Routh Reduction by Stages

This paper deals with the Lagrangian analogue of symplectic or point reduction by stages. We develop Routh reduction as a reduction technique that preserves the Lagrangian nature of the dynamics. To do so we heavily rely on the relation between Routh reduction and cotangent symplectic reduction. The main results in this paper are: (i) we develop a class of so called magnetic Lagrangian systems and this class has the property that it is closed under Routh reduction; (ii) we construct a transformation relating the magnetic Lagrangian system obtained after two subsequent Routh reductions and the magnetic Lagrangian system obtained after Routh reduction w.r.t. to the full symmetry group.


Introduction
It is not a surprise that the bigger part of the recent literature on the geometric description of mechanical systems deals in some way or another with symmetry. The benefits of exploiting symmetry are indeed self-evident: The equations of motion of a mechanical system which exhibits a symmetry can be reduced to a new set of equations with fewer unknowns, possibly easier to solve. In particular, the Hamiltonian description of such systems has attracted most of the attention in the literature, and the important role played by Poisson manifolds and symplectic structures has been strongly emphasized (see e.g. [12] and references therein). Less well-known is the process of symmetry reduction for Lagrangian systems. Much like on the Hamiltonian side, there are in fact two different paths that lead to two different Lagrangian reduction theories. Roughly speaking, the invariance of the Lagrangian leads via Noether's theorem to a set of conserved quantities (the momenta). Whether or not one takes these conserved quantities into account in the reduction process leads to either the Routh or the Lagrange-Poincaré reduction method (see e.g. [4,5,6,10,14,15]).
In this paper we deal with Routh's reduction procedure. In a way, one restricts the attention to only those solutions of the system with a prescribed value of momentum. The price one has to pay is that the new symmetry group (after restriction) is in general only a subgroup of the symmetry group of the original Lagrangian system. In [10] it was pointed out that Routh reduction can be interpreted as a special case of symplectic reduction, which moreover preserves the Lagrangian nature of the system.
In the current paper we will investigate those aspects of the theory that are related to reduction in multiple stages. In case the symmetry group of the system has a normal subgroup, one can indeed first perform a Routh reduction by means of this subgroup. It is then natural to ask whether the reduced system is invariant under some residual, as yet to be determined group, and whether a subsequent second Routh reduction leads to results which are equivalent to direct reduction by the full group. The answer to the same question, but for the above mentioned Lagrange-Poincaré reduction theory, is by now well-known [4]. It can be understood in the following sense: After each reduction step one remains in the category of so-called 'Lagrangian systems on Lie algebroids' and symmetry reduction can be phrased in terms of Lie algebroid morphisms (see e.g. [5] and the references therein). Equivalently, one may use a connection to decompose the Lie algebroid structure at each stage. The corresponding category is then the so-called category of 'Lagrange-Poincaré bundles' from [4], while the corresponding 'Lagrange-Poincaré morphisms' do the reduction.
It is natural to ask whether or not such a category of systems exists also for Routh reduction (it is mentioned as an open problem in [12]). The candidate we propose (in Section 3) is the class of what we call 'magnetic Lagrangian systems'. We show in Sections 4 and 5 that this class has the property that it is closed under Routh reduction, i.e. that after each step of a reduction in stages the reduced system remains in the class of magnetic Lagrangian systems. As in [10], we heavily rely on a generalized version of cotangent symplectic reduction. In Section 6 we apply the framework of magnetic Lagrangian systems to reduction in several stages, where the first stage consists of Routh reduction by a normal subgroup of the overall symmetry group. We end the paper with some illustrative examples in the final section.

Preliminaries on Routh reduction
Definition 1. A Lagrangian system is a pair (Q, L) with Q a manifold and L a function on T Q. The manifold Q is the configuration space and L is called the Lagrangian. The dimension of Q is called the number of degrees of freedom of the Lagrangian system. A Lagrangian system is of mechanical type if for arbitrary v q ∈ T q Q, the Lagrangian can be written as with ·, · Q a Riemannian metric on Q and V a function on Q, called the potential energy of the Lagrangian. The function 1 2 v q , v q Q is called the kinetic energy. We are interested in the Euler-Lagrange equations associated to a Lagrangian system. These equations are necessary conditions for a curve q(t) : I ⊂ R → Q to extremize the action integral I L(q(t),q(t))dt. In a local coordinate chart (q i ) on Q the Lagrangian L is a function of (q i , v i ) and the Euler-Lagrange equations are d dt Routh reduction is a reduction technique for Lagrangian systems that are invariant under the action of a symmetry group. In classical textbooks, it is a step-by-step procedure that describes the construction of a new Lagrangian function with fewer degrees of freedom. Before we can formulate Routh reduction, we fix notations for the action of a Lie group on an arbitrary manifold M and related concepts.
Actions of Lie groups and principal bundles. When a Lie group G is given, g denotes the Lie algebra of G and exp the exponential map from g to G. The adjoint action of G on g is denoted by Ad and the coadjoint action of G on g * by Ad * .  Every right action Ψ M gives rise to a left action Φ M : Φ M g (m) = Ψ M g −1 (m). We only consider right actions. This is not a true restriction since one may reformulate the main results for left actions if needed.
The orbit space M/G of a free and proper action is a manifold and π : M → M/G carries the structure of a principal G-bundle. Throughout the paper [m] G denotes a point in the orbit space M/G, i.e. it is the orbit through m ∈ M . Every tangent vector in the kernel of T π is of the form ξ M (m) for some ξ ∈ g. These vectors are called vertical and form a distribution which we call the vertical distribution V π = ker T π.
Definition 3. A principal connection on a principal G-bundle is a g-valued 1-form A satisfying two conditions: 1) it is equivariant, i.e. Ψ * g A = Ad g −1 · A for any g ∈ G, and 2) for ξ ∈ g arbitrary, A(ξ M ) = ξ.
The kernel of A determines a right invariant distribution on M which is a complement of the vertical distribution. It is therefore called the horizontal distribution of A and is typically denoted by H A ⊂ T M . On the other hand, any right invariant distribution H satisfying H ⊕ V π = T Q determines a principal connection, see [8].
We denote by f, A , where f is a g * -valued function on M , the 1-form on M pointwise defined by The cotangent vector f, Routh reduction.
Definition 4. Let (Q, L) denote a Lagrangian system and assume that the configuration space is equipped with an action Ψ Q .
If a mechanical Lagrangian system is G-invariant then the kinetic energy metric ·, · Q and the potential energy V are both invariant under the pull-back of Ψ Q g , for arbitrary g.
1. For a Lagrangian system (Q, L), the Legendre transform FL : The Lagrangian is hyperregular if FL is a diffeomorphism.
2. For a G-invariant Lagrangian system (Q, L), the momentum map J L : T Q → g * is the map (ψ Q ) * • FL, i.e. for arbitrary v q ∈ T Q and ξ ∈ g 3. Given a G-invariant mechanical Lagrangian system (Q, L), and a point q in Q. The inertia tensor I q is a metric on g defined by The momentum map associated to a G-invariant Lagrangian system satisfies the following two important properties: Every G-invariant mechanical Lagrangian system is G-regular. To show this, remark that for a mechanical Lagrangian system Hence J L | vq (ξ) = J L | vq (0)+I q (ξ) is an affine map, whose linear part is determined by the inertia metric on g.
Proposition 1. Consider a G-invariant and G-regular Lagrangian system (Q, L) and fix a regular value µ ∈ g * of the momentum map J L . Let G µ denote the isotropy subgroup of µ w.r.t. the coadjoint action of G on g * , i.e. g ∈ G µ iff Ad * g µ = µ.
The first statement is a direct consequence of the equivariance of J L . We postpone a proof of the second statement (Proposition 7, page 15).
We are now ready to describe a preliminary version of Routh reduction where the symmetry group G is Abelian and the bundle Q → Q/G is trivial, so that we can choose a connection A with vanishing curvature. Since G is Abelian, we have that the isotropy subgroup G ν for every ν ∈ g * is the entire group G, and in particular T (Q/G) × Q/G Q/G ν is nothing but T (Q/G).
Theorem 1 (Routh reduction -Abelian version). Let G be Abelian and let A be a connection on Q with vanishing curvature. The Routh reduction procedure of a G-invariant and G-regular Lagrangian system (Q, L) consists of the following steps.
1. Fix a regular value µ of the momentum map J L and consider the submanifold J −1 L (µ). 2. Compute the restriction of the G-invariant function L − µ, A to the level set J −1 L (µ). Let L denote its projection to the quotient manifold T (Q/G) ∼ = J −1 L (µ)/G. Every solution of the Euler-Lagrange equations of the Lagrangian system (Q/G,L) is the projection of a solution of the Euler-Lagrange equations of the original system (Q, L) with momentum µ. Conversely, every solution of the Euler-Lagrange equations of the Lagrangian system (Q, L) with momentum µ projects to a solution of the Euler-Lagrange equations of the system (Q/G,L).
One may also find a description of global Abelian Routh reduction in [3]. Example: the spring pendulum. The system consists of a point particle with mass m moving in a horizontal plane, and attached to the origin by means of a spring with spring constant k. We choose polar coordinates (r, θ) for this system, so that the mechanical Lagrangian is given by L = 1 2 m(ṙ 2 + r 2θ2 ) − 1 2 kr 2 . This system is clearly invariant under translations in the θ-direction, given by Ψ a (r, θ) = (r, θ + a). The momentum map for this action is J L = mr 2θ and the Euler-Lagrange equations are Let us fix a regular value 0 = µ = mr 2θ for the momentum map and let A = dθ be the standard connection with vanishing curvature. The Routhian is obtained from The Euler-Lagrange equation forL is mr = −kr + µ 2 mr 3 and solutions of this equations are in correspondence to solutions of the Euler-Lagrange equations for L with momentum µ.
Routh reduction can be extended to more general contexts, including non-Abelian group actions [14], Lagrangians of non-mechanical type [2,6], Lagrangians invariant up to a total time derivative [10] and Lagrangians that are not G-regular [11]. Additional complications arise in these cases: for instance, if the bundle Q → Q/G is not trivial, the reduced system is typically subjected to an additional force term associated to the curvature of the chosen connection. On the other hand, if the symmetry group is not Abelian, the quotient space J −1 L (µ)/G µ no longer has the structure of a tangent bundle; instead it is diffeomorphic to T (Q/G) × Q/G Q/G µ . The interpretation of the reduced system as a Lagrangian system then requires additional definitions, which are postponed to the following section.
Our main interest in this paper is Routh reduction by stages, where the assumption is that the Lagrangian system obtained after applying a first Routh reduction carries additional symmetry, so that we can reapply Routh reduction. The system obtained after one Routh reduction is formulated on a fibred product of the type T (Q/G) × Q/G Q/G µ . This is the fibred product of a bundle Q/G µ → Q/G with the tangent bundle to the base space. This observation is our main motivation to extend Routh reduction to Lagrangian systems defined on such fibred products: in order to develop Routh reduction by stages, we have to be able to reduce Lagrangian systems obtained after a first reduction.
Routh reduction is closely related to symplectic reduction on the associated cotangent bundles (see [10,14]). In the next section we will introduce the concept of a "magnetic Lagrangian system" and we will emphasize its symplectic formulation. The concept of magnetic Lagrangian systems is the analogue of Hamiltonian systems one encounters in magnetic cotangent bundle reduction [12].

