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Bifurcation of relative equilibria in mechanical systems with symmetry

The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and construct a collection of implicitly defined functions and reduced equations describing the set of relative equilibria in a neighborhood of a given relative equilibrium. The structure of the reduced equations is studied in a few relevant situations. In particular, a persistence result of Lerman and Singer [LS98] is generalized to the framework of Abelian proper actions. Also, a Hamiltonian version of the Equivariant Branching Lemma and a study of bifurcations with maximal isotropy are presented. An elementary example is presented to illustrate the use of this approach.


Introduction
The systematic analysis of bifurcations of relative equilibria was greatly stimulated about fifteen years ago by specific applications with nonconservative vector fields, namely the secondary bifurcations from nontrivial equilibria in hydrodynamical systems such as Couette-Taylor flows and Rayleigh-Bénard convection in a spherical shell. The problem was attacked analytically by Chossat and Iooss [CI85], and more qualitatively by Rand [R82]. A major success of the analytical approach was obtained by Iooss [I86], who classified the possible patterns bifurcating from a group orbit of equilibria in a system with symmetry O(2). In Moutrane [M88], the bifurcation of rotating waves, which are relative equilibria with a single drift frequency, was investigated in the problem of the onset of convection in a system with spherical symmetry. However it was Krupa [K90] who first developed a general theory for the bifurcation from relative equilibria. The basic tool he used was the Invariant Slice Theorem of Palais (see [Pal61,B72]). If G is a Lie group acting properly on the manifold M , the Slice Theorem establishes for each m ∈ M an isomorphism between a tubular neighborhood of the orbit G · m and the normal bundle with base G · m and fiber the normal slice N m to the tangent space to G · m at m. It was shown by Field [F80] and then by Krupa that any G-equivariant vector field X ∈ X(M ) admits in the tubular neighborhood a decomposition into the sum of two vector fields: one, X N , defined on the normal bundle, and the other, X T , defined on the tangent bundle to G · m. Krupa showed that the dynamical information, in particular the bifurcation properties for a parameter dependent family of vector fields, is entirely contained in X N .
The analysis of relative equilibria of conservative systems has played a central role in the development of geometric mechanics, ranging from the classic work of Riemann [R860] and Routh [R882], [R884] to Smale's seminal work [S70]. However, the use of local singularity theory methods, rather than explicit calculations or global topological methods, in the analysis of conservative systems is relatively recent [HM83,GoS87,LMR87,L87,DMM92,MS93]. Bifurcations of relative equilibria of Lagrangian systems and canonical Hamiltonian systems, i.e. Hamiltonian systems on cotangent bundles, with the canonical symplectic structure and a lifted group action, have been studied by Lewis et al. [LSMR92] and Lewis [L93, L94] using the reduced energy-momentum method developed in [SLM91] and [L92]. This approach uses the locked Lagrangian, the generalization of Smale's augmented potential to Lagrangian systems and their Hamiltonian analogs, to characterize relative equilibria as critical points of functions on the configuration manifold parameterized by elements of the algebra g of the symmetry group G. A key component of the reduced energy-momentum method is the decomposition of the tangent space T q Q of the configuration manifold Q at a point q into the tangent space g · q to the group orbit and an appropriate complement consisting of so-called 'internal' variations. The associated decomposition of the relative equilibrium equations into 'rigid' and 'internal' equilibrium conditions is analogous to the decompositions introduced by Field [F80] and Krupa [K90] in the context of general equivariant vector fields. The 'rigid' condition can be used to determine a submanifold of 'candidate relative equilibria'; imposing the remaining equilibrium conditions on this submanifold determines the relative equilibria.
Our goal is the development in the symplectic category of a decomposition tool analogous to that of Krupa that will take into account the additional structure present at the kinematical level in Hamiltonian systems, without assuming all the ingredients utilized in the reduced energymomentum method. Given that many Hamiltonian systems are constructed on symplectic manifolds that are not cotangent bundles, such a tool is of much interest. The analog of the Invariant Slice Theorem in the symplectic category is given by the Marle-Guillemin-Sternberg normal form [Mar85,GS84a,GS84b] (we will refer to it as the MGS-normal form) so, in principle, one could work as in Krupa [K90], using this normal form instead of the Slice Theorem. This does not seem to be the best way to proceed, since to search for relative equilibria of Hamiltonian systems one does not need to work with the Hamiltonian vector field; there are scalar functions, the augmented Hamiltonians, whose critical points are precisely the relative equilibria. Guided by Krupa's normal bundle decomposition for equivariant vector fields and the MGS-normal form, in Section 2 we will construct a slice mapping with which we can decompose the critical point equations determining the relative equilibria into a system of four equations. These split critical point equations are analyzed in Section 3 in a neighborhood of a given a relative equilibrium m e . Using the Implicit Function Theorem and Lyapunov-Schmidt reduction, we can construct a local submanifold containing all relative equilibria sufficiently near the group orbit of m e . The remaining equilibrium conditions, called the reduced critical point equations, on this submanifold can be analyzed using standard techniques from bifurcation theory. In Section 3.1 we study the equivariance properties of the reduced critical point equations. In Section 3.2 we construct a slice mapping with respect to which one of the reduced critical point equations admits a simpler solution.
In Section 4 we use the reduced critical equations and a slice mapping constructed via the MGS-normal form to study the persistence of a family of relative equilibria in a neighborhood of a non-degenerate relative equilibrium when the symmetry group of the system is Abelian. In particular, we generalize to proper group actions a result from Lerman and Singer [LS98] originally proven for compact groups. This result was already presented in [O98].
In Section 5 we study bifurcations from a degenerate relative equilibrium and find Hamiltonian analogs of bifurcation theorems for solutions with maximal isotropy which were first stated in the non-conservative context, namely the Equivariant Branching Lemma of Vanderbauwhede [V80] and Cicogna [Ci81], and a theorem for bifurcation of solutions with maximal isotropy group of complex type [M94,CKM95].
Finally we provide a simple application of the method described here to wave resonances. This example is, for the most part, well known; it is included to illustrate the implementation of the method, rather than to provide new information.

Relative equilibria as critical points
Let G be a Lie group acting smoothly on the manifold M and let X ∈ X(M ) be a smooth Gequivariant vector field on M with flow F t . We say that the point m e ∈ M is a relative equilibrium of the vector field X if there exists an element ξ of the Lie algebra g of G, called a generator of the relative equilibrium, such that F t (m e ) = exp(t ξ) · m e . Suppose now that the manifold M is symplectic and that the vector field X is Hamiltonian, with associated G-invariant Hamiltonian h ∈ C ∞ (M ). In addition, assume that the Lie group action of G on M is Hamiltonian, with associated equivariant momentum map J : M → g * . These hypotheses imply that the Hamiltonian vector field X h of h and its flow F t are G-equivariant. In this framework, the search for relative equilibria reduces to the determination of the critical points of a certain class of functions. Indeed, a classical result ([AM78, page 307] and [A78, page 380]) states that a point m e ∈ M is a relative equilibrium of X h with generator ξ ∈ g if and only if m e is a critical point of the augmented Hamiltonian h ξ := h − J ξ . Thus, our algorithm is intended to identify the pairs (m e , ξ) ∈ M × g such that Dh ξ (m e ) = 0. (1) Note that if m e has nontrivial continuous symmetry, i.e. g me = {ζ ∈ g : ζ M (m e ) = 0} = {0}, then the generator of m e is not unique. If ξ is a generator of a relative equilibrium m e , then for any ζ ∈ g me , ξ + ζ is also a generator.
