Linear instability of relative equilibriafor n-body problems in the plane

# Linear instability of relative equilibria for n-body problems in the plane

## Abstract

Following Smale, we study simple symmetric mechanical systems of point particles in the plane. In particular, we address the question of the linear and spectral stability properties of relative equilibria, which are special solutions of the equations of motion.

Our main result is a sufficient condition to detect spectral (hence linear) instability. Namely, we prove that if the Morse index of an equilibrium point with even nullity is odd, then the associated relative equilibrium is spectrally unstable. The proof is based on some refined formulæ for computing the spectral flow.

As a notable application of our theorem, we examine two important classes of singular potentials: the -homogeneous one, with , which includes the gravitational case, and the logarithmic one. We also establish, for the -homogeneous potential, an inequality which is useful to test the spectral instability of the associated relative equilibrium.

MSC Subject Class: Primary 70F10; Secondary 37C80.

Keywords: linear instability, relative equilibria, spectral flow, partial signatures, -body problem, -homogeneous potential, logarithmic potential.

1

## 1 Introduction

Simple mechanical systems are a special class of Hamiltonian systems in which the Hamiltonian function can be written as the sum of the potential and kinetic energies. The search for special orbits, such as equilibria and periodic orbits, and the understanding of their stability properties are amongst the major subjects in the whole theory of Dynamical Systems.

In 1970, in one of his famous papers [25], S. Smale, following the ideas sketched out by E. Routh in [24], examined the stability of relative equilibria of simple mechanical systems with symmetries. For a general system of this kind, a relative equilibrium is a dynamical fixed point (i.e. an equilibrium point) in the reduced phase space obtained by quotienting the original phase space by the symmetry group. Thus, generally speaking, relative equilibria are the analogue of fixed points for systems without symmetry (whence their great importance), yet they can also be viewed as one-parameter group orbits. Of course, the larger the symmetry group is, the richer the supply of relative equilibria becomes. For a system of particles in the plane described in the coordinates of the centre of mass, subject to the action of the rotation group  — like the one that we examine here — relative equilibria are solutions in which the whole system rotates with constant angular velocity around the barycentre. For this reason they are also called dynamical motions in steady rotation.

Given a relative equilibrium, it is natural to investigate its stability properties in order to understand the dynamical behaviour of the orbits nearby. Two of the main methods used to study the stability of relative equilibria are the Energy-Casimir method and the Energy-Momentum method; however, even when applicable, they do not give any information about the instability without further investigation. One of the few feasible methods to study the matter of stability is to show that the Hamiltonian , or some other integral, has a maximum or minimum at a critical point: if the maximum or minimum is isolated then is a Lyapunov function and the equilibrium point is stable. Unfortunately, in the -body context, it is easy to see (cf. [19, page 86]) that this approach never works in the case of relative equilibria, and for this reason it is hopeless to try to prove their stability (or instability). Instead of that, we concentrate here on the notions of linear and spectral stability (see Subsection 2.2 for their definition): we linearise the Hamiltonian system around a relative equilibrium and analyse its features. This involves the computation of the spectrum of a Hamiltonian matrix, which is symmetric with respect to both axes in the complex plane. A direct consequence of this fact is that relative equilibria are never asymptotically stable.

In studying symmetric systems of particles it is usual to introduce the so-called augmented potential , which is equal to the potential of the system plus a term coming from the centrifugal forces (cf. [15] and references therein). The reason is that relative equilibria are precisely the critical points of this modified potential (see [25]).

Our main result reads as follows (see Theorem 4.1 and Section 4 for a more precise statement, further details and the proof).

###### Theorem.

Let be a critical point of the augmented potential and assume that it has even nullity. If is odd, then the relative equilibrium corresponding to is spectrally unstable.

An immediate consequence is the following.

###### Corollary.

Let be a critical point of the augmented potential. If its Morse index or its nullity are odd then the corresponding relative equilibrium is linearly unstable.