Magnetic Lagrangian systems
A magnetic Lagrangian system is a Lagrangian system with configuration space the total space of a bundle ǫ : P → Q and where the Lagrangian is independent of the velocities tangent to the fibres of ǫ. Additionally the system is subjected to a force term that is of magnetic type. It might help to keep in mind that in the case of a Routh reduced Lagrangian system, P corresponds to Q/G µ and the fibration ǫ is given by the projection Q/G µ → Q/G. Definition 7. A magnetic Lagrangian system is a triple (ǫ : P → Q, L, B) where ǫ : P → Q is a fibre bundle, L is a smooth function on the fibred product T Q × Q P and B is a closed 2-form on P . We say that P is the configuration manifold of the system and that L is the Lagrangian.
A coordinate chart (q i , p a ), i = 1, . . . , n = dim Q, a = 1, . . . , k = dim P − dim Q, adapted to the fibration ǫ : P → Q determines a coordinate chart (q i , v i , p a ) on T Q × Q P , and the Lagrangian L is then a function depending on (q i , v i , p a ). By definition, L is independent of the velocities in the fibre coordinates p a and therefore it determines a singular Lagrangian when interpreted as a function on T P . Locally, the Euler-Lagrange equations for this singular Lagrangian are d dt Here we used the following coordinate expression of the 2-form B is 1 2 B ij dq i ∧ dq j + B ia dq i ∧ dp a + 1 2 B ab dp a ∧ dp b . These Euler-Lagrange equations have a geometric interpretation (see Proposition 2). First we introduce additional notations. Definition 8. Assume a magnetic Lagrangian system (ǫ : P → Q, L, B) is given.
1. T P Q denotes the fibred product T Q × Q P and (v q , p), where v q ∈ T Q and p ∈ P such that ǫ(p) = q, is a point in T P Q. Similarly, T * P Q denotes the fibred product T * Q × Q P and (α q , p), with ǫ(p) = q, is an arbitrary element in T * P Q. 2. V ǫ denotes the distribution on P of tangent vectors vertical to ǫ.
3.ǫ : T P → T P Q is the projection fibred over P that maps v p ∈ T P onto (T ǫ(v p ), p) ∈ T P Q.
5. ρ 2 : T P Q → P is the projection that maps (v q , p) ∈ T P Q onto p ∈ P .
6. ǫ 1 : T * P Q → T * Q is the projection that maps (α q , p) ∈ T * P Q onto α q ∈ T * Q. 7. ǫ 2 : T * P Q → P is the projection that maps (α q , p) ∈ T * P Q onto p ∈ P . 8. The Legendre transform FL : for arbitrary w q ∈ T q Q.

