Persistence of stationary motion under explicit symmetry breaking perturbation

Explicit symmetry breaking occurs when a dynamical system having a certain symmetry group is perturbed to a system which has strictly less symmetry. We give a geometric approach to study this phenomenon in the setting of Hamiltonian systems. We provide a method for determining the equilibria and relative equilibria that persist after a symmetry breaking perturbation. In particular a lower bound for the number of each is found, in terms of the equivariant Lyusternik–Schnirelmann category of the group orbit.


Introduction
When we talk about symmetries, we either refer to the symmetry of a physical law (dynamical equations) or the symmetry of a physical state (solution of these equations). The symmetry or symmetry group of a physical law (or a physical state) is defined to be the group of transformations which leave these equations (or this solution) invariant. Explicit symmetry breaking is defined as a process of perturbing symmetric dynamical equations such that the resulting equations have a lower symmetry group. Any physical law observed in nature can be thought as a perturbation of a physical law having a bigger symmetry group. However the more symmetric a dynamical system is, the more simple its solutions are. In fact, complicated and interesting dynamical behaviours require low symmetry groups. For example, Lauterbach et al. [6,15,19] show that some periodic solutions of an unperturbed dynamical system persist under symmetry breaking perturbations and become heteroclinic cycles.
The lack of symmetries of a perturbed system can be due for example to the presence of terms whose origin is different from case to case. As explained in Brading and Castellani [5], such terms can be introduced artificially in order to match with theoretical or experimental observations. For example in quantum field theory, the Lagrangian for weak interactions is constructed so that the parity-symmetry and the charge-parity symmetry are violated, making the theory in the line with experimental observations. Besides, quantization processes might also be a cause for the appearance of such terms which are the so-called quantum anomalies. In this case, the terms are not artificially introduced but they appear after a renormalization procedure.
The dynamical systems we focus on are Hamiltonian systems. Some aspects of explicit symmetry breaking phenomena are studied by several authors including Ambrosetti et al. [1], Grabsi, Montaldi and Ortega [13] and Gay-Balmaz and Tronci [12]. Phase spaces of Hamiltonian systems are symplectic manifolds and the symmetries of such systems are encoded into Lie group actions on those manifolds. A symplectic manifold is a smooth manifold M equipped with a non-degenerate closed two-form ω. A (proper) action of a Lie group G on M is canonical if it is smooth and it preserves ω. A class of canonical group actions on symplectic manifolds are Hamiltonian. To those actions we can associate a Noether conserved quantity expressed in term of a momentum map Φ G : M → g * , where g * is the dual of the Lie algebra of G. This notion generalizes the notion of angular momentum in classical mechanics, when the phase space is T * R 3 , acted on by the group of rotations SO (3). By a Hamiltonian (proper) G-manifold, we mean a quadruple (M, ω, G, Φ G ) as described above, where Φ G : M → g * is equivariant with respect to the coadjoint action Ad * : (g, µ) ∈ G × g * → Ad * g −1 µ ∈ g * . The dynamics is governed by a Hamiltonian h which is a G-invariant real-valued function defined on M . The ring of such functions is denoted C ∞ (M ) G . The non-degeneracy of ω implies that, associated to any Hamiltonian h ∈ C ∞ (M ) G , there is a unique vector field X h defined by ι X h ω = −dh.
Since the action of G on M is canonical and h is G-invariant, the integral curve ϕ t (m) of X h starting at m ∈ M satisfies ϕ t (g · m) = g · ϕ t (m) for all g ∈ G. The resulting Hamiltonian equations are thus G-equivariant and we say that G is the symmetry group of (1). We study the effect of a small Hamiltonian perturbation of these equations, which is invariant with respect to a subgroup of G.
Section 3 is devoted to the question of persistence of equilibria. In this case, the required non-degeneracy condition on an equilibrium m ∈ M of h is a particular case of Morse-Bott condition, when the critical manifold of h is the group orbit G · m (cf. Definition 3.1). We show that at least a certain number of H-orbits of equilibria persist under a small H-perturbation, in a tubular neighbourhood of G · m (cf. Theorem 3.1 and Corollary 3.2). This number is the positive integer Cat H (G/G m ), which is the H-equivariant Lyusternik-Schnirelmann category of the group orbit. We present applications of our result, including the problem of an ellipse-shaped planar rigid body moving in a planar irrotational, incompressible fluid with zero vorticity and zero circulation around the body.
