Morse index and stability of the planar N-vortex problem

This paper concerns the investigation of the stability properties of relative equilibria which are rigidly rotating vortex configurations sometimes called vortex crystals, in the N-vortex problem. Such a configurations can be characterized as critical point of the Hamiltonian function restricted on the constant angular impulse hypersurface in the phase space (topologically a pseudo-sphere whose coefficients are the circulation strengths of the vortices). Relative equilibria are generated by the circle action on the so-called shape pseudo-sphere (which generalize the standard shape sphere appearing in the study of the N-body problem). Inspired by the planar gravitational $N$-body problem, and after a geometrical and dynamical discussion, we investigate the relation intertwining the stability of relative equilibria and the inertia indices of the central configurations generating such equilibria. In the last section we apply our main results to some symmetric three and four vortices relative equilibria.

1 Introduction and description of the problem e study of vortex dynamics can be traced back to Helmholtz's work on hydrodynamics in 1858 [Hel1858] and it plays an important role in the study of superfluids, superconductivity, and stellar systems. Its Hamiltonian formulation could be dated back to Kirchhoff in the plane, and later on generalized by Routh in [Rou1880] and then Lim [Lim43] to general domains in the plane. In this paper, we are interested to the problem in the first order Hamiltonian system of the form Γ iżi (t) = J∇ zi H(z(t)) i ∈ 1, . . . , N. (1.1) Here J := 0 1 −1 0 is the standard symplectic matrix in the Euclidean plane. e Hamiltonian function H is Here Γ 1 , . . . Γ N ∈ R \ {0} are the vorticities or vortex strengths. e Hamiltonian it is defined on the configuration space F N (R 2 ) := z ∈ R 2N z i = z j for i = j of the N (coloured) points in the plane. It is clear by the definition that H(z 1 , . . . , z N ) becomes singular if |z i − z j | → 0 for some i = j. Se ing G(w 1 , w 2 ) = − log |w 1 − w 2 | then the Hamiltonian can also be wri en as H(z) = i<j Γ i Γ j G(z i , z j ) and it is usually called hydrodynamic Green's function. As already observed, the Hamiltonian system in Equation (1.1) appear as singular limit equations in problems from physics. More precisely in fluid dynamics is derived from the Euler equation and for instance in superconductivity H appears as renormalized energy for Ginzburg-Landau vortices. Concerning the existence and stability properties of periodic solutions of the N-vortex problem given in Equation (1.1), the literature is quite broad and we refer the interested reader to [New01] and references therein. Among the simplest periodic orbits of the planar N -vortex problem are the relative equilibria. ese configurations of vortices rotates rigidly about their center of vorticity and sometimes are referred to as vortex crystals and are frequently observed in natural phenomena (e.g. the hurricanes). Relative equilibria are crucial in deeply understanding the intricate dynamics of this singular Hamiltonian problem and as the name suggested are rest-points in a suitable rotating coordinate system. As we'll discuss in Section 2, relative equilibria can be characterized as critical points (or more precisely critical orbits) of the restriction of the Hamiltonian to the angular impulse unitary (pseudo-)sphere of the phase space. Otherwise stated such a rigid configurations are generated through a rotation with angular velocity ω of a special critical configuration of the system usually called central configuration. (Cfr. Section 2 for further details).
A natural and classical problem is to understand how the spectral properties of these central configurations or more precisely the inertia indices of the Hessian of H at these configurations reflect on the dynamical properties of the generated vortex crystal (through rotation) like, for instance, spectral or linear stability properties etc. is problem is very classical in the gravitational N -body problem in which central configurations are characterized as critical points of the self-interacting potential on the shape sphere (which is the base space of the circle bundle) whose total space is the inertia ellipsoid. ere is a long standing conjecture due to Moeckel stating that a linearly stable relative equilibrium must be a nondegenerate minimum of the Newtonian potential restricted to the shape sphere. e other direction is false even for other class of weakly a racting singular potentials. (Cfr., for instance, [HLS14,BJP16]). e investigation of the relation between the stability properties of a relative equilibrium and the spectral properties of the central configuration generating such an equilibrium in the N-vortex problem is pre y different. Despite of the fact that the circulation strengths could have any sign (in the classical gravitational N -body problem they correspond to the masses which are all positive), the Hessian of the Hamiltonian computed at a central configuration has some commutativity properties with respect to the Poisson matrix K induced by J that greatly simplify the problem. (Cfr. Lemma 2.5, for further details). Such a property was observed by Roberts in its interesting paper of Roberts in [Rob13]. In the aforementioned paper, in fact, the author was able among others to characterized in the case of positive circulation strengths, the linearly stable relative equilibria of the N -vortex problem as nondegenerate minima of the Hamiltonian H restricted to the shape sphere.
is result was the starting point of our analysis and motivated us to investigate what is the effect of mixed sign circulation strengths, which a er all, are very common in the applications.
Before describing our main results we start to observe that this indefinite case, the stability analysis is much more delicate. is situation, as we'll try to clarify, reflects somehow the difficulties and it is the paradigm of the difference between the Riemannian and the Lorentzian world.

