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Point vortices on the hyperbolic plane

We investigate some properties of the dynamical system of point vortices on the hyperboloid. This system has noncompact symmetry SL(2, R) and a coadjoint equivariant momentum map J. The relative equilibrium conditions are found and the trajectories of relative equilibria with non-zero momentum value are described. We also provide the classification of relative equilibria and the stability criteria for a number of cases, focusing on N=2, 3. Contrary to the system on the sphere, relative equilibria with non-compact momentum isotropy subgroup are found, and are used to illustrate the different stability types of relative equilibria.


INTRODUCTION
Relative equilibria in systems of point vortices have previously been considered in detail on the plane and on the sphere. A thorough historical summary of research of these studies can be found in [1,3,12,24] for the plane, and [14] for the sphere.
On the other hand, the case of point vortices on the hyperbolic plane has only been treated in [3,7,10,11,13,21,22]. As on the plane and sphere, the governing equations of the system of point arXiv:1403.2138v1 [math.DS] 10 Mar 2014 vortices on the Hyperbolic plane are Hamiltonian. Y. Kimura [13] gives a uniform formulation for vortex motion on the sphere (positive curvature) and on the hyperbolic plane (negative curvature). Deforming the phase space rather than the dynamics S. Boatto [7], J. Montaldi and T. Tokieda [22] show how the curvature affects the stability conditions of a ring with N vortices: as the curvature decreases Lyapunov stability is confirmed for a given radius of ring. Rings of vortices on the hyperbolic plane are also mentioned in [3], and a more in-depth study is provided by S. Hwang and S.-C. Kim [11]. In this paper, Hwang and Kim present the relative equilibrium conditions for rings of vortices on the hyperbolic plane, and also conclude that any two point vortex configuration is a relative equilibrium.
The fixed and relative equilibria of three point vortices on the hyperbolic plane were first presented by Hwang and Kim in [10]. In this present paper we recover these relative equilibrium conditions using the symmetries of the system; the basic result is that relative equilibria fall into two broad classes: either the configurations form equilateral triangles or the three points lie on a geodesic (we call these geodesic relative equilibria). This is entirely analogous to the situation on the plane or the sphere.
Our principal aim is to study the stability of these relative equilibria, and one of the motivations for this is that the symmetry group SL(2, R) is not compact. The three conserved quantities form the components of the momentum map J, and the symmetry properties of this map allow one to divide relative equilibria with non-zero momentum into three principal classes: elliptic, parabolic and hyperbolic, and this plays an important role in questions of stability.
We first present the simple system of two point vortices in Sec. 3.1 in which every motion is a relative equilibrium, in order to illustrate the difference between the three classes of relative equilibrium. This system is mentioned in [11,13] (the latter for the particular case of a vortex dipole).
Finally, in Sec. 4 we discuss the different types of stability results for two and three point vortices. We show that every two point vortex configuration is stable relative to SL (2, R). However, there is a finer notion of stability, namely stability relative to the subgroup SL(2, R) µ (this is the isotropy subgroup for the momentum value µ), and this only holds when the momentum value is elliptic, which in turn is true if the vortex strengths are of the same sign or, if they are of opposite sign, the vortices are not too far apart, see Corollary 4.7 For the stability of relative equilibria of three point vortices, we find remarkably that an equilateral three vortex configuration has the exact same stability conditions of those for systems on the plane and on the sphere, namely that they are stable whenever i < j Γ i Γ j > 0; here again stability is relative to the subgroup SL(2, R) µ . For geodesic relative equilibria the results are incomplete due to the complexity of the equations. We prove in Theorem 3.4 that the momentum value of any geodesic relative equilibrium is either zero or elliptic, and in Section 4.2 provide some graphs showing the stability regions for isosceles configurations.