Extending Theorem 3.1 and Corollary 3.2 to the case of relative equilibria is a bit more challenging because we must take into account the conservation of momentum. This case is treated in Section 4. Whereas equilibria are just critical points of the Hamiltonian function h, relative equilibria are critical points of the restriction of this same function to a level set Φ −1 G (µ) of the momentum map. Let m ∈ M be one of those critical points. The element ξ ∈ g playing the role of a Lagrange multiplier is called the velocity of m, which is in general not unique when the action is not free. For that reason, we refer to a relative equilibrium as a pair (m, ξ) ∈ M × g. We denote the underlying Lagrange function associated to ξ by h ξ .
A standard definition says that a relative equilibrium (m, ξ) of h is nondegenerate if the Hessian of h ξ at m is a non-singular quadratic form when restricted to some symplectic subspace N 1 ⊂ T m M , called the symplectic slice at m. If the perturbations h λ are invariant with respect to the full symmetry group G, this notion of non-degeneracy is enough to guarantee the persistence of a relative equilibrium. This is no longer the case if h λ has a smaller symmetry group than the one of h and we require a stronger non-degeneracy condition on the relative equilibrium (cf. Definition 4.1). In Grabsi, Montaldi and Ortega [13] a step in that direction is taken, when the symmetry group is a torus that breaks into a subtorus. In addition, the group actions in consideration are assumed to be free. We extend their result to non-free actions and non-abelian symmetry groups.
A necessary condition for a relative equilibrium of h to persist under an H-perturbation is that the velocity ξ belongs to h, the Lie algebra of H. If the non-degeneracy condition on (m, ξ) ∈ M × h holds, and modulo some technicalities, the least number of H µ -orbits of relative equilibria with velocity close to ξ, which persist under a small H-perturbation in some This is the content of Theorem 4.3 and Corollary 4.4. As an example, we show that the Hamiltonian of the spherical pendulum can be thought as an S 1 -perturbation of the SO(3)-invariant Hamiltonian h on T * S 2 governing the co-geodesic motions on the sphere. The relative equilibria of h project to the great circles on S 2 . Only one of those circles persists in the good neighbourhood under this S 1 -perturbation, provided the gravity encoded in the parameter is sufficiently small.

Preliminaries
In this section we introduce an equivariant version of the Lyusternik-Schnirelmann Theorem and a symplectic local model for a Hamiltonian proper G-manifold near a group orbit.

Equivariant Lyusternik-Schnirelmann Theorem
In their original paper [20], Lyusternik and Schnirelmann introduce a numerical homotopy invariant of a topological space M that they denote Cat(M ). They define it to be the least number of open subsets of M , whose inclusion is nullhomotopic, that are required to cover M . They show that if M is a closed (i.e. compact without boundary) C 2 -manifold, then any function f ∈ C 1 (M ) has at least Cat(M ) critical points. The infinite dimensional case has been studied by Schwartz [31] when M is a complete C 2 -Hilbert manifold, and f satisfies a suitable compactness condition. The equivariant analogue has been proved by Fadell [9] and Marzantowicz [25] in the case when G is a compact Lie group. The extension to proper Lie group actions can be found in Ayala, Lasheras and Quintero [2]. They define Cat G (M ) to be the least number of G-invariant open subsets of M , which admits a G-deformation retract onto a G-orbit, that are required to cover M . If (M, G) is a proper G-manifold, there is a G-invariant Riemannian metric on M . Given a function f ∈ C ∞ (M ) G the associated gradient vector field ∇f is G-equivariant. We say that m ∈ M is a critical point of f if ∇f (m) = 0. Denote by C c (f ) the set of critical points m such that f (m) = c.
The extension of the Lyusternik-Schnirelmann Theorem to non-compact manifolds requires some additional assumptions on M and f . (ii) f is bounded below.
(iii) If (x n ) n∈N ⊂ M is a sequence such that the associated sequence of images f (x n ) is bounded and ∇f (x n ) converges to zero, then there exists a sequence (g n ) n∈N ⊂ G such that the sequence (g n · x n ) n∈N contains a convergent subsequence in M .
Condition (iii) is a compactness condition which implies that f restricted to its set of critical orbits modulo G is a proper map. In particular for any real number c, the subset C c (f ) is a closed bounded subset of M . Since M is complete, the Hopf-Rinow Theorem implies that C c (f ) is compact in M .