Main results
Our first result, provides a characterization of the spectral stability of a relative equilibrium z in terms of a spectral condition on the central configuration ξ, no ma er how the signs of the circulations are. Before stating and describing our first result, we pause by recalling what stability notion we are talking about. Being the Hamiltonian H invariant under translations and rotations this implies, among others, that 0 (having algebraic multiplicity 2) as well as ±ωi are Floquet characteristic multipliers, arising precisely from these symmetries.
What is natural to do is to define the linear stability properties of a relative equilibrium by ruling out the eigenvalues coming from these conservation laws. More precisely, we define linear stability by restricting to a complementary subspace of the invariant space defined by the above symmetries.
Definition 1.1. A relative equilibrium z will be termed non-degenerate provided the remaining 2n − 4 eigenvalues of the matrix B are not vanishing. A non-degenerate relative equilibrium is • spectrally stable if the nontrivial eigenvalues are purely imaginary • linearly stable if, in addition, the restriction of the stability matrix B to W ⊥ has a block-diagonal Jordan form with blocks 0 β i −β i 0 .
Remark 1.2. Otherwise stated a relative equilibrium z is spectrally stable if all Floquet multipliers (the eigenvalues of the monodromy matrix) belongs to the unit circle (centered at the origin) U in the complex plane. Furthermore if the monodromy matrix is also diagonalizable, then z is linearly stable. In this last case, in fact, the monodromy matrix can be factorized as direct symplectic sum of rotations or which is the same, it belongs to the maximal compact Lie subgroup of Sp(2N − 4).
T 1. A non-degenerate relative equilibrium z (with angular velocity ω) generated by the central configuration ξ is spectrally stable if and only if for every eigenvalue µ of M −1 Γ D 2 H(ξ) one of the following alternative holds where M Γ := Γ i I 2 δ ij (where I 2 denotes the 2 × 2 identity matrix and δ ij is the Kronecker delta). In particular it is non-degenerate if and only if µ = ±ω. e idea for proving this result is essentially based on the relation between the matrix M −1 Γ D 2 H(ξ) and the so-called stability matrix, namely the matrix which is responsible of the stability of the relative equilibrium which is defined by where K is the Poisson matrix, i.e. a 2 × 2 block diagonal matrix in which each non-vanishing block is given by J.
It is worth to observe that, no ma er how the signs of the circulations are, the matrix appearing in eorem 1 i.e. M −1 Γ D 2 H(ξ) is M Γ -symmetric namely is symmetric with respect to the scalar product induced by M Γ (cfr. Definition 2.11 for further details). However, if the circulations are all positive such a scalar product is positive definite and this implies that the spectrum of M −1 Γ D 2 H(ξ) is diagonalizable in the orthogonal group and its spectrum is real. For mixed signs circulations, this is not true anymore, and in fact the spectrum of M −1 Γ D 2 H(ξ) will be, in general, not real anymore. is reflects the indefinite Krein structure behind and among others responsible of the presence of Jordan blocks that are intimately related to the spectral stability properties of the relative equilibrium. In conclusion eorem 1 represents the generalization of [Rob13, eorem 3.1] in the case of mixed signs circulations. e matrix B is M Γ -Hamiltonian, namely is Hamiltonian with respect to the symplectic form ω Γ which is represented by K with respect to the M Γ -scalar product.
Our second main result relates the spectrally stability properties of a relative equilibrium and the inertia indices of the central configuration generating it.
T 2. Let z be non-degenerate relative equilibrium generated by the central configuration ξ and let A(ξ) := D 2 H(ξ) + ωM Γ . We assume that z is spectrally stable. en the following result holds.
• Case of positive angular velocity ω • Case of negative angular velocity ω Given N -vortices in the plane, we define total vortex angular momentum L as L := i<j Γ i Γ j . us, if the vortex strengths are all positive, then L > 0. However, when vorticities are different in signs, then L could be of any sign or even vanishes. Analogously to the moment of inertia in the N -body problem, it is possible to define the so-called angular-impulse of the N -vortex problem as follows and, as we will see in the sequel, it will be crucial in order to give a variational interpretation to the central configurations, miming the analogous interpretation in the N -body problem. As already observed, for a relative equilibrium z generated by ξ, the angular velocity is constant and it is given by us in the case of mixed signs circulations the angular velocity could have any sign what that cannot happen in the case of constant sign circulations. e proof of eorem 1 and eorem 2 will be given in Section 4. In the last section, we analyze some interesting symmetric central configuration; more precisely, the equilateral triangle and the rhombus (sometimes called kite) central configuration.
e equilateral triangle central configuration in the three-vortex problem is obtained by placing three vortices of any strength at the vertices of an equilateral triangle. Synge in his celebrated paper published in 1949 (cfr. [Syn49] ), proved that the corresponding relative equilibrium is linearly stable if and only if L > 0.
Starting from this we get information on the a central configuration knowing the stability of the induced relative equilibrium. More precisely, let us given three circulations Γ 1 , Γ 2 , Γ 3 placed at the following points and we let c = 3 i=3 Γ i z i . Assuming that L > 0 and se ing ξ = (ξ 1 , ξ 2 , ξ 3 ), for ξ i = ξ i − c then we conclude that About the kite central configuration, it is know (cfr. [Rob13] for further details) that there exist two families of relative equilibria where the configuration is a rhombus. Set Γ 1 = Γ 2 = 1 and Γ 3 = Γ 4 = m, where m ∈ (−1, 1] is a parameter. Place the vortices at z 1 = (1, 0), z 2 = (−1, 0), z 3 = (0, y) and z 4 = (0, −y), forming a rhombus with diagonals lying on the coordinate axis.
is configuration is a central configuration provided that (1.2) e angular velocity is given by Taking plus sign in Equation (1.2) yields a solution for m ∈ (−1, 1] that always has ω > 0. We refer to this solution as rhombus A. Taking − in Equation (1.2) yields a solution for m ∈ (−1, 0) having ω > 0 for m ∈ (−2 + √ 3, 0), but ω < 0 for m ∈ (−1, −2 + √ 3). We refer to this solution as rhombus B. Assuming that z is the relative equilibrium generated by the rhombus central configuration ξ.