The energy
(Here the contraction of an element (α q , p) ∈ T * P Q with (v q , p) ∈ T P Q is defined naturally as (α q , p), (v q , p) = α q , v q .) 10. By means of the Legendre transform we can pull-back the presymplectic 2-form ǫ * The latter is denoted by Ω L,B (Here ω Q = dθ Q , with θ Q the Poincaré-Cartan 1-form on T * Q and a presymplectic 2-form is understood to be a closed 2-form, not necessarily of constant rank.) Proposition 2. Given a curve p(t) in P , and let γ(t) denote the curve in T P Q equal to (q(t), p(t)) ∈ T P Q with q(t) = ǫ(p(t)). The curve p(t) in P is a solution to the Euler-Lagrange equations for the magnetic Lagrangian system (ǫ : P → Q, L, B) iff γ(t) is a solution to the presymplectic equation Locally, the presymplectic equation coincides with the previously mentioned Euler-Lagrange equations, since 2) is of mechanical type if L(v q , p) = 1 2 (v q , p), (v q , p) ρ 1 − V (p) with ·, · ρ 1 is a metric on the vector bundle ρ 1 : T P Q → P and V is a function on P .
Note that if B is nondegenerate, then the typical fibre of P necessarily has to be evendimensional. In a local coordinate chart the nondegeneracy condition on B is expressed by det B ab = 0. Proposition 3. If a magnetic Lagrangian system (ǫ : P → Q, L, B) is hyperregular, the 2-form Ω L,B = FL * (ǫ * 1 ω Q + ǫ * 2 B) determines a symplectic structure on T P Q.
Proof . Assume that the magnetic Lagrangian system is hyperregular. The nondegeneracy of Ω L,B is easily checked if we work in a coordinate chart adapted to the fibration: Since FL is a diffeomorphism, the 1-forms d ∂L ∂v i , dq i and dp a provide pointwise a basis on T * (T P Q). One can now use standard arguments to prove the nondegeneracy.
We conclude that a hyperregular magnetic Lagrangian system has a symplectic structure although the Lagrangian itself is singular when interpreted as a function on T P . The energy E L is the Hamiltonian. This is an important observation. Remark 1. Throughout the paper we only consider magnetic Lagrangian systems that are hyperregular. This is not a true restriction. The results remain valid for general magnetic Lagrangian systems: instead of relating Routh reduction to symplectic reduction, it is possible to relate it to presymplectic reduction [7].
Remark 2. We conclude this section with a remark on equivalent magnetic Lagrangian systems. Roughly speaking, two Lagrangian systems are equivalent if the resulting dynamics coincide, more specifically if they produce the same Euler-Lagrange equations. It is well known that for a Lagrangian system the Euler-Lagrange equations do not change when the Lagrangian is augmented with a total time derivative of a function on the configuration space. For magnetic Lagrangian systems this gauge freedom can be extended.
Consider a 1-form α along ǫ, i.e. a section of ǫ 2 : T * P Q → P , or in other words a linear function on T P Q given by (v q , p) → α(p), (v q , p) . When α : P → T * P Q is composed with T * ǫ • ǫ 1 : T * P Q → T * P , it determines a 1-form on P which, with a slight abuse of notation, we denote by ǫ * α. Now consider the function L ′ (v q , p) = L(v q , p) − α(p), (v q , p) and the gyroscopic 2-form B ′ = B + dǫ * α. Together they define a new magnetic Lagrangian system (ǫ : P → Q, L ′ , B ′ ) which is equivalent to (ǫ : P → Q, L, B). This is a reformulation of a well-known result in classical mechanics, saying that an exact gyroscopic force can be taken into account by means of a velocity dependent potential [19].
Proof . We show that any solutions to the Euler-Lagrange equations of (ǫ, L ′ , B ′ ) is a solution to the Euler-Lagrange equations of (ǫ, L, B). We work in a local coordinate neighborhood (q i , p a ) as before. We fix a coordinate expression for It now remains to substitute the definition of L ′ and B ′ , i.e. L ′ = L − α i v i and 4 Magnetic Lagrangian systems with symmetry

Symplectic reduction: a brief introduction
An overall reference for this section is e.g. [12]. Let (M, ω) be a symplectic manifold on which G acts on the right, Ψ M : M × G → M . Given a function f : M → g * , then f ξ for ξ ∈ g denotes the real valued function on M defined by f ξ (m) = f (m), ξ .

The action Ψ
3. If M is connected, the non-equivariance 1-cocycle σ of the momentum map J equals where m is arbitrary in M .
The definition of σ is independent of m (see [1]). Recall that a 1-cocycle with values in g * statisfies, for g, h ∈ G arbitrary, Definition 11. Let σ be a 1-cocycle with values in g * . The affine action of G on g * with 1-cocycle σ is given by (g, µ) → Ad * g µ + σ(g −1 ) for arbitrary µ ∈ g * and g ∈ G.
The momentum map J is equivariant with respect to the affine action with 1-cocyle from Definition 10: . For a fixed element µ ∈ g * , the Lie group G µ < G denotes the isotropy subgroup of µ w.r.t. the affine action, i.e. g ∈ G µ if µ = Ad * g µ + σ(g −1 ) or equivalently if Theorem 2 (symplectic reduction SR). Let (M, ω) be a symplectic manifold, with G acting canonically on M . Let J be a momentum map for this action with non-equivariance cocycle σ. Assume that µ is a regular value of J, and denote by G µ the isotropy group of µ under the affine action of G on g * . Then the pair The Hamiltonian vector field X h is tangent to J −1 (µ) and the corresponding vector field on J −1 (µ) is π µ -related to the Hamiltonian vector field X hµ on M µ .

Invariant magnetic Lagrangian systems and momentum maps
In order to define invariant magnetic Lagrangian systems, we start from a free and proper action of a Lie group G on both P and Q such that they commute with ǫ: for arbitrary g ∈ G, In other words, G acts on ǫ : P → Q by bundle automorphisms. The projections of the principal bundles are denoted by π Q : Q → Q/G and π P : P → P/G. These actions induce lifted right actions on T P Q and T * P Q: ) for (α q , p) ∈ T * P Q and g ∈ G. Definition 12. A magnetic Lagrangian system (ǫ : Let (ǫ : P → Q, L, B) be a G-invariant hyperregular magnetic Lagrangian system and consider the symplectic 2-form ǫ * 1 ω Q + ǫ * 2 B on T * P Q. Proposition 4. The action Ψ T * P Q is canonical and the Legendre transform FL : To show the equivariance of FL we use its definition: let g ∈ G, v q , w q ∈ T Q be arbitrary, then In order to define a momentum map, we introduce a concept that is similar to the so-called Bg-potential in [12].
Recall that δ ξ is defined pointwise as δ ξ (p) = δ(p), ξ . If B is G-invariant and admits a Bg-potential δ, then it follows that d((Ψ P g ) * δ ξ − δ Adgξ ) = 0. If P is connected, this induces a g * -valued 1-cocycle on G: . For the following proposition, recall from Definition 2 that (ψ Q ) * : T * Q → g * is the dual of the infinitesimal action of G on Q.
Proof . The map (ψ Q ) * : T * Q → g * is an equivariant momentum map for the symplectic manifold (T * Q, ω Q ). It is straightforward that the combined map J δ = (ψ Q ) * • ǫ 1 − δ • ǫ 2 is a momentum map for the lifted action on T * P Q w.r.t. the symplectic form ǫ * 1 ω Q + ǫ * 2 B with non-equivariance cocycle −σ δ . The rest of the statement follows by construction.
We conclude that the symplectic structures (T P Q, Ω L,B ) and (T * P Q, ǫ * 1 ω Q + ǫ * 2 B) associated to a G-invariant and hyperregular magnetic Lagrangian system with a Bg-potential δ admit a momentum map with cocycle −σ δ and are amenable to symplectic reduction. 5 Routh reduction for magnetic Lagrangian systems 5