The Symplectic Tube Theorem
The Symplectic Tube Theorem is used to study the local dynamics and the local geometry of a Hamiltonian proper G-manifold (M, ω, G, Φ G ). It states essentially that every m ∈ M admits a G-invariant neighbourhood, which is G-equivariantly symplectomorphic to a neighbourhood of the zero section of a symplectic associated bundle. This construction provides tractable semiglobal coordinates for M near G-orbits. Those coordinates are sometimes referred as slice coordinates. This theorem was obtained by Guillemin and Sternberg [14] and by Marle [21], for canonical Lie group actions with equivariant momentum maps. It has been extended independantly by Ortega and Ratiu [28] and Bates and Lerman [4], for general canonical Lie group actions. Schmah [30] and Perlmutter et al. [29] studied the case when M is a cotangent bundle.
We briefly recall the construction underlying the Symplectic Tube Theorem. The reader is referred to Ortega and Ratiu (cf. [28] Chapter 7) or Cushman and Bates (cf. [28] Appendix B Section 3.2) for details. Let m ∈ M with momentum µ = Φ G (m). Denote by G m and G µ the stabilizers of m and µ respectively and by g m and g µ their respective Lie algebras. The stabilizer G m is compact by properness of the action of G on M . We can thus split g µ and g into a direct sum of G m -invariant subspaces g µ = g m ⊕ m and g = g m ⊕ m ⊕ n.
We denote by g · m the tangent space at m of G · m. Elements of g · m are vectors of the form x M (m) := d dt t=0 exp(tx) · m, where x ∈ g and exp : g → G is the group exponential. The tangent space T m M can be decomposed into a direct sum of four G m -invariant subspaces defined as follow: (i) T 0 := ker (DΦ G (m)) ∩ g · m = g µ · m.
(ii) T 1 := n · m which is a symplectic vector subspace of (T m M, ω(m)).
(iii) N 1 is a choice of G m -invariant complement to T 0 in ker (DΦ G (m)). It is a symplectic subspace of (T m M, ω(m)) and is called the symplectic slice. The linear action of G m on N 1 is Hamiltonian with momentum map Φ N 1 : There is an isomorphism f : N 0 → m * given by f (w), y = ω(m) (y M (m), w) for every w ∈ N 0 and y ∈ m.
Since N 1 is a G m -invariant subspace, there is a well-defined action of G m on the product G × m * × N 1 given by This action is free and proper by freeness and properness of the action on the G-factor. The orbit space Y is thus a smooth manifold whose points are equivalence classes of the form [(g, ρ, ν)]. The group G acts smoothly and properly on Y , by left multiplication on the G-factor. Let m * 0 ⊂ m * and (N 1 ) 0 ⊂ N 1 be G m -invariant neighbourhoods of zero in m * and N 1 , respectively. Then is a neighbourhood of the zero section in Y . It comes with a symplectic form We call the triplet (ϕ, Y 0 , U ) a symplectic G-tube at m and we also say that . Besides the momentum map Φ G : M → g * can be expressed in terms of the slice coordinates:

Symmetry breaking for equilibria
The aim of this section is to give an estimate of the number of H-orbits of equilibria that persist under a small H-perturbation of some G-invariant Hamiltonian h. This is the content of Corollary 3.
Assume N is some vector space and f : G × N → R is a smooth function. We denote by d N f the partial derivative(s) of f with respect to the Nvariable(s). By abuse of notations we write D 2 f for the Hessian of f and D 2 N f for the Hessian with respect to the N -variables.
In other words, the Hessian is non-singular in the directions normal to the group orbit.
If m ∈ M is a G-nondegenerate equilibrium of h then so is any p ∈ G · m, by G-invariance. For the same reason, the tangent space T p (G · m) is contained in ker (D 2 h(p)) for any p ∈ G·m. Definition 3.1 is a particular case of Morse-Bott non-degeneracy when G·m is the critical manifold of h (cf. [7] Appendix E.2). Note that Condition (ii) implies that the critical manifold G · m is isolated in the sense that there exists a tubular neighbourhood of G · m that does not contain any other critical points of h.

Persistence of equilibria
We say that a closed subgroup H ⊂ G is co-compact (in G) if the left multiplication of H on G is co-compact i.e. the orbit space H \ G under this action is compact (as a topological space). We can now state the main result of this section.