The main goal of this section is the decomposition of the relative equilibrium equation (1) into a systems of four equations, each defined on a space determined by the geometry of the problem.
Assume that m e is a relative equilibrium with generator ξ and momentum µ := J(m e ). Let g me denote the Lie algebra of the isotropy subgroup of m e and g µ the Lie algebra of the isotropy subgroup of µ. Choose complements q of g µ in g and m of g me in g µ , so that (2) The symbols i and P with appropriate subscripts will denote the natural injections and projections according to the splittings (2). For instance i gm e : g me → g = g me ⊕ m ⊕ q is the canonical injection of g me into g and P gm e : g = g me ⊕ m ⊕ q → g me extracts the g me component of any vector in g.
Definition 2.1 Let V be a vector space and U ⊂ m * × V be an open neighborhood of (0, 0) ∈ m * × V . A smooth mapping Ψ : U ⊂ m * × V → M is said to be a slice mapping at the point m e ∈ M if it is a diffeomorphism onto its image satisfying the following conditions: (SM3) The pullback j := J • Ψ : U → g * of the momentum map satisfies for all δη ∈ m * and δv ∈ V .
In the following proposition we explicitly construct a slice mapping Ψ at a point m e in a finite dimensional manifold M .
Proposition 2.2 Let ψ : U ⊂ X → M be a coordinate chart around a point m e in a finite dimensional manifold M and let V and W be subspaces of the vector space X satisfying is an isomorphism.
Let V ′ and W ′ be neighborhoods of the origin in V and W such that V ′ × W ′ ⊂ U and set U : Proof Property (SM1) follows trivially from (i). Property (SM3) follows from (ii), (iii) and the definition of Ψ.

Remark 2.3
If symmetry is broken in a neighborhood of m e , then g me · Ψ(η, v) is typically nontrivial. In this case, the first two conditions alone do not guarantee that the rigid condition ad * ξ j(η, v) = 0 is satisfied. However, if all four conditions are satisfied, then D H − j ξ (Ψ(η, v)) = 0; in particular, Remark 2.4 Note that in order to split the critical point equation (1) into (7), only property (SM2) of the slice mapping was utilized. As we shall see in the following section, property (SM3) simplifies the analysis of the equations (7). Equations (RE1) and (RE3) are, by construction, nondegenerate in the sense that implicit solutions to these equations always exist. Thus the bifurcation analysis is carried out only on the equations obtained by substituting the solutions of (RE1) and (RE3) into (RE2) and (RE4).

The reduced critical point equations
In this section we start with a relative equilibrium m e with generator ξ ∈ g and, using the Implicit Function Theorem and Lyapunov-Schmidt reduction (see for instance [GoS85]), we derive a minimal set of mappings and equations determining the relative equilibria in a neighborhood of m e . We will call the remaining equations the reduced critical point equations. We proceed in three steps.
Here we used property (SM3) of the slice mapping Ψ.
Since δγ → i * q (ad * δγ µ) is an isomorphism between q and q * , we conclude that the partial derivative D q F 1 (0) is an isomorphism. Thus the Implicit Function Theorem implies that there is a function γ : for all (η, v, α, β) ∈ U 1 . In other words, we have found a m * × V × g me × m-parameter family of points that satisfy part (RE1) of the split critical point equations. Set Step 2. In this step we assume that the subspace m is reflexive, that is, m * * ≃ m. We now construct a m * × V × g me -parameter family of points satisfying the relative equilibrium equations (RE1) and (RE3) by applying the Implicit Function Theorem to (RE3), solving for the m component of the family of points constructed in Step 1. Let F 2 : U 1 ⊂ m * × V × g me × m → m * * ≃ m be the mapping defined by F 2 (η, v, α, β) := D m * F (η, v, ω 1 (η, v, α, β)). Since we intend to solve the equation F 2 = 0 for the m parameter using the Implicit Function Theorem, we compute D m F 2 (0, 0, 0, 0). Given arbitrary δβ ∈ m and δη ∈ m * , follows from property (SM3) of the slice map and the formula (8) for the generator ω 1 . Hence D m F 2 (0) is the identity map. The Implicit Function Theorem thus implies that there is a function β : Step 3. We now treat the (RE4) component of the relative equilibrium equation. We use the standard Lyapunov-Schmidt reduction procedure of bifurcation theory to partially solve (RE4). Let L : V → V * denote the linear transformation satisfying for all v and w ∈ V . Set V 0 := ker L and choose closed subspaces V 1 ⊂ V and V 2 ⊂ V * such that To guarantee the existence of the complements V 1 and V 2 in infinite dimensions, we assume that L is a Fredholm operator. Let P : V * → V 2 denote the projection determined by the decomposition of V * . Note that since the operator L is such that for all v and w ∈ V , L v, w = L w, v , the spaces V 0 and V 2 can be naturally identified by choosing an inner product (when V is infinite dimensional we still can do it provided that V is a Hilbert space). Define Using the Implicit Function Theorem once more, we can solve the equation since (I − P)L = L and V ⊂ ker Dj(0). Thus D V1 F 3 (0) is an isomorphism of V 1 onto range L and the Implicit Function Theorem guarantees the existence of a neighborhood U 3 of (0, 0, 0) ∈ m * ×V 0 ×g me , and a local function v 1 : U 3 → V 1 such that for any (η, v 0 , α) ∈ U 3 . Define the generator map Ξ : U 3 → g, B : U 3 → V 2 , and ρ : U 3 → m by In a sufficiently small neighborhood U 3 of the origin any solution (η, v 0 , α) of the equations On the other hand, any relative equilibrium m sufficiently near m e in the slice Ψ(m * × V ) satisfies m = Ψ(η, v 0 + v 1 (η, v 0 , α)) for some solution (η, v 0 , α) of (B1) and (B2); any generator ξ of m satisfies ξ − Ξ(η, v 0 , α) ∈ g m . Equations (B1) and (B2) will be usually referred to as the bifurcation equation and the rigid residual equation respectively. Let R : h × m * × V 0 → m * × V 0 be the mapping that groups both equations, that is, We will refer to the equality as the reduced critical point equations.