.1 Magnetic cotangent bundle reduction
In this section we study the reduction of the symplectic manifold (T * P Q, ǫ * 1 ω Q + ǫ * 2 B) determined from the magnetic 2-form B of a G-invariant hyperregular magnetic Lagrangian system with Bg-potential δ. From Proposition 5, it follows that the function J δ is a momentum map with non-equivariance cocycle −σ δ . As usual, G µ denotes the isotropy subgroup of µ ∈ g * for the affine action.
Proposition 6. Fix a connection A Q on the bundle π Q : Q → Q/G and a regular value µ of the momentum map J δ .
Proof . Recall the pointwise definition of µ + δ, A P : given v p ∈ T P , then µ + δ, We continue with the proof.
1. This is straightforward from the definition of σ δ . 2. The g * -valued 1-form A P determines a principal connection if it satisfies A P (ξ P ) = ξ for all ξ ∈ g, and if (Ψ P g ) * A P = Ad g −1 · A P for all g ∈ G: A P (ξ P )(p) = A Q (ǫ(p))(T ǫ(ξ P (p))) = A Q (ǫ(p))(ξ Q (ǫ(p))) = ξ and 3. The pull-back 1-form (Ψ P g ) * µ + δ, A P equals It is projectable to P/G µ since the contraction with any fundamental vector field ξ P with ξ ∈ g µ vanishes: where we used the fact that ξ ∈ g µ , or equivalently ad * ξ µ = i ξ Σ δ . 5. Similar to cotangent bundle reduction, we use a 'shift map' to construct the required diffeomorphism. Recall that V π Q denotes the bundle of tangent vectors vertical to π Q : Q → Q/G. The subbundle V 0 π Q of T * Q is defined as the annihilator of V π Q . Below we introduce a shift map φ A µ between J −1 δ (µ) and V 0 P π Q = V 0 π Q × Q P and we show that it is equivariant w.r.t. to actions of G µ obtained by restriction of Ψ T P Q to J −1 δ (µ) and V 0 P π Q . This is sufficient for φ A µ to project to a diffeomorphism [φ A µ ] between the quotient spaces. This provides us with the desired diffeomorphism because the quotient of V 0 P π Q is well known: Def inition of the shift map. In the following we use a slight abuse of notations: if we write µ + δ(p), A Q (q) for some p ∈ P and q = ǫ(p), then this is the cotangent vector in Equivariance of the shift map. For arbitrary g ∈ G µ , The projection of φ A µ is thus well defined, and it is denoted by In the following proposition we put a symplectic structure on the fibred product T * P/Gµ (Q/G) = T * (Q/G) × Q/G P/G µ . The notations are similar to those on T * P Q.
Definition 15. We define the projections The canonical symplectic 2-form on T * (Q/G) is denoted by ω Q/G .
Completely analogous to the construction on T * P Q, we introduceǫ * 1 ω Q/G + ǫ * 2B as a 2-form on T * P/Gµ (Q/G). Theorem 3 (Generalized magnetic cotangent bundle reduction). Let µ denote a regular value of the momentum map J δ for the symplectic structure (T * P Q, ǫ * 1 ω Q +ǫ * 2 B) with isotropy subgroup G µ , and let A Q be a principal connection on Q → Q/G. Then the symplectically reduced manifold Although this result is a straightforward extension of standard magnetic cotangent bundle reduction in [12], we include a proof because its method turns out to be useful later on.
We use the fact that ω µ is uniquely determined by i * µ ω Q = π * µ ω µ , with i µ : J −1 δ (µ) → T * P Q the natural inclusion and π µ : J −1 δ (µ) → M µ the projection to the quotient space. Due to the uniqueness property, it is therefore sufficient to show that The left-hand side of (2).
The right-hand side of (2). We first show the equality of two 1-forms: and v q ∈ T q Q denotes the projection of V (αq,p) . Then, the first 1-form equals The second 1-form evaluated on this tangent vector gives The two 1-forms are identical and after taking the exterior derivative and adding ǫ * 2 B, we obtain that the right-hand side of (2) equals .
one easily verifies that the left-hand side equals the right-hand side in (2).
We conclude with a result on the behavior of a symplectomorphism under symplectic reduction. Assume two symplectic manifolds (M, Ω) and (M ′ , Ω ′ ) and a symplectomorphism f : M → M ′ (i.e. a map for which f * Ω ′ = Ω) are given. We assume in addition that both M and M ′ are equipped with a canonical free and proper action of G. Let J : M → g * and J ′ : M ′ → g * denote corresponding momentum maps for these actions on M and M ′ respectively. We say that f is equivariant if f (mg) = f (m)g for arbitrary m ∈ P , g ∈ G. Note that the non-equivariance cocycles for J and J ′ are equal up to a coboundary. Without loss of generality we may assume that f * J ′ = J and that the non-equivariance cocycles coincide. This in turn guarantees that the affine actions on g * coincide and that the isotropy groups of an element µ ∈ g * coincide for both affine actions. Finally, fix a regular value µ ∈ g * for both J and J ′ .