The main tool that we use in the proof of this theorem is the spectral flow (in the very elementary case of Hermitian matrices). We recall that this is a well-known integer-valued homotopy invariant of paths of self-adjoint Fredholm operators introduced by M. F. Atiyah, V. K. Patodi and I. M. Singer in [3]. In finite-dimensional situations it is nothing else but the difference of the Morse index at the endpoints (see Section 3 for its definition and Appendix A for its main properties). Up to perturbation, non-degeneracy and transversality conditions, this invariant can be computed in terms of the so-called crossing forms, which, intuitively speaking, counts in an appropriate way the net number of eigenvalues crossing the value in a transversal way. In our setting this need not be the case; however, the third author developed in other papers (see for instance [9], or Appendix A for a short description) a non-perturbative analysis of the non-transversal intersections. The reason behind the choice of a non-perturbative technique lies in the fact that, in general, perturbative methods preserve global invariants but completely destroy the local information concerning the single intersection. By means of this theory, based on what has been termed partial signatures, we have been able to prove Theorem 4.1.

The main applications of our result (see Section 5) are directed towards the -homogeneous and the logarithmic potentials, although also some other interesting classes can be reformulated in our framework, such as the Lennard-Jones interaction potential. Our theorem offers indeed a unifying viewpoint of all these quite different situations, since the property that it unravels descends only from the rotational invariance of the mechanical system. All of these potentials are extensively studied in literature: the -homogeneous ones are the natural generalisation of the gravitational attraction () and they are employed in different atomic models, whilst the logarithmic potential naturally arises when looking from a dynamical viewpoint at the stationary helicoidal solutions of the -vortex filaments model, which is popular and useful in Fluid Mechanics. See [22, 26, 7] and references therein for the homogeneous cases, [21] for the logarithmic one and [4] for a general overview.

To detect a relative equilibrium in an -body-type problem means to determine a moving planar central configuration of the bodies which solves Newton’s equations and in which the attractive force is perfectly balanced by the centrifugal one. This is currently the only way known to obtain exact solutions, albeit finding central configurations amounts to solving a system of highly nonlinear algebraic equations and is therefore very hard (see [19] for the Newtonian case and [8] for the -homogeneous one).

Being invariant under the symmetry group of Euclidean transformations and admitting linear momentum, angular momentum and energy as first integrals, -body-type problems are highly degenerate. This in particular yields Jacobians with nullity (cf. [16, 17] for the gravitational force), but only in an inertial reference frame: indeed, if we move (as we do) to a suitable uniformly rotating coordinate system (so that the relative equilibrium becomes an effective equilibrium) six out of the eight eigenvalues produced by the first integrals depend on the angular velocity. This is not surprising at all, since linear stability properties strongly depend on the choice of the frame of the observer. For this reason, studying the case , R. Moeckel in [19] defined the linear and spectral stability by ruling out all the eigenvalues linked to this kind of degeneracy. In the same context, K. R. Meyer and D. S. Schmidt concluded in [18] a deep study of the linearised equations: in particular, they introduced a suitable system of symplectic coordinates in which the matrices are block-diagonal, with one block representing the translational invariance of the problem and another one carrying the symmetries induced by dilations and rotations. These two submatrices generate the eight eigenvalues responsible of degeneracy, whilst a third (and last) block contains all the information about stability, in the sense mentioned above. We observe that an analogous decomposition holds also for the potentials that we examine (see Subsection 5.2).

In this picture, it is worthwhile to mention a conjecture on linear stability stated by Moeckel, which we report here.

Moeckel’s Conjecture (cf. [2, Problem 16]) — In the planar Newtonian -body problem, the central configuration associated with a linearly stable relative equilibrium is a non-degenerate minimum of the potential function restricted to the shape sphere (i.e. the -quotient of the ellipsoid of inertia).

This conjecture is still unproved; however, X. Hu and S. Sun have made some progress. More precisely, they showed in [11] that if the Morse index or the nullity of a central configuration (viewed as a critical point of the potential restricted to the shape sphere) are odd, then the corresponding relative equilibrium is linearly unstable. Therefore the central configurations giving rise to linearly stable relative equilibria should correspond to a critical point with even Morse index and nullity. The main result in [11] is the first attempt towards the understanding of the relationship (if there is any) between two dynamics: the gradient flow on the shape sphere and Hamilton’s equations in the phase space.

The contribution of our paper in this setting is twofold:

1. We provide a complete and detailed proof of the result on linear instability proved in [11] and we extend it to a very general class of interaction potentials by using spectral flow techniques.

2. We prove a result on spectral instability by means of the theory of partial signatures previously developed in [9]. Note that our Corollary 5.8 is actually the main result in [11] (written there in the gravitational setting only).

Moeckel’s Conjecture can thus be adapted to the class of potentials that we study; accordingly, we reformulate it as follows:

Conjecture — In planar -symmetric mechanical systems, a critical point of the augmented potential associated with a linearly stable relative equilibrium, is a non-degenerate minimum.

