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On the symmetry breaking phenomenon

We investigate the problem of symmetry breaking in the framework of dynamical systems with symmetry on a smooth manifold. Two cases will be analyzed: general and Hamiltonian dynamical systems. We give sufficient conditions for symmetry breaking in both cases.


Introduction
In this paper we investigate the problem of symmetry breaking in the framework of dynamical systems with symmetry on a smooth manifold. Two cases will be analyzed: general and Hamiltonian dynamical systems. For general vector fields we translate the problem locally to a symmetry breaking problem on a vector space where the techniques are more developed and easier to apply because of the existence of a Hilbert basis of invariant polynomials for representations of compact Lie groups. The symmetric steady state and Hopf bifurcations are considered as examples. Sufficient conditions for the existence of symmetry breaking bifurcations are known in this case. In the Hamiltonian context we find sufficient conditions for the existence of symmetry breaking bifurcations for Hamiltonian vector fields on symplectic vector spaces. The Hamiltonian steady state (passing of eigenvalues of the linearization on the imaginary axis at the origin) and the Hamiltonian Hopf bifurcations are analyzed. It turns out that the same methods as in the general case work, because in the Hamiltonian context one can work only with the invariant polynomials of the action and ignore the equivariant polynomials. In addition, it is shown how to locally translate these Hamiltonian bifurcation problems from the manifold to the tangent space at the bifurcation point. This is done by using the equivariant Darboux theorem.

Some basic results from the theory of group actions
We shall need a few fundamental results form the theory of group actions which we now briefly recall. For proofs and further information see [5], [7], [11], [20] and for the subsection on compact group representations see [12], [13], [23].
2.1. Twisted products. Let G be a Lie group and H ⊂ G a Lie subgroup. Supose that H acts on the left on a manifold A. The twisted action of H on the product G × A is defined by h · (g, a) = (gh, h −1 · a), h ∈ H, g ∈ G, a ∈ A.
Note that this action is free and proper by the freeness and properness of the action on the G-factor. The twisted product G × H A is defined as the orbit space (G × A)/H of the twisted action. The elements of G × H A will be denoted by [g, a], g ∈ G, a ∈ A. The twisted product G × H A is a G-space relative to the left action defined by g ′ · [g, a] = [g ′ g, a]. Also, the action of H on A is proper if and only if the G-action on G × H A is proper. The isotropy subgroups of the G-action on the twisted product G × H A satisfy G [g,a] = gH a g −1 , g ∈ G, a ∈ A.

2.2.
Slices. Throughout this paragraph it will be assumed that Ψ : G × Q → Q is a left proper action of the Lie group G on the manifold Q. This action will not be assumed to be free, in general. For q ∈ Q we will denote by H := G q := {g ∈ G | g · q = q} the isotropy subgroup of the action Ψ at q. We shall introduce also the following convenient notation: if K ⊂ G is a Lie subgroup of G (possibly equal to G), k its Lie algebra, and q ∈ Q, then k · q := {η Q (q) | η ∈ k} is the tangent space to the orbit K · q at q. A tube around the orbit G · q is a G-equivariant diffeomorphism ϕ : G × H A → U , where U is a G-invariant neighborhood of G · q and A is some manifold on which H acts. Note that the G-action on the twisted product G × H A is proper since the isotropy subgroup H is compact and, consequently, its action on A is proper. Hence the G-action on G × H A is proper. Let S be a submanifold of Q such that q ∈ S and H · S = S. We say that S is a slice at q if the map is a tube about G · q, for some G-invariant open neighborhood of G · q. Notice that if S is a slice at q then g · S is a slice at the point g · q. The following statements are equivalent: (i) There is a tube ϕ : G × H A → U about G · q such that ϕ([e, A]) = S.
(ii) S is a slice at q.
(iii) The submanifold S satisfies the following properties: (a) The set G · S is an open neighborhood of the orbit G · q and S is closed in G · S.
G·S is an open neighborhood of G·q and there is an equivariant smooth retraction r : G·S → G·q of the injection G · q ֒→ G · S such that r −1 (q) = S.
If Q is a Riemannian manifold then B can be chosen to be a G q -invariant neighborhood of 0 in (g · q) ⊥ , the orthogonal complement to g · q in T q Q. In this case U = G · Exp q (B), where Exp q : T q Q → Q is the Riemannian exponential map.
2.3. Type submanifolds and fixed point subspaces. Let G be a Lie group acting on a manifold Q. Let H be a closed subgroup of G. We define the following subsets of Q : All these sets are submanifolds of Q. The set Q (H) is called the (H)-orbit type submanifold , Q H is the H-isotropy type submanifold , and Q H is the H-fixed point submanifold . We will collectively call these subsets the type submanifolds. We have: • Q H is closed in Q; If Q is a vector space on which H acts linearly, the set Q H is called in the physics literature the space of singlets or the space of invariant vectors.
