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...

Повний опис

Збережено в:

Бібліографічні деталі
Дата:	2011
Автори:	Langerock, B., Mestdag, T., Vankerschaver, J.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2011
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/148086
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Routh Reduction by Stages / B. Langerock, T. Mestdag, J. Vankerschaver // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 20 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-148086
record_format	dspace
spelling	irk-123456789-1480862019-02-17T01:27:41Z Routh Reduction by Stages Langerock, B. Mestdag, T. Vankerschaver, J. 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. 2011 Article Routh Reduction by Stages / B. Langerock, T. Mestdag, J. Vankerschaver // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 20 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37J05; 37J15; 52D20 DOI: http://dx.doi.org/10.3842/SIGMA.2011.109 http://dspace.nbuv.gov.ua/handle/123456789/148086 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	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.
format	Article
author	Langerock, B. Mestdag, T. Vankerschaver, J.
spellingShingle	Langerock, B. Mestdag, T. Vankerschaver, J. Routh Reduction by Stages Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Langerock, B. Mestdag, T. Vankerschaver, J.
author_sort	Langerock, B.
title	Routh Reduction by Stages
title_short	Routh Reduction by Stages
title_full	Routh Reduction by Stages
title_fullStr	Routh Reduction by Stages
title_full_unstemmed	Routh Reduction by Stages
title_sort	routh reduction by stages
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/148086
citation_txt	Routh Reduction by Stages / B. Langerock, T. Mestdag, J. Vankerschaver // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 20 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT langerockb routhreductionbystages AT mestdagt routhreductionbystages AT vankerschaverj routhreductionbystages
first_indexed	2025-07-12T18:12:04Z
last_indexed	2025-07-12T18:12:04Z
_version_	1837465787961442304
fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 109, 31 pages Routh Reduction by Stages Bavo LANGEROCK †‡§, Tom MESTDAG † and Joris VANKERSCHAVER †? † Department of Mathematics, Ghent University, Krijgslaan 281, S22, B9000 Ghent, Belgium E-mail: bavo.langerock@ugent.be, tom.mestdag@ugent.be, joris.vankerschaver@ugent.be ‡ Belgian Institute for Space Aeronomy, Ringlaan 3, B1180 Brussels, Belgium § Department of Mathematics, K.U. Leuven, Celestijnenlaan 200 B, B3001 Leuven, Belgium ? Department of Mathematics, University of California at San Diego, 9500 Gilman Drive, San Diego CA 92093-0112, USA Received June 16, 2011, in final form November 22, 2011; Published online November 29, 2011 http://dx.doi.org/10.3842/SIGMA.2011.109 Abstract. This paper deals with the Lagrangian analogue of symplectic or point reduc- tion 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 mag- netic 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. Key words: symplectic reduction; Routh reduction; Lagrangian reduction; reduction by stages 2010 Mathematics Subject Classification: 37J05; 37J15; 52D20 Contents 1 Introduction 2 2 Preliminaries on Routh reduction 2 3 Magnetic Lagrangian systems 6 4 Magnetic Lagrangian systems with symmetry 9 4.1 Symplectic reduction: a brief introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2 Invariant magnetic Lagrangian systems and momentum maps . . . . . . . . . . . . . . . . 