We cast some light on this question with Theorem 4.1 and with Theorem 5.6 in the special case of -body-type problems.

Furthermore, following the approach of G. E. Roberts in [23], we are able to give a sufficient condition for spectral instability of a relative equilibrium (at least in the -homogeneous case) in terms of the potential evaluated at a central configuration. It is in fact rather foreseeable that the linear stability depends also on the homogeneity parameter (see Subsection 5.4 and cf. Corollary 5.10 for a precise statement).

###### Theorem.

Let be a central configuration. If the following inequality holds

 n∑i,j=1i2n+α−4α Uα(¯x)

then the arising relative equilibrium is linearly unstable.

We conclude this section by pointing out that no sufficient condition for detecting the linear or spectral stability has been found thus far. This question is addressed in a forthcoming paper [5], where we are trying to establish in a precise way the stability properties of the relative equilibria by using some symplectic and variational techniques, mainly based on the Maslov index, index theorems and topological invariants.

## 2 Description of the problem: setting and preliminaries

In this section we briefly outline the basic definitions and properties of simple mechanical systems with symmetry, as well as their reduction to the quotient space.

Consider the Euclidean plane endowed with the usual inner product and let be positive real numbers which can be thought of as masses. The configuration space of point particles with masses , with , will therefore be a suitable subset (equipped with its Euclidean inner product, which we denote again by ). For any position vector , with (column vector) for every , we can define a norm in through the moment of inertia:2

 I(q)\coloneqq∥q∥2M\coloneqq⟨Mq,q⟩=n∑i=1mi|qi|2,

where is the diagonal mass matrix , is the identity matrix and denotes the Euclidean norm of in .

A simple mechanical system of point particles on is described by a Lagrangian function of the form

 L(q,˙q)\coloneqqK(q,˙q)+U(q),

where is the kinetic energy of the system and is its potential function. This Lagrangian thus equals the difference between the kinetic energy and the potential energy (); in our case we have .

Using the mass matrix , Newton’s equations can be written as the following second-order system of ordinary differential equations on :

 M¨q=∇U(q), (2.1)

which can of course be transformed into a first-order system as follows. Let us introduce the Hamiltonian function , defined by

 H(q,p)\coloneqq12⟨M−1pT,pT⟩−U(q).

Here , with (row vector) for all , is the linear momentum conjugate to . The Hamiltonian system associated with (2.1) is the first-order system of ordinary differential equations on the phase space given by

 {˙q=∂pH=M−1pT˙pT=−∂qH=∇U(q). (2.2)

We shall consider simple mechanical systems with an -symmetry, meaning that the group acts properly on through isometries that leave the potential function unchanged. It follows that the Lagrangian and the Hamiltonian are -invariant under the natural lift of this action to and to , respectively.

### 2.1 Relative equilibria

Among all the solutions of Newton’s Equations (2.1), as already observed, the simplest are represented by a special class of periodic solutions called relative equilibria.

In the following and throughout all this paper, the matrix

 J2n\coloneqq(0−InIn0)

will denote the complex structure in , but it will always be written simply as , its dimension being clear from the context.

Let be the matrix representing the rotation in the plane with angular velocity . In order to rewrite Hamilton’s Equations (2.2) in a frame uniformly rotating about the origin with period , we employ the following symplectic change of coordinates:

 {x\coloneqqR(t)qyT\coloneqqR(t)pT

where is the block-diagonal matrix . Since a symplectic change of variables preserves the Hamiltonian structure, in these new coordinates System (2.2) is still Hamiltonian and transforms as follows:

 {˙x=∂yˆH=ωKx+M−1yT˙yT=−∂xˆH=∇U(x)+ωKyT (2.3)

where is the block-diagonal matrix and is the new Hamiltonian function given by

 ˆH(x,y)\coloneqq12⟨M−1yT,yT⟩−U(x)+ω⟨Kx,yT⟩. (2.4)

From the physical point of view, the term involving is due to the Coriolis force.