Theorem 2.3. (The stratification theorm) Let Q be a smooth manifold and G be a Lie group acting properly on it. The connected components of the orbit type manifolds Q (H) and their projections onto the orbit space Q (H) /G constitute a Whitney stratification of Q and Q/G, respectively. This stratification of Q/G is minimal among all Whitney stratifications of Q/G.
The proof of this result, that can be found in [7] or [22], is based on the Slice Theorem and on a series of extremely important properties of the orbit type manifolds decomposition that we enumerate in what follows. We start by recalling that the set of conjugacy classes of subgroups of a Lie group G admits a partial order by defining (K) (H) if and only if H is conjugate to a subgroup of K. Also, a point q ∈ Q in a proper G-subspace of Q (or its corresponding G-orbit, G · q) is called principal if its corresponding local orbit type manifold is open in Q. The orbit G · q is called regular if the dimension of the orbits nearby coincides with the dimension of G · q. The set of principal and regular orbits will be denoted by Q princ /G and Q reg /G, respectively. Using this notation we have: • For any q ∈ Q there exists a neighborhood U of q that intersects only finitely many connected components of finitely many orbit type manifolds. If Q is compact or a linear space where G acts linearly, then the G-action on Q has only finitely many distinct connected components of orbit type manifolds. • For any q ∈ Q there exists an open neigborhood U of q such that (G q ) (G x ), for all x ∈ U .
In particular, this implies that dim G · q ≤ dim G · x, for all x ∈ U .

2.4.
General facts about compact Lie groups representations. Let G be a compact Lie group acting on a finite dimensional vector space V . The compactness of G implies the existence of a Ginvariant inner product ·, · on V , so one can always assume that the G-representation is orthogonal. Denote by P G (V ) the ring of G-invariant real valued polynomials on V . The Hilbert-Weyl theorem implies that P G (V ) is finitely generated over R and it can be shown that one can choose the generators to be homogeneous polynomials. The elements {θ 1 , ..., θ l } of such a basis over R are called invariant generators. The following theorem of Schwarz gives a method of writing a G-equivariant smooth map in terms of invariants and equivariants.
Theorem 2.4. Let V and W be two G-representation spaces. The set of G-equivariant polynomial maps from V to W is a finitely generated P G (V )-module. Let {F 1 , ..., F k } be one of its minimal bases. Then, if F is a smooth G-equivariant map from V to W , there exist smooth functions f 1 , ..., f k on R l such that The next theorem gives an identification between the orbit space of a linear action of a compact Lie group G on a vector space V and a semialgebraic subset of R l . Theorem 2.5. Let π be the Hilbert map V ∋ v → (θ 1 (v), ..., θ l (v)) ∈ R l . Then π induces a homeomorphism between the orbit space V /G and Im(π).
The invariant generators are intimately related to the orthogonal space to the G-orbits.
The following theorem is proved in [3]. Two useful consequences of this fact are the orthogonal sum decomposition Since the orbit space is the image of a finite dimensional vector space by a polynomial map, the Tarski-Seidenberg theorem guarantees that it is a semialgebraic set. Thus, as a semialgebraic set, V /G has a canonical stratification. A theorem of Bierstone implies that this canonical semialgebraic stratification is the same as the stratification by orbit types obtained by projection of those from V . Since the orbit space V /G inherits a smooth structure from V and G, but in general is not a manifold, we have to define the meaning of a smooth vector field on V /G. Definition 2.7. Let X be a smooth vector field defined on a neighborhood U of 0 ∈ R l and tangent to the strata of π(V ) ∩ U . Then X is said to be smooth vector field on V /G in a neighborhood of the origin.
If one projects a smooth G-equivariant vector field on the orbit space, one obtains a smooth vector field on V /G in the sense defined above. When we speak of a vector field on V /G we mean a vector field which is tangent to each stratum of V /G. The next step gives a method for projecting a G-equivariant vector field on the orbit space. For doing this, let F be a smooth G-equivariant vector field on the finite dimensional vector space V . Let {θ 1 , ..., θ l } be a set of generators for P G (V ) and let {F 1 , ..., F k } be a set of generators for the P G (V )-module of G-equivariant polynomials maps from V to V . By the Schwarz theorem we can write F in terms of invariants and equivariants as where the functions f i are smooth functions on R l . To obtain the projection of this vector field on the orbit space V /G, one has to compute the derivatives of the invariant generators θ i with respect to time: Since the expression grad v (θ j ), F i (v) is G-invariant one can rewrite it in terms of the invariant generators. Thus the projection F of F on the orbit space V /G has the expression

Formulation of Symmetric Systems on Vector Spaces
This section presents a method of translating a bifurcation problem for a general vector fields family with G-symmetry from a smooth manifold to a vector space. We begin by quickly recalling the necessary notations and statements for a left smooth proper Lie group action Φ : G × Q → Q, where Q is a smooth manifold and G is a Lie group with Lie algebra g.
for all q ∈ Q and g ∈ G. If X is G-equivariant, then G is said to be a symmetry group of the dynamical systemq = X(q).