Remark 3.1 Notice that even though the critical point equations (1) determining the relative equilibria in our situation can be naturally understood as a gradient equation, this analytic feature is not in general available for the reduced version (11) of these equations. A particular case where the gradient character of (1) is preserved by the reduction procedure is when the relative equilibrium m e that we start with has total isotropy, that is, it is actually an equilibrium and G me = G. Notice that in this case m = q = {0} since g me = g. Therefore, the rigid residual equation (B2) is trivial and then, as we will show, the bifurcation equation (B1) is a gradient equation. The strategy that we will take follows very closely the one introduced in [GMSD95].
If m = q = {0}, then any coordinate chart ψ : U ⊂ X → M such that ψ(0) = m e is a slice mapping at m e , with V = X, and the critical point equations (RE1)-(RE4) collapse to the single equation In this situation only the third step of the general procedure, the Lyapunov-Schmidt reduction, is nontrivial.
We fix an inner product ·, · on V and denote by ∇ V H − j ξ (v) the usual gradient of H − j ξ with respect to ·, · , i.e.
If m e is a relative equilibrium with generator ξ, the relative equilibria around m e are given by the zeroes of the map F : for any v and w ∈ V . Note that the mapping L is a self-adjoint operator; hence if we set V 0 = ker L and V 1 = range L, then V has the orthogonal decomposition and v 1 ∈ V 1 , and apply the Implicit Function Theorem to the equation The remaining equation that is, the bifurcation equation, is Indeed, note that for any w ∈ V 0 , P is self-adjoint, and (13) is satisfied.

The equivariance properties of the reduced critical point equations
The symmetries of the relevant equations play an important role the solution of a bifurcation problem (see for instance [GSS88]). We will see that if the G-action on M is proper, then the relative equilibrium equations (B1) and (B2) can be constructed so as to be equivariant with respect to the induced action of G me ∩ G ξ on m * × V 0 . Here G ξ denotes the isotropy subgroup of the generator ξ ∈ g of the relative equilibrium m e ∈ M with respect to the adjoint action of G on g.
An equivariant slice mapping is a mapping Ψ : U ⊂ m * × V → M satisfying Definition 2.1 and the condition (ESM) The subspace m * of g * is Ad * (Gm e ∩G ξ ) -invariant and the slice mapping Ψ : U ⊂ m * ×V → M is G me ∩ G ξ -equivariant with respect to the coadjoint action of G me ∩ G ξ on m * and some action of G me ∩ G ξ on V .
Note that since the group G me ∩ G ξ is compact and fixes (0, 0) ∈ m * × V , the open neighborhood U of (0, 0) ∈ m * × V in (ESM) can always be chosen to be G me ∩ G ξ -invariant.
Proposition 3.2 If the group G acts properly on M and the coordinate chart ψ : U ⊂ X → M with ψ(0) = m e is equivariant with respect to some action of G me ∩ G ξ on X, then the subspaces m, q, V , and W can be taken to be G me ∩ G ξ invariant. For these choices, the slice mapping constructed in Proposition 2.2 is G me ∩ G ξ -equivariant.
Proof First, we show that G me ∩ G ξ -invariant decompositions g = g me ⊕ m ⊕ q and X = V ⊕ W exist. Note that the isotropy subgroup G me is compact, since the action of G on M is assumed to be proper; consequently the subgroup G me ∩ G ξ is also compact. This guarantees the existence of a Ad (Gm e ∩G ξ ) -invariant inner product on g, which we can use to determine a Ad (Gm e ∩G ξ ) -invariant decomposition g = g me ⊕ m ⊕ q of the Lie algebra.
The orthogonal complement to g µ · m e in ker T me J with respect to a G me ∩ G ξ -invariant inner product is an invariant subspace. Hence the preimage with respect to the equivariant map T 0 ψ of this orthogonal complement is a G me ∩ G ξ -invariant subspace of X; we choose this subspace as the vector space V in Definition 2.1. Analogously, the space W can be chosen to be invariant under the action on X.
Given these choices of subspaces, the action of G me ∩ G ξ on M induces a well-defined action on m * × V via the slice map. Equivariance of the momentum map, the coordinate chart, and the projection P m imply that the slice map Ψ is equivariant.
Recall that the relative equilibrium equations were obtained using two consecutive applications of the Implicit Function Theorem (Steps 1 and 2) followed by the Lyapunov-Schmidt reduction procedure (Step 3). It is well known that if the Implicit Function Theorem is applied to an equation F = c determined by an equivariant map F and a fixed point c of the group action, then the resulting implicitly defined function is also equivariant. In addition, if the Lyapunov-Schmidt reduction procedure is applied to such an equation using invariant subspaces, then the resulting functions and equations will be equivariant. (See, e.g., [GoS85,GSS88] for precise statements and proofs of these results). Using these fundamental results, we now show that, given appropriate choices of slice maps and subspaces, the generator map Ξ and the functions B and ρ determining the reduced relative equilibrium equations are equivariant with respect to the induced G me ∩ G ξ action on m * × V × g me . Proposition 3.3 If the spaces m, q, V , and W are G me ∩ G ξ invariant and the slice mapping is G me ∩ G ξ -equivariant, then the maps Ξ, v 1 , B, ρ, and F are all G me ∩ G ξ -equivariant.
Proof It suffices to show that the functions F 1 , F 2 , and F 3 given in steps 1, 2, and 3 are G me ∩ G ξequivariant. We first consider the mapping F 1 : U × g me × m × q → q * introduced in Step 1. For arbitrary g ∈ G me ∩ G ξ : Thus F 1 is G me ∩ G ξ -equivariant and, hence, the implicitly defined functions γ and ω 1 are also G me ∩ G ξ -equivariant. An analogous verification can be carried out for the mapping F 2 in Step 2, allowing us to conclude that the function ω 2 is also G me ∩ G ξ -equivariant.
To establish the invariance (respectively equivariance) of the spaces and maps constructed in Step 3, we first note that H − j ξ is G me ∩ G ξ -invariant, since the augmented Hamiltonian h − J ξ is G ξ -invariant and the slice map Ψ is G me ∩ G ξ -equivariant. Equivariance of the map F , and hence invariance of the subspaces ker F and range F , follows immediately from the invariance of H − j ξ . The compactness of the group G me ∩ G ξ allows us to choose G me ∩ G ξ -invariant complements V 1 and V 2 to ker F and range F . (See for instance [GSS88, Proposition 2.1].) With these choices, the canonical projection P and the function F 3 are equivariant. Consequently the function v 1 , as well as the generator map Ξ and the reduced relative equilibrium equations are equivariant, as required.

Treatment of the rigid residual equation
In this section we consider some situations in which the rigid residual map is either trivial or at least fairly simple. For example, if G is Abelian, then the full rigid equation ad * ξ J(m e ) = 0 is trivial. Hence, the rigid residual equation is obviously satisfied as well. If G is not Abelian, an appropriate choice of a slice map Ψ : m * × V → M can be useful in identifying the solutions of the residual rigid equation. We will present a few cases in which these helpful choices are possible.
Given a relative equilibrium m e of h : We shall say that a subspace We now prove that, generically, the rigid equation ρ can be reduced by an appropriate choice of slice map to an equation on g µ .