Notation
At last, let us introduce some notation that we shall use henceforth without further reference. We have already mentioned that I stands for the angular impulse , however, the similar symbol I X or just I will denote the identity operator on a space X and we set for simplicity I k := I R k for k ∈ N. We denote throughout by the symbol # T (resp. # −T ) the transpose (resp. inverse transpose) of the operator #. Mat(m, n; K) stands for the space of m×n matrices in the field K and if m = n we just use the short-hand notation Mat(m; K). σ(#) denotes the spectrum of the linear operator #. We denote throughout by J the standard symplectic matrix J := 0 1 −1 0 . U denotes the unit circle in the complex plane namely the set of all complex numbers of modulus 1. If Z is a finite dimensional vector space. We denote by L (Z) the vector space of all linear operators on Z.
Acknowledgements e second named author wishes to thank all faculties and staff of the Mathematics Department in the Shandong University (Jinan) for providing excellent working conditions during his stay.
For i ∈ {1, . . . , N }, let Γ i ∈ R \ {0} representing the vortex strength of the N -point vortex z i ∈ V . We will assume throughout that the total circulation Γ := N i=1 Γ i is nonzero. e center of vorticity is then well-defined as c := Γ −1 N i=1 Γ i z i . Let H : V N → R be the function defined as follows In what follows we refer to H as the N -vortex Hamiltonian function. Denoting by I N the N × N identity matrix, we define the matrix of circulations as the real 2N × 2N matrix given by and the symplectic matrix K : Its complement in V N is the collision set It is immediate to check that the restriction of H to F N (V ) is indeed a smooth function.

Central configurations
denote the circulation scalar product of v and w, where v i · w i denotes the standard Euclidean product in V of the i-th component of v and w. We observe that, if the vortex strengths are all positive, then ·, · Γ is, actually, an inner product equivalent to the Euclidean one; otherwise, is an indefinite (non-degenerate) scalar product. 1 We also notice that v, w Γ = w T M Γ v where · T denotes the transpose with respect to the Euclidean product. Given I 0 ∈ R, we define the pseudo-sphere S N (V ) and we'll refer to as circulation (pseudo)-sphere or circulation sphere for short as In particular the circulation sphere is equal to the sphere (with respect to the circulation scalar product) in V N with collisions removed; thus Remark 2.1. It is worth noticing that if vortex strengths are all positive, then the pseudo-sphere is in general an ellipsoid (thus, topologically a sphere) and if are all equal it reduces to the round sphere. In the general case, however, it is a (non-compact) quadric.
A central configuration for the the N -vortex problem is a (non-collision) configuration ξ ∈ F N (V ) with the property that exists ω ∈ R such that ∇H(ξ) + ω∇I(ξ) = ∇H(ξ) + ωM Γ (ξ) = 0. (2.2) for each j = 1, . . . , N . It is easy to check that C is an isometry with respect to the ·, · Γ . In fact, it holds We observe that H(Cξ) = H(ξ) and by the computation performed in Equation (2.3), we conclude immediately that I(ξ ′ ) = I(ξ). By these two fact readily follows that if ξ is a central configuration then also Cξ is a central configuration. Now, by using Equation (2.2), we can conclude that Cξ = ξ and hence ξ = 0. us, if ξ is a central configuration, then its center of vorticity c = 0. As consequence of this discussion and without leading in generalities in the sequel we'll restrict to the reduced phase space which is the 2(N − 1)-dimensional subspace of V N defined by By using once again Equation (2.2) it follows in fact, that a central configuration can be seen as a critical point of the Hamiltonian function restricted to a level surface of the angular impulse in which ω acts as a Lagrangian multiplier. Note that, ifz is a central configuration, so is λz for any scalar λ. In this case, the parameter ω must be scaled by a factor 1/λ 2 .
We observe that by the invariance property of the Hamiltonian function as well as of the angularimpulse, we get that central configurations are not isolated and appears in a continuous family. To eliminate such a degeneracy, it is customary to fix a scaling (e.g. I = I 0 ) and to identify central configurations that are identical under rotations.
Remark 2.2. It is also worth noticing that the linear stability properties of such rigid motions are not affected by such rotation.
We also define the following sets S c N := S N ∩ X and S c N := S N ∩ X.
By the above discussion, in particular we get that if z is a central configuration then z ∈ X. However, in principle, a critical point of the restriction of H| S c N is not necessarily a critical point of H| SN . However since the Hamiltonian function is C-invariant and being X the space fixed by the action of the (compact Lie) orthogonal group of V N , it follows that any critical point of the restriction of H| S c N is indeed a critical point of H| SN (cf. [Pal79], for further details). However, as already observed critical points of H| S c N are not isolated. In fact, if z 0 is a critical point of H| S c N , then e ϑK z 0 is, for every ϑ. In order to eliminate this further degeneracy, we consider the quotient spaces S c N := S c N /S 1 and S c N := S c N /S 1 and we'll refer to the shape sphere without collision and the shape sphere respectively. It is worth noticing that both are the orbit space of the circle action on the spheres S c N and S c N , respectively. In what follows, we'll refer to a central configuration as the critical point of H S c N in order to distinguish from critical points of H S c N usually called relative equilibria.