Definition 3.2.
A relative equilibrium of a G-equivariant vector field X is a point q e ∈ Q at which the values of X and of the infinitesimal generator ξ Q of some element ξ ∈ g, called velocity of the relative equilibrium, coincide, that is, where exp : g → G is the Lie group exponential map. A relative equilibrium q e is said to be asymmetric if g qe = {0}, and symmetric otherwise.
From the definition above and the uniqness of the integral curves of a vector field with given initial condition, it follows that the dynamical orbit q(t) of X staring at the relative equilibrium q e , that is, q(0) = q e , coincides with the curve exp tξ · q e := Φ(exp tξ, q e ) given by the group action. Note that if ξ is a velocity of the relative equilibrium q e , then so is ξ + η, where η ∈ g qe := {ζ ∈ g | ζ Q (q e ) = 0 qe }. The Lie subalgebra g qe is called the symmetry algebra of q e . Denote for each g ∈ G by Ad : G × g → g the adjoint representation of G on g; for each g ∈ G, Ad g is the Lie algebra isomorphism obtained by taking the derivative of the conjugation isomorphism on G.
If q e is a relative equilibrium with velocity ξ ∈ g, then, for any g ∈ G, Φ g (q e ) is also a relative equilibrium with velocity Ad g ξ.
which proves the statement.
By one of the remarks following Theorem 2.3, we can choose a G-invariant tubular open neigborhood W qe of the orbit G · q e such that the isotropy subgroup G x for any x ∈ W qe is conjugate to a (possibly non-strict) subgroup of G qe . The next lemma will show that we can do better.
and, by construction, contains the point q.
Properness of the G-action on Q implies that the isotropy subgroup G qe is compact. By the lemma above we can conclude that there exists an open G qe -invariant neighborhood W 1 qe of q e contained in W qe . Thus the local study of the symmetry breaking from G qe to a smaller symmetry group can be done by restricting the action to W 1 qe . Next, assume that Q carries a G-invariant Riemannian metric. The Riemannian exponential map Exp : T Q → Q is also G-equivariant, relative to the naturally lifted G-action on T Q. Since Exp qe : T qe Q → Q is a local diffeomorphism around the origin, we can further shrink the neighborhood W 1 qe (keeping the G qe -invariance), till it is contained in the image under Exp qe of the neighborhood of the origin where Exp qe is a diffeomorphism. Denote this shrunk G qe -invariant neighborhood by U qe . The inverse map of the Riemannian exponential on U qe , Log qe : Proposition 3.5. Let (Q, ·, · ) be a smooth Riemannian manifold, q e ∈ Q, G a Lie group acting properly and smoothly on Q by isometries, and X ∈ X(Q) a smooth G-equivariant vector field. Then, in the notations above, the vector field Exp * qe X| Uq e ∈ X(V 0q e ) is a smooth G qe -equivariant vector field on the open neighborhood V 0q e of the origin 0 qe in the vector space T qe Q.
Proof. The proof is a simple consequence of the fact that Exp qe : V qe ⊂ T qe Q → U qe ⊂ Q and Log qe : U qe → V qe are G qe -equivariant diffeomorphisms inverse to each other. Indeed, since X is Gequivariant, its restriction X| Uq e is G qe -equivariant. Therefore, for any h ∈ G qe and v qe ∈ V qe we have , which proves that Exp * qe X| Uq e is a G qe -equivariant vector field on V 0q e . Proposition 3.6. Let (Q, ·, · ) be a smooth Riemannian manifold, G a Lie group acting properly on Q by isometries, and X λ ∈ X(Q), λ ∈ R, a family of smooth G-equivariant vector fields. Suppose that q e ∈ Q is a (relative) equilibrium for X λ0 with symmetry G qe . Assume the family of vector fields Exp * qe X λ ∈ X(V 0q e ) has the property that there exist ǫ > 0 and a continuous curve γ : [λ 0 , λ 0 +ǫ] → V 0q e , γ(λ 0 ) = 0 qe , such that all the points on the curve γ(λ) for λ > λ 0 are relative equilibria of Exp * qe X λ with isotropy subgroups conjugate to H ⊂ G qe . Then the curve Exp qe •γ : [λ 0 , λ 0 + ǫ] → U qe is formed by relative equilibria of X λ with isotropy subgroups conjugate to H for all λ > λ 0 and to G qe for λ = λ 0 .