9 5 Routh reduction for magnetic Lagrangian systems 11 5.1 Magnetic cotangent bundle reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2 Routh reduction for magnetic Lagrangian systems . . . . . . . . . . . . . . . . . . . . . . 14 5.3 Reduction of magnetic Lagrangian systems on Lie groups . . . . . . . . . . . . . . . . . . 18 6 Routh reduction by stages 19 6.1 Symplectic reduction by stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.2 Routh reduction by stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7 Examples 25 7.1 Elroy’s Beanie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7.2 Rigid bodies on the Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 References 30 mailto:bavo.langerock@ugent.be mailto:tom.mestdag@ugent.be mailto:joris.vankerschaver@ugent.be http://dx.doi.org/10.3842/SIGMA.2011.109 2 B. Langerock, T. Mestdag and J. Vankerschaver 1 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 re- duction 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 ‘La- grangian 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 con- nection 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. 2 Preliminaries on Routh reduction Definition 1. A Lagrangian system is a pair (Q,L) with Q a manifold and L a function on TQ. 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 Routh Reduction by Stages 3 of mechanical type if for arbitrary vq ∈ TqQ, the Lagrangian can be written as L(vq) = 1 2 〈〈vq, vq〉〉Q − V (q), with 〈〈·, ·〉〉Q a Riemannian metric on Q and V a function on Q, called the potential energy of the Lagrangian. The function 1 2 〈〈vq, vq〉〉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), q̇(t))dt. In a local coordinate chart (qi) on Q the Lagrangian L is a function of (qi, vi) and the Euler–Lagrange equations are d dt ( ∂L ∂vi ) − ∂L ∂qi = 0, q̇i = vi, i = 1, . . . , n = dimQ. 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∗. Definition 2. 1. A right action of a group G on a manifold M is denoted by ΨM : G×M →M ; (g,m) 7→ ΨM g (m) = mg. Throughout this paper we only consider free and proper actions of Lie groups on manifolds. 2. The infinitesimal action is given by ψMm : g→ TmM ; ξ 7→ d/dε\|0(m exp εξ). 3. For any element ξ ∈ g one can consider the fundamental vector field ξM , defined pointwise as ξM (m) = ψMm (ξ). 4. The map defined pointwise as the dual to ψMm is denoted by (ψM )∗ : T ∗M → g∗. 5. The lifted action ΨTM on TM of ΨM is given by ΨTM : G × TM → TM ; (g, vm) 7→ TΨM g (vm). 6. The lifted action ΨT ∗M on T ∗M is given by ΨT ∗M : G × T ∗M → T ∗M ; (g, αm) 7→ T ∗ΨM g−1(αm). With these notations, ξTM denotes a fundamental vector field on TM determined by the lifted action. It follows that ξTM is the complete lift of ξM . Every right action ΨM gives rise to a left action ΦM : ΦM g (m) = ΨM 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 Tπ is of the form ξM (m) for some ξ ∈ g. These vectors are called vertical and form a distribution which we call the vertical distribution V π = kerTπ. Definition 3. A principal connection on a principal G-bundle is a