Relative equilibria
A system of N point vortices (in the plane) with vortex strength Γ i = 0 and positions z i ∈ V evolves according to the phase flow induced by the following Hamiltonian system where the Hamiltonian function H is defined in Equation (2.1), and ∇ i denotes the two-dimensional partial gradient with respect to z i . We will assume throughout that (H) the total circulation Γ = N i=1 Γ i is nonzero and that the center of vorticity c = 0. In short-hand notation Equation (2.4), could be rewri en in the following form where K is the real 2N × 2N matrix given by A special class (maybe the easiest) of periodic solutions for this problem is given by the rigid motions of the system around its center of mass. Such a motions are termed relative equilibria. More precisely we introduce the following definition.
Definition 2.3. We term relative equilibrium (RE, for short) any T := 2π/|ω| periodic solution of Equation (2.5), namely Remark 2.4. By this definition, as already observed, it follows that a relative equilibrium is a periodic solution in which each point vortex uniformly rotates with angular velocity ω = 0 around (its common center of vorticity represented by) the origin.
By a direct computation and by using Equation (2.6), it follows that the central configuration ξ generating a relative equilibrium satisfy the following equation us in a properly rotating frame a relative equilibrium is nothing but a central configuration.
e following result points out some crucial properties of the Hamiltonian function H that will be useful later on and we refer the interested reader to [Rob13, Lemma 2.3] for the proof.
Lemma 2.5. e Hamiltonian H has the following three properties: Remark 2.6. As we will see later on, property (iii) plays a crucial role in the investigation of the linear stability for relative equilibria. For all of the same sign vorticity strengths, such a condition reduces the problem to the investigation of the spectrum to a 2 × 2 symmetric matrix or equally well a complete factorization of the characteristic polynomial into even quadratic factors.
is property doesn't hold for relative equilibria of the N -body problem and in fact a challenging longstanding still open problem is to establish a precise relation between the dynamical properties of the relative equilibria and the spectral properties of central configurations originating them.
Differentiating with respect to z the equality appearing at first item in Lemma 2.5, we get Since ξ is a central configuration and by using once again Equation (2.2), we immediately get ∇H(ξ) = −ωM Γ ξ and by summing up we get the equality Equation (2.7) together with property (iii) Lemma 2.5 shows that BKx = 0. e equation of motions given in Equation (2.5), in a uniformly rotating frame with angular velocity ω reduces to In fact, let w(t) := e ωKt z(t); thus by a direct computation, we get where the commutativity properties of M Γ with respect to K and e K were tacitly used. In particular, a rest point of the Hamiltonian vector field appearing in Equation (2.8) is a relative equilibrium, as expected.
Remark 2.7. We observe that if the circulations have mixed sign then ω could be of any sign (meaning that the vortices can rotate clockwise or counterclockwise with respect to the center of vorticity). In fact, by taking the scalar product with respect to ξ in Equation (2.2) as well as invoking the first claim in Lemma 2.5, we get that where the last equality directly follows by using the Euler theorem on positively homogeneous functions a er observing that I is homogeneous of degree 2. Now, the claim follows by observing that a priori L could be of either positive or negative.
Remark 2.8. In a more geometrical way the Hamilton equations in the uniformly rotating frame are nothing but the Hamilton equation on the cotangent bundle T * S c N with the symplectic form induced by the standard symplectic form whose Hamiltonian vectorfield (i.e. the symplectic gradient) is defined by e variational equation associated to the Hamiltonian system given in Equation (2.8) is In particular if w(t) = e ωKt ξ is a relative equilibrium solution at the central configuration ξ, and the admissible variations belongs to the tangent along the fibers of the principal S 1 -bundle, then In fact, since H is invariant under rotation, it follows that H z(t) = H ξ . Now, by differentiating twice this last equality for w(t) = e ωKt ξ (here the admissible variations are of the form e ωKt u for u ∈ T S c N ), we get Inserting the expression given in Equation (2.10) into Equation (2.8) and se ing η = e ωKt ξ, we geṫ Following Roberts in [Rob13] we introduce the following definition.
is termed the stability matrix of the relative equilibrium z generated by ξ.
Definition 2.11. Let (R k , N ) be a (maybe indefinite) non-degenerate scalar product space on the real vector space R k and let G ∈ Mat(k, R). e matrix G is termed a N -symmetric matrix if where · T denotes the transpose with the respect to the Euclidean product. e matrix R ∈ Sp(2n, Ω) is termed N -Hamiltonian, if R T P N + N P R = 0 where P represents the symplectic form Ω with respect to the Euclidean product.
Remark 2.12. It is worth noticing that if N = I then the definitions of N -symmetric (resp. N -Hamiltonian) matrix, reduces to the standard definition of symmetric (resp. Hamiltonian) matrix with respect to the Euclidean scalar product (resp. canonical symplectic structure).
By a direct calculation follows that M −1 is last claim directly follows by Equation (2.7). Otherwise stated, the matrix B(ξ) is Hamiltonian with respect to the vortex symplectic form defined by Remark 2.13. We pause the exposition by introducing the following remark that explain why the mixed sign circulations case is really completely different from the constant sign circulations. It is well-known that the product of two symmetric matrices is symmetric as soon as the two matrices commute.
us, in general, the matrix M −1 Γ D 2 H(ξ)(ξ) whatever ξ is, and no ma er how the signs of Γ i arem not be symmetric. Clearly, if all circulations are equal then M Γ is just a multiple of the identity and, of course, is symmetric. However, if the circulations strengths Γ i are all positive, then the matrix M Γ is positive definite and in particular its spectrum is real. For mixed signs circulation strengths, however, M Γ -is (nondegenerate) but indefinite and its spectrum [GLR05, eorem 5.1.1, pag.74] is not necessarily real, anymore and this fact is responsible among others of some technicalities as well as a deep change in the dynamics of the problem.
We conclude this section by showing a nice and important block matrix structure of the Hessian matrix D 2 H(ξ).
is property comes from item (iii) in Lemma 2.5. Let ξ be a central configuration; thus ξ = (ξ 1 , ξ 2 , . . . , ξ n ) ∈ R 2n and let ξ ij = (ξ i − ξ j )/r ij . A direct computation shows that (2.11) Note that A ij = A ji and, for i = j where ξ = (x, y). e fact that J commutes with each A ij gives another proof of the fact that D 2 H(ξ) and K anti-commute.
Proof. From the conservation of the center of vorticity, by using Equation (2.11) and by a straightforward calculation we get that is implies that us s ∈ ker D 2 H(ξ). Furthermore, by the last claim in Lemma 2.5, we have D 2 H(ξ)Ks = −KD 2 H(ξ)s = 0 which implies that also Ks ∈ ker D 2 H(ξ).
e proof of the second claim directly follows by Equation (2.7) together with property (iii) Lemma 2.5. In fact, by a direct computation we get is concludes the proof.
By the first claim in Lemma 2.14 it then follows that the restriction of the stability matrix to the spectrum of B is precisely {±ωi}. For a given relative equilibrium z with corresponding central configuration ξ, let W = span{ξ, Kξ}. As already proved in Lemma 2.14 this is an invariant subspace for B and the restriction of B to W is given by In what follows, we denote by W ⊥ ⊂ V N the M Γ -orthogonal complement of W , that is, Lemma 2.16. e vector space W ⊥ has dimension 2n−2 and is invariant under B. If L = 0, then W ∩W ⊥ = {0}.
Proof. For the proof of this result, we refer the interested reader to [Rob13, Lemma 2.6].
As long as L = 0, Lemma 2.16 allows us to define the linear stability with respect to the M Γ -orthogonal complement of the subspace W . us, a relative equilibrium is spectrally (resp. linearly) stable if the restriction of the matrix B onto W ⊥ is spectrally (resp. linearly) stable according to Definition 1.1. However instead of working on a reduced phase space (eliminating the rotational symmetry), it is more convenient to work on the full phase space we investigate the linear stability properties of the orbits in the full space. However, in this case, by the invariance properties of H, it readily follows that 4 is the minimal possible nullity (or kernel dimension) of the corresponding linear differential operator.
3.1 Spectral properties of the stability matrix e aim of this subsection is to study the relation intertwining the spectrum of the stability matrix B and the spectrum of M −1 Γ D 2 H(ξ) (and hence of A). e first result, that we recall here for the sake of the reader, was proved by Roberts in [Rob13]. e following result relates the spectrum of B with the spectrum of the matrix M −1 Γ D 2 H(ξ) and will be a key ingredient for the stability analysis.