Proof. Let X λ ∈ X(U qe ) be one of the vector fields restricted to the G qe -invariant open neighborhood U qe . Suppose that γ(λ) is a relative equilibrium of the vector field Exp * qe X λ . This means that there exists an element ξ(λ) ∈ g qe ⊂ g such that If we apply to the above equality T γ(λ) Exp qe and use the G qe -equivariance of the exponential map we get This means that Exp qe (γ(λ)) is a relative equilibrium of X λ . So we have proved that the curve Exp qe •γ is formed by relative equilibria of X λ for any λ ∈ [λ 0 , λ 0 + ǫ]. Because the exponential map is a G qeequivariant diffeomorphism on the neighborhoods considered, we have G Exp qe (γ(λ)) = G γ(λ) , and so all the points on the curve γ(λ) with the possible exception of γ(λ 0 ) are conjugate to H.
Since the G qe -equivariant family of vector fields Exp * qe X λ ∈ X(V 0q e ) are defined on an open G qeinvariant neighborhood of the origin in the vector space T qe Q, we can arbitrarily extend their restrictions to a smaller G qe -invariant neighborhood of the origin to smooth G qe -equivariant vector fields X λ ∈ X(T qe Q) on the whole space T qe Q. However, on T qe Q we have an orthogonal representation of the compact group G qe and thus, as explained earlier, the vector fields X λ ∈ X(T qe Q) can be expressed in terms of the invariant and equivariant polynomials on T qe Q. Thus to study the local symmetry breaking phenomenon around the point q e ∈ Q, one can translate the problem to the vector space T qe Q and study this problem for the G qe -equivariant family X λ ∈ X(T qe Q).

Stationary and Hopf Bifurcation with Symmetry
We shall analyze now the local symmetry breaking phenomenon around 0 qe ∈ T qe Q for the family of G qe -equivariant vector fields X λ ∈ X(T qe Q) under the following two nondegeneracy conditions: • Stationary bifurcation with symmetry: where σ X : (−ǫ, ǫ) → R is a smooth map satisfying σ X (0) = 0 and σ ′ X (0) = 0. In addition, it is assumed that the compact group G qe acts absolutely irreducibly on T qe Q, i.e., that only scalar multiples of the identity commute with the G qe -action.
• Hopf bifurcation with symmetry: where σ X , ρ X : (−ǫ, ǫ) → R are smooth maps satisfying σ X (0) = 0, σ ′ X (0) = 0, and ρ X (0) = 0. In addition, it is assumed that the vector space T qe Q carries a natural complex structure J Tq e Q and that the compact group G qe acts on this complex vector space T qe Q irreducibly (as a complex representation). In this case the manifold Q is necessarily of even dimension.
In the following will denote by C ∞ (T qe Q × R, T qe Q) Gq e o the class of G qe -equivariant vector fields that verifies one of the above nondegeneracy conditions. Let {θ 1 (v qe ), ..., θ l (v qe )} be a set of generators for P Gq e (T qe Q) and let {F 1 , ..., F k } be a set of generators for the P Gq e (T qe Q)-module of G qe -equivariant maps from T qe Q to itself. Then by the Schwarz theorem we have where the functions f i are smooth functions on R l . The projection of this family of vector fields on the orbit space (T qe Q)/G qe has the expression: According to the orbit space projection of the vector field X λ we introduce the maps .., f k (0, λ)) ∈ R k and define the sets: where Π : T qe Q → (T qe Q)/G qe is the orbit space projection and H ⊂ G qe is an isotropy subgroup of the G qe -action on T qe Q. Recall that E is a semialgebraic subset of R l × R k . Denote by Σ its canonical Whitney stratification. This stratification induces a Whitney semialgebraic stratification Σ (H) on each E (H) . Let B denote the Whitney stratification Σ (Gq e ) of E (Gq e ) = {0} × R k . Next we will define a notion of G qe -transversality introduced by Koenig and Chossat [13] that was inspired by Bierstone [4] and Field [8].