Proof -Let m ∈ M be a G-nondegenerate equilibrium of h whose stabilizer is denoted by K := G m . Following the notations of Section 2 the product space N := m * ×N 1 is a K-vector space and it is isomorphic to some K-vector space complementary to g · m in T m M . By the Symplectic Tube Theorem 2.2 and the G-nondegeneracy of m, we can choose a K-invariant (ii) The only critical points of h in U are on G · m.
In that model the point m reads [(e, 0)] and the H-pertubation is identified In particular the map satisfies d N h(e, 0) = 0 and its derivative with respect to the N 0 -variables, evaluated at (e, 0), is non-vanishing. The Implicit Function Theorem implies the existence of a neighbourhood V 1 × W 1 of (0, e) in R × G such that, for By By G-invariance of h, (6) holds when replacing (e, 0) by any (g, 0) ∈ G × N 0 . We apply the previous argument for every (g, 0) with g / ∈ H and use the compacity of H \G to extract a finite collection of open subsets be an open interval containing 0 ∈ R. By uniqueness of each φ i , we can glue them together to define an H-invariant smooth function For every fixed parameter λ ∈ V , It thus descends to a function f λ ∈ C ∞ (G/K) H given by where [g] K denotes a coset in G/K. For any pair β = (λ, This function has a non-degenerate 2 critical point at 0 ∈ N 0 . Indeed by (8) and Moreover, its Hessian D 2h β (0) is non-singular, because non-degeneracy is a stable condition 4 . By the Morse Lemma (cf. Lemma 2.2 in [27]) there is a local coordinate system ν β = (ν 1 , . . . , ν ), defined in a neighbourhood where . Since the functions defining (10) are H-invariant, the identity (11) holds on (N β , ν β ) when re- We thus obtain a collection of Morse charts (N β , ν β ) indexed on V × (H\G/K). The compacity of H\G is used next to extract a finite number of Morse charts (N β i , ν β i ) for i = 1, . . . , r. Using a partition of unity, we construct a local coordinate system ν = ( We may define a smooth map ψ : Replacing (13) in (10) yields where Let U ⊂ M be the G-invariant neighbourhood of m whose symplectic local model is G × K N 0 . In particular if λ ∈ V , the critical points of h λ in U are in one-to-one correspondence with those of the function f λ ∈ C ∞ (G/K) H defined in (9). Note that if G is compact, the (OP S) condition in Corollary 3.2 is automatically satisfied. Indeed, any compact manifold is automatically complete and the compactness condition on f λ is fulfilled.
where D n is the dihedral group of order 2n. The perturbed Hamiltonian h λ has 2n critical points whose coordinates are ( π n k, 0) for k = 0, . . . , 2n − 1, which form a regular 2n-gone as shown in Figure 1 for the case n = 3. Since G/G m = O(2)/ r 0 is topologically a circle, we find Cat H (G/G m ) = 2 (cf. [25] Corollary 1.17). Since G is compact, condition (OP S) of Corollary 3.2 is automatically satisfied. There are thus two H-orbits of equilibria of h which persist, each of them being a regular n-gone (cf. Figure 2).

Dynamics of a 2D rigid body in a potential flow
We apply the result of Corollary 3.2 to the problem of a planar rigid body B of mass m moving in a planar irrotational, incompressible fluid with zero vorticity and zero circulation around the body. The motion is governed by Kirchhoff equations [17]. Classical treatments of the problem can be found in Lamb [18] and Milne-Thomson [26]. The configuration space of the bodyfluid system is a submanifold Q of the product SE(2) × Emb vol (F 0 , R 2 ), where SE(2) is the special Euclidean group describing the motion of the body, and Emb vol (F 0 , R 2 ) is the space of volume-preserving embeddings of the fluid reference space F 0 in R 2 . The symmetry group of this system is the direct product of SE(2) (group of uniform body-fluid translations and rotations) and the particle relabeling symmetry group (volume-preserving diffeomorphisms of F 0 ). Since these actions commute, the system can be reduced by the process of symplectic reduction by stages (cf. Marsden et al. [23]). The Hamiltonian of the system is invariant under the particle relabeling symmetry group. Geometrically, eliminating the fluid variables amounts to carry out a symplectic reduction by this group. The particle relabeling symmetry group acts on T * Q in a Hamiltonian fashion. The associated momentum map has two components corresponding to the vorticity and the circulation. The reduction at zero momentum corresponds to a fluid with zero circulation and zero vorticity. In this case, the symplectic reduced space is identified with T * SE(2), endowed with the canonical symplectic form and the SE(2)-invariant reduced Hamiltonian is the sum of the kinetic energy of the body-fluid system by the addition of the so-called "added masses", and the kinetic energy of the body. Those added masses depend only on the body's shape and not on the mass distribution. The reader is refered to Kanso et al. [16] and Vankerschaver et al. [32] for details. Since SE(2) acts symplectically on T * SE(2), the dynamics can be reduced a second time using Poisson reduction and thereby the reduced motion is governed by the Kirchhoff equations that are the Lie-Poisson equations on the dual Lie algebra se(2) * .
