_{i j}^{k}}{ω_{i j} {(x)}^{2}} \\ \frac{\partial^{2} U}{\partial x_{k} \partial x_{l}} & = 2 \sum_{i < j} m_{i} m_{j} \frac{w_{i j}^{k} w_{i j}^{l}}{ω_{i j} {(x)}^{3}} \end{array}

where $w_{i j} = (w_{i j}^{1}, w_{i j}^{2}, . . ., w_{i j}^{n - 1})$ and $x = (x_{1}, x_{2},,, x_{n - 1})$ . Hence, the Hessian of $U$ at $x \in C_{0}$ is the following positive deﬁnite quadratic form in the variable $t = (t_{1}, t_{2}, . . ., t_{n - 1})$

H U (x) (t) = \frac{1}{2} \sum_{k, l = 1}^{n - 1} \frac{\partial^{2} U}{\partial x_{k} \partial x_{l}} (x) t_{k} t_{l} = \sum_{i < j} m_{i} m_{j} \frac{ω_{i j} {(t)}^{2}}{ω_{i j} {(x)}^{3}}

On the other hand, $U (x) > 0$ on $C_{0}$ and $U (x) \to \infty$ as $x$ approaches the walls of the chamber and is bounded away from the origin. It follows that $U$ has a unique critical point in the spherical chamber, namely a minimumspoint $x_{0}$ . _

The induced congruence group action on $ℝ^{n - 1}$ is simply the group $O (1) = \{\pm 1\}$ acting by inversion $x \to - x$ , and therefore antipodal points on the sphere $S^{n - 2}$ represent collinear $n$ -conﬁgurations with the same shape :

S^{n - 2} \to S^{n - 2} ∕ O (1) = ℝ P^{n - 2} \subset M_{n}^{*}

(154)

In particular, a pairs of antipodal chambers is mapped to the same chamber in $ℝ P^{n - 2}$ , and consequently the latter space is divided into $n! ∕ 2$ chambers, each containing a unique critical point of $U$ as a function on $ℝ P^{n - 2}$ . This completes the proof of the enumeration result originally due to Moulton.

Remark 6.2. The above central conﬁguration solutions $(a_{1,} a_{2}, . . ., a_{n})$ can be distinguished by the ordering $a_{i_{1}} > a_{i_{2}} > . . . . > a_{i_{n}}$ (modulo inversion of order), and the minimumspoint $x_{0}$ in the fundamental chamber $C_{0}$ is the solution with the ordering (152). However, for a given mass distribution $(m_{1}, m_{2}, . . ., m_{n})$ the solution satisfying $a_{i_{1}} > a_{i_{2}} > . . . . > a_{i_{n}}$ can also be found by the same procedure, namely as the minimumspoint $x_{0}$ in the fundamental chamber $C_{0}$ corresponding to the permuted mass distribution $(m_{i_{1}}, m_{i_{2}}, . . ., m_{i_{n}})$ . In particular, for each string of equal masses $m_{i} = m_{j} = . . = m_{k}$ the set of solutions is invariant under permutations of $a_{i}, a_{j}, . ., a_{k}$ .

Let us brieﬂy consider the explicit numerical calculation of the collinear central conﬁgurations by calculating the critical points of $U$ on $S^{n - 2}$ . Observe that the identity (151) with the additional condition $I = 1$ may be interpreted as an application of the classical Lagrange multiplier method with a constraint. Similarly, in the present case where we restrict to collinear conﬁgurations, we seek the $n!$ solutions of the system

\nabla U (x) = λ x, |x| = 1

(155)

where $λ = - U (x)$ (by Euler’s formula) and

\nabla U (x) = - \sum_{i < j} m_{i} m_{j} \frac{w_{i j}}{ω_{i j} {(x)}^{2}}

The condition for a solution is the vanishing of the component of the gradient tangential to the sphere, namely

\nabla U (x) + U (x) x = 0

(156)

As a simpleminded algorithm for ﬁnding a solution we construct a sequence of points $y_{k} \in S^{n - 2}$ by ”moving” on the sphere in the direction opposite to the vector (156), as follows

y_{k + 1} = \frac{(1 - U (y_{k})) y_{k} - \nabla U (y_{k})}{|(1 - U (y_{k})) y_{k} - \nabla U (y_{k})|},

starting from an initial point $y_{1}$ in a given chamber. In general, the sequence will stay in the chamber and converge to the minimumspoint of $U .$ The algorithm can certainly be made more eﬀective, but we shall not discuss these matters.

Another approach is to introduce new variables $t_{i}$ linearly related to $x_{1}, x_{2}, ., x_{n - 1}$ via the simple roots.

t_{i} = ω_{i, i + 1} (x) = a_{i} - a_{i + 1}, i = 1, 2, . . ., n - 1

For example, in the case $n = 3$ the system (155) reads

\begin{align} 0 & = - \frac{m_{1}}{{(t_{1})}^{2}} + \frac{1 - m_{1}}{{(t_{2})}^{2}} + \frac{m_{1}}{{(t_{1} + t_{2})}^{2}} - λ t_{2} & (157) \\ 0 & = \frac{m_{2}}{{(t_{1})}^{2}} + \frac{m_{3}}{{(t_{1} + t_{2})}^{2}} - λ [(1 - m_{1}) t_{1} + m_{3} t_{2}] \end{align}

as compared to the initial system (151) which has three dependent equations similar to the above ones. In fact, the ﬁrst equation of system (151) and (157) are identical. By elimination of $λ$ we obtain a homogeneous equation of degree 5 which has a unique solution (modulo scaling) with both $t_{i} > 0$ . Thus it can be reduced to an inhomogeneous equation of degree 4, whereas Euler’s approach led to the solution of a 5th order inhomogeneous equation $P (ω) = 0$ (cf. [25], $§$ 14). The latter equation is, in fact, obtained from the system (157) when we write $(t_{1} + t_{2}) = t,$ $t_{1} = ω t, t_{2} = (1 - ω) t$ and eliminate the resulting multiplier $λ^{'} =$ $λ t^{3}$ .

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Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU) N-7491 Trondheim, Norway

E-mail address: eldars@math.ntnu.no

Received October 30, 2006

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