Proposition 3.4 If the complement q to g µ in g is g µ -invariant, then given any slice map Ψ : U → M at m e , there exists a map φ :Ũ → q such that 1. the mapΨ :Ũ ⊂ U → M given bỹ is also a slice map 2. the associated generator mapΞ takes values in g µ 3. the pullbackj := J •Ψ of the momentum map takes values in µ + q • 4. φ(0, 0) = 0 and Dφ(0, 0) = 0.
If the original slice mapping is G me ∩ G ξ -equivariant, thenΨ is equivariant.
Proof We obtain the map φ through yet another application of the Implicit Function Theorem.
Using φ :Ũ ⊂ m * × V → q and (14), we see that the pullbackj of the momentum map satisfies for all (η, v, α, β) ∈Ũ 1 . Thus executing Step 1 of Section 3 using the modified slice mappingΨ yields a mappingγ : for any (η, v, α, β) ∈Ũ 1 .γ ≡ 0 clearly satisfies this equation; hence it is the unique solution of the equation F 1 ≡ 0 given by the Implicit Function Theorem. Thus steps 2 and 3 yield the generator mapΞ Suppose now that the slice map Ψ has the property (ESM). Note that for any Equivariance of R implies that φ, and henceΨ, are equivariant.
If the hypotheses of Proposition 3.4 are satisfied, the rigid residual equation involves only elements of g µ and g * µ . Specifically, if we let [ , ] µ denote the Lie bracket on g µ and J µ : M → g * µ denote the momentum map associated to the action of G µ on M , namely J µ = i * gµ J, then ρ satisfies for all β ∈ m. In particular, if g µ is Abelian, then ρ is identically zero. Thus we have established the following corollary.
Corollary 3.5 Let m e be a relative equilibrium with momentum µ = J(m e ). If g µ is Abelian and there exists a g µ -invariant complement to g µ in g, then there is a slice map with respect to which the rigid residual map ρ is identically zero.
Another approach to the search for solutions of the rigid residual equation is to restrict this search to fixed point subspaces corresponding to subgroups of the symmetry group of ρ. More explicitly, suppose that the hypotheses of Proposition 3.4 are satisfied and that we start with an equivariant slice map Ψ. In that case, Proposition 3.3 guarantees that ρ is G me ∩ G ξ -equivariant and satisfies (16). Equivariance implies that for any Lie subgroup K ⊂ G me ∩ G ξ , the map ρ maps the set of fixed points of K into the set of fixed points of K in m * . Hence all zeroes of the restriction of ρ to (m * ) K ×V K 0 ×g K me are also zeroes of ρ (here the superscript K denotes the subspace of K-fixed points with respect to the relevant action). In other words, we can look for the solutions of the rigid residual equation by searching the zeroes of its restrictions to different sets of K-fixed points, with K and arbitrary subgroup of G me ∩ G ξ which, in principle, should be easier, since the dimension of the system has been lowered without introducing additional complexity into the equations.
If the restriction of the Lie bracket of the Lie algebra g µ to g K µ is trivial, then the entire subspace The equivariance ofΞ and J µ implies that ζ ∈ g K µ and ν ∈ (g due to the hypothesis on the Lie bracket on g K µ . The arbitrary character of ξ ∈ m K implies that ρ(η, v 0 , α) = 0.
Thus we have then proved the following Proposition 3.6 Let m e be a relative equilibrium with momentum µ = J(m e ) and generator ξ ∈ g.
If there exists a g µ -invariant complement to g µ in g, then for any subgroup K ⊂ G me ∩ G ξ for which the restriction of the Lie bracket of the Lie algebra g µ to the set of fixed points g K µ is trivial, there is a slice mapΨ with respect to which the entire subspace (m * ) K × V K 0 × g K me consists of zeroes of the rigid residual equation ρ.
(See [RdSD97] for persistence results on nondegenerate Hamiltonian relative equilibria valid under conditions of this sort.)

Persistence in Hamiltonian systems with Abelian symmetries
In this section we will focus on the relative equilibria of Hamiltonian systems with Abelian symmetry groups. Specifically, we consider a Hamiltonian system (M, ω, h, G, J : M → g * ), where ω is the symplectic structure of M , the Abelian group G acts properly on M with associated momentum map J, and the Hamiltonian h is G-invariant. Let m e ∈ M be a relative equilibrium with generator ξ and momentum µ = J(m e ). Since the adjoint and coadjoint actions of an Abelian group are trivial, G µ = G and the rigid residual equation (B2) is trivially satisfied. We also assume that the bifurcation equation (B1) is trivial, i.e. that m e is a nondegenerate relative equilibrium, with In this situation Steps 1 through 3 in Section 3 guarantee the existence of a m * × g me -parameter family of relative equilibria persisting from m e , whose dimension and structure we will study in what follows. We use the word persistence as opposed to the word bifurcation, given that the latter is customarily used to indicate a qualitative change in the family of relative equilibria as a given parameter is varied. This is analytically reflected in the need for a nontrivial Lyapunov-Schmidt reduction procedure in order to write the bifurcation equations. We shall see that in the case at hand no such tool will be necessary.
In this section we will use a very special slice mapping based on the Marle-Guillemin-Sternberg normal form [Mar85, GS84a, GS84b] (we will refer to it as the MGS-normal form), that we briefly describe. The following exposition includes without proof the details of the MGS-normal form that will be needed in our discussion. For additional information the reader should consult the above mentioned original papers or [RdSD97,O98,OR99a].
We start by introducing the main ingredients of the MGS construction. Even though we are in the Abelian case we will present, for future reference, the general case. First, the properness of the G-action implies that the isotropy subgroup G me is compact. Second, the vector space V me := T me (G·m e ) ω /(T me (G·m e ) ω ∩T me (G·m e )) = ker T me J/T me (G µ ·m e ) is called the symplectic normal space, which is a symplectic vector space with the symplectic normal form ω Vm e defined by  Ad Gm e -invariant inner product on g consider the inclusions m * ⊂ g * µ ⊂ g * . Then, the manifold can be endowed with a symplectic structure ω Y with respect to which the left G- Since we intend to prove general statements about relative equilibria of Hamiltonian systems with Abelian symmetries, the previous theorem allows us to reduce the problem to the study of systems of the form (Y, ω Y ). Indeed, we will assume that the MGS-normal form is constructed around the relative equilibrium m e represented by [e, 0, 0] in "MGS coordinates". It can be easily shown that the map given by is a slice mapping at the point [e, 0, 0] for D(h Y − J ξ Y ), which is the representation given by the MGS-normal form of D(h − J ξ ).