For the sake of simplicity we will assume that the body B is shaped as an ellipse with semi-axes of length A > B > 0. We will use the formulae and follow the notations of Fedorov et al. [10]. At the center of mass of B we attach a frame {E 1 , E 2 } that is aligned with the symmetry axes of the body. Its position is related at any time to a fixed space frame {e 1 , e 2 } by an element of SE (2). An element of the Lie algebra ξ ∈ se(2) is identified with a vector whereθ ∈ R is the angular velocity of B and (v 1 , v 2 ) T ∈ R 2 is the linear velocity of its center of mass, expressed in the body's frame. In this setting the body has kinetic energy with I B := diag(I B , m, m), where I B is the moment of inertia of the body about its center of mass. The kinetic energy of the fluid is given by where is the tensor of added masses, and ρ is the fluid density. In the absence of external forces, the Lagrangian of the body-fluid system L : T SE(2) → R is given by L = T B + T F . It defines a Riemannian metric on SE(2) with respect to which the motion of the body B is geodesic. Since L does not depend on the group variables, it is SE(2)-invariant and can thus be reduced to the function : se(2) → R given by with ξ as in (15). An element ν of the dual Lie algebra se(2) * is identified with a one by three matrix (x, α 1 , α 2 ). The dual pairing ·, · between se(2) * and se (2) is thus given by We perform the Legendre transform FL : ξ ∈ se(2) → ((I B + I F )ξ) T ∈ se(2) * to obtain the reduced Hamiltonian h : se(2) * → R defined by The Lie-Poisson equations on se(2) * that describe the motion of the bodyfluid system areν = ad * δh δν ν.
where ad * ξ ν is identified with (α 1 v 2 − α 2 v 1 ,θα 2 , −θα 1 ). This problem turns out to exhibit symmetry breaking phenomena from different points of view: These two approaches are the same from a group theoretical point of view. Contrary to Example 1, the Hamiltonian in consideration will not be perturbed by adding some potential energy. In this case, there is no potential energy involved, only the metric is perturbed giving rise to a modified kinetic energy. Let us now discuss the two cases mentioned above.
Adding a fluid to the system amounts to look at the variation of the parameter 7 6 In fact, the full symmetry group should be O(2)×Z 2 since Z 2 acts on the x-component by swapping the sign. However since this discrete part does not contribute in the further application, we do not take it into account. 7 We could simply consider ρ as being the parameter, but in this case the parameter would not be dimensionless and we want to avoid this.
This gives rise to the perturbed Hamiltonian h λ (ν) = 1 2 ν · I λ ν with where and c 3 = π(B 2 +md) 4d are fixed constants encoding the datas of the system. The perturbed Hamiltonian reads h λ (ν) = 1 2 and has symmetry H = D 2 , the dihedral group of order four 8 . This perturbation coincides with h when λ = 0 and the function (λ, ν) → h λ (ν) is smooth. Therefore, h λ is an H-pertubation of h. The symmetry is broken because the fluid influences the motion of the body if it is elliptical. If the body is circular (A = B), or if it moves in the vacuum, its center of mass would move at constant velocity and it would rotate at constant angular speed.
(ii) We carry out another kind of perturbation: rather than perturbing the rigid body motion by adding a fluid to the system, we start with a circular planar rigid body (A = B) in a fluid and break the symmetry by changing the body shape into an ellipse. The unperturbed Hamiltonian is given by where d 2 = ρπB 2 4 , ν := (x, α 1 , α 2 ), I := (I B + I F ) −1 and A = B in the definition of I F . The Hamiltonian is invariant with respect to G = O (2). For each c ∈ R, the level sets h(ν) = c also describe spheroids in R 3 .