Canonical forms and invariant splitting of the phase space
is subsection is devoted to study the relation between the invariant subspaces of B (which are crucial for reducing the operator B and the generalized eigenspaces of A Γ ). Lemma 3.3 that we state below for the sake of the reader, was proved in [Rob13, Lemma 2.5].  Consequently, p(λ) has a quadratic factor of the form λ 2 + ω 2 − µ 2 .
(b) Suppose that v = v 1 + i v 2 is a complex eigenvector of M −1 Γ D 2 H(ξ) corresponding to the complex eigenvalue µ = α + i β.
en the span of the four vectors {v 1 , v 2 , Kv 1 , Kv 2 } is a real invariant subspace of B and the restriction of B to this subspace is given by  Consequently, p(λ) has a quartic factor of the form (λ 2 + ω 2 − µ 2 )(λ 2 + ω 2 − µ 2 ).
Proof. e proof of this result follows by a direct computation by using Lemma 3.1. (Cf. [Rob13, Lemma 2.5]).
In the case of mixed signs circulations, the matrix M −1 Γ D 2 H(ξ) is M Γ -symmetric with respect to an indefinite scalar product and this, among others, in particular implies that the spectrum is not real and M −1 Γ D 2 H(ξ) and hence A Γ are not semi-simple. In order to decompose the full space into B invariant subspaces it is then crucial to understand in which manner Lemma 3.3 can be carried over in this more general situation we are dealing with.
is is essentially the content of Lemma 3.4 and Proposition 3.5, below.
Lemma 3.4. Let {v i } l i=0 be a Jordan chain of A Γ with eigenvalue ν, namely Proof. Since KM −1 Γ D 2 H(ξ) = −M −1 Γ D 2 H(ξ)K (as directly follows by the third item in Lemma 2.5), then by a direct computation, we get is concludes the proof. Lemma 3.4 provides a constructive way to reduce the operator B (by decomposing the whole space into B-invariant subspaces).
is an invariant space for B.