Before stating the following theorem, we recall from elementary differential geometry the basic notion of the rank of a surface given in parametric form. Let g : U ⊂ R n → R m be a parameterization of a surface S in R m . Given a value u ∈ R n of the parameter, the rank of the surface S g(u) at the point g(u) ∈ R m is the rank of the Jacobian of the function g at u. If this rank is constant, the Fibration Theorem [AMR88, Theorem 3.5.18] guarantees that S is a submanifold of R m and its rank coincides with the dimension of S as a manifold on its own. Theorem 4.3 Let (M, ω, h) be a Hamiltonian system, and G be an Abelian Lie group acting properly on M in a globally Hamiltonian fashion, with invariant momentum map J : M → g * . Suppose that h is G-invariant and that the Hamiltonian system determined by h has a nondegenerate relative equilibrium at the point m e ∈ M , with generator ξ ∈ g. Set H := G me and µ = J(m e ) ∈ g * . Then there is a surface S of relative equilibria through m e that can be locally expressed as using the MGS normal form Y constructed around the orbit G · m e . Here v : m * × h → V me is a smooth function such that v(0, 0) = 0 and rank(Dv(η, α)) = dim H − dim H v(η,α) . The rank, rank S [g,η,v(η,α)] , of the surface S at the relative equilibrium [g, η, v(η, α)] equals Proof The surface S of relative equilibria is constructed in Steps 1 through 3 of Section 3, taking as slice mapping the map Ψ(η, v) = [e, η, v] constructed with the help of the MGS-normal form. Indeed, since the nondegeneracy of m e and the Abelian character of G imply that (B1) and (B2) are trivially satisfied, there is a neighborhood U ⊂ m * × h of the point (0, 0) and functions v : U → V me and Ξ : U → g such that for any (η, α) ∈ U, the point [e, η, v(η, α)] ∈ Y ≃ M is a relative equilibrium of the system (M, ω, h) with generator Ξ(η, α) ∈ g. At the same time, since the Lie group G is Abelian and the Hamiltonian flow F t associated to h is G-equivariant, it is easy to verify that if the point [e, η, v(η, α)] is a relative equilibrium with generator Ξ(η, α) ∈ g then, for any g ∈ G, the point [g, η, v(η, α)] is also a relative equilibrium with the same generator. In order to prove (19), we compute Dv(η, α) by implicit differentiation of the equation F 3 (η, v(η, α), α) = 0 defining the function v in Step 3. Note that in this case the space V 0 is trivial and we have dropped the subscript from v 1 . Note that q is trivial in the Abelian case and hence ω 2 (η, v, α) = ω 1 (η, v, α, β(η, v, α)) = ξ + α + β(η, v, α).
For u ∈ V me , for arbitrary δα ∈ h, if we set α t = α + t δα, we have The last equality follows from the identity which implies that By hypothesis, the quadratic form is nondegenerate; therefore, since nondegeneracy is an open condition, is nondegenerate for any (η, α) ∈ m * × h sufficiently close to (0, 0). Hence the rank of D h v(η, α) equals the rank of DJ Vm e (v(η, α)) at a point (η, α) ∈ m * × h sufficiently close to (0, 0). Thus as required. The expression (19) for the rank of the surface S at a relative equilibrium [g, η, v(η, α)] is a straightforward consequence of the formula 23 for the rank of D h v(η, α). The rank of S at [g, η, v(η, α)] is the rank of the parameterization of the surface S. The map S has rank at [g, η, v(η, α)], as required.
As a corollary to the previous theorem, we can formulate a generalization of a result due to E. Lerman and S. Singer [LS98], originally stated for toral actions, to proper actions of Abelian Lie groups. This result was already presented in [O98]. Proof The manifold Σ is the submanifold of the surface S, obtained by setting the parameter α ∈ h equal to zero; in other words The submanifold Σ is a smooth manifold, since (19) implies that it has constant rank 2(dim G − dim H); that is, the map with image Σ is a local constant rank map around (e, 0) ∈ G × m * with rank equal to 2(dim G − dim H), which implies that the surface Σ is locally a manifold through the relative equilibrium m e , of dimension 2(dim G − dim H). (See, for instance, [AMR88, Theorem 3.5.18].) The symplectic nature of Σ can be verified in a straightforward manner. Indeed, we will check that if i : Σ ֒→ Y is the natural inclusion then the pair (Σ, ω Σ ), with ω Σ = i * ω Y , is a symplectic submanifold of (Y, ω Y ). Let π : G × m * × V me → G × H (m * × V me ) be the canonical projection. Note that every vector in T [g,η,v(η,0)] Σ can be written as T (g,η,v(η,0)) π(T e L g · ζ, δη, D m * v(η, 0) · δ), for some ζ ∈ g and δη ∈ m * . The two-form ω Σ is clearly closed. In order to prove that it is nondegenerate, let's suppose that the vector T (g,η,v(η,0)) π(T e L g · ζ, δη, D m * v(η, 0) · δη ′ ) is such that for every ζ ′ ∈ g and δη ′ ∈ m * . We will show that this implies that T (g,η,v(η,0)) π(T e L g ·ζ, δη, D m * v(η, 0)· δη) = 0 necessarily. Using ω Σ = i * ω Y and the explicit expression of the symplectic form ω Y associated to the MGS normal form (see the previously quoted original papers, as well as [RdSD97, OR99, O98]), we can write (25) in the form for any δη ′ ∈ m * and ζ ′ ∈ g. If we fix δη ′ = 0 and let ζ ′ be arbitrary, we obtain If we now fix ζ ′ = 0 and let δη ′ be arbitrary, we obtain ζ ∈ h, which, together with (26), guarantees that T (g,η,v(η,0)) π(T e L g · ζ, δη, D m * v(η, 0) · δη) = 0, as required.
In the remainder of this section we will show that the persistence phenomena described by Theorem 4.3 and Corollary 4.4 preserve stability. More specifically, we will show that if the relative equilibrium m e is stable, then the entire local symplectic manifold Σ given by Corollary 4.4 consists of stable relative equilibria. First, we recall the definition of nonlinear stability of a relative equilibrium: Definition 4.5 Let (M, ω, h, G, J : M → g * ) be a Hamiltonian system with symmetry and let G ′ be a subgroup of G. A relative equilibrium m e ∈ M is called G ′ -stable, or stable modulo G ′ , if for any G ′ -invariant open neighborhood V of the orbit G ′ · m e , there is an open neighborhood U ⊆ V of m e , such that if F t is the flow of the Hamiltonian vector field X h and u ∈ U , then F t (u) ∈ V for all t ≥ 0.
Before recalling the stability result that we will use in our discussion we introduce the following notation. Suppose that we fix a splitting of g as in (2). If ξ = ξ 1 + ξ 2 , with ξ 1 ∈ g me and ξ 2 ∈ m, is a generator of the relative equilibrium m e , then the unique element ξ 2 ∈ m is called the orthogonal generator of m e with respect to the splitting (2).
We now state the following theorem whose proof can be found in [LS98] or in [OR99]. Let m e ∈ M be a relative equilibrium such that J(m e ) = µ ∈ g * , g * admits an Ad * Gµ -invariant inner product, H := G me , and ξ ∈ Lie(N Gµ (H)) is its orthogonal generator, relative to a given Ad H -invariant splitting. If the quadratic form is definite for some (and hence for any) subspace W such that then m e is a G µ -stable relative equilibrium. If dim W = 0, then m e is always a G µ -stable relative equilibrium. The quadratic form D 2 (h − J ξ )(m e )| W ×W , will be called the stability form of the relative equilibrium m e .