We perturb the body shape by setting λ where B > 0 is fixed and A ≥ B > 0 varies. This gives rise to the perturbed Hamiltonian h λ (ν) = ν · I λ ν with where d 1 = ρπB 4 4 . The perturbed Hamiltonian is thus given by h λ (ν) = 1 2 and is again symmetric with respect to the action of H = D 2 . In this case, if there was no fluid (ρ = d 2 = 0), no symmetries would have been broken.
Since the reduced motion is governed by the Lie-Poisson equations (20), it is constrained to the coadjoint orbits of SE (2). As shown in [24] (Chapter 14.6), almost all of them are cylinders (the singular orbits consist of points on the vertical dashed line in Figure 3). In both cases, the level sets of h λ are ellipsoids and those of h = h 0 are spheroids. Their intersections with a coadjoint orbit are shown in Figure 3. In particular, the circle of equilibria of h (in red in Figure 3) breaks into four fixed points of h λ , two of which are connected by four heteroclinic cycles.
Let us go back to the first case we discussed above with h λ as in (23). We will apply Corollary 3.2 to predict the existence of the four fixed points that persist (cf. Figure 3). The Fréchet derivative of h λ is Therefore, the Lie-Poisson equations (20) reduce to Setting λ = 0 in (28), we see that the fixed points of h = h 0 are either of the form (0, α 1 , α 2 ) with (α 1 , α 2 ) ∈ (R 2 ) * , or of the form (x, 0, 0) which correspond to points on the singular coadjoint orbit. Let µ := (0, α 1 , α 2 ) with α 2 1 + α 2 2 = 1 be a fixed point of the unperturbed hamiltonian h. The isotropy subgroup of µ is G µ = r ϑ where r ϑ is a reflection in the plane. The quotient G/G µ = O(2)/ r ϑ is topologically a circle yielding Cat D 2 (S 1 ) = 2. The four fixed points appearing in Figure 3 are the two H-orbits that persist.

Symmetry breaking for relative equilibria
In this section, we extend Theorem 3.1 and Corollary 3.2 to the case of relative equilibria which is more subtle for two reasons: firstly we must take into account the conservation of momentum, and secondly for a non-zero velocity the so-called augmented Hamiltonian no longer has symmetry G.
We start by briefly recalling some standard facts about relative equilibria, the reader is invited to consult the book of Marsden [22] (Chapter 4) for a more detailed exposition. Given a Hamiltonian proper G-manifold (M, ω, G, Φ G ), a relative equilibrium of a Hamiltonian h ∈ C ∞ (M ) G is a pair (m, ξ) ∈ M × g such that X h (m) = ξ M (m). Equivalently, if (m, ξ) is a relative equilibrium of h, then m is a critical point of the augmented Hamiltonian , ξ , which is a G ξ -invariant function which depends linearly on ξ. A standard fact about relative equilibria is that the velocity ξ and the momentum µ = Φ G (m) commute i.e. ξ ∈ g µ . Note that, if the isotropy group G m is non trivial and (m, ξ) is a relative equilibrium of h, then (m, ξ + η) is also a relative equilibrium of h, for any η ∈ g m . Moreover if (m, ξ) is a relative equilibrium of h then so is (g · m, Ad g ξ) for every g ∈ G. In general a relative equilibrium is said to be non-degenerate if the Hessian D 2 h ξ (m) is a non-singular quadratic form, when restricted to the symplectic slice N 1 at m relative to the G-action. However, this definition of non-degeneracy is not enough to guarantee that a relative equilibrium of some h ∈ C ∞ (M ) G persists under an H-perturbation. For that reason, we need a stronger version of non-degeneracy.

Induced momentum map
Let H be a closed subgroup of G. The dual of the inclusion of Lie algebras i h : h → g is the projection i * h : g * → h * and is given by Let (ϕ, G × Gm (m * 0 × (N 1 ) 0 ) , U ) be a symplectic G-tube at m as in Theorem 2.2. Linearising ϕ −1 at m yields a linear symplectomorphism For x + y ∈ g m ⊕ m and z ∈ n we have N 1 ) is the orbit map. By definition, the subspace ker (DΦ H (m)) consists of the elements where the normal form for the momentum map is given by Theorem 2.3. As required −ad * z µ + f (w) ∈ h • since the kernel of i * h is equal to h • .