ese two last equalities imply that the subspace generated by
is concludes the proof.
Notation 3.6. We introduce the following where p denotes by the order of this matrix.
If ν ∈ σ(A Γ ) we denote by E ν the (real) generalized spectral space corresponding to the eigenvalue ν.
Directly from Proposition 3.5 and by using notation above, we get that the restriction of B onto the subspace E ν := E ν ⊕ E ν−2ω can be represented by the following 2l × 2l matrix that in block matrix form can be wri en as follows

A Γ -invariant M Γ -orthogonal decomposition
Let us start to introduce the following symmetric matrix e rest of this section a bit technical in its own and the basic idea behind is to establish the behavior of the restriction of A onto some subspaces constructed through the spectral subspaces (maybe generalized spectral subspaces) of A Γ which, as consequence of Corollary 3.8, are A Γ -orthogonal. e next result provide a sufficient condition in order the generalized spectral subspaces relative to different and not conjugated eigenvalues to be M Γ -orthogonal. Lemma 3.7. Suppose that ν 1 , ν 2 ∈ σ(A Γ ). If ν 1 = ν 2 , then for every v 1 ∈ E ν1 and v 2 ∈ E ν2 we have Proof. We split the proof into two steps. First step. We assume that v 1 , v 2 are eigenvectors relative to the eigenvalues ν 1 and ν 2 respectively; thus, we have A Γ v i = ν i v i , i = 1, 2. So, we have since ν 1 = ν 2 , we get desired result. Second step. We assume that v 1 , v 2 are generalized eigenvectors and we consider the Jordan chains where v 0 1 = v 0 2 = 0. By arguing as above, we get that So, taking the difference between the two equalities in Equation (3.2), we get where the first equality follows by the fact that A Γ is M Γ -symmetric and {0, 1, . . . , p} and ∀ j ∈ {0, 1, . . . , q}. (3.4) To conclude the proof we argue by induction. Let i + j = k. So Equation (3.4) is trivially true fork = 0. Now, we suppose Equation (3.4) holds true for i + j = k l and we want to prove that it is true for i + j = k = l + 1. Now, by taking into account Equation (3.3) and being ν 1 = ν 2 , it readily follows that Equation (3.4) holds true. is concludes the proof.
In particular generalized eigenspaces relative to different and not conjugated eigenvalues areÂ Γ orthogonal.
For ν ∈ σ(A Γ ), we set Notation 3.9. We introduce the following notation. Given any subspace X ⊂ C 2n , we denote by n − ( A Γ | X ) (resp. n + ( A Γ | X )), the dimension of the maximal negative (resp. positive) spectral subspace of the restriction of the quadratic form A Γ ·, · onto X.
By the previous discussion, we can decompose the C 2n = R 2n ⊗ C C into A Γ -invariant , M Γ -orthogonal subspaces; thus we have C 2n = I ν1 ⊕ · · · ⊕ I ν l , where ν i are all distinct eigenvalues of A Γ with ℑν i 0.
Proof. We start to observe that ker A Γ ⊆ E 0 = ker A Γ 2 . Now, arguing by contradiction, we assume that A Γ ·, · | Iν is degenerate for some ν = 0. us there exists u ∈ I ν and u0 =, such that Since C 2n is the direct sum of all different I ν ′ , where ν ′ ∈ σ(A Γ ), this implies (by invoking by Lemma 3.7) that A Γ u, v = 0 for all v ∈ C 2n . So u ∈ ker A Γ and in hence u ∈ ker A Γ = E 0 . us ν = 0 which is a contradiction. is concludes the proof. e next result shed some light on the relation between the dimension of I ν and the Morse index .
Lemma 3.11. Let ν ∈ σ(A Γ ), ℑν > 0 and let m ν ∈ N be its algebraic multiplicity. en 2 Actually if M Γ is positive definite it can be proved that also the converse inclusion holds Proof. By using Lemma 3.7, the quadratic form A Γ ·, · on I ν = E ν ⊕ Eν can be represented in the block matrix form by 0 Y Y T 0 for some Y ∈ Mat(ν; C). Moreover, by Lemma 3.10 we infer that Y is non-degenerate and by this fact the conclusion readily follows.
By taking into account Corollary 3.8 as well as the additivity property of the inertia indices of A with respect to the direct sum decomposition of the space into A Γ -orthogonal subspaces, it follows that i=0 be a Jordan chain for for the generalized eigenspace E ν ; thus By invoking Lemma 3.4, {Kv i } l i=0 is a Jordan chain for the generalized eigenspace E 2ω−ν relative to 2ω−ν. More explicitly, the restriction of A Γ into the subspace generated by {Kv i } l i=0 is given by Γ l (2ω − ν, −1). e next two results, Lemma 3.12 and Lemma 3.13 will be very useful later on for computing the inertia indices of A ν Γ in terms of that of M ν Γ . In Lemma 3.12 we investigate such a relation by restricting on a single Jordan block. In Lemma 3.13 we assume that there exists two different Jordan blocks corresponding to the same eigenvalue.
Lemma 3.12. Let 0 = ν ∈ σ(A Γ ) ∩ R and let {v i } l i=0 be a Jordan chain for for the generalized eigenspace E ν . Under the previous notation we get that