Analogous to the map (4.3) we define
This map induces the following sets parameterized by the isotropy type subgroups of the G qe -action on T qe Q: The sets E (H) and A (H) are semialgebraic sets in T qe Q × R k and R k respectively. The next proposition gives an relation between the pair of subsets Proposition 4.2. Let Π : T qe Q → (T qe Q)/G qe be the projection onto the orbit space and H and isotropy type of the G qe -action on T qe Q. Then: Proof. The proof is as in [12]. To show the first equality let v qe ∈ Π −1 (θ). Then vq e is generated by {grad vq e θ j (v qe ), for j ∈ {1, . . . , l}}, we must have F (v qe , t) ∈ g qe · v qe . Thus (θ, t) ∈ E (H) ⇐⇒ (v qe , t) ∈ E (H) . To prove the second equality let t ∈ A (H) . Then there exists (v n , t n ) ∈ E (H) such that v n → 0 and t n → t. By the first equality, this implies that (Π(v n ), t n ) ∈ E (H) and the continuity of Π implies that t ∈ A (H) . Conversely, let t ∈ A (H) . Then there exists a sequence (θ n , t n ) ∈ E (H) whose limit is (0, t).
Because the action of G qe on T qe Q is linear we have that Π −1 (0) = 0 and hence t ∈ A (H) .

Remark 4.3.
If H is conjugated to a proper subgroup of G qe we have codim R k A (H) ≥ 1.
The next theorem, due to Koenig and Chossat [13], gives a sufficient criterion for the existence of symmetry breaking branches of relative equilibria for the family of vector fields X λ in a neighborhood of 0 qe . For the proof of this theorem we will need the following result of Field and Richardson [9]: Theorem 4.5. Let H be an isotropy group conjugate to a proper subgroup of G qe and X λ the projection on the orbit space (T qe Q)/G qe of the family of vector fields X λ ∈ C ∞ (T qe Q × R, T qe Q)    4.6. In the above theorem the condition X λ ∈ C ∞ (T qe Q × R, T qe Q) Gq e o is used to insure that f 1 (0, 0) = 0 and ∂ ∂λ f 1 (0, λ) λ=0 = 0, where f 1 is the first coefficient in the expansions (4.1) or (4.2). In both cases considered here, this is equivalent to σ X (0) = 0 and σ ′ X (0) = 0. Now we give the proof of Theorem 4.4.
Proof. (a) Suppose that the systemv qe = X λ (v qe ) admits relative equilibria with isotropy subgroup conjugated to H in a neighborhood of 0 qe . This means that the projection of this vector field has zeroes whose isotropy subgroups are conjugate to H. By Theorem 4.5 this implies that codim R k A (H) ≤ 1 and γ(0) ∈ A (H) . This is in contradiction with codim  [14], we decompose locally in a tubular neighborhood of the orbit G qe · v λ , the G qe -equivariant vector field X λ into its tangential and normal components. Since the projection of this vector field to the orbit space has c(λ) as equilibrium we have that X λ (v λ ) ∈ g qe · v λ . This means that the local normal component of the vector field vanishes. So, the solution of the Cauchy probleṁ w qe = X λ (w qe ), w qe (0) = v λ , is given by w qe (t) = g(t) · v λ , where g(t) = exp(ξ λ t), for small enough t ≥ 0 and ξ λ ∈ g qe . The existence for each λ of a flow invariant maximal torus of rank n H on which the solution exp(ξ λ t) · v λ is generically dense is guaranteed by Theorem 4.1 in [14]. Hence we have obtained a branch w λ qe (t) = exp(ξ λ t) · v λ of periodic or quasi-periodic solutions all of whose points have isotropy type conjugate to H.
Using Proposition 3.6, we can deduce the same result as the one in Theorem 4.4 for a one-parameter family of vector fields X λ defined in a small neighborhood around the point q e ∈ Q. The resulting bifurcation result will give a family of tori bifurcating from q e whose symmetry is broken to that of the a priori given subgroup H G qe . The bifurcating solutions are periodic or quasi-periodic curves lying on these tori. The method given by Proposition 3.6 allows one to always translate bifurcation results obtained in the linear setting to analogous theorems on the manifolds itself.
In the case of the Hopf bifurcation, several additional remarks are in order. First, the symmetry group of the problem can be considered to be of the form G × S 1 . This follows from a normal form result (see e.g. [10], Chapter XVI) which ensures that there is a natural S 1 -action on T qe Q such that the family of vector fields X λ are G × S 1 -equivariant. Second, given k ∈ N and a G-equivariant family of vector fields X λ on T qe Q, there is a C k λ-dependent change of variables on T qe Q that maps X λ into new family of vector fields X λ which is G × S 1 -equivariant and that coincides with X λ up to order k. Third, one can interpret this result as saying that close enough to 0 qe the S 1 -relative equilibria of the G × S 1 -equivariant family X λ are mapped by this change of variables to the set of periodic solutions of X λ . Similarly, the G × S 1 -relative equilibria of X λ are mapped by this change of variables to G-relative periodic orbits of X λ .