g-valued 1-form A satisfying two conditions: 4 B. Langerock, T. Mestdag and J. Vankerschaver 1) it is equivariant, i.e. Ψ∗gA = Adg−1 · A for any g ∈ G, and 2) for ξ ∈ g arbitrary, A(ξM ) = 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 HA ⊂ TM . On the other hand, any right invariant distribution H satisfying H ⊕ V π = TQ 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 vm 7→ 〈f(m),A(m)(vm)〉 ∈ R. The cotangent vector 〈f,A〉(m) ∈ T ∗mM is often denoted by 〈f(m),A(m)〉. In particular, if f = µ ∈ g∗ is constant, then 〈µ,A〉 is a 1-form on M . Routh reduction. Definition 4. Let (Q,L) denote a Lagrangian system and assume that the configuration space is equipped with an action ΨQ. The Lagrangian system (Q,L) is G-invariant if L is invariant under the lifted action ΨTQ, i.e. L(ΨTQ g (vq)) = L(vq) for arbitrary vq ∈ TQ and g ∈ G. 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 ΨQ g , for arbitrary g. Definition 5. 1. For a Lagrangian system (Q,L), the Legendre transform FL : TQ → T ∗Q is the fibre derivative of L, i.e. for arbitrary vq, wq ∈ TQ 〈FL(vq), wq〉 = d du ∣∣∣∣ u=0 L(vq + uwq). The Lagrangian is hyperregular if FL is a diffeomorphism. 2. For a G-invariant Lagrangian system (Q,L), the momentum map JL : TQ → g∗ is the map (ψQ)∗ ◦ FL, i.e. for arbitrary vq ∈ TQ and ξ ∈ g 〈JL(vq), ξ〉 = 〈FL(vq), ξQ(q)〉. 3. Given a G-invariant mechanical Lagrangian system (Q,L), and a point q in Q. The inertia tensor Iq is a metric on g defined by Iq(ξ, η) = 〈〈ξQ(q), ηQ(q)〉〉Q. The momentum map associated to a G-invariant Lagrangian system satisfies the following two important properties: 1. It is conserved along the solutions of the Euler–Lagrange equations, i.e. if q(t) is a solution to the Euler–Lagrange equations, then d dt (JL(q̇(t))) = 0. 2. It is equivariant w.r.t. the action ΨTQ and the coadjoint action Ad∗ on g∗, i.e. JL(ΨTQ g (vq)) = Ad∗g(JL(vq)) for arbitrary g ∈ G and vq ∈ TQ. Definition 6. Given a G-invariant Lagrangian system (Q,L) and an arbitrary vector vq ∈ TQ. 1. The map JL\|vq : g→ g∗ is the map ξ 7→ JL(vq + ξQ(q)). 2. The Lagrangian system (Q,L) is G-regular if JL\|vq is a diffeomorphism for every vq ∈ TQ. Routh Reduction by Stages 5 Every G-invariant mechanical Lagrangian system is G-regular. To show this, remark that for a mechanical Lagrangian system 〈JL\|vq(ξ), η〉 = 〈〈vq, ηQ(q)〉〉Q + 〈〈ξQ(q), ηQ(q)〉〉Q . Hence JL\|vq(ξ) = JL\|vq(0)+Iq(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 re- gular value µ ∈ g∗ of the momentum map JL. Let Gµ denote the isotropy subgroup of µ w.r.t. the coadjoint action of G on g∗, i.e. g ∈ Gµ iff Ad∗gµ = µ. 1. The submanifold iµ : J−1 L (µ) → TQ is Gµ-invariant and the restricted action of Gµ on J−1 L (µ) is free and proper. 2. The quotient manifold J−1 L (µ)/Gµ is diffeomorphic to the fibred product T (Q/G) ×Q/G Q/Gµ over Q/G. The first statement is a direct consequence of the equivariance of JL. 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 Gν for every ν ∈ g∗ is the entire group G, and in particular T (Q/G)×Q/G Q/Gν 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 JL 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 L̃ 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, L̃) is the pro- jection of a solution of the Euler–Lagrange equations of the original system (Q,L) with mo- mentum µ. Conversely, every solution of the Euler–Lagrange equations of the Lagrangian sys- tem (Q,L) with momentum µ projects to a solution of the Euler–Lagrange equations of the system (Q/G, L̃). 