Non-degeneracy condition and regularity condition
We now state a stronger version of non-degeneracy of a relative equilibrium. This intersection is the subspace q · m ⊂ g · m where q is an H m -invariant complement to g µ in the "symplectic orthogonal" The non-singularity of D 2 h ξ (m) along g·m depends only on that of D 2 φ ξ G (m) which has symmetry group G ξ . The condition is a consequence of the following lemma whose proof is available in [11] (cf. Chapter 6): Lemma 4.2 (cf. Fontaine [11]). Let (M, ω, G, Φ G ) be a Hamiltonian proper G-manifold. Let m ∈ M with momentum µ = Φ G (m) and an element ξ ∈ g µ . If g is semi-simple then the Hessian D 2 φ ξ G (m) restricted to g · m is singular only along (g ξ + g µ ) · m.
Therefore if an equilibrium (m, ξ) ∈ M × g of some h ∈ C ∞ (M ) G with momentum µ = Φ G (m) is α-nondegenerate in the sense of Definition 4.1, then g ξ has trivial intersection with q. In Theorem 4.3 we show that a number of orbits of relative equilibria of h persist under H-perturbations. Such relative equilibria must have their velocity ξ in h µ . We assume an additional regularity assumption This says essentially that µ needs to be "more regular" (cf. Fontaine [11] Definition 6.2.2). However if ξ ∈ h µ , this assumption depends on the embedding of h → g as shown in the example below. This is not a problem for us because isomorphic Lie algebras have different underlying Lie groups.
Example 2. In this example we show when condition (R) holds for g = so(4) and a subalgebra isomorphic to h = so(3). The Lie algebra g is identified with the set of pairs (x, a) ∈ R 3 × R 3 with Lie bracket The dual Lie algebra g * consists of pairs (χ, ρ) ∈ R 3 × R 3 which satisfy The linearized coadjoint action of g on g * is given by Lie subalgebras isomorphic to so(3). Elements of h = so(3) are identified with vectors x ∈ R 3 . We consider two Lie subalgebra of g isomorphic to h, namely Regularity condition. Given a fixed momentum µ := (χ, ρ) ∈ g * , the stabilizer Lie subalgebra is by (31). We show below whether condition (R) is satisfied for our two different choices of Lie subalgebras isomorphic to h.

Persistence of relative equilibria
We are now ready to state an equivalent version of Theorem 3.1 for relative equilibria. The proof follows the same steps as Theorem 3.1. For that reason some details have been skipped.
. Let α be the restriction of µ to h. We assume that there is a function f η λ ∈ C ∞ (G µ /G m ) Hµ whose critical points are in one-to-one correspondence with those of h η λ in U .
Proof -Let (m, ξ) ∈ M × h be an α-nondegenerate relative equilibrium h, where α is the restriction of the momentum µ = Φ G (m) to h. By assumption Φ −1 H (α) is a smooth manifold on which G µ ⊂ H α acts canonically and properly.
Let K = G m and consider the K-vector space N := N 1 ⊕M, where N 1 is a symplectic slice at m relative to the G-action, and M is as in Proposition 4.1. By construction N is isomorphic to some K-vector space complementary to g µ · m in T m Φ −1 H (α) . By the Tube Theorem (cf. [28] Theorem 2.3.28), there is a K-invariant neighbourhood N 0 ⊂ N of zero, such that (ii) The only critical points of h ξ in U are on G µ · m.
In that model, the point m corresponds to [(e, 0)] and the augmented Hamil- According to the proof of Theorem 3.1, the critical points of h ξ λ are in bijective correspondence with those of the lift We may thus work with h ξ λ instead of h ξ λ . We define a (left) action of the direct product G µ × K on G µ × N 0 by (h, k) · (g, ν) = (hgk −1 , k · ν).