Hamiltonian Steady State and Hamiltonian Hopf Bifurcation with Symmetry
In this section we study a bifurcation problem similar to the one in Theorem 4.4 in the context of a Hamiltonian system on a symplectic vector space. To obtain the symmetry breaking result the Hamiltonian nature of the vector field plays a crucial role. One cannot deduce Theorem 5.3 below from the generic results in [13] because the non-degeneracy conditions are not satisfied.
Let (V, ω) be a symplectic vector space on which the compact Lie group G acts linearly and symplectically. In this case it is known that the map G × V → V is smooth. Let F λ : V → R, λ ∈ R, be a family of smooth G-invariant functions. If 0 ∈ V is an equilibrium (with symmetry G) of the Hamiltonian vector field X F0 , we are interested in finding conditions that guarantee the existence of smooth branches of tori bifurcating from zero such that all the points on these tori have symmetry conjugate to an a priori fixed proper symmetry subgroup H. Analogous to the cases considered in the previous section we shall analyze here the following situations: • Hamiltonian steady state bifurcation with symmetry: where σ : (−ǫ, ǫ) → R is a smooth map satisfying σ(0) = 0 and σ ′ (0) = 0. In addition, it is assumed that the vector space V carries a natural complex structure J and that the compact Lie group G acts on this complex vector space V irreducibly (as a complex representation). In this case a basis for gl(V ) G , the endomorphisms of V which commute with the G-action, is formed by {I, J}. This implies that a symplectic matrix in gl(V ) G should be a real multiple of J. In this case, we have passing of eigenvalues of DX F λ (0) on the imaginary axis at the origin as λ crosses zero.
In the following will denote by C ∞ (V × R, V ) G oo the class of G-equivariant Hamiltonian vector fields that verifies one of the above non-degeneracy conditions.
As opposed to the general case, if the vector field is Hamiltonian, it turns out that one needs to use only invariant polynomials, as the proposition below shows.
Proposition 5.1. Let (V, ω) be a symplectic representation space of the compact Lie group G. Denote by {θ 1 , θ 2 , . . . , θ l } a set of generators of P G (V ). If F : V → R is a smooth G-invariant function then the associated Hamiltonian vector field X F has the following expression in canonical coordinates, in terms of invariant polynomials and their gradients: whereF : R l → R is given by: F (x) =F (θ 1 (x), . . . , θ l (x)), for any x ∈ V.
Proof. The proof follows directly from the definition of the Hamiltonian vector field.
The next step gives a method for projecting a G-equivariant Hamiltonian vector field on the orbit space. To do this, let F be a G-invariant smooth function and X F its associated Hamiltonian vector field. Therefore, X F is G-equivariant on (V, ω). Note that J grad θ i are equivariant polynomials. Thus, by the proposition above we can write X F in terms of invariants and equivariants as where the functionF is smooth on R l . To obtain the projection of this vector field on the orbit space V /G, one has to compute the derivatives of the invariant generators θ j with respect to time: The projection X F of X F on the orbit space V /G has thus the expression · · · · · · · · · {θ l , θ 1 } · · · · · · 0   .
As in the Riemannian case, one search for symmetry breaking zeroes of the projected vector field. To do this, we need a few preliminary results. Define the following maps associated to the family of smooth where Π : V → V /G is the orbit space projection and H ⊂ G is an isotropy subgroup of the G-action on V . Recall that E is a semialgebraic subset of R 2l . Denote by Σ the canonical Whitney stratification of E. This stratification induces a Whitney semialgebraic stratification Σ (H) on each E (H) . In particular, Σ G stratifies E (G) = {[0]} × R l . Next we recall the notion of G-transversality introduced by Koenig and Chossat [13] that was in turn inspired by Bierstone [4] and Field [8]; see Definition 4.1. In the present case, this says that the projection to the orbit space V /G of a G-equivariant smooth family of vector fields X F λ ∈ X(V ) is transverse to [0] ∈ V /G at λ = 0 ∈ R if γ : R → R l is transverse at λ = 0 to the stratum of Σ G containing γ(0), that is, im T 0 γ + T γ(0) S = R l , where S is the stratum of Σ G containing γ(0). Analogous to the map (5.3) we define This map induces the following sets parameterized by the isotropy type subgroups of the G-action on V : The sets E (H) and A (H) are semialgebraic sets in V × R l and R l , respectively. A proof analogous to that We shall prove below that if H = G, then codim R l A (H) ≥ 1.
Proof. From (5.3), it follows that the i 0 -component of the map Φ is given by Let θ := (θ 1 , . . . , θ l ) ∈ V /G ⊂ R l . If (θ, t) ∈ E (H) , the above relation becomes This implies that Dividing this relation by {θ i0 , θ i1 }(x) and letting x → x 0 one concludes that t i1 = 0. This proves that A (H) ⊂ t ∈ R l | t i1 = 0 . The identity A (G) = R l follows directly from the definitions.