GEOMETRY & EQUATIONS OF MOTION
We begin by recalling the hyperboloid model we use for the hyperbolic plane. Alternative models, such as the Poincaré disc and the upper half plane are of course equivalent, but the hyperboloid model lends itself to a more straightforward representation of the momentum map.
Hyperboloid model The hyperboloid model H 2 of the hyperbolic plane is represented by the upper sheet of the 2-sheeted hyperboloid, with the Riemannian metric d s 2 This metric induces the hyperbolic inner product 〈·, ·〉 H 2 between X 1 = x 1 , y 1 , z 1 and X 2 = x 2 , y 2 , z 2 given by and the hyperbolic cross product Any geodesic of this model is given by the curve of intersection of H 2 with a plane through the origin [8,10]. The hyperbolic distance d (X 1 , X 2 ), between X 1 and X 2 ∈ H 2 , is naturally defined as the hyperbolic length of the path on the geodesic connecting these two points. A well known result [8] relates the hyperbolic inner product to the hyperbolic distance by Symmetry group of H 2 The symmetry group of the hyperbolic plane is SL(2, R). Explicitly, the action in the hyperboloid model is given as matrix multiplication using the map where M (3, R) ⊂ GL(3, R) is the group of normalised Möbius transformations presented in [17]. That is given g ∈ SL (2, R) then g · X = g X . It is well-known that the action of SL (2, R) on the hyperboloid H 2 ⊂ R 3 is transitive and proper (see for example [23, Lemma 3.1.18] for a proof).
The Lie algebra of SL (2, R), denoted by sl(2,R), is given by the set of 2 × 2 real matrices with zero trace [5,9,19]. Therefore is a basis of sl(2,R). Furthermore, one can identify the dual space sl(2,R) * with the same set of trace zero 2 × 2 matrices using the natural pairing Hence is the basis of sl(2,R) * dual to the basis B of sl(2,R).
(A) Each sheet of the twosheeted hyperboloids is the coadjoint orbit for aμ inside of C, that is det µ > 0 (µ is elliptic).
(B) The null-cone C without the origin is the union of the two coadjoint orbits forμ = 0 such that det µ = 0 (µ is parabolic). The origin itself is the coadjoint orbit ofμ = 0.
(C) The coadjoint orbit forμ outside of C, that is det µ < 0 (µ is hyperbolic), is a hyperboloid of one sheet.

Coadjoint action of SL(2, R)
The non-degeneracy of the trace pairing (4) implies that the adjoint and coadjoint actions of SL (2, R) on sl(2,R) and sl(2,R) * are equivalent. Direct calculations show that the coadjoint action of R ∈ SL (2, R) on µ ∈ sl(2,R) * is given by In the following theorem we show that every µ = 0 in sl(2,R) * has a coadjoint isotropy subgroup SL (2, R) µ which is a 1-parameter subgroup generated by a Möbius transformation. We call this the type of µ: µ is elliptic, hyperbolic or parabolic for SL (2, R) µ generated by an elliptic, hyperbolic or parabolic Möbius transformation, respectively. . Let µ ∈ sl(2,R) * and C = X ∈ R 3 | det X = 0 . Then the coadjoint isotropy subgroups SL (2, R) µ and coadjoint orbits are classified as follows: 1. If det µ > 0 then SL (2, R) µ ∼ = SO(2, R), the type of µ is elliptic, and the coadjoint orbit is one sheet of the hyperboloid of two sheets shown in Figure 1a.
Proof. The case µ = 0 is trivial. For µ = 0, the proof consists in showing that given X 1 with the same sign of determinant then G µ ∼ = G X 1 .
Consider µ with positive determinant, that is, the vectorμ is inside the null-cone C. The nullcone C is asymptotic to H 2 and is the boundary of all vectors of this type. Therefore the line (A)μ inside of C.
The proof of the remaining case, µ ∈ g such that det µ < 0, consists of the same arguments.
The symplectic leaves are the connected components of the coadjoint orbits, hence the hyperboloid model H 2 is the symplectic leaf that corresponds to the coadjoint orbit of µ = µ 1 , µ 2 , µ 3 ∈ R 3 ∼ = sl(2,R) * with det µ = 1 and µ 3 > 0. We now use the map (3) to analyse the coadjoint action of SL (2, R) µ =0 on H 2 . This map not only preserves the hyperbolic inner product (1) but also for any On other hand, let Pμ := X ∈ R 3 | X ,μ H 2 = 〈μ,μ〉 H 2 denote the hyperbolic normal plane passing throughμ itself, Pμ will help us to describe the group orbit of Ad * µ =0 . Before that we state the following Lemma which is a consequence of (7), and the fact that the inner product (1) is invariant under the action of B .
Sinceμ itself is in Pμ, an important consequence of this lemma is that Pμ ∩ H 2 remains invariant under the coadjoint action of SL (2, R) µ for every µ = 0 ∈ H 2 . It is remarkable that to any µ = 0 in sl(2,R) * , the intersection Pμ ∩ H 2 is a conic (ellipse, hyperbola or parabola) related to its isotropy group, and coincides with the type of µ as defined in Theorem 2.1. This can be seen in Figure 2.
In conclusion, for µ = 0 the group orbit of µ is contained in the conic resulting of Pμ ∩ H 2 .