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 2m(ṙ2 + r2θ̇2)− 1 2kr 2. This system is clearly invariant under translations in the θ-direction, given by Ψa(r, θ) = (r, θ + a). The momentum map for this action is JL = mr2θ̇ and the Euler–Lagrange equations are mr̈ −mrθ̇2 + kr = 0, d dt (mr2θ̇) = 0. Let us fix a regular value 0 6= µ = mr2θ̇ for the momentum map and let A = dθ be the standard connection with vanishing curvature. The Routhian is obtained from L̃(r, ṙ) = (L− µθ̇)\|mr2θ̇=µ = 1 2 mṙ2 − 1 2 kr2 − 1 2 µ2 mr2 . 6 B. Langerock, T. Mestdag and J. Vankerschaver The Euler–Lagrange equation for L̃ is mr̈ = −kr + µ2 mr3 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)/Gµ no longer has the structure of a tangent bundle; instead it is diffeomorphic to T (Q/G)×Q/G Q/Gµ. 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/Gµ. This is the fibred product of a bundle Q/Gµ → 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]. 3 Magnetic Lagrangian systems A magnetic Lagrangian system is a Lagrangian system with configuration space the total space of a bundle ε : P → Q and where the Lagrangian is independent of the velocities tangent to the fibres of ε. 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/Gµ and the fibration ε is given by the projection Q/Gµ → Q/G. Definition 7. A magnetic Lagrangian system is a triple (ε : P → Q,L,B) where ε : P → Q is a fibre bundle, L is a smooth function on the fibred product TQ×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 (qi, pa), i = 1, . . . , n = dimQ, a = 1, . . . , k = dimP − dimQ, adapted to the fibration ε : P → Q determines a coordinate chart (qi, vi, pa) on TQ ×Q P , and the Lagrangian L is then a function depending on (qi, vi, pa). By definition, L is independent of the velocities in the fibre coordinates pa and therefore it determines a singular Lagrangian when interpreted as a function on TP . Locally, the Euler–Lagrange equations for this singular Lagrangian are d dt ( ∂L ∂vi ) − ∂L ∂qi = Bij q̇j + Biaṗa, i = 1, . . . , n, − ∂L ∂pa = −Biaq̇i + Babṗb, a = 1, . . . , k. Here we used the following coordinate expression of the 2-form B is 1 2Bijdq i ∧ dqj + Biadqi ∧ dpa + 1 2Babdp a ∧ dpb. These Euler–Lagrange equations have a geometric interpretation (see Proposition 2). First we introduce additional notations. Routh Reduction by Stages 7 Definition 8. Assume a magnetic Lagrangian system (ε : P → Q,L,B) is given. 1. TPQ denotes the fibred product TQ ×Q P and (vq, p), where vq ∈ TQ and p ∈ P such that ε(p) = q, is a point in TPQ. Similarly, T ∗PQ denotes the fibred product T ∗Q ×Q P and (αq, p), with ε(p) = q, is an arbitrary element in T ∗PQ. 2. V ε denotes the distribution on P of tangent vectors vertical to ε. 3. ε̂ : TP → TPQ is the projection fibred over P that maps vp ∈ TP onto (Tε(vp), p) ∈ TPQ. 4. ρ1 : TPQ→ TQ is the projection that maps (vq, p) ∈ TPQ onto vq ∈ TQ. 5. ρ2 : TPQ→ P is the projection that maps (vq, p) ∈ TPQ onto p ∈ P . 6. ε1 : T ∗PQ→ T ∗Q is the projection that maps (αq, p) ∈ T ∗PQ onto αq ∈ T ∗Q. 7. ε2 : T ∗PQ→ P is the projection that maps (αq, p) ∈ T ∗PQ onto p ∈ P . 