An equilibrium for System (2.3) must satisfy the conditions

 {ωKx+M−1yT=0∇U(x)+ωKyT=0,

which, taking into account that and that , can be rewritten as

 {yT=−ωMKxM−1∇U(x)+ω2x=0. (2.5)

Setting now , it is easy to see that the Hamiltonian defined in (2.4) coincides with the augmented Hamiltonian function

 HΞ(x,y)\coloneqqKΞ(x,y)−UΞ(x),

where

 KΞ(x,y)\coloneqq12∥∥M−1yT+Ξx∥∥2M

is the augmented kinetic energy and

 (2.6)

is called the augmented potential function. In terms of these augmented quantities, System (2.5) becomes

 {yT=−MΞx∇UΞ(x)=0

and we have the following definition.

###### Definition 2.1.

The point is a relative equilibrium for Newton’s Equations (2.1) with potential if both the following conditions hold:

1. ;

2. is a critical point of the augmented potential function .

Let us now consider the autonomous Hamiltonian System (2.3) in : by grouping variables into , it can be written as follows:

 ˙z(t)=−J∇ˆH(z(t)). (2.7)

Linearising it at the relative equilibrium , we obtain the linear autonomous Hamiltonian system

 ˙z(t)=−JBz(t), (2.8)

where is the constant symmetric matrix given by

 B\coloneqq(−D2U(¯x)ΞTΞM−1). (2.9)

### 2.2 Linear and spectral stability for autonomous Hamiltonian systems

We now recall some basic definitions and well-known facts about the linear stability of autonomous Hamiltonian systems, starting with the definition of the symplectic group and its Lie algebra. The reader is invited to consult, for instance, [1] for more details.

The (real) symplectic group is the set

 Sp(2n,R)\coloneqq\SetS∈Mat(2n,R)STJS=J.

Symplectic matrices correspond to symplectic automorphism of the standard symplectic space , where is the standard symplectic form represented by via the standard inner product of , i.e.  for every .

By differentiating the equation and evaluating it at the identity matrix, we find the characterising relation of the Hamiltonian matrices: the Lie algebra of the symplectic group is defined as

 sp(2n,R)\coloneqq\SetH∈Mat(2n,R)HTJ+JH=0,

and its elements are called Hamiltonian or infinitesimally symplectic.

###### Remark 2.2.

Since is a matrix Lie group and is its Lie algebra, the exponential map coincides with the usual matrix exponential, and therefore we have that is a Hamiltonian matrix if and only if is symplectic. It follows that if and only if .

The next proposition recollects the symmetries of the spectra of Hamiltonian and symplectic matrices.

###### Proposition 2.3.

The characteristic polynomial of a symplectic matrix is a reciprocal polynomial. Thus if is an eigenvalue of a real symplectic matrix, then so are , , .

The characteristic polynomial of a Hamiltonian matrix is an even polynomial. Thus if is an eigenvalue of a Hamiltonian matrix, then so are , , .

###### Proof.

See [17, Proposition 3.3.1]. ∎

###### Remark 2.4.

It descends directly from Proposition 2.3 that the spectrum of a Hamiltonian matrix is, in particular, symmetric with respect to the real axis of the complex plane. Moreover, has always even (possibly zero) algebraic multiplicity as a root of the characteristic polynomial of .

We now present the definition of spectral and linear stability for Hamiltonian matrices, in view of the fact that these are the ones on which we shall focus in our analyses.

###### Definition 2.5.

A Hamiltonian matrix is said to be spectrally stable if , whereas it is linearly stable if and in addition it is diagonalisable.

This concept is easily adapted to symplectic matrices by using the exponential map, as explained in Remark 2.2, and by remembering that the imaginary axis of the complex plane is the Lie algebra of the unit circle in the same plane (cf. Remark 2.4). Indeed, a symplectic matrix is said to be spectrally stable if and, as before, the property of linear stability requires in addition the diagonalisability of .

A linear autonomous Hamiltonian system in has the form

 ˙ζ(t)=JAζ(t), (2.10)

where is a symmetric matrix. Being it autonomous, its fundamental solution can be written in the explicit form

 γ(t)\coloneqqexp(tJA).

The definition of spectral and linear stability for this kind of systems is given in accord with Definition 2.5.

###### Definition 2.6.

The linear autonomous Hamiltonian System (2.10) is spectrally (resp. linearly) stable if the symplectic matrix corresponding to its fundamental solution at time is spectrally (resp. linearly) stable. We say that System (2.10) is degenerate if , or equivalently if , and non-degenerate otherwise.

We conclude the subsection by reporting a criterion for linear stability of symplectic matrices, in order to complete our brief recollection of definitions and results on this topic. We also point out that we are not aware of any existing proof of this lemma. In the following, the symbol will denote the norm of a bounded linear operator from the Hilbert space to itself.