Phase space
We now return to the system of N point vortices on the hyperbolic plane H 2 . LetX i be the vector from the origin in R 3 to the i th vortex X i ∈ H 2 with nonzero vorticity Γ i . A candidate for the manifold of the dynamical system of N point vortices consists of N copies of the hyperboloid H 2 × · · · × H 2 . However, configurations that lead to infinite energy must be avoided, and this is obtained by discarding the set of collisions Hence, the phase space is given by M = H 2 × · · · × H 2 \ ∆. Neglecting collisions of vortices guarantees the action of SL (2, R) in M to be free provided N 2.
Poisson structure The evolution equations for the system of point vortices are Hamiltonian, and the symplectic/Poisson structure on the phase space is given as follows.
We consider the Lie-Poisson bracket as the Lie-Poisson structure on the hyperboloid H 2 . Hence, given µ ∈ sl(2,R) * , and F , G two functions defined on sl(2,R) * , we have that d F µ and dG µ ∈ (sl (2, R) * ) * ∼ = sl(2,R) and We define the Poisson structure on M as where · denotes the Euclidean product [21,23]. Therefore, denoting the positive part of (9) for the ith copy of H 2 that containsX i by ω H 2 (·, ·) i , we construct a symplectic structure on M by This symplectic form and the Poisson structure above are compatible in the usual way.

Momentum map and its equivariance
, that is the vector fields of J and of the infinitesimal generator of M are equal, defines a momentum map J : for all ξ ∈ sl(2,R) and z ∈ M . Therefore the map given by is a momentum map for the symplectic manifold (M , ω, SL (2, R)). Alternatively, the momentum map can be defined in terms of the symplectic form by where X ∈ M , w ∈ T X M and ξ ∈ g. Whenever the symmetry group is semisimple the momentum map of a symplectic manifold can be chosen to be coadjoint equivariant, as was shown by J.-M. Souriau [29]. Since the KKS form is invariant and it defines the symplectic structure (10), the momentum map (11) does satisfy this equivariance, that is for all g ∈ SL (2, R), X ∈ M . By Noether's theorem the momentum map is a conserved quantity under the flow of every invariant Hamiltonian, particularly the Hamiltonian representing the total energy of this dynamical system. We adopt the Hamiltonian constructed by Y. Kimura in [13], which in terms of the hyperbolic inner product is given by Note that if all vorticities have the same sign, as two point vortices in H 2 get closer, i.e. as the hyperbolic distance between them tends to 0, the total energy H tends to ∞ [21], as expected when a collision occurs. On other hand if two points get far apart, the hyperbolic distance tends to ∞ and the contribution of their interaction to the total energy is 0. From the Lie-Poisson structure we derive the differential equations governing this dynamical systemẊ with r ∈ {1, .., N }. This equation differs from the differential equations derived by Y. Kimura in [13] by a factor of 2 in the denominator, which is carried by the choice of the basis B of the Lie algebra (2).