Proof. By direct computation we infer that
By taking the difference of the first and last members in the previous equations, we get Being v 0 = 0, we infer also that M Γ v i , v j = 0 for every 1 j l−i and i = 1, · · · , l−1.
is concludes the proof.
Lemma 3.13. Let 0 = ν ∈ σ(A Γ ) ∩ R and we assume that {v i } p i=0 and {w j } q j=0 are two Jordan chains for the generalized eigenspaces relative to the same eigenvalue ν and such that v 0 = w 0 = 0; furthermore we assume that p q. en we have Since v 0 = w 0 = 0 and being p q, then we have M Γ v i , w j = 0 for 1 i + j q. is concludes the proof.

Proof of main results
is section is devoted to prove the main results of this paper. e first result provides a characterization of the spectral stability of a relative equilibrium z in terms of a spectral condition on the central configuration ξ.
e proof of this result result direct follows by the first claim in Lemma 3.2. In fact, µ is an eigenvalue of M −1 Γ D 2 H(ξ) if and only if λ = µ 2 − ω 2 is an eigenvalue of B. By definition, z is spectrally stable if and only if the spectrum of B is purely imaginary or which is the same that µ 2 − ω 2 0. is concludes the proof. e next result shed provides a cler relation intertwining the spectral condition on the central configuration generating the relative equilibrium seen as critical point of the Hamiltonian on the shape pseudosphere and the dynamical (stability) properties of it. Roberts in eorem 3.3 of [Rob13] characterizes linearly stable relative equilibria in terms of the minimality properties that the central configuration (originating such an equilibrium) possesses. eorem 4.1 (Roberts 2013). We assume that for every j, Γ j > 0. en a relative equilibrium z is linearly stable if and only if it is a non-degenerate minimum of H restricted to the shape-pseudo-sphere.
us, by eorem 4.1, the linear stability of a relative equilibrium is equivalent to the fact that the central configuration generating it has a vanishing Morse index and it is non-degenerate (meaning that the kernel dimension of the Hessian of the Hamiltonian restricted to the shape pseudo-sphere vanishes identically). However this result is valid only under the assumption that all circulations have the same sign. In eorem 2, by using the analysis performed in the previous sections, we are able to remove the condition on the circulations' sign admi ing any kind of (non-vanishing) circulation and we provide a relation between the spectral stability of a relative equilibrium and the Morse index of the central configuration generating it. As Corollary of this result, we complement the aforementioned eorem 4.1.
Before giving the proof of this result, we observe that if all circulations strengths have all the same sign (for instance, positive), then M Γ is positive definite (thus n − (M Γ ) = 0) and by Equation (2.9) in particular ω is positive. us by the first claim of eorem 2, we conclude that ξ is a minimum (maybe degenerate).
Corollary 4.2. If Γ j > 0 for all j, and we assume that z is a spectral stable non-degenerate relative equilibrium. en the central configuration ξ is a (maybe degenerate) minimum of H. Proof.
e proof of the first claim follows by the above discussion.
Before providing the proof of eorem 2, we start proving the following technical result.
Lemma 4.3. Let ν be a non-zero real eigenvalue of matrix A Γ , then we have that Proof. We assume that there exist two different Jordan blocks Γ 1 (ν), Γ 2 (ν) corresponding to same eigenvalue ν and, as before, we denote by {v i } p i=0 and {w j } q j=0 the Jordan chains corresponding to these Jordan blocks. Let us consider the following matrix block decomposition By Lemma 3.12, one immediately get that the p × p block A 1 is given by: It is readily seen that the matrix A1 given in Equation (4.1) can be wri en in equivalent form, as follows A1 = M1Γp(ν, 1). (4.2) Analogously, we have that A3 = M3Γq(ν, 1). (4.3) By Lemma 3.13, one gets that the p × q block A2 is given by so, as before, by Equations (4.4) and (4.5) imply that A2 = M2Γq(ν, 1). (4.6) Similarly for the term us (4.2), (4.3), (4.6) and finally (4.7) imply that Case 1. If ν > 0, we define the (analytic) path of symmetric matrices pointwise given by If an eigenvalue of f (t) (resp. g(s)) changes sign, than det f (t) = 0 (resp. det(g(s) = 0). However, it is immediate to see that this cannot occur. We observe that the composition of the two paths f and g is a continuous path joining the matrices M ν Γ matrix to A ν Γ . By this argument it then follows that both matrices belong to the same connected component and in particular the inertia indices coincide; thus in symbols, we have n− A(ν) = n− MΓ(ν) , Case 2. If ν < 0, as before, we define the path of symmetric matrices where s ∈ [−1, 0]. Arguing as before, we get Proof of eorem 2. Since z is non-degenerate and spectrally stable relative equilibrium, then by invoking Lemma 3.2, we get that We notice that A Γ (ξ) = D 2 H(ξ)ξ + ωM Γ ξ = M Γ M −1 Γ D 2 H(ξ) + ωI ξ = 0 and from property (iii) of Lemma 2.5 we have If W = span(ξ, Kξ), then we have In conclusion, we get where sign stands for the sign. Let us now consider theÂ Γ -orthogonal M Γ -orthogonal invariant decomposition of the full (complexified) phase space C 2n .