###### Lemma 2.7.

A matrix is linearly stable if and only if

 supm∈N∥Sm∥L(R2n)<+∞.
###### Proof.

If is linearly stable, then in particular it is similar to a diagonal matrix through an invertible matrix , so that we have

 supm∈N∥Sm∥L(R2n)=supm∈N∥∥P−1DmP∥∥L(R2n)⩽∥∥P−1∥∥L(R2n)∥P∥L(R2n)supm∈N∥Dm∥L(R2n)=∥∥P−1∥∥L(R2n)∥P∥L(R2n)<+∞,

where the last equality holds true because all the eigenvalues of (and hence those of ) lie on the unit circle.

Vice versa, if is not linearly stable then it is not spectrally stable or it is not diagonalisable (or both). If it is spectrally unstable there exists, by definition, at least one eigenvalue , and we can assume, by the properties of the spectrum of symplectic matrices, that . Writing in its Jordan form (possibly diagonal) and computing yields on the diagonal a power , whose modulus diverges as . Hence . If is not diagonalisable, then there exists at least one Jordan block of size (say) relative to the eigenvalue . Its -th power has the form

 ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣λmmλm−10200λmmλm−10…066020λmmλm−1030λm⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,

and therefore even in this case (regardless of the fact that or not) the norm of tends to as goes to . ∎

## 3 Auxiliary results

In this section we present the lemmata and the propositions needed in the proof of the main results in Section 4. We first introduce some notation and definitions; for further properties we refer to Appendix A.

### 3.1 Notation and definitions

Let be, throughout all this paper, a finite-dimensional complex Hilbert space (we shall specify its dimension when needed). We denote by the Banach algebra of all (bounded) linear operators and by the subset of all (bounded) linear self-adjoint operators on . For a subset , the writing indicates the set of all invertible elements of .

###### Definition 3.1.

For any , we define its index , its nullity and its coindex as the numbers of its negative, null and positive eigenvalues, respectively. Its extended index and the extended coindex are defined as

 n−\textupext(T)\coloneqqn−(T)+ν(T),n+\textupext(T)\coloneqqn+(T)+ν(T).

The signature of is the difference between its coindex and its index:

 sgn(T)\coloneqqn+(T)−n−(T).
###### Remark 3.2.

We shall refer to the index of a self-adjoint operator also as its Morse index, which will be denoted by .

###### Definition 3.3.

Let be a topological space, a subspace and , with . We denote by the set of all continuous paths with endpoints in . Instead of we simply write . Two paths are said to be (free) homotopic if there is a continuous map which satisfies the following properties:

1. , ;

2. , for all .

The set of homotopy classes in this sense is denoted by .

###### Remark 3.4.

Note that the endpoints are not fixed along the homotopy; however, they are allowed to move only within .

Taking into account [13, Corollary 3.7], we are entitled to give the following definition:

###### Definition 3.5.

Let , with , and let . We define its spectral flow on the interval as:

 sf(T,[a,b])\coloneqqn+\textupext(T(b))−n+\textupext(T(a)).
###### Remark 3.6.

It is worthwhile noting that

 sf(T,[a,b])=n−(T(a))−n−(T(b)).

We now switch to introduce the key notion of crossing.

###### Definition 3.7.

Let , with , and let . A crossing instant (or simply a crossing) for the path is a number for which is not injective. We define the crossing operator (also called crossing form) of with respect to the crossing by

 Γ(T,t∗)\coloneqqQ˙T(t∗)Q∣∣kerT(t∗), (3.1)

where denotes the orthogonal projection onto the kernel of . A crossing is called regular if the crossing form is non-degenerate. We say that the path is regular if each crossing for is regular.

###### Remark 3.8.

The computation of the spectral flow of a path of operators involves the signature of the crossing form. We point out here that we actually refer to the signature of the quadratic form associated with the linear map defined in (3.1), that is, we make the following implicit identification. Given an endomorphism on a vector space , it is associated in a natural way with a bilinear form defined by

 BΓ(u,f)\coloneqqf(Γu),

where is an element of the dual space of . Since one can then define

 BΓ(u,v)\coloneqqvTΓu.

The quadratic form associated with is thus the quadratic form associated with . This is the justification for the abuse of language and notation that the reader will encounter throughout the paper.