RELATIVE EQUILIBRIA
A relative equilibrium X e is a group orbit that is invariant under the dynamics. Since the momentum value µ = J (X e ) is a conserved quantity (by Noether's theorem), the level sets J −1 µ are invariant under the flow of the Hamiltonian, that is H J −1 µ ⊂ J −1 µ . This also means that we must restrict our attention to the action of G µ , implying that a configuration X e is a relative equilibrium if G µ · X e remains invariant.

TWO POINT VORTICES
For a two point vortex configuration X e = (X 1 , X 2 ) ∈ M , the Implicit Function theorem implies dim J −1 µ = 1. We additionally know that G µ · X e ⊂ J −1 µ , so they must be equal, hence G µ · X e is indeed invariant. In conclusion, any two point vortex configuration X e is a relative equilibrium. Furthermore, it is easy to see that a two vortex configuration X e = (X 1 , X 2 ) ∈ M has non-zero momentum value µ = J (X e ). Thus, as shown in Figure 3, the trajectories of X 1 and X 2 are the conics determined by the determinant of the momentum value µ.
For example the vortex dipole Γ 1 = −Γ 2 = 1, treated before by Kimura [13] and recently by Hwang and Kim [11], X e has momentum value µ with determinant less than zero. Therefore G µ is related to an hyperbolic Möbius transformation, and the trajectories of the vortices are on hyperbolas parallel to each other (Figure 3c).

THREE POINT VORTICES
On the case of more than two vortices the conditions for the relative equilibrium existence are not straightforward. Considering the orbit space M /SL (2, R), an invariant group orbit is just a point that is invariant under the dynamics. Thus a relative equilibrium X e is an equilibrium point in the reduced space. Given that the level sets J −1 µ are invariant under the flow of Hamiltonian, a relative equilibrium must be a critical point of the augmented Hamiltonian H J −1 (µ) . This is a well-known result given by many authors before, a proof of it can be found in [21]. Hence if it exists ξ ∈ sl(2,R) such that X e is a critical point of H ξ (X e ) = H (X e ) − µ, ξ , then X e is a relative equilibrium.
Evidently we require a relative equilibrium X e restrained to H 2 , we include this restriction with the addition of a Lagrange multiplier to H ξ . Having said that, for the system of N point vortices, X e = (X 1 , . . . , X N ) is a relative equilibrium if it is a critical point of whereX r = X 1 r , X 2 r , X 3 r ,X s = X 1 s , X 2 s , X 3 s ∈ H 2 , with vortex strengths Γ r and Γ s respectively, and λ r is the corresponding Lagrange multiplier.
Therefore the general condition for relative equilibria is given as the solutions of the following equation of angular velocity ξ where L pr denotes X p , X r An important observation is that the angular velocity ξ satisfieṡ X r =ξ× H 2X r . This means that a relative equilibrium X e rotates "hyperbolically" around ξ as found by Hwang and Kim in Proposition 2 of [10].
Classification In this section we provide the classification of relative equilibria of three point vortices. The following result is obtained by solving (16) foř Any other set of three point vortices is equivalent to this one by hyperbolic rotations. Since the dynamics are preserved by that type of transformation, the same relative equilibrium conditions follow for any other X . Proof. The result is obtained by calculating the solutions of (16) for configurations of the form given in (17) (see [23] for details).