A relative equilibrium satisfying the hypotheses of Theorem 4.6 is said to be formally stable. Note that in the Abelian case all the adjoint invariance requirements in the statement of the previous theorem are trivially satisfied. We now state our stability persistence result.
Proposition 4.7 Under the conditions of Corollary 4.4, suppose that the relative equilibrium m e is formally (and consequently nonlinearly) stable; that is, it has an orthogonal generator ξ ∈ m with respect to the splitting (2) such that the quadratic form is definite for some (and hence for any) subspace W such that ker DJ(m e ) = W ⊕ T me (G · m e ), then the symplectic manifold Σ of relative equilibria passing through m e can be chosen (by taking, if necessary, a sufficiently small neighborhood of m e in the submanifold Σ of Corollary 4.4) to consist exclusively of nonlinearly stable relative equilibria.
Proof Recall that the symplectic manifold Σ consists of points of the form [g, η, v(η, 0)] ∈ Y , with η ∈ m * sufficiently close to 0, which are relative equilibria with generator ξ + β(η, 0). Since ξ ∈ m is by hypothesis an orthogonal generator with respect to the splitting (2), and the function β maps into m, the generator ξ + β(η, 0) is also an orthogonal generator for the relative equilibrium [g, η, v(η, 0)] ∈ Y . Hence, in order to prove the Proposition it suffices to show that the quadratic form . Using the expression of the momentum map in the MGS-coordinates described in Proposition 4.1, it is easy to verify that ker DJ ([g, η, v(η, 0) (η, 0)))), where Φ g denotes the G-action in MGS coordinates (see Proposition 4.1) and Ψ is the slice mapping introduced in (18). This identity singles out the space (η, 0)))) as a choice for W [g,η,v(η,0)] . We are now in position to study the definiteness of the stability form of the relative equilibrium [g, η, v(η, 0)], using as W [g,η,v(η,0)] the space just mentioned. Indeed, The formal stability of m e implies that the quadratic form is definite, therefore, since definiteness is an open condition, for any η ∈ m * close enough to 0, is also definite. Since ker DJ Vm e (v(η, 0)) in expression (27) is a subset of V me , the definiteness of the stability form of relative equilibrium [g, η, v(η, 0)] is guaranteed for small enough η ∈ m * , as required.

Bifurcation of relative equilibria with maximal isotropy
Consider the symmetric Hamiltonian system (M, ω, h, G, J : M → g * ), where the Lie group G acts properly on the manifold M . Let m e ∈ M be a relative equilibrium with momentum µ = J(m e ). In contrast to the previous section, we will assume here that the relative equilibrium m e is degenerate, that is, there is a generator ξ ∈ g and a nontrivial vector subspace V 0 ⊂ T me M for which This hypothesis implies that in writing the reduced critical point equations the Lyapunov-Schmidt reduction will be nontrivial and there will be the possibility of genuine bifurcation.
In this section we will focus on the study of the bifurcation equation (B1); that is, we will assume that the rigid residual equation is satisfied and therefore the relative equilibria near m e correspond to the zeroes of (B1).
In the framework of general dynamical systems, the bifurcation of relative equilibria with isotropy group K, out of a degenerate (i.e. nonhyperbolic) isolated equilibrium, is generic 1 if K is maximal and satisfies an additional property, e.g. has an odd-dimensional fixed-point subspace in the space V 0 on which the bifurcation equation is defined, or has an even dimensional fixed-point subspace together with a non-trivial S 1 action. The famous Equivariant Branching Lemma (see, e.g., [GSS88]), belongs to the former case, while the latter appears in a work of Melbourne (see [M94,CKM95]). We shall see that both results have a counterpart in the symmetric Hamiltonian case, although being Hamiltonian is a nongeneric property from the general dynamical systems point of view. When searching for relative equilibria, the generator (α ∈ g) or momentum (η ∈ g * ) serves as a bifurcation parameter, in addition to any physical control parameters present in system. Due to the "rigidity" of these geometric "parameters", care must be taken when adapting the bifurcation theorems to relative equilibria of Hamiltonian systems.
As a final preliminary remark, we point out the fact that our theorems will be stated for bifurcation from a general relative equilibrium, not just from one pure (isolated) equilibrium. In the latter case, the gradient character of the bifurcation equation (see Remark 3.1) simplifies the arguments (see Remark 5.5).

A Hamiltonian Equivariant Branching Lemma
In the situation described above, let m e ∈ M be a relative equilibrium satisfying the degeneracy hypothesis (28). As we saw in Proposition 3.3, the bifurcation equation (B1) can be constructed so as to be G me ∩ G ξ -equivariant, which implies that for any subgroup K ⊂ G me ∩ G ξ , B can be restricted to the K-fixed point subspaces in its domain and range; hence we can find solutions of B by finding the solutions of Assume now that K ⊂ G me ∩ G ξ is a maximal isotropy subgroup of the G me ∩ G ξ -action on V 0 and, moreover, that dim(V K 0 ) = 1. Under this hypothesis we will look for pairs (η, v 0 ) ∈ (m * ) Note that dim(V K 0 ) = 1 implies that (see for instance [B72]) Recall that L acts naturally on (m * ) K and on V K 0 , and that B K is L-equivariant. Depending on the character of the L-action, the first terms in the Taylor expansion of (29) can be written as for some vector κ ∈ (m * ) K and some constant c that are generically nonzero. In both instances these expressions generically allow us to solve for v 0 in terms of the other variables via the Implicit Function Theorem, giving us saddle-node type branches if L ≃ {id} and a pitchfork bifurcation if L ≃ Z 2 (see [GSS88] for arguments of this sort). More explicitly, we have proved the following result.
Theorem 5.1 (Equivariant Branching Lemma) Let m e ∈ M be a relative equilibrium of the Hamiltonian system (M, ω, h, G, J : M → g * ), where the Lie group G acts properly on the manifold M . Suppose that there is a generator ξ ∈ g and a nontrivial vector subspace V 0 ⊂ T me M for which Then, generically, for any subgroup K ⊂ G ξ ∩ G me for which dim(V K 0 ) = 1 and the rigid residual equation is satisfied on (m * ) K × V K 0 × {0}, a branch of relative equilibria with isotropy subgroup K bifurcates from m e . If N G ξ ∩Gm e (K)/K ≃ {Id}, the bifurcation is a saddle-node; if N G ξ ∩Gm e (K)/K ≃ Z 2 , it is a pitchfork.
We will illustrate this result with an example in the following section.