As in the proof of Theorem 3.1, we can use the Implicit Function Theorem and the compacity of H µ \G µ to get an H µ -invariant smooth function φ η λ : G µ → N 0 , depending on parameters (λ, η) taken in a neighbourhood V ⊂ R × h of (0, ξ), satisfying Example 3 (Torus action). As a first application, we recover the result of Grabsi, Montaldi and Ortega [13] for compact abelian groups and free actions. Let (M, ω, T n , Φ T n ) be a Hamiltonian T n -manifold where T n is a n-dimensional torus acting freely on M and let T r be a subtorus of T n . Assume h ∈ C ∞ (M ) T n has an α-nondegenerate relative equilibrium (m, ξ) ∈ M × t r with momentum µ = Φ T n (m) and where α = µ t r . As T n and T r are abelian, condition (R) always hold. By compactness of T n , condition (OP S) is automatic and then any T r -perturbation h λ with λ small enough has at least Cat T r (T n ) T r -orbit of relative equilibria with velocity closed to ξ in a neighbourhood of T n · m in Φ −1 T r (α). Since T n acts freely on T r by left multiplication, Cat T r (T n ) = Cat(T n /T r ) = Cat(T n−r ).

The spherical pendulum
As an application of Corollary 4.4, we consider the case of the spherical pendulum whose Hamiltonian is viewed as a perturbation of the Hamiltonian governing the motion of an unit mass point constrained to move on the surface of S 2 . Endow R 3 with the standard inner product ·, · and let e 1 , e 2 , e 3 be the standard basis. The phase space for the spherical pendulum is the Hamiltonian proper G-manifold (T * S 2 , ω, G, Φ G ) where G = SO(3) acts on T * S 2 = (x, y) ∈ S 2 × R 3 | x, y = 0 by matrix multiplication A · (x, y) = (Ax, Ay). The associated momentum map Φ G : T * S 2 → R 3 is Φ G (x, y) = x × y.
Let H = SO(2) be the subgroup of rotations about the e 3 -axis with Lie algebra h, the one-dimensional vector space generated by e 3 . We can think about the Hamiltonian of the spherical pendulum h λ (x, y) = 1 2 y 2 + λ x, e 3 as an H-perturbation of the G-invariant Hamiltonian h(x, y) = 1 2 y 2 . The relative equilibria of h λ are the pairs ((x, y), ξ) ∈ T * S 2 × h such that where φ ξ H (x, y) := x × y, ξ . A straightforward calculation shows that a relative equilibrium satisfies the equations λ + ξ 2 x, e 3 = 0 and y = ξ × x. (36) The relative equilibria ((x, y), ξ)) of the unperturbed Hamiltonian h are such that x moves along a great circle on S 2 and those of h λ are such that x describes a circular trajectory at fixed height in the lower hemisphere of S 2 . Therefore the only H µ -orbit that has a chance to persist under h λ is the one when x moves along the equator. Indeed the point m = (e 1 , se 2 ) and ξ = se 3 , with s > 0, define a G-nondegenerate relative equilibrium of the unperturbed Hamiltonian h, with momentum µ = se 3 . Observe that G µ = SO(2) is the group of rotations about the e 3 -axis. Embedding S 1 in S 2 as the equator allow us to view the orbit G µ ·m as the set of perpendicular pairs (x, y) ∈ S 1 × S 1 s where S 1 s is the equator of the sphere of radius s. The (stable) relative equilibria of h λ are of the form ((x, y), η) with η = re 3 for some r ∈ R and with x ∈ S 2 having uniform circular motion at constant negative height − λ r 2 . Since x ∈ S 2 we must have |λ| < r 2 . Given x we can calculate y from (36). Setting α = s we get The last equation can be solved for r and we find two solutions, one positive and one negative. The condition |λ| < r 2 implies that the only valid solution is the positive one. Since the distance |r −s| is controlled by λ, the velocity η is close to ξ as λ is sufficiently small. We conclude that for λ small enough, h λ has exactly one H µ -orbit of relative equilibria in a neighbourhood of G µ · m in Φ −1 H (α) with velocity close to ξ. For this example, the assumptions of Theorem 4.3 are all satisfied. As expected, we have Cat Hµ (G µ /G m ) = Cat SO(2) (S 1 ) = 1.
Bifurcation diagram. If (x, y) satisfies (36) for some ξ, then it is a critical point of rank one of the energy momentum map F λ = (h λ , Φ H ). If we let ξ varies, s is viewed as a parameter and the boundary of F λ (T * S 2 ) is the convex region in R 2 defined by the curve The critical points of rank zero (when ξ = 0) are sent on F λ (±e 3 , 0) = (±λ, 0). The point (−e 3 , 0) is of type elliptic-elliptic whereas the point (e 3 , 0) is of type focus-focus (cf. Cushman and Duistermaat [8] and Vu Ngoc and Sepe [33]).