By Lemma 4.3, we have that
Case 2: ν ∈ {ix + ω|x ∈ R} ⊂ C \ R. By using Lemma 3.11, we already know that n − ( A ν Γ ) = m ν , where m ν is the algebraic multiplicities of the eigenvalue ν. Moreover, from Corollary 3.8, the matrix M ν Γ can be represented as follows is concludes the proof.

Some symmetric examples
is section is devoted to the application of eorem 2 to some specific examples of relative equilibria in the planar N -vortex problem.

e equilateral triangle
We begin with the well-known equilateral triangle solution in the three-vortex problem. Placing three vortices of any strength at the vertices of an equilateral triangle yields a relative equilibrium.
Proof. As proved by author in [Syn49], we get that ξ = (ξ 1 , ξ 2 , ξ 3 ) is a linearly stable iff L > 0; moreover, the angular velocity ω = Γ 3 . We note that the vortex positions ξ i = ξ i − c have center of vorticity at the origin.
By an explicit calculation, we get Summing up all computations we get Let a := Γ Γ 2 1 + Γ 2 2 + Γ 2 3 + 3Γ 1 Γ 2 Γ 3 , then the signature of the quadratic form M Γ ξ, ξ coincides with that of the quadratic form a.
We distinguish the following four cases.
ird case. We assume that Γ i < 0 only for one index i. In this case, without losing in generalities, we may assume that Γ 3 < 0. Since L > 0, then we have that We claim that M Γ ξ, ξ > 0. In order to prove this , as already observed, it is enough to prove that a > 0.
We observe that a could be re-wri en as follows We let Since Γ 1 > 0, us the signature of the quadratic form a agrees with the sign of the function b defined below b(x, y) := (1 + x + y)(1 + x 2 + y 2 ) + 3xy, Fix x ∈ [0, +∞), differentiating (1 + x + y)(1 + x 2 + y 2 ) + 3xy with respect to y, yields 3y 2 + 2y(x + 1) + 3x + x 2 + 1 = 3 y + x + 1 3 is implies that y → b(x, y) is a monotone increasing function (with respect to y) thus the infimum is ge ing precisely at − x 1 + x . Se ing y = − x 1 + x , then we get By this, we immediately get that for every x ∈ (0, +∞) and − x 1 + x < y < 0, the function b is positive or which is the same that the quadratic form a is positive definite, hence M Γ ξ, ξ > 0.
Fourth case. We assume now that Γ i > 0 only for one index i. Arguing precisely as before we get M Γ ξ, ξ < 0. In this case, however as direct consequence of Equation (2.9) s well as of the fact that L > 0 and M Γ ξ, ξ < 0, we get that ω < 0. According to eorem 2, we have that n − ( A Γ ) = n + (M Γ ) = 2.

e rhombus families
From paper [Rob13], we know that there exist two families of relative equilibria where the configuration is a rhombus. Set Γ 1 = Γ 2 = 1 and Γ 3 = Γ 4 = m, where m ∈ (−1, 1] is a parameter. Place the vortices at z 1 = (1, 0), z 2 = (−1, 0), z 3 = (0, y) and z 4 = (0, −y), forming a rhombus with diagonals lying on the coordinate axis. is configuration is a central configuration provided that e case m < −1 or m > 1 can be reduced to this setup through a rescaling of the circulations and a relabeling of the vortices. ere are two solutions depending on the sign choice for y 2 .
Taking plus sign in Equation (5.1) yields a solution for m ∈ (−1, 1] that always has ω > 0. We will call this solution rhombus A. Taking − in Equation (5.1) yields a solution for m ∈ (−1, 0) having ω > 0 for m ∈ (−2 + √ 3, 0), but ω < 0 for m ∈ (−1, −2 + √ 3). We will call this solution rhombus B. In the aforementioned paper, author computed the nontrivial eigenvectors of M −1 Γ D 2 H(ξ); in particular he proved that they are reals for every value of m and are given by 1. Rhombus A is linearly stable for −2 + √ 3 < m ≤ 1 . At m = −2 + √ 3 the relative equilibrium is degenerate. For −1 < m < −2 + √ 3 , rhombus A is unstable and the nontrivial eigenvalues consist of a real pair and a pure imaginary pair.
2. Rhombus B is always unstable. One pair of eigenvalues is always real. e other pair of eigenvalues is purely imaginary for −1 < m < m * and real for m * < m < 0 , where m * is the only real root of the cubic 9m 3 + 3m 2 + 7m + 5. At m = m * , rhombus B is degenerate.
where the algebraic multiplicity of ω is two. Moreover where s is the vector defined in Lemma 2.14. us, by using Lemma 3.7, we get the following M Γorthogonal direct sum decomposition where I 0 = span{ξ}, I 2ω = span{Kξ}, I ω = span{s, Ks}, I ω+µ1 = span{v 1 }, By a straightforward calculations, we get that By the rhombi classifications, we distinguish the two cases.
Rhombus A central configuration. We assume that the central configuration corresponds to rhombus A. en, as already observed, we have