Bifurcation with maximal isotropy of complex type
In what follows we will use a strategy similar to the one introduced by Melbourne [M94] in the study of general equivariant dynamical systems, to drop the hypothesis on the dimension of V K 0 in the Equivariant Branching Lemma. Our setup will be the same as in Theorem 5.1 but in this case we will be looking at maximal complex isotropy subgroups K of the G me ∩ G ξ -action on V 0 , that is, maximal isotropy subgroups K for which Notice that in such cases V K 0 has even dimension. As in the previous section, we will use the equivariance properties of the bifurcation equation in order to restrict the search for its solutions to the K-fixed space (m * ) K × V K 0 × g K me . Moreover, we will consider only solutions of the form (0, v 0 , α) ∈ (m * ) K × V K 0 × p, where p is some Ad NG me ∩G ξ (K)invariant complement to k in Lie(N Gm e ∩G ξ (K)). Note that (30) implies that p ≃ l ≃ R.
We now show that the adjoint action of N Gm e ∩G ξ (K) on p is trivial. The canonical projection π : N Gm e ∩G ξ (K) → L is a group homomorphism; hence the commutivity of L implies that π ghg −1 = π(g)π(h)π(g) −1 = π(h) for any g, h ∈ N Gm e ∩G ξ (K). In particular, for any g ∈ N Gm e ∩G ξ (K) and α ∈ Lie(N Gm e ∩G ξ (K)), which implies that Ad g −id maps Lie(N Gm e ∩G ξ (K)) into ker(T e π) = k. Since p∩k = {0} and p is Ad NG me ∩G ξ (K) -invariant, it follows that (Ad g − id) | p = 0 for all g ∈ N Gm e ∩G ξ (K), i.e. that the adjoint action on p is trivial.
Theorem 5.2 Let m e ∈ M be a relative equilibrium of the Hamiltonian system (M, ω, h, G, J : M → g * ), where the Lie group G acts properly on the manifold M . Suppose that there is a generator ξ ∈ g and a nontrivial vector subspace V 0 ⊂ T me M for which Suppose that the fixed point set V Gm e ∩G ξ 0 = {0}. Then for each maximal complex isotropy subgroup K of the G me ∩ G ξ -action on V 0 such that [Lie(N Gm e ∩G ξ (K)), g K me ] = 0 and each Ad NG me ∩G ξ (K) -invariant complement p to k in Lie(N Gm e ∩G ξ (K)) such that the rigid residual equation ρ(0, v 0 , α) = 0 is satisfied for all v 0 ∈ V K 0 and α ∈ p, there are generically at least 1 2 dim V K 0 (respectively 1 4 dim V K 0 ) branches of relative equilibria bifurcating from m e if N Gm e ∩G ξ (K)/K ≃ S 1 (respectively S 1 × Z 2 ).
Proof Let B : U 3 ⊂ m * × V 0 × g me → V 2 be the bifurcation equation corresponding to the reduced critical point equations constructed around m e using the MGS-slice mapping introduced in (18). The equivariance of this slice mapping guarantees that B is G me ∩ G ξ -equivariant; hence any solutions of are solutions of B. As we stated above, we will restrict our search to solutions in the set {0} × V K 0 × p, where p is some Ad NG me ∩G ξ (K) -invariant complement to k. Identify V 0 and V 2 using an invariant inner product and defineB K : for any v 0 , u ∈ V K 0 and α ∈ p. The equivariance properties of B and the triviality of the action on p imply thatB K satisfies the following equivariance condition: Note that as a corollary to this property we havẽ since for all g ∈ N Gm e ∩G ξ (K), g ·B K (0, α) =B K (0, α) and, consequently, the isotropy subgroup of B K (0, α) contains N Gm e ∩G ξ (K) and hence it strictly contains K. By the maximality of K as an isotropy subgroup, the isotropy subgroup ofB K (0, α) is necessarily G me ∩ G ξ . However, given that by hypothesis V Gm e ∩G ξ 0 = {0}, we haveB K (0, α) = 0, as claimed. We find the solution branches by first finding an open ball B r (0) about the origin in V K 0 and a function α : B r (0) → p satisfying then usingB K and α to define a family of vector fields on the unit sphere in V K 0 . Standard topological arguments show that these vector fields have the requisite number of equilibria, which correspond to solutions of the original equations.
As the first step in finding the function α we compute the Taylor expansion ofB K . As a result of the Lyapunov-Schmidt reduction and of (33), we can writẽ where L(α) is a linear operator such that L(0) = 0, and g(v 0 , α) is such that g(0, α) = 0, D v0 g(0, α) = 0 for all α. Moreover, a lengthy but straightforward computation shows that where L 1 (0) = L ′ 1 (0) = 0. We now show that if we identify V 0 and V 2 by means of an invariant inner product, then there exists a constant k ∈ N * such that where I V K 0 denotes the identity on V K 0 . Indeed, note that and hence for any v, w ∈ V me We now restrict our attention to elements v, w ∈ V K me . Recall that since V me is symplectic, the vector subspace V K me is symplectic with a canonical L action; hence for any α ∈ l and v ∈ V me there is an infinitesimally symplectic transformation A α such that α · v = A α v. The equivariant version of the Williamson normal form due to Melbourne and Dellnitz [MD93], implies the existence of a basis in which A α and ω V K me admit simultaneous matrix representations consisting of three diagonal blocks corresponding to the subspaces E R , E C , and E H of V K me on which L acts in a real, complex, and quaternionic fashion, respectively. Moreover, in this basis the restrictions of A α and ω V K me to E C take the form: for some natural numbers k 1 , . . . , k q . The signs in these two equalities are consistent, that is, they are either both positive or both negative (in all that follows we will focus only in the positive case). These expressions follow directly from the tables in [MD93] and the absence of nilpotent parts in A α , which is dictated by the requirement that A α be the zero matrix when α = 0. By hypothesis K is a maximal isotropy subgroup of the G me ∩ G ξ -action on V 0 for which Using this expression in (35), we obtain (34) and hencẽ where L 1 (·) is of order higher than |α| and g(·, α) is of order higher than v 0 . It follows that in the equation we can factor out v 0 2 and then apply the implicit function theorem to obtain a unique function α : B r (0) → p, for some r > 0, near the solution (0, 0).
Using this function we can define a one parameter family of L-equivariant vector fields X ǫ on S 2n−1 by X ǫ (u) =B K (ǫ u, α(ǫ u)).
The zeroes of these vector fields correspond to solutions of the bifurcation equation. Since L acts freely on S 2n−1 , X ǫ determines a smooth vector fieldX ǫ on S 2n−1 /L; the Poincaré-Hopf theorem implies thatX ǫ generically has at least The following lemma proves that X ǫ (u) is always orthogonal to the tangent space l · u of the L-orbit of u, i.e. X ǫ (u), ζ S 2n−1 (u) = 0 for any u ∈ S 2n−1 and ζ ∈ l. Hence the equilibria ofX ǫ correspond to orbits of equilibria of X ǫ , which in turn determine orbits of solutions of the bifurcation equation.