8. The Legendre transform FL : TPQ → T ∗PQ maps (vq, p) ∈ TPQ to (αq, p) ∈ T ∗PQ where αq ∈ T ∗qQ is determined from 〈αq, wq〉 = d du ∣∣∣∣ u=0 L(vq + uwq, p), for arbitrary wq ∈ TqQ. 9. The energy EL is a function on TPQ defined by EL(vq, p) = 〈FL(vq, p), (vq, p)〉 − L(vq, p). (Here the contraction of an element (αq, p) ∈ T ∗PQ with (vq, p) ∈ TPQ is defined naturally as 〈(αq, p), (vq, p)〉 = 〈αq, vq〉.) 10. By means of the Legendre transform we can pull-back the presymplectic 2-form ε∗1ωQ+ε∗2B on T ∗PQ to a presymplectic 2-form FL∗(ε∗1ωQ + ε∗2B) on TPQ. The latter is denoted by ΩL,B (Here ωQ = dθQ, with θQ 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 TPQ equal to (q̇(t), p(t)) ∈ TPQ with q(t) = ε(p(t)). The curve p(t) in P is a solution to the Euler–Lagrange equations for the magnetic Lagrangian system (ε : P → Q,L,B) iff γ(t) is a solution to the presymplectic equation iγ̇(t)Ω L,B(γ(t)) = −dEL(γ(t)). Locally, the presymplectic equation coincides with the previously mentioned Euler–Lagrange equations, since ΩL,B = d ( ∂L ∂vi ) ∧ dqi + 1 2 Bijdqi ∧ dqj + Biadqi ∧ dpa + 1 2 Babdpa ∧ dpb, dEL = vid ( ∂L ∂vi ) + ∂L ∂vi dvi − dL = vid ( ∂L ∂vi ) − ∂L ∂qi dqi − ∂L ∂pa dpa. Definition 9. A magnetic Lagrangian system (ε : P → Q,L,B) 1) is hyperregular if FL is a diffeomorphism and if the restriction of B to V ε is nondegenerate, 2) is of mechanical type if L(vq, p) = 1 2 〈〈(vq, p), (vq, p)〉〉ρ1 − V (p) with 〈〈·, ·〉〉ρ1 is a metric on the vector bundle ρ1 : TPQ→ P and V is a function on P . Note that if B is nondegenerate, then the typical fibre of P necessarily has to be even- dimensional. In a local coordinate chart the nondegeneracy condition on B is expressed by detBab 6= 0. 8 B. Langerock, T. Mestdag and J. Vankerschaver Proposition 3. If a magnetic Lagrangian system (ε : P → Q,L,B) is hyperregular, the 2-form ΩL,B = FL∗(ε∗1ωQ + ε∗2B) determines a symplectic structure on TPQ. Proof. Assume that the magnetic Lagrangian system is hyperregular. The nondegeneracy of ΩL,B is easily checked if we work in a coordinate chart adapted to the fibration: ΩL,B = d ( ∂L ∂vi ) ∧ dqi + 1 2 Bijdqi ∧ dqj + Biadqi ∧ dpa + 1 2 Babdpa ∧ dpb. Since FL is a diffeomorphism, the 1-forms d ( ∂L ∂vi ) , dqi and dpa provide pointwise a basis on T ∗(TPQ). 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 TP . The energy EL 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 ε, i.e. a section of ε2 : T ∗PQ → P , or in other words a linear function on TPQ given by (vq, p) 7→ 〈α(p), (vq, p)〉. When α : P → T ∗PQ is composed with T ∗ε ◦ ε1 : T ∗PQ → T ∗P , it determines a 1-form on P which, with a slight abuse of notation, we denote by ε∗α. Now consider the function L′(vq, p) = L(vq, p) − 〈α(p), (vq, p)〉 and the gyroscopic 2-form B′ = B + dε∗α. Together they define a new magnetic Lagrangian system (ε : P → Q,L′,B′) which is equivalent to (ε : P → Q,L,B). Lemma 1. The magnetic Lagrangian systems (ε, L′,B′) and (ε, L,B) are equivalent. 