Routh reduction for magnetic Lagrangian systems
In Proposition 5 we have introduced the momentum map J L,δ = J δ • FL for a G-invariant hyperregular magnetic Lagrangian system (ǫ : P → Q, L, B), i.e.
We know from Proposition 4 that the Legendre transform FL is equivariant. After restriction to J −1 L,δ (µ) it reduces to a symplectic diffeomorphism between the symplectically reduced spaces J −1 L,δ (µ)/G µ and J −1 δ (µ)/G µ (see also Theorem 4). The following diagram summarizes these previous observations. We are now only three steps away from a description of Routh reduction for magnetic Lagrangian systems. The final goal is to describe the symplectic reduced manifold (J −1 L,δ (µ)/G µ , Ω µ ) as a symplectic manifold associated with a new magnetic Lagrangian system, with Lagrangian say L µ .
1. The first step is the construction of a diffeomorphism ∆ µ : This is crucial because the Hamiltonian dynamics determined by the Euler-Lagrange equations reduce to Hamiltonian dynamics on a manifold of the form T P/Gµ (Q/G). If we can characterize this reduced dynamics as the Euler-Lagrange equations of a magnetic Lagrangian system with configuration manifold P/G µ → Q/G, we have developed a Routh reduction technique for magnetic Lagrangian systems. This characterization consists of the two remaining steps mentioned before: 2. We show that the composition [φ A µ ] • [(FL) µ ] • ∆ µ equals the fibre derivative of some Lagrangian functionL on T P/Gµ (Q/G), and this implies that the symplectic 2-form ∆ * µ Ω µ on T P/Gµ (Q/G) is of the form which is the symplectic structure associated to the reduced magnetic Lagrangian system (P/G µ → Q/G,L,B); 3. We show that the reduction of the energy E L in the symplectic reduction scheme is precisely the energy EL on T P/Gµ (Q/G). This guarantees that the Euler-Lagrange equations associated to the reduced LagrangianL are related to the Euler-Lagrange equations for L.
The new reduced LagrangianL is what we call the Routhian (it is often alternatively denoted by R or R µ ). This is summarized in the diagram in Fig. 3. Step 1: the def inition of ∆ µ : . We will define the map ∆ µ as the inverse of Π µ . In general however, Π µ is not invertible. The following definition is the analogue of G-regularity for a classical Lagrangian.
Definition 16. The Lagrangian L of a G-invariant magnetic Lagrangian system is called Gregular if the map J L,δ | (vq ,p) : g → g * ; ξ → J L,δ (v q + ξ Q (q), p) is a diffeomorphism for all (v q , p) ∈ T P Q.
Every mechanical G-invariant magnetic Lagrangian system is G-regular because J L,δ | (vq ,p) is an affine map modeled on the metric on g obtained from the kinetic energy metric.
and p ∈ [p] Gµ . Due to the G-regularity of L, there exists a unique element ξ in g such that J L,δ (v q + ξ Q (q), p) = µ. The orbit of the point (v q + ξ Q (q), p) under the G µ -action determines an element in J −1 L,δ (µ)/G µ . We will show that this construction is independent of the chosen point (v q , p), and therefore defines a map ∆ µ : T P/Gµ (Q/G) → J −1 L,δ (µ)/G µ which is the inverse to Π µ . To show that the construction is independent of the chosen point (v q , p) in T P Q, we choose any other point . The point is of the form (v ′ q , p ′ ) = (Ψ T Q g (v q + η Q (q)), pg) for some g ∈ G µ and η ∈ g. We now repeat the previous construction applied to (v ′ q , p ′ ): we consider a unique element ξ ′ ∈ g for which J L,δ (Ψ T Q g (v q + η Q (q)) + ξ ′ Q (qg), pg) = µ. Due to the equivariance of J L,δ , we conclude that .
The previous proposition guarantees that for a given G-regular Lagrangian and a bundle adapted coordinate chart (x r , g m ) in Q → Q/G and (x r , g m , p a ) in P → Q, the functions (x r ,ẋ r , g m , p a ) determine a coordinate chart in J −1 L,δ (µ). This is useful in the following.
Step 2: the reduced symplectomorphism. We will now introduce a new LagrangianL on T P/Gµ (Q/G) with the property that its Legendre transform FL : As is already described in [10], the structure of the LagrangianL is completely determined by the cotangent bundle reduction scheme, and in particular by the shift map To make this statement more precise, we remark that The latter is the fibre derivative of a new function on T P Q, namely L(v q , p)− µ+δ(p), A Q (q)(v q ) . This function is clearly G µ -invariant.
Definition 17. The functionL is defined as the pull-back under ∆ µ of the quotient map of the restriction to J −1 L,δ (µ) of the function Proof . Fix elements (v [q] G , [p] Gµ ) ∈ T P/Gµ (Q/G) and fix a representative (v q , p) ∈ J −1 L,δ (µ) in the orbit ∆ µ (v [q] G , [p] Gµ ) ∈ J −1 L,δ (µ)/G µ . By definition of the maps involved, we have Next we study the fibre derivative of the mapL. Fix a point (w [q] G , [p] Gµ ) and we compute We construct a curve u → ζ(u) in J −1 L,δ (µ) that projects onto the curve L,δ (µ) → P (recall that ρ 2 : T P Q → P is the projection onto the second factor in the fibred product).
The existence of such a curve is best shown in a specific coordinate chart (see above) for J −1 L (µ): (x r ,ẋ r , g m , p a ) where the index r = 1, . . . , dim Q/G, m = 1, . . . , dim G and a = 1, . . . , dim P − dim Q. Note that (x r ,ẋ r ) is a coordinate chart in T (Q/G). In these coordinates, we let (v q , p) = (x r 0 ,ẋ r 0 , g m 0 , p a 0 ) and w [q] G = (x r 0 , w r 0 ). We define the curve ζ(u) to be the curve u → (x r 0 ,ẋ r 0 + uw i 0 , g m 0 , p a 0 ). Then the tangent vector to T ρ 1 (ζ(0)) is the vertical lift of some Finally, from the definition ofL and the fibre derivative FL we obtain This concludes the proof.
Step 3: the reduced energy Hamiltonian. The third and last step concerns the specific reduced dynamics. We have to relate the energy E L on T P Q to the energy of the RouthianL on T P/Gµ (Q/G). In the following Lemma we again use the notations from Theorem 2 applied to the reduction of the symplectic structure on (T P Q, Ω L,B ).
Lemma 3. The energy EL is the reduced Hamiltonian, i.e. it satisfies: . Using the result from Lemma 2 this concludes the proof.
Routh reduction. The previous three steps are summarized in the following theorem.
Theorem 5 (Routh reduction for magnetic Lagrangian systems). Let (ǫ : P → Q, L, B) be a hyperregular, G-invariant and G-regular magnetic Lagrangian system and let δ be a Bg-potential of the magnetic term B with 1-cocycle σ δ .
1. Let µ ∈ g * be a regular value of the momentum map J L,δ and let G µ be the isotropy subgroup of µ w.r.t. the affine action on g * with 1-cocycle −σ δ , i.e. g ∈ G µ if and only if µ = Ad * g µ − σ δ (g −1 ). 2. Fix a connection A Q on π Q : Q → Q/G and let A P be the corresponding connection on P → P/G. Compute the restriction of the G µ -invariant function L(v q , p) − µ + δ(p), A Q (q)(v q ) to J −1 L,δ (µ) and letL be its quotient to T P/Gµ (Q/G) ∼ = J −1 L,δ (µ)/G µ . 3. ComputeB as the projection to P/G µ of the 2-form B + d µ + δ, A P .
This reduced magnetic Lagrangian system is hyperregular and every solution p(t) ∈ P to the Euler-Lagrange equations for (ǫ, L, B) with momentum µ projects under P → P/G µ to a solution of the Euler-Lagrange equations for (ǫ µ : P/G µ → Q/G,L,B). Conversely, every solution to the Euler-Lagrange equations for (ǫ µ : P/G µ → Q/G,L,B) is the projection of a solution to the Euler-Lagrange equations for (ǫ, L, B) with momentum µ.
It is possible to say more about the structure ofB and its relation to the connection A Q . This and reconstruction aspects fall out of the scope of this paper. We refer to [11] where these topics are described in more detail. They carry over to this more general framework in a straightforward way.

Reduction of magnetic Lagrangian systems on Lie groups
Consider a magnetic Lagrangian system on P = Q = G, i.e. the configuration space is a Lie group G. We start from a function ℓ on g and with it we associate a Lagrangian L on T G by right multiplication L(g, v g ) = ℓ(v g g −1 ). By definition L is invariant under the right action of G on itself. We assume that a magnetic 2-form B is given which is invariant under right multiplication and admits a Bg-potential δ : G → g * . We first rephrase some definitions in this specific setting.
3. We use the right identification of T G with G×g, i.e. (g, v g ) is mapped to (g, v g g −1 ) ∈ G×g.
The right action of G on T G equals right multiplication in the first factor of G × g under this identification.
4. We use the Maurer-Cartan principal connection on G → G/G: A(g)(v g ) = g −1 v g . In the right identification, the connection corresponds to the map (g, ξ) ∈ G × g → Ad g −1 ξ ∈ g.

Routh reduction by stages
In reduction by stages, we study the reduction of a G-invariant system (symplectic or Lagrangian) under the action of the full group G and under the induced action w.r.t. a normal subgroup K ✁ G. We shall adopt as far as possible the notations used in [12]. A detailed construction of the following definitions is found in this reference.
1. The Lie-algebra of K is K and i denotes the injection i : K → g with dual i * : g * → K * .
2. The group G acts on K by restriction of the adjoint action. The induced action of G on K * is denoted by the same symbol Ad * : G × K * → K * .
3. µ denotes an element in g * and ν ∈ K * . Then G µ is the isotropy subgroup of µ under the Ad * -action of G on g * ; G ν is the isotropy subgroup of ν under the Ad * -action of G on K * obtained as the dual of the restricted Ad-action of G on K; and K ν is the isotropy of ν w.r.t. to standard coadjoint action of K on K * . These groups satisfy G ν ∩ K = K ν and K ν is normal in G ν .
4. g ν and K ν denote the Lie algebras of G ν and K ν respectively.Ḡ ν denotes the quotient group G ν /K ν and its Lie algebra equalsḡ ν = g ν /K ν .
In [12] symplectic reduction by stages is performed under the condition of a so-called 'stages hypothesis'. An element µ ∈ g * is said to satisfy the stages hypothesis if for any µ ′ ∈ g * satisfying µ| K = µ ′ | K = ν and µ| gν = µ ′ | gν =ν, there exists an element k ∈ K ν and g ∈ (G ν )ν such that Ad * kg µ ′ = µ. The stages hypothesis is a condition on a chosen momentum value and depends only on the symmetry group G. It was already clear in [12] that the hypothesis is automatically satisfied if G is a central extension or if G is a semi-direct product group. In the recent contribution [16] it has been pointed out that the hypothesis is in fact always satisfied, and that it can be taken out of the reduction by stages statements altogether. Taking advantage of this result, in this paper, we will not make further reference to the stages hypothesis.