As last piece of information, we point out that in the rest of the paper we shall denote the matrix by .

### 3.2 Relationships among linear stability, spectral flow and partial signatures

Here are the properties and facts that we shall exploit later to prove our main theorem. In this subsection we identify the Hilbert space with and consider the affine path defined by

 D(t)\coloneqqA+tG,

where is a real symmetric matrix (hence is Hamiltonian). Without different indication, it will be understood that , and are as defined above.

Thanks to the identification , we implicitly fix the canonical basis of and therefore every operator in is represented by a complex Hermitian matrix.

We explicitly note that the spectral flow does not depend on the particular inner product chosen but only on the associated quadratic form (see [10]).

###### Lemma 3.9.

Assume that is linearly stable.

Then if is singular there exist and such that

1. The instant is the only crossing for the path on ;

2. for all ;

3. is an even number.

If is non-singular there exists such that

1. for all ;

2. is an even number.

###### Proof.

Since is symmetric, the matrix is Hamiltonian. Therefore its spectrum is symmetric with respect to the real axis of the complex plane and (which is equal to because is an isomorphism) is even-dimensional, being diagonalisable. Furthermore, due to the Krein properties of (see Subsection A.3), the crossing form is always non-degenerate on each eigenspace .

Hence the hypotheses of Proposition A.5 or of Corollary A.6 (depending whether is invertible or not) are fulfilled and this proof reduces to the corresponding one in Appendix A. ∎

###### Proposition 3.10.

Assume that is an isolated (possibly non-regular) crossing instant for the path . Then, for small enough,

 sf(D,[t∗−ε,t∗+ε])=sgnB1,

where

 B1\coloneqq⟨G⋅,⋅⟩∣∣Ht∗

and is the generalised eigenspace given by

 Ht∗\coloneqq4n⋃j=1ker(GA+t∗I)j.
###### Proof.

We observe that for

 D(t)=t(1tA+G)\eqqcolont˜D(1t).

Clearly, the spectral flow is invariant by multiplication of a path for a positive real-analytic function:

 sf(D,[t∗−ε,t∗+ε])=sf(˜D,[1t∗−ε,1t∗+ε]).

Using now Proposition A.11, with and , we obtain the thesis (observe that the difference in sign to the local contribution to the spectral flow is due to the change of variable ). ∎

We now prove the main result of this section by means of the theory of partial signatures (see Subsection A.2).

###### Theorem 3.11.

If is spectrally stable, then is even.

###### Proof.

If we write

 D(t)=−J(JA−itI)

we see that is a crossing instant for if and only if

 it∗∈σ(JA)∩i[0,+∞).

Indeed, since is an isomorphism,

 kerD(t)=ker(JA−itI)∀t∈[0,+∞),

and thus there is a bijection between the set of crossing instants of and the set of pure imaginary eigenvalues of of the form . Being an affine path, it is real-analytic, and the Principle of Analytic Continuation implies that every crossing (be it regular or not) is isolated, because it can be regarded as a zero of the (real-analytic) map .

Let us examine the strictly positive crossings. By Proposition 3.10, in a suitable neighbourhood with radius around a crossing we see that

 sf(D,[t∗−δ,t∗+δ])=sgnB1,

where and are as in the aforementioned proposition. Furthermore, by the general theory of the Krein signature (see Subsection A.3), for any crossing the restriction of the Krein form to each generalised eigenspace is non-degenerate. In particular, Remark A.12 yields

 sf(D,[t∗−δ,t∗+δ])≡dimHt∗mod2. (3.2)

for every strictly positive crossing instant .

When turning our attention to the instant , we have to distinguish two situations: one where is singular and one where it is not. Let us start with the former and assume that is non-invertible, so that is a crossing for the path . Since this is isolated, by arguing as in the proof of (T2) in Proposition A.5 we can find and such that the path has only as crossing instant on and for every . Thus, recalling Remark 3.6 and the fact that , we obtain

 sf(D,[ε,T])=n−(D(ε))−n−(D(T))=n−(D(ε))−2n. (3.3)

We observe that the dimension of the generalised eigenspace (which coincides with the algebraic multiplicity of the eigenvalue ) is even, being Hamiltonian. Intuitively speaking, then, since the Krein form is non-degenerate on this subspace, the null eigenvalues move from as leaves ; and since its signature at the initial instant is (by Krein theory, see Appendix A, page A.3), they split evenly: half become positive and half negative. This justifies the choice of so small that

 n−(D(ε))=n−(A)+dimH02. (3.4)