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 (ε, L′,B′) is a solution to the Euler–Lagrange equations of (ε, L,B). We work in a local coordinate neighborhood (qi, pa) as before. We fix a coordinate expression for α = αi(q j , pa)dqi. The Euler–Lagrange equations for (ε, L′,B′) equal, with q̇i = vi and ṗa = va d dt ( ∂L′ ∂vi ) − ∂L′ ∂qi = B′ijvj + B′iava, i = 1, . . . , n, −∂L ′ ∂pa = −B′iavi + B′abvb, a = 1, . . . , k. It now remains to substitute the definition of L′ and B′, i.e. L′ = L− αivi and B′ij = Bij + ( ∂αj ∂qi − ∂αi ∂qj ) , B′ia = Bia − ∂αi ∂pa , B′ab = Bab. � Routh Reduction by Stages 9 4 Magnetic Lagrangian systems with symmetry 4.1 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, ΨM : M ×G→M . Given a function f : M → g∗, then fξ for ξ ∈ g denotes the real valued function on M defined by fξ(m) = 〈f(m), ξ〉. Definition 10. 1. The action ΨM is said to be canonical if (ΨM g )∗ω = ω for all g ∈ G. 2. A map J : M → g∗ is a momentum map if iξMω = −dJξ, for ξ ∈ g arbitrary. 3. If M is connected, the non-equivariance 1-cocycle σ of the momentum map J equals σ : G→ g∗ : g 7→ J ( mg−1 ) −Ad∗g−1(J(m)), 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, σ(gh) = σ(g) + Ad∗g−1σ(h). 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) 7→ Ad∗gµ+ 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: J(mg) = Ad∗gJ(m) + σ(g−1). For a fixed element µ ∈ g∗, the Lie group Gµ < G denotes the isotropy subgroup of µ w.r.t. the affine action, i.e. g ∈ Gµ if µ = Ad∗gµ+ σ(g−1) or equivalently if µ− J(mg) = Ad∗g(µ− J(m)). 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 Gµ the isotropy group of µ under the affine action of G on g∗. Then the pair (Mµ, ωµ) is a symplectic manifold, with Mµ = J−1(µ)/Gµ and with ωµ a 2-form on Mµ uniquely determined from i∗µω = π∗µωµ, with iµ : J−1(µ)→M and πµ : J−1(µ)→Mµ = J−1(µ)/Gµ. Any Hamiltonian h on M which is invariant under the action of G induces a function hµ on Mµ satisfying π∗µhµ = i∗µh. The Hamiltonian vector field Xh is tangent to J−1(µ) and the corresponding vector field on J−1(µ) is πµ-related to the Hamiltonian vector field Xhµ on Mµ. 4.2 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 ε: for arbitrary g ∈ G, ε ◦ΨP g = ΨQ g ◦ ε. (1) In other words, G acts on ε : P → Q by bundle automorphisms. The projections of the principal bundles are denoted by πQ : Q → Q/G and πP : P → P/G. These actions induce lifted right actions on TPQ and T ∗PQ: 10 B. Langerock, T. Mestdag and J. Vankerschaver 1) ΨTPQ g (vq, p) := (TΨQ g (vq),Ψ P g (p)) for (vq, p) ∈ TPQ and g ∈ G, 2) Ψ T ∗PQ g (αq, p) := (T ∗ΨQ g−1(αq),Ψ P g (p)) for (αq, p) ∈ T ∗PQ and g ∈ G. Definition 12. A magnetic Lagrangian system (ε : P → Q,L,B) is G-invariant if 1) B is invariant under the action of ΨP : (ΨP g )∗B = B for all g ∈ G, 2) the Lagrangian L is invariant under the lifted action ΨTPQ on TPQ, i.e. L ◦ΨTPQ g = L for all g ∈ G. Let (ε : P → Q,L,B) be a G-invariant hyperregular magnetic Lagrangian system and consider the symplectic 2-form ε∗1ωQ + ε∗2B on T ∗PQ. Proposition 4. The action ΨT ∗PQ is canonical and the Legendre transform FL : TPQ → T ∗PQ is equivariant, i.e. FL ◦ ΨTPQ g = Ψ T ∗PQ g ◦ FL. As a consequence the action ΨTPQ is canonical w.r.t. ΩL,B. Proof. Note that ε1 ◦Ψ T ∗PQ g = T ∗ΨQ g−1 ◦ ε1 and ε2 ◦Ψ T ∗PQ g = ΨP g ◦ ε2. Since the action ΨT ∗Q g = T ∗ΨQ g−1 on T ∗Q is canonical w.r.t. ωQ and since B is a G-invariant 2-form on P , we conclude that ΨT ∗PQ is canonical. To show the equivariance of FL we use its definition: let g ∈ G, vq, wq ∈ TQ be arbitrary, then 〈FL ( ΨTPQ g (vq, p) ) ,ΨTPQ g (wq, p)〉 = d du ∣∣∣∣ u=0 L(ΨTPQ g (vq + uwq)) = d du ∣∣∣∣ u=0 L(vq + uwq) = 