Symplectic reduction by stages
Theorem 6 (Symplectic reduction by stages [12]). Let (M, ω) be a symplectic manifold with a canonical G-action Ψ M with an equivariant momentum map J G .
1. Fix a regular value µ ∈ g * of the momentum map and perform symplectic reduction to obtain the symplectic manifold (M µ , ω µ ).
2. The restriction of the action Ψ M to K is canonical and the map J K = i * • J G : M → K * determines an equivariant momentum map for this induced action. Fix a regular value ν of J K and perform symplectic reduction to obtain the symplectic manifold (M ν , ω ν ).

The level set
The groupḠ ν = G ν /K ν acts on M ν by projecting the restricted action of G ν on J −1 K (ν). This induced action Ψ Mν is free, proper and canonical on (M ν , ω ν ). Assume that K ν is connected.

Fix an elementν in g *
ν such that the restriction ofν| Kν equals ν| Kν . There is a well-defined momentum map JḠ ν : M ν →ḡ * ν for the induced action Ψ Mν . This momentum map is determined from J G andν and has a non-equivariance cocycle: JḠ ν satisfies 6. Let ρ ∈ḡ * ν be a regular value for the momentum map JḠ ν and let (Ḡ ν ) ρ be the isotropy subgroup of ρ w.r.t. the affine action ofḠ ν onḡ * ν . Perform symplectic reduction to obtain the symplectic manifold If ρ is chosen such that (r ′ ν ) * ρ = µ| gν −ν, then there exists a symplectic diffeomorphism For our purpose it is also important to understand the reduction of a G-invariant Hamiltonian h on M . We assume that all conditions in Theorem 6 are satisfied. First note that, by definition of the momentum maps, we have an inclusion j µ of J −1 G (µ) in J −1 K (ν). Recall that we use π ν for the projection J −1 K (ν) → M ν . It was shown in [12] that the image of π ν • j µ : . Moreover, this map is equivariant w.r.t. the action of G µ on J −1 G (µ) and (Ḡ ν ) ρ on J −1 Gν (ρ) (this makes sense, since G µ projects to a subset of (Ḡ ν ) ρ ). The quotient of π ν • j µ is the symplectic diffeomorphism F mentioned in the previous theorem (see also Fig. 4).

Routh reduction by stages
Routh reduction by stages is symplectic reduction by stages applied to the symplectic structure of the initial Lagrangian system. In this section we show that the symplectic structures and energy hamiltonians in the different stages can in fact be associated to specific magnetic Lagrangians systems, and eventually gives us Routh reduction by stages. The symplectic reduction by stages then provides us a diffeomorphism relating the solutions of the different Euler-Lagrange equations for the Lagrangian systems in the final stages. We start with a hyperregular Lagrangian L, invariant under the action of a Lie group G. We assume that this Lagrangian satisfies a regularity condition which is more stringent than mere G-regularity.
Definition 19. The Lagrangian L is said to be G-hyperregular if for any v q ∈ T Q and any subspace K ′ < g with injection i ′ : Lagrangians of mechanical type are G-hyperregular. Let K be a normal subgroup of G, K the Lie algebra of K and i : K → g the canonical injection. Due to the hyperregularity the invariant Lagrangian L is both G-and K-hyperregular, and both G-and K-invariant. By definition of J L , the map i * • J L = i * • (ψ T Q ) * • FL is the momentum map for the K-action.
Theorem 7 (Routh reduction by stages). Assume (Q, L) is a hyperregular, G-hyperregular and G-invariant Lagrangian system. Let K denote a normal subgroup of G.
1. Let µ ∈ g * be a regular value of the momentum map J L and A 0 a G-connection on Q. Let (Q/G µ → Q/G, L 0 , B 0 ) be the magnetic Lagrangian system obtained by performing Routh reduction with respect to G.
2. Fix a regular value ν ∈ K * of the momentum map i * • J L for the K-action and a Kconnection A 1 on Q. Assume that A 1 is G-equivariant w.r.t. the action of G on Q and K * . Consider the magnetic Lagrangian system (Q/K ν → Q/K, L 1 , B 1 ) obtained by performing Routh reduction with respect to K.
3.Ḡ ν acts on Q/K ν and Q/K by projecting the induced action of G ν on Q. These induced actions are free and proper.
Proof . 1 and 2 are obtained by applying Routh reduction. 3 follows from [12, p. 152]: we know that the quotient groupsḠ ν = G ν /K ν acts in a free and proper way on the quotient space Q/K ν . The groupḠ ν is a subgroup ofḠ and acts freely and properly and Q/K. We now show 4.
Lemma 5. If the connection A 1 is chosen such that it is equivariant w.r.t. the action of the full group G, i.e. if then the magnetic Lagrangian system (Q/K ν → Q/K, L 1 , B 1 ) isḠ ν -invariant and δ 1 is a B 1ḡνpotential.
Proof . We first show that L 1 isḠ ν -invariant. For that purpose, we choose an arbitraryḡ ∈Ḡ ν and let g ∈ G ν be a representative. Similar we choose a point (v [q] K , [q] Kν ) ∈ T Q/Kν (Q/K) such that it is the projection of v q ∈ (i • J L ) −1 (ν) ⊂ T Q. By definition of the quotient action on T Q/Kν (Q/K), the action ofḡ on an element (v [q] K , [q] Kν ) equals the projection of v q g. We now check the invariance of L 1 at an arbitrary point in T Q/Kν (Q/K): Next, we check theḠ ν -invariance of B 1 . Recall that B 1 is the projection to Q/K ν of the 2-form d ν, A 1 on Q. We first consider the equivariance of this 2-form under G ν . Let g ∈ G ν be arbitrary, then We thus obtain G ν -invariance for d ν, A 1 , and we may conclude that (Ψ Q/Kν g The third and final step is the definition of the B 1ḡν -potential. We consider an element ξ = [ξ] Kν ∈ḡ ν = g ν /K ν and let ξ ∈ g ν be a representative. Then, by definition of B 1 , the 1-form iξ Q/Kν B 1 is the projection to Q/K ν of the 1-form i ξ Q d( ν, A 1 ) on Q (i.e. ξ Q projects toξ Q/Kν ). Again we concentrate on the 1-form on Q: Since ν, A 1 is G ν -invariant, we conclude that i ξ Q d ν, A 1 = −d i ξ Q ν, A 1 . The exact 1-from on the right gives a strong hint of the structure of theḡ ν -potential. Assume now that we fixed an element ν ∈ g * ν such that ν| Kν = ν| Kν . The function δ on Q, defined by −δ ξ (q) = ν, A 1 (q)(ξ Q (q)) − ν, ξ is our candidate for the B 1ḡν -potential. This statement makes sense provided that δ ξ projects to a function on Q/K ν and that it only depends on the equivalence classξ = ξ + K ν of ξ ∈ g ν . The latter is a straightforward consequence of the fact that A 1 is a principal K-connection. The K ν -invariance is more involved, and we rely on a result in [12]. For any k in K ν , we have Therefore δ ξ is constant on the orbits of K ν in Q if ν, A 1 (q)((ξ − Ad k ξ) Q (q)) vanishes for all k. To show this we introduce a function f on K ν given by f (k) = ν, A 1 (q)((ξ − Ad k ξ) Q (q)) and we use similar arguments as in [12, p. 156]. If we can show that f (e) = 0, df | e = 0 and f (k 1 k 2 ) = f (k 1 ) + f (k 2 ) for arbitrary k 1,2 ∈ K ν , we may conclude that f = 0 (since K ν is assumed connected).
The first condition f (e) = 0 is trivial. To check the second condition: let κ ∈ K ν be arbitrary, then Above, we have used the fact that K ν is normal in G ν and that, as a consequence, the Lie bracket [κ, ξ] is in K ν . Therefore the contraction of the corresponding fundamental vector field with A 1 is precisely [κ, ξ]. Next, we check the third condition and compute f (k 1 k 2 ). Given the identity and the fact that This completes the proof: the g * ν -valued function δ is shown to be projectable to aḡ * ν -valued function on Q/K ν . This is the sought-after potential δ 1 : for arbitrary q ∈ Q, we have Symplectic structure of (Q/K ν → Q/K, L 1 , B 1 ) and symplectic reduction by stages. Lemma 6. Apply symplectic reduction by stages to the symplectic structure associated to the G-invariant Lagrangian system (Q, L). Identify the symplectically reduced manifold M ν with the symplectic structure on T Q/Kν (Q/K) induced by the magnetic Lagrangian system (Q/K ν → Q/K, L 1 , B 1 ). Then: 1. The action Ψ T Q/Kν (Q/K) ofḠ ν on T Q/Kν (Q/K) is precisely the induced action on the first reduced space M ν in symplectic reduction by stages.
Proof . 1. The momentum map for the K action is precisely J K := i * K • J L , with J L : T Q → g * . By definition, the induced action ofḠ ν on J −1 K (ν)/K ν is obtained by projecting the action of G ν on J −1 K (ν). If we take into account that we realize the quotient manifold J −1 K (ν)/K ν as T Q/Kν (Q/K), the induced action on T Q/Kν (Q/K) is obtained by projection of the action of G ν on T Q under the projection T Q → T Q/Kν (Q/K). This is precisely the action we have introduced above.
2. The induced momentum map JḠ ν is defined in the following way (we consider it directly as a function on T Q/Kν (Q/K) instead of on M ν ): where ξ ∈ g ν is arbitrary and projects toξ ∈ḡ ν , v q projects to (v [q] K , [q] Kν ). By definition of the momentum map of the magnetic Lagrangian system (Q/K ν → Q/K, L 1 , B 1 ), we have We now show that the right-hand side of (5) equals the right-hand side of (4). We therefore use the definition of FL 1 and δ 1 as being the projection of maps upstairs: Finally, before we can reapply Routh reduction for the second stage, we need to check that L 1 isḠ ν -regular.
Proof . We have to show that, for any (v [q] K , [q] Kν ) the map . Let η denote an arbitrary element inḡ * ν . Due to the assumed G-hyperregularity, there is a unique ξ ∈ g ν such that k * ν (J L (v q + ξ Q (q))) = (r ′ ν ) * η + ν. The projectionξ = r ′ ν (ξ) of ξ defines the inverse element for η, since it is such that Symplectic and Routh reduction by stages. Summarizing the above lemmas, we conclude that the magnetic Lagrangian system (Q/K ν → Q/K, L 1 , B 1 ) is amenable to Routh reduction and that the symplectic structure and momentum map associated to this Lagrangian system correspond to the symplectic structure and momentum map encountered in symplectic reduction by stages. If ρ is chosen such that the compatibility relation (r ′ ν ) * ρ = µ| gν −ν holds, then from symplectic reduction by stages we have that the symplectic structures associated to (Q/G µ → Q/G, L 0 , B 0 ) and ((Q/K ν )/(Ḡ ν ) ρ → (Q/K)/Ḡ ν , L 2 , B 2 ) are symplectically diffeomorphic by means of the symplectic diffeomorphism F introduced earlier. From Proposition 8 it follows that F * E L 2 = E L 0 and therefore the corresponding Hamiltonian vector fields are F -related. We define a map τ : Q/G µ → (Q/K ν )/(Ḡ ν ) ρ as τ ([q] Gµ ) = [q] Kν (Ḡν )ρ . The map is well-defined since G µ is a subgroup of G ν and since r ν (G µ ) ⊂ (Ḡ ν ) ρ . Lemma 8. The symplectic diffeomorphism F is fibred over τ .