Adapting the Proposition 5.16 of Field and Richardson [9] to the family of vector fields X F λ , yields the following result. Note that X F λ ∈ C ∞ (V × R, V ) G oo implies the condition from the Remark 4.6, this condition being equivalent to σ(0) = 0 and σ ′ (0) = 0. We shall make some remarks regarding the Hamiltonian Hopf bifurcation. We have made the hypothesis that the symmetry group of the problem is of the form G × S 1 . This always holds if we assume only that the system has a compact Lie group G of symmetries represented on the symplectic vector space V . The reason for this is that in the case of the Hamiltonian Hopf bifurcation there is a natural S 1action on V given by the exponential of the semisimple part of the linearization on the resonance space corresponding to ν 0 . More precisely, let (V, ω, F λ ) be a λ-parameter family (λ ∈ R in a neighborhood of the origin) of G-Hamiltonian systems such that F 0 (0) = 0, dF 0 (0) = 0, and the G-equivariant infinitesimally symplectic linear map DX F0 (0) is nonsingular and has ±iν 0 as eigenvalues. Let (U ν0 , ω| Uν 0 ) be the resonance space of DX F0 (0) with primitive period T ν0 := 2π/ν 0 . Recall that the resonance space is the direct sum of the real eigenspaces of DX F0 (0) corresponding to eigenvalues of the form ±kiν 0 with k = 1, 2, . . . . A theorem of Vanderbauwhede and van der Meer [26] states that for each k ≥ 0 one can find a C k -mapping ψ : U ν0 × R → V and a C k+1 -function F λ : U ν0 × R → R such that ψ(0, λ) = 0, for all λ, D Uν 0 ψ(0, 0) = I Uν 0 , and F λ is a G × S 1 -invariant function that coincides with F λ up to order k + 1. In the same paper it is shown (Theorem 3.2) that if λ is close to 0 then the S 1 -relative equilibria of the G × S 1 -invariant Hamiltonian F λ that lie in a sufficiently small neighborhood of the origin in U ν0 are mapped by ψ(·, λ) to the set of periodic solutions of (V, ω, F λ ) in a neighborhood of 0 ∈ V with periods close to T ν0 . Thus, on can replace the search of periodic orbits of (V, ω, F λ ) by the search of S 1 -relative equilibria of the G × S 1 -invariant family of Hamiltonian systems (U ν0 , ω| Uν 0 , F λ ). Similarly, the G × S 1 -relative equilibria of X F λ are mapped by ψ(·, λ) to G-relative periodic orbits of X F λ .

Bifurcation of relative equilibria in Hamiltonian systems on symplectic vector spaces
In this section we will prove a bifurcation result in symplectic vector spaces in the context of Hamiltonian systems. We consider a symplectic representation space (V, ω) of a compact Lie group G and let {θ 1 , . . . , θ l } a Hilbert base. We will denote by V (H) ⊂ V the principal stratum of the action of G on V and by Π : V → V /G the orbit space projection. Recall that each connected component of V (H) is an open dense subset of the vector space V . We consider a family of G-invariant smooth functions F λ : V → R, λ ∈ R, such that there exists λ 0 ∈ R and v 0 ∈ V (H) a relative equilibrium for the Hamiltonian vector field X F λ 0 . We will prove that under a certain non-degeneracy condition we have the existence of a smooth branch of relative equilibria (starting from v 0 ) for the Hamiltonian vector fields X F λ for λ close to λ 0 . To do this we recall some of the techniques necessary to find the expression of the projected Hamiltonian vector field on V /G . More precisely, if F λ (x) =F λ (θ 1 (x), . . . , θ l (x)), for any x ∈ V , whereF λ : R l → R is smooth, then the j-component of the projection on the orbit space V /G of the Hamiltonian vector field X F λ is given by: Searching for relative equilibria of a vector field is the same as searching for equilibria for the projected vector field on the orbit space. Thus, we reduced the problem to find branches in λ of solutions of the system: ∂F λ ∂θ i (θ 1 , . . . , θ l ){θ j , θ i } = 0, for j ∈ {1, . . . , l}.
To do this, we introduce the function g : (V /G ⊂ R l ) × R → R l , g = (g 1 , . . . , g l ), where the components g j are defined by: As v 0 ∈ V (H) is a relative equilibrium of X F λ 0 this implies that θ 0 =: Π(v 0 ) is an equilibrium point for the projection on the orbit space V /G ⊂ R l of the Hamiltonian vector field X F λ 0 . This means that θ 0 ∈ V /G ⊂ R l is a solution of the equation g(θ, λ 0 ) = 0. The next theorem will give a sufficient condition that guarantees the existence of a smooth curve of solutions of the equation g(θ, λ) = 0, in an open neighborhood of the point (θ 0 , λ 0 ).