Remark 5.4 Note that Theorem 5.2 provides a (generic) lower bound for the number of branches of critical points of bifurcating from m e . In fact, if m e has nontrivial isotropy, then in many situations a sheet of critical points bifurcates from m e , rather than a finite number of one dimensional branches. An example of this phenomenon is given in §6. A continuous curve of bifurcation points with nontrivial isotropy appears in many other symmetric Hamiltonian systems, including the Lagrange top and the Riemann ellipsoids. (See, for example, [LSMR92], [L93], [L94], and [L98].) In [L93] it is shown that for Lagrangian systems with S 1 symmetry this phenomenon occurs under conditions that are generic within that class of systems.
Remark 5.5 There are two cases in which the theorem that we just proved applies in a particularly straightforward manner. First, suppose that the relative equilibrium m e is such that its momentum value µ = J(m e ) has an Abelian isotropy subgroup G µ . In such situation we have automatically that [Lie(N Gm e ∩G ξ (K)), g K me ] = 0 for any K ⊂ G me ∩ G ξ ⊂ G µ and also, using the techniques introduced in Section 3.2 (see especially Corollary 3.5), condition (ii) on the rigid residual equation can be easily dealt with.
Another case of interest is when m e is actually a pure equilibrium whose isotropy is the entire symmetry group G, i.e. the G-orbit of m e reduces to m e itself. Note that in that case m = q = {0} and therefore the rigid residual equation just does not exist. Also, the condition [Lie(N Gm e ∩G ξ (K)), g K me ] = 0 in the statement of the theorem is not necessary in that case since the bifurcation equation is variational (see Remark 3.1) and therefore the associated vector field is orthogonal to the G-orbits, and a fortiori to the N Gm e ∩G ξ (K))-orbits in V K 0 . It is interesting to note that in this case, the Equivariant Branching Lemma stated in Theorem 5.1 is not applicable, because the parameter η is now missing.

An example from wave resonance in mechanical systems
When a Hamiltonian mechanical system has two natural modes with frequencies in the approximate ratio p : q, p and q integers, one says it has a p : q internal resonance. When in addition the system possesses rotational as well as reflectional invariance, and the natural modes are wavy, that is, they break the rotational invariance in the form of m waves for mode p, say, and n waves for mode q, then the reflectional invariance implies the coexistence of 4 complex amplitudes associated to these modes, namely z 1 e miϕ and z 2 e −miϕ for mode p, and z 3 e niϕ and z 4 e −niϕ for mode q. The case with (p, q) = (m, n) = (1, 2) is of particular interest, for example, in the analysis of certain water wave problems in a cylindrical geometry, and has been studied with particular attention in [CD95]. In this work, the method of investigation was the projection of the vector field in normal form on the orbit space of the group action which here is G = O(2) × S 1 , where the S 1 term comes from the normal form assumption. The simple form of the group action allowed, by this method, a fairly detailed description of the possible dynamics near the trivial state. In particular, various kinds of relative equilibria were described in this way. This is therefore a good example to test our approach to the bifurcation analysis of relative equilibria.
A straightforward change of coordinates in the group SO(2) × S 1 allows us to define the toral part of the action of G as where (z 1 , z 2 , z 3 , z 4 ) ∈ C 4 are the complex amplitudes of the system. The reflection acts by permutation of z 1 with z 2 and of z 3 with z 4 . Then, it was shown in [B90] that the general form of a G-invariant, real smooth Hamiltonian h is and h is invariant under the (simultaneous) permutation of X 1 with X 2 , X 3 with X 4 and U 1 with U 2 . The Lie algebra g of G acts on C 4 by (ξ 1 , ξ 2 ) ∈ R 2 → (iξ 1 z 1 , iξ 2 z 2 , 2iξ 1 z 3 , 2iξ 2 z 4 ) (39) and the momentum map J is defined by where we have identified g * with R 2 . We now write the relative equilibrium equation, D(h − J ξ )(m) = 0 in complex coordinates. We set ξ = (ξ 1 , ξ 2 ) and Note that these coefficients are real and satisfy the relations a 2k (X 1 , X 2 , X 3 , X 4 , U 1 , U 2 ) = a 2k−1 (X 2 , X 1 , X 4 , X 3 , U 2 , U 1 ), k = 1, 2, and b 2 (X 1 , X 2 , X 3 , X 4 , U 1 , U 2 ) = b 1 (X 2 , X 1 , X 4 , X 3 , U 2 , U 1 ). Thus We can use the symmetries of the system (41) to easily identify a branch of relative equilibria; we will then use the results of the previous sections to find other branches of relative equilibria bifurcating from this branch. Note that the subgroup H = Z 2 × S 1 of the toral group is an isotropy subgroup of G, with fixed-point subspace Fix(H) = {(0, 0, z 3 , 0) | z 3 ∈ C}. This space is invariant under the map D h − J ξ ; specifically, D h − J ξ (0, 0, z 3 , 0) = (0, 0, (a 3 − 2ξ 1 )z 3 , 0).
There are two kinds of bifurcation points: 1. Bifurcation at λ + 1 or λ − 1 = 0. Because these are simple eigenvalues, the Lyapunov-Schmidt procedure yields one-dimensional bifurcation equations. The conditions of the Hamiltonian Equivariant Branching Lemma are met, hence we can conclude the existence of a bifurcated branch of relative equilibria parameterized by η ∈ R at each of these points. Note that Z 2 acts as −Id on the eigenvectors associated with these eigenvalues. Thus it follows that the bifurcation is of pitchfork type. The isotropy group of these solutions still contains S 1 . Therefore these relative equilibria fill 1-tori, i.e. are still periodic solutions for the Hamiltonian vector field.
Note that in this case, C can be taken as the bifurcation parameter. This is however the same thing as taking η, since W is defined as the subspace {(0, 0, C + η, 0)} in C 4 .
2. Bifurcation at λ 2 = 0 or λ 4 = 0. In both of these cases, the eigenvalue is double and therefore the space V 0 determined by the Lyapunov-Schmidt procedure is two dimensional. Note that SO(2) acts nontrivially on ker(A 2 ) and ker(A 4 ). Therefore the isotropy subgroup is maximal of complex type in both cases. Applying Theorem 5.2 yields at least one branch of circles of relative equilibria in each case. In fact, there is a two-parameter family (modulo symmetry) of relative equilibria containing (RE I ). These solutions live on 2-tori and are quasi-periodic whenever the ratio of the two components of the generator is irrational. What distinguishes these two families, aside from the fact that they bifurcate at different values of ξ 2 , is their symmetry: the isotropy of the solutions bifurcating in the z 4 direction is Z 2 , while it reduces to the trivial group for those bifurcating in the z 2 direction.
Note that while the bifurcations associated to λ ± 1 = 0 generically occur only at isolated values of C, the bifurcations associated to λ 2 = 0 and λ 4 = 0 occur for any value of C satisfying the nondegeneracy condition a 2 (0, 0, C 2 , 0) = a 4 (0, 0, C 2 , 0), since the second component ξ 2 of the generator at z e can always be chosen to equal a 2 (z e ) or a 4 (z e ).