Elroy's Beanie
This system appears in e.g. [13]. It consists of two planar rigid bodies that are connected in their center of mass. The system moves in the plane and it is subject to some conservative force with potential V . The configuration space is SE(2) × S 1 , with coordinates (x, y, θ, ψ). Here (x, y) is the position of the center of mass, θ is the rotation of the first rigid body, and ψ the relative rotation of the second body w.r.t. the first. The kinetic energy of the system is SE(2)-invariant and we will suppose that the potential is invariant as well. This means in fact that only the relative position of the two bodies matters for the dynamics of the system. The Lagrangian is of the form The Euler-Lagrange equations of the system are, in normal form, The symmetry group and the principal connection. An element of SE(2), the special Euclidean group, can be represented by a matrix of the form The identity of the group is (x = 0, y = 0, θ = 0) and the multiplication is given by (x 1 , y 1 , θ 1 ) * (x 2 , y 2 , θ 2 ) = (x 2 cos θ 1 − y 2 sin θ 1 + x 1 , x 2 sin θ 1 + y 2 cos θ 1 + y 1 , θ 1 + θ 2 ).
The matrices and a basis of invariant vector fields iŝ One can easily verify that the Lagrangian is invariant under the SE(2)-action. There is a trivial principal connection on P = Q = SE(2) × S 1 → Q/SE(2) = S 1 , which locally takes the form (dx + ydθ)e 1 + (dy − xdθ)e 2 + dθe 3 .
The momentum map J L is given by: In what follows we perform two Routh reductions on the Lagrangian. The first reduction is done w.r.t. the full symmetry group SE(2), and the second reduction w.r.t. the Abelian normal subgroup R 2 .
Full reduction. Let µ = µ 1 e 1 + µ 2 e 2 + µ 3 e 3 be a generic element in g * . An element ξ = ξ 1 e 1 + ξ 2 e 2 + ξ 3 e 3 of the isotropy algebra g µ satisfies So if we suppose that µ 1 and µ 2 do not both vanish -we will set µ 1 = 1 from now on -then a typical element of g µ is ξ = ξ 1 (e 1 + µ 2 e 2 ). Since g µ is 1-dimensional, G µ is of course Abelian. A convenient way to describe the manifold P/G µ = SE(2)/G µ locally is by considering a coordinate transformation (x ′ , y ′ , θ ′ ) in the group coordinates such that the vector field associated to an element in g µ becomes a coordinate vector field: in the new coordinates, we should have ∂ x ′ = ∂ x + µ 2 ∂ y . This is obtained by the following transformation Then clearly (y ′ , θ) is a coordinate chart on SE(2)/G µ . And simultaneously, we have that (y ′ , θ, ψ) is a coordinate chart on the reduced configuration manifold P/G µ = (SE(2) × S 1 )/G µ , and the fibration P/G µ → Q/G = S 1 is locally represented by (y ′ , θ, ψ) → (ψ). For a more systematic treatment on appropriate coordinate changes, we refer to [6]. We now compute the Routhian L 0 and the 2-form B 0 . Following [19], a convenient way to compute the (unreduced) Routhian for mechanical Lagrangians is by using 2 where p i is the momentum in the ith coordinate. We have: In the last step we have left out some constant terms. The reduced Lagrangian is then obtained by taking the quotient w.r.t. the action of G µ . This is done by applying the coordinate transformation introduced above. We get: which is clearly independent of x ′ . The 2-form B 0 is obtained by reducing the 2-form Using the coordinate change we get B 0 = dy ′ ∧ dθ.
In this example the Routhian L 0 depends on the velocity corresponding to the coordinate ψ on S 1 , but is independent of the velocities corresponding to the two remaining coordinates (y ′ , θ). With the above, the reduced Euler-Lagrange equations take the forṁ y ′ = 0, Note that the second order equation in ψ decouples from the first order equations, and that these two first order equations are the momentum equations rewritten in normal form. Abelian reduction. We now perform first Routh reduction w.r.t. the Abelian symmetry group R 2 of translations in the x and y direction. Let us denote the symmetry group by K = R 2 and study the quotient spaces. We will use the same notations as before: the Lie algebra elements e 1 , e 2 denote a basis for the subalgebra K of K in g. The momentum map for this action is now J K = i * • J L = mẋe 1 + mẏe 2 . We choose ν to be the projection of the momentum µ we had used in the full reduction: let ν = e 1 + µ 2 e 2 ∈ g * . Since K is Abelian, K ν = K and the quotient space is SE(2) × S 1 /R 2 = S 1 × S 1 . If we choose A 1 = dxe 1 + dye 2 to be the trivial connection, we simply get B 1 = 0. The Routhian L 1 can now be obtained from where we ignored again some constant terms. The Routh reduced system is now a standard Lagrangian system on S 1 × S 1 with Lagrangian L 1 = 1 2 I 1θ 2 + 1 2 I 2 (θ +ψ) 2 − V (ψ) (see the paragraphs on Abelian Routh reduction). Its equations of motion arë For this example there is actually no second stage: the groupḠ ν = G ν /K ν is the trivial one {e}, and the vector spaceḡ ν = g ν /K ν is only the zero vector. So, there is no second momentum map to take into account, and there is no further symmetry to quotient out.
In the reduction by stages process we have not made use of µ 3 . We now show that the two ways of reducing the system are equivalent.
This diffeomorphism maps the G µ -reduced system on the K ν -reduced system, as is obvious from the respective equations of motion.