Remark 3.2.
As mentioned in the introduction, a geodesic on the hyperboloid model H 2 is the intersecting curve of a plane through the origin with H 2 . Hwang and Kim [10] point out that contrary to the system of point vortices on a sphere, it is not possible to have an equilateral configuration in a geodesic of the hyperboloid model. Therefore, for an equilateral configuration Although not mentioned explicitly by Hwang and Kim, the formulae (17) − (19) in [10] leads to the same result of relative equilibria derived here, where we have used the geometric approach of symmetric Hamiltonian systems. Furthermore,X 1 ,X 2 andX 3 are linearly independent in R 3 , implying that J (X e ) = Γ 1 X 1 + Γ 2 X 2 + Γ 3 X 3 = 0 for every equilateral configuration. Thus the group orbit of the momentum of an equilateral relative equilibrium is one of the conics described in Section 2.
The next two theorems present the conditions (if any) on the vorticities Γ for relative equilibria of three point vortices. The complete proofs can be found in [23], they consist in finding any restrictions on Γ for the existence of the angular velocity ξ in (16). . Every equilateral configuration X e of point vortices (X 1 , X 2 , X 3 ) in M is a relative equilibrium. The angular velocity of X e is given by Note that L = sinh 2 (d (X i , X j )) as follows from (2). Since for equilateral configurations J = 0, it follows from (18) that these are never equilibria (it is also easy to see this geometrically from first principles). Moreover, the momentum value µ of a geodesic relative equilibrium is either zero or elliptic.
Proof. The proof of expression (19) is derived by computing the relative equilibrium conditions (16) forX 1 = x 1 , 0, 1 + x 2 1 , X 2 = (0, 0, 1) , in terms of L i j where x 1 = L 12 , x 3 = L 23 . For the final part, we first consider isosceles geodesics configurations, that is x 1 = x 3 . Straightforward calculations show that for Equation (19) to be satisfied Γ 1 = Γ 3 must hold. Conversely, substituting L 13 in terms of L 12 and L 23 in (19) with Γ 1 = Γ 3 leads to L 12 = L 23 . Under this vorticity condition, the determinant of the momentum value µ = J (X 1 , therefore det µ > 0 for all Γ 2 = −2Γ 1 1 + x 2 1 , otherwise µ = 0. Suppose now that x 1 = x 3 and Γ 1 = Γ 3 , the determinant of the momentum value is Recall that x 1 > 0 and x 3 > 0, hence µ is ellip- It is remarkable that for any geodesic configuration of three vortices X e there can always be found a set of vorticities Γ such that X e is a relative equilibrium. Conversely, for given values of the vorticities there is a 1-parameter family of inequivalent geodesic relative equilibria. Equilibria It is interesting to ask which of the relative equilibria are in fact (fixed) equilibria. This was answered by Hwang and Kim [10], who showed that a necessary and sufficient condition for an equilibrium is that where i , j , k is a cyclic permutation of 1, 2, 3, and this is only A particular case is the isosceles geodesic equilibrium whose stability we will consider again at the end of the paper. A calculation using (19) shows that if L 12 = L 23 (isosceles case) then Γ 3 = Γ 1 for a relative equilibrium, and, as shown in the proof of Theorem 3.4 above, its momentum value µ is elliptic if a = 〈X 1 ,X 2 〉 H 2 = Γ 2 2Γ 1 , otherwise µ = 0. Taking the hyperbolic inner product of the condition (22) withX 2 and combining with the value of a just given leads to Therefore, X e with Γ 1 = Γ 3 and Γ 2 given by (23) is an isosceles equilibrium configuration indeed. Furthermore, the momentum value of X e is elliptic.

STABILITY OF RELATIVE EQUILIBRIA
Before presenting any of our stability results, we begin by recalling the notions of (nonlinear) stability for relative equilibria symmetric Hamiltonian systems. Let G be the group of symmetries, and J : P → g * be the momentum map.
The first notion is G-stability of a relative equilibrium: this is the usual definition of Lyapunov stability but using G-invariant open sets. Since the dynamics on M projects to dynamics on the orbit space P /G (or shape space), and the relative equilibrium projects to an equilibrium point, this is equivalent to Lyapunov stability of this projected equilibrium. Specifically, a relative equilibrium x e is G-stable if for every G-invariant neighbourhood V of x e there exists a G-invariant neighbourhood U of x e such that any trajectory intersecting U lies entirely within V .
Since in addition, the momentum is conserved, one can also study stability with a level-set of the momentum, J −1 (µ) (for the appropriate value of µ). The system on this level set is invariant under G µ (by definition of G µ ), and so the natural notion of stability is Lyapunov stability relative to G µ on this level set; this is called leafwise stability.
A finer notion of stability was introduced by Patrick [25], and this is stability relative to a subgroup G of G, and in particular the subgroup G µ (but here the stability is relative to all perturbations of the initial condition, not just those with the same momentum value). The definition is the same as that above, with G replaced by G .
It is straightforward to show that if x e is G -stable, then it is also G-stable (see [23], Proposition 5.4.2 for details).
Before progressing further, the definitions of split and regular points in momentum space are needed: we follow the definitions given in [15,26]. Definition 4.1. Let µ ∈ g * and G 0 µ the identity component of G µ . One says that µ is split if there exists a G 0 µ -invariant complement n µ to g µ in g.