On the other hand, we have

 4n=2 ∑\mathclapit∗∈σ(JA)∩i(0,+∞) dimHt∗+dimH0,

or, equally well,

 2n−dimH02= ∑\mathclapit∗∈σ(JA)∩i(0,+∞) dimHt∗. (3.5)

By Equation (3.2) and by the concatenation axiom defining the spectral flow, we get

 sf(D,[ε,T])≡ ∑\mathclapit∗∈σ(JA)∩i(0,+∞) dimHt∗mod2, (3.6)

and comparing (3.5) and (3.6) we infer

 sf(D,[ε,T])≡−dimH02≡dimH02mod2. (3.7)

Equations (3.3) and (3.4) also yield

 sf(D,[ε,T])≡n−(A)+dimH02mod2, (3.8)

and from the last two congruences (3.7) and (3.8), we finally conclude that

 n−(A)≡0mod2.

In the case where is invertible, the initial instant is not a crossing and therefore we can repeat the previous discussion in a simpler way, by considering the spectral flow directly on the interval (cf. Corollary A.6). ∎

The following corollary is a direct consequence of Theorem 3.11; however, since the case is much simpler and does not require in fact the partial signatures, we give an independent proof. In this special case in which the matrix is diagonalisable the result can be proved directly by arguing as in Proposition A.11 and by taking into account the local contribution to the spectral flow as discussed in Lemma A.3.

###### Corollary 3.12.

If is invertible and is linearly stable, then is even.

###### Proof.

First we observe that the second assumption implies that there is a bijection between the crossing instants and the pure imaginary eigenvalues of of the form for positive real . Let us then compute the crossing form in correspondence of a crossing : by definition it is given by

 Γ(D,t∗)\coloneqqQ˙D(t∗)Q∣∣kerD(t∗)=QGQ∣∣kerD(t∗),

where is the orthogonal projection onto the kernel of . Note that the linear map coincides (in the sense of Remark 3.8) with the quadratic Krein form:

 Γ(D,t∗)[u]=⟨Gu,u⟩,∀u∈Eit∗(JA),

since for every crossing . By Krein theory and by the fact that is diagonalisable, for any crossing instant the Krein form is non-degenerate on each eigenspace and by Proposition A.5 there exists such that for every . Thus we get

 sf(D,[0,T])=n−(A)−n−(D(T))=n−(A)−2n.

Since is diagonalisable we have

 4n=2∑\mathclapλ∈σ(JA)∩i(0,+∞) dimEλ,

or, which is the same,

 2n=∑\mathclapλ∈σ(JA)∩i(0,+∞) dimEλ

Equation (A.5) applied to the path yields

 sf(D,[0,T])≡∑\mathclapλ∈σ(JA)∩i(0,+∞) dimEλmod2

and we conclude that

 n−(A)≡0mod2.\qed
###### Remark 3.13.

We observe that Corollary 3.12 can be proved without using the technique of partial signatures also in the case where is not invertible. In order to take care of the crossing instant it is enough to argue as in the proof of Theorem 3.11, with the only difference that, assuming diagonalisability, coincides with the kernel of (and, consequently, the kernel of ).

## 4 Main theorem

We state and prove here the main result of our research, concerning the relationship between the Morse index of a critical point and the spectral instability of an associated relative equilibrium.

Consider the matrix defined in (2.9) and set

 N\coloneqq(IMΞ0I)(−D2U(¯x)ΞTΞM−1)(I0MΞTI)=(−(D2U(¯x)+ω2M)00M−1). (4.1)

Observe that is precisely the Hessian of the augmented potential evaluated at its critical point and define then the nullity and the Morse index of as:

 ν(¯x) \coloneqqν(D2UΞ(¯x)), i\textupMorse(¯x) \coloneqqi\textupMorse(D2UΞ(¯x)).

Thus we have the following theorem.

###### Theorem 4.1.

Let be a critical point of the augmented potential function defined in (2.6) and assume that is even. If is odd, then the relative equilibrium corresponding to is spectrally unstable.

###### Proof.

Let and define the path as

 D(t)\coloneqqB+tG

with , as in the previous section. We prove the contrapositive of the statement, that is, we show that if the relative equilibrium corresponding to the given critical point is spectrally stable then its Morse index is even. Thus, assuming spectral stability, Theorem 3.11 immediately yields

 n−(B)≡0mod2.