Theorem 6.1. Let (V, ω) be a symplectic representation space of a compact Lie group G and Π : V → V /G be the orbit space projection. Let F λ : V → R, λ ∈ R, be a family of G-invariant smooth functions. Suppose that v 0 ∈ V is a relative equilibrium of the Hamiltonian vector field X F λ 0 which belongs to the principal stratum of the G-action on V . Denote θ 0 =: Π(v 0 ). If d θ g(θ 0 , λ 0 ) is non-degenerate, then there exists an open subset I of R containing λ 0 and a smooth curve γ : λ ∈ I → γ(λ) ∈ V /G ⊂ R l , such that γ(λ) ∈ V /G ⊂ R l lifts to relative equilibria of X F λ and γ(λ 0 ) = θ 0 ∈ V /G ⊂ R l .
Proof. The proof is a consequence of the implicit function theorem and of the fact that v 0 ∈ V (H) ⊂ V is an element of the principal stratum; the connected component containing it is an open subset of V .
Note that in this case the only thing we can say about the symmetry of the relative equilibria on the branch is that for λ small enough all of them are of principal type. This theorem can be used in the following way to study bifurcating branches of relative equilibria from a given one with isotropy K. Let v 0 ∈ V be a relative equilibrium with isotropy K. Since G is compact, so is K and hence the subspace V K of K-fixed vectors is a symplectic subspace of (V, ω). Let us assume that the normalizer N (K) of K in G is closed, which implies that N (K)/K is a compact Lie group that acts naturally on V K . Note that the restriction of this action on the K-invariant open subset V K is free. Then we can apply the previous theorem to V K , N (K)/K, and the restrictions of the vector fields X F λ to the open subset V K of V K . The corresponding function g from before will be restricted to the open set V K /(N (K)/K) and, under the same non-degeneracy hypothesis, we obtain λ-branches of relative equilibria for X F λ consisting of points with isotropy exactly K.

From symplectic vector spaces to manifolds
In this short section we will give a method of translating the results obtained in the previous two sections from symplectic vector spaces to symplectic manifolds. As all the symmetry breaking results in sections 5 and 6 are of local type we will make use of the Equvariant Darboux Theorem to pass locally from a symplectic manifold to a symplectic vector space.
Theorem 7.1. Suppose that Γ is a compact Lie group acting on a finite-dimensional symplectic manifold Q and let ω be a Γ-invariant symplectic form on Q. Let q ∈ Q be a Γ-invariant point. Then there are open Γ-invariant neighborhoods U ⊂ Q of q and V ⊂ T q Q of 0 q ∈ T q Q and a Γ-invariant symplectic diffeomorphism f : (U, ω) → (V, ω(q)), such that f (q) = 0 q . This theorem follows from the G-relative Darboux Theorem (see, for example, [20]). Theorem 7.2. Let Q be a smooth manifold and ω 0 and ω 1 two symplectic forms on it. Let G be a Lie group acting properly on Q and symplectically with respect to both ω 0 and ω 1 . Assume that ω 0 (g · q)(v g·q , w g·q ) = ω 1 (g · q)(v g·q , w g·q ) for all g ∈ G and v g·q , w g·q ∈ T g·q Q. Then there exist two open G-invariant neighborhoods U 0 and U 1 of G · q and a G-equivariant diffeomorphism Ψ : U 0 → U 1 such that Ψ | G·q = Id and Ψ * ω 1 = ω 0 . Theorem 7.1 is proved as follows. On the symplectic manifold (Q, ω) consider around the point q an open neighborhood W ⊂ Q contained in a Darboux chart. Thus, on this neighborhood the symplectic form ω 1 is the pull back of the symplectic form ω(q) on T q Q. Recall that ω(q) is Γ-invariant, where Γ acts by linearization of the original action on T q Q. Since the point q is Γ-invariant we can shrink the neighborhood W to a Γ invariant open neighborhood B ⊂ W of q. So, on the Γ-invariant manifold B we have two Γ-invariant symplectic forms: ω and ω 1 . Now using the Γ-relative Darboux theorem we get the existence of two open Γ-invariant neighborhoods U 0 ⊂ B and U 1 ⊂ B of q and a Γ-equivariant diffeomorphism Ψ : U 0 → U 1 such that Ψ | Γ·q = Id and Ψ * ω = ω 1 .
The nonlinear version of the results from sections 5 and 6 can be stated in the following theorem.