Definition 4.2.
A point µ ∈ g * is regular if dim g ν = dim g µ for every ν in a neighbourhood of µ.
Proposition 3.2.5 of [23] shows that for any µ ∈ sl(2,R) * the isotropy subgroups are connected so G 0 µ = G µ , and following on from this Proposition 3.2.7 proves that µ is split if and only if µ is not parabolic. The regularity condition of every non-zero µ ∈ sl(2,R) * is demonstrated in Proposition 3.2.8. Patrick related formal stability with the concept of G µ stability for regular points of Hamiltonian systems with compact symmetry in [25]. This important result motivated E. Lerman and S. F. Singer [15] to drop the regularity condition for proper group actions and prove G µ -stability for split relative equilibria, provided formal stability and the existence of a G µ invariant inner product on g * .
Evidently, the topological properties of regularity and splitness of µ play an important role on the G µ stability analysis. Patrick, Roberts and Wulff [26] give some important consequences of these properties. In the next section we put these results in context with the system of point vortices on the hyperboloid H 2 .
The paper of Patrick, Roberts and Wulff [26] presents an excellent survey of stability results for systems with symmetry. Keeping in mind the regularity and splitness properties of this dynamical system, we use their results to compute the stability conditions of two and three point vortices. Proof. In Section 2, we recalled that the action of SL (2, R) on M = H 2 × · · · × H 2 \∆ is free and proper for N 2. Hence the conditions of the following result of Patrick, Roberts and Wulff are satisfied.
1. If µ is regular, then g µ is Abelian and g * /G is Hausdorff at G · µ.

Let µ be split, then
The same authors provide the following results of G and G µ stability based on the previous results of [4,15,16,20,25,26].  As mentioned before, any two point vortex configuration has non-zero momentum value, hence µ is regular. Consequently g * /G is Hausdorff at G · µ. By the Witt decomposition [6,28] and the Implicit Function Theorem, the dimension of the symplectic normal space N 1 must be zero, so the assumptions of Corollary 4.5 are satisfied.
By Corollary 4.6, if either µ is additionally an elliptic momentum value, the stronger result of SL (2, R) µ is also obtained as stated in the next corollary. Proof. The momentum value µ of X e is elliptic for