Now, by Sylvester’s Law of Inertia, we observe that

 n−(B)=n−(N),

where is given by (4.1), and since , it directly follows that

 i\textupMorse(¯x)≡0mod2.\qed

The next corollary is an immediate consequence of the previous theorem.

###### Corollary 4.2.

Let be a critical point of the augmented potential function . If or are odd then the corresponding relative equilibrium is linearly unstable.

###### Remark 4.3.

Assuming linear stability we have that , which is even due to the diagonalisability of .

## 5 An important application: n-body-type problems

With reference to the notation and the setting outlined in the beginning of Section 2, we define two -body-type problems by specifying two potential functions as follows. For each pair of indices , , we let denote the collision set of the -th and -th particles

 Δij\coloneqq\Setq∈R2nqi=qj;

we call the collision set (by definition, then, is a union of hyperplanes) and the (collision-free) configuration space.

On this set (which is a cone in ) we define the potential functions (generally denoted by ) as

 Uα(q) \coloneqqn∑i,j=1i

From now on, unless otherwise specified, every reference to the contents of Section 2 will be intended as concerning these two potential, i.e. we consider .

###### Remark 5.1.

Note that for one finds the gravitational potential of the classical -body problem. Moreover, the logarithmic potential can be considered as a limit case of the -homogeneous3 one, in the following sense:

 Uα(q)−1α∼Ulog(q),α→0+,

for every . Nevertheless, it displays quite a different behaviour with respect to , as we shall show.

Since the centre of mass of the system moves with uniform rectilinear motion, without loss of generality we can fix it at the origin, that is we can set . We thus consider the reduced (collision-free) configuration space as follows:

 ˆX\coloneqq\Setq∈Xn∑i=1miqi=0.
###### Remark 5.2.

We observe that the Hamiltonian flow of System (2.2) is well defined on but it is not complete on , due to the existence of solutions for which the potential escapes to infinity in a finite time. This happens, for instance, for initial conditions leading to a collision between two or more particles.

### 5.1 Central configurations and relative equilibria

We recall here some well-known facts about central configurations and fix our notation. For further references in the classical gravitational case, we refer to [19].

Let , with . We call a (planar) central configuration if there is some smooth real-valued function , with for all , such that

 q(t)\coloneqqr(t)¯q (5.2)

is a (classical) solution of Newton’s Equations (2.1). Here represents the constant shape of the configuration, while its time-depending size. Substituting (5.2) into (2.1) we obtain:

-homogeneous case:

 ¨rM¯q=r−(α+1)∇Uα(¯q).

Taking the scalar product with in both sides of the above equality and applying Euler’s theorem on homogeneous functions, we get , where

 λα\coloneqqαUα(¯q)I(¯q). (5.3)

Logarithmic case:

 ¨rM¯q=r−1∇Ulog(¯q).

Taking again the scalar product with as before, we get , where

 λlog\coloneqq−⟨∇Ulog(¯q),¯q⟩I(¯q).

A straightforward computation shows that , so that

 λlog=MI(¯q). (5.4)
###### Remark 5.3.

It is worthwhile noting that in the logarithmic case the Lagrange multiplier depends only on the size of the central configuration (via the moment of inertia) and not on its shape.

In both cases, a central configuration satisfies the central configurations equation

 M−1∇U(¯q)+λ¯q=0, (5.5)

where (resp. ) when (resp. ). Thus we can also look at a central configuration as a special distribution of the bodies in which the acceleration vector of each particle lines up with its position vector, and the proportionality constant is the same for all particles. Equation (5.5) is a quite complicated system of nonlinear algebraic equations and only few solutions are known.

Let us now introduce the ellipsoid of inertia (also called the standard ellipsoid)

 S\coloneqq\Setq∈ˆXI(q)=1.

If is a central configuration, then so are and , for any and any block-diagonal matrix with blocks given by a fixed matrix in . We observe that the rescaled configuration solves a system analogous to (5.5) obtained by replacing with and with . Because of these facts, it is standard practice to count central configurations by fixing a constant (the “scale”: this actually means to work on ) and to identify all those which are rotationally equivalent. This amounts to take the quotient of the configuration space with respect to homotheties and rotations about the origin, or, which is the same, to consider the so-called shape sphere

 S\coloneqqS/SO(2).

Note that the second equation of System (2.5) (with