Rigid bodies on the Heisenberg group
As a second example of Routh reduction by stages, we discuss the dynamics of a rigid body immersed in a potential flow with circulation [20]. We assume that the body is circular, and in this case the equations of motion are given by d dt Here Γ represents the circulation and M is a (non-diagonal) mass matrix, which incorporates the inertia and added masses of the body. The right-hand side of the equations of motion represents the so-called Kutta-Joukowski lift force, a gyroscopic force due to circulation [9,17].
While this system is extremely easy to integrate, it nevertheless exhibits all the interesting geometric characteristics of more complicated examples. As we show below, the configuration space for this system is the Heisenberg group, arguably the simplest non-trivial central extension group, and the procedure of reduction by stages demonstrated here can be applied equally well to more complicated central extensions, such as the oscillator group (describing the dynamics of rigid bodies of arbitrary cross section in circulatory flow) and the Bott-Virasoro group describing the KdV equation.
In this context, the Heisenberg group H is the Euclidian space R 3 , equipped with the multiplication (x, y, s) · (x ′ , y ′ , s ′ ) = x + x ′ , y + y ′ , s + s ′ + 1 2 (xy ′ − yx ′ ) , using the definition of H given in [18]. The coordinates (x, y) describe the center of the circular disc, while the coordinate s corresponds to the flux of the fluid around the body. The conjugate momentum corresponding to s will be the circulation of the fluid. The Heisenberg group can alternatively be described as the central extension of R 2 constructed by means of the cocycle B((x, y), (x ′ , y ′ )) = 1 2 (xy ′ − yx ′ ). We now introduce the infinitesimal cocycle C : R 2 × R 2 → R, given by For the relation between B and C, see for instance [12]. The Lie algebra h of H can then be identified with R 3 with the bracket The dual Lie algebra h * can again be identified with R 3 , with coordinates (p x , p y , p) and duality pairing (p x , p y , p), (v x , v y , v) = p x v x + p y v y + pv. We now induce the following quadratic Lagrangian ℓ on h: and we define L on H by left extension: L(g,ġ) = ℓ(g −1ġ ), or explicitly L(x, y, s;ẋ,ẏ,ṡ) = 1 2 [ẋ,ẏ] M ẋ y The Euler-Poincaré equations obtained from ℓ are given by After setting p = Γ, we obtain the equations (7). These equations also coincide with the Euler-Lagrange equations obtained from the Lagrangian (8).
Reduction with respect to R. The center of H is the normal subgroup isomorphic to R which consists of all elements of the form (0, 0, s), where s ∈ R. We first perform Routh reduction with respect to the left action of this subgroup on H. On the principal bundle H → H/R ∼ = R 2 we consider the connection one-form given at the identity by A(e)(v x , v y , v) = v, and extended to the whole of H by left translation. Explicitly, we have A(x, y, s) = ds − 1 2 (xdy − ydx).
Since the structure group R is Abelian, the curvature of A is given by B = dA = −dx ∧ dy.
Similarly, the momentum map J L : T H → R for the R-action on the tangent bundle T H is given by J L (x, y, z;ẋ,ẏ,ż) =ṡ − (xẏ − yẋ)/2, so that J −1 L consists of all points (x, y, s;ẋ,ẏ,ṡ) withṡ = Γ + (xẏ − yẋ)/2. The isotropy subgroup R Γ = R acts on this level set by translations in the s-direction, so that the reduced velocity space is given by The symplectic form on the reduced space can easily be computed, and is explicitly given by Adẋ ∧ dx + B(dẋ ∧ dy + dẏ ∧ dx) + Cdẏ ∧ dy − Γdx ∧ dy, where A, B, C are the entries of the mass matrix M. The last term of the symplectic form, −Γdx ∧ dy, is the curvature term of the connection, paired with Γ ∈ R. Finally, a quick computation shows that the reduced Lagrangian is just the kinetic energy Lagrangian on T R 2 : up to constant terms. Second reduction. We now perform reduction with respect to the remaining symmetry group, H/R ∼ = R 2 , using the results from Section 5.3. We have a left invariant magnetic Lagrangian system on the group R 2 , with Lagrangian (9) and magnetic form B Γ = −Γdx ∧ dy. The potential δ : R 2 → R 2 corresponding to the latter is given by δ(x, y) = Γ −y x , and the momentum map is therefore J 2 (x, y,ẋ,ẏ) = M(ẋ,ẏ) T − δ(x, y). The non-equivariance 2-cocycle of the momentum map is Σ δ = B Γ . The affine action of R 2 on itself is given by (x, y) · (p x , p y ) = (p x − Γy, p y + Γx). If we fix a momentum value (λ, µ) ∈ R 2 , the isotropy group R 2 (λ,µ) of the affine action consists of just the zero element, and consequentially the twice-reduced space J −1 2 (λ, µ)/R 2 (λ,µ) is nothing but R 2 .
The reduced Euler-Lagrange equations (3) in the case of a left action, assume the following form d dt with (p x , p y ) T = M(v x , v y ) T , and these are nothing but the equations (7).