THREE POINT VORTICES
Given that the dimension of symplectic normal space N 1 is two, when this is a symplectic slice to a three-vortex relative equilibrium X e with non-zero momentum value µ, the stability study is not as trivial as for two vortices. S. Pekarsky and J. E. Marsden introduce a symplectic slice for three point vortices on the sphere in [27]. We note that a symplectic slice, for the system of N point vortices on the hyperboloid H 2 , can be obtained following their construction and using the hyperbolic geometry of the hyperboloid model H 2 . The symplectic normal spaces N 1 , used to compute the stability conditions of equilateral and geodesic configurations follow from this theorem, the proof of it can be found in [23]. We treat the case of µ = 0 at the end of this section.
Proof. From Theorem 4.8 with D 1 =X 1 +X 2 and D 2 =X 2 +X 3 , generate a symplectic slice N 1 to X e . We know that any equilateral configuration has non-zero and, therefore, regular momentum value µ. Thus the SL (2, R)stability condition follows from Proposition 4.4 together with Corollary 4.5, by testing definiteness of the Hessian of H ξ (14) restricted to N 1 at X e . Moreover, µ is either elliptic, parabolic or hyperbolic, with determinant Thus for µ parabolic or hyperbolic only (25) holds, that is x e is SL (2, R) unstable. Consequently, every parabolic or hyperbolic x e is SL (2, R) µ unstable. If µ is elliptic, the SL (2, R) µ stability results follow from Corollary 4.6.
Remark 4.11. The G-stability conditions for three equilateral vortices on the hyperboloid coincide with those for the system on the plane [2] and on the sphere [21,27].
Aref [2] showed that a three point vortex configuration on the plane, a relative equilibrium with i = j Γ i Γ j = 0 is marginally stable. Meanwhile, for the system on the sphere, Marsden, Pekarsky and Shkoller [18] performed numerical integrations and observed changes of the stability for The conjecture that a Hamiltonian bifurcation occurs has also been mentioned in the references [21,27].
We have performed numerical integrations in Maple for equilateral configurations on the hyperboloid, which suggest that a bifurcation occurs at i = j Γ i Γ j = 0. Note that this is actually the equation of a cone as shown in Figure 4, where the stability depends only on the choice of Γ's. The configurations for which i = j Γ i Γ j < 0 have a set of Γ's outside this cone, so these points are SL (2, R)unstable. On other hand, any equilateral configuration with Γ's inside the cone is a SL (2, R) µ -stable relative equilibrium, and follows the trajectory of an ellipse rotating around its momentum value and is therefore periodic.   If A (Γ 1 , Γ 2 , a) > 0 then X e is SL (2, R) µ -stable. Conversely if A (Γ 1 , Γ 2 , a) < 0 then X e is SL (2, R)unstable. In addition, if X e is also an equilibrium point, then X e is SL (2, R) µ -stable for all a ∉ (−1.191, −1.106). Finally, if a = Γ 2 2Γ 1 then µ = 0 and hence not a regular point.

Isosceles geodesic relative equilibria
Proof. In the proof of Theorem 3.4 we showed that det µ > 0 for all a = Γ 2 2Γ 1 , otherwise µ = 0. By Theorem 4.8, a symplectic normal space N 1 to X e is generated by where D 1 =X 1 +X 2 , D 2 = (0, 1, 0) and k = − x 2 1 + 1. Given that µ is regular and elliptic, the stability condition (27) is derived from testing the definiteness of the Hessian of (14) restricted to N 1 .
Recall from Section 3.2, that an isosceles geodesic relative equilibrium is also in equilibrium provided Γ 2 = Γ 1 a 1−a . Given that a < −1 for all X e , we obtain A (Γ 1 , Γ 2 , a) > 0 for a ∉ (−1.191, −1.106). The symplectic slice of µ = 0 is trivial, this can be easily shown by dimension count. Since SL (2, R) µ=0 = SL (2, R) the dimension of sl(2,R) µ · X e is 3, and by the Implicit Function Theorem the dimension of a symplectic slice at X e must be zero. As discussed in Example 4 of [26], a configuration with µ = 0 is trivially leafwise stable and, G-stable if the angular velocity ξ points into the null-cone C. Simple calculations show that the angular velocity of a geodesic configuration with momentum value µ = 0 is always elliptic, hence µ = 0 is SL (2, R)-stable.
Despite the complexity of (27), we can get an idea of the stability of X e by looking at the values a, Γ 1 , Γ 2 in Figure 5, where the stability regions for Γ 1 = Γ 3 = 1 are plotted. In Figure 5b the stability conditions can be seen with more detail for vortices that are close to each other, that is for small values of a. The dashed blue line represents a = Γ 2 2Γ 1 , in which case µ = 0 and X e is SL (2, R)-stable as the angular velocity ξ is elliptic.
Geodesic configuration with three different lengths Additional information of this general case can be found in [23]. For this type of configuration the computation of the Hessian is rather involved, and further analysis is required to give a conclusion on the stability criteria of a relative equilibrium X e . Nevertheless it is of particular use the fact that as for the isosceles case, a relative equilibrium must have either elliptic or zero momentum value.