〈FL(vq, p), (wq, p)〉. � In order to define a momentum map, we introduce a concept that is similar to the so-called Bg-potential in [12]. Definition 13. A g∗-valued function δ on P is a Bg-potential for the magnetic 2-form B if iξPB = dδξ for any ξ ∈ g. Recall that δξ is defined pointwise as δξ(p) = 〈δ(p), ξ〉. If B is G-invariant and admits a Bg-potential δ, then it follows that d((ΨP g )∗δξ − δAdgξ) = 0. If P is connected, this induces a g∗-valued 1-cocycle on G: σδ : G→ g∗; g 7→ (ΨP g−1)∗δ(p)−Ad∗g−1 · δ(p), with p arbitrary in P . The infinitesimal version of this 1-cocycle is denoted by Σδ(ξ, η) = −〈Teσδ(ξ), η〉. It is a 2-cocycle on the Lie algebra and satisfies Σδ(ξ, η) = −B(ξP , ηP )− δ[ξ,η] = ξP (δη)− δ[ξ,η]. For the following proposition, recall from Definition 2 that (ψQ)∗ : T ∗Q→ g∗ is the dual of the infinitesimal action of G on Q. Proposition 5. The map Jδ = (ψQ)∗◦ε1−δ◦ε2 is a momentum map for the symplectic manifold (T ∗PQ, ε ∗ 1ωQ + ε∗2B) with non-equivariance 1-cocycle −σδ. Due to the equivariance of FL, the map JL,δ = Jδ ◦ FL is a momentum map for the symplectic manifold (TPQ,Ω L,B) with non- equivariance 1-cocycle −σδ. Proof. The map (ψQ)∗ : T ∗Q → g∗ is an equivariant momentum map for the symplectic manifold (T ∗Q,ωQ). It is straightforward that the combined map Jδ = (ψQ)∗ ◦ ε1 − δ ◦ ε2 is a momentum map for the lifted action on T ∗PQ w.r.t. the symplectic form ε∗1ωQ + ε∗2B with non-equivariance cocycle −σδ. The rest of the statement follows by construction. � We conclude that the symplectic structures (TPQ,Ω L,B) and (T ∗PQ, ε ∗ 1ωQ + ε∗2B) associated to a G-invariant and hyperregular magnetic Lagrangian system with a Bg-potential δ admit a momentum map with cocycle −σδ and are amenable to symplectic reduction. Routh Reduction by Stages 11 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 ∗PQ, ε ∗ 1ωQ+ ε∗2B) 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 Jδ is a momentum map with non-equivariance cocycle −σδ. As usual, Gµ denotes the isotropy subgroup of µ ∈ g∗ for the affine action. Proposition 6. Fix a connection AQ on the bundle πQ : Q → Q/G and a regular value µ of the momentum map Jδ. 1. g ∈ Gµ iff µ+ δ(pg) = Ad∗g(µ+ δ(p)). 2. AP = ε∗AQ is a principal connection on πP : P → P/G. 3. The 1-form 〈µ+ δ,AP 〉 on P is Gµ-invariant. 4. The 2-form B + d(〈µ+ δ,AP 〉) is Gµ-invariant and reducible to a 2-form on P/Gµ. 5. The quotient manifold J−1 δ (µ)/Gµ is diffeomorphic to T ∗P/Gµ(Q/G). Proof. Recall the pointwise definition of 〈µ+δ,AP 〉: given vp ∈ TP , then 〈µ+δ,AP 〉(p)(vp) := 〈µ + δ(p),AP (p)(vp)〉. Below, 〈µ + δ(p),AP (p)〉 ∈ T ∗pP denotes the cotangent vector 〈µ + δ,AP 〉(p). We continue with the proof. 1. This is straightforward from the definition of σδ. 2. The g∗-valued 1-form AP determines a principal connection if it satisfies AP (ξP ) = ξ for all ξ ∈ g, and if (ΨP g )∗AP = Adg−1 · AP for all g ∈ G: AP (ξP )(p) = AQ(ε(p))(Tε(ξP (p))) = AQ(ε(p))(ξQ(ε(p))) = ξ and (ΨP g )∗AP = (ΨP g )∗ε∗AQ = ε∗(ΨQ g )∗AQ = Adg−1 · ε∗AQ = Adg−1 · AP . 3. The pull-back 1-form (ΨP g )∗〈µ+ δ,AP 〉 equals 〈µ+ δ(pg), ((ΨP g )∗AP )(p)〉 = 〈Ad∗g(µ+ δ(p)),Adg−1 · AP (p)〉. 4. That B + d(〈µ + δ,AP 〉) is Gµ-invariant is a straightforward consequence of 3. It is projectable to P/Gµ since the contraction with any fundamental vector field ξP with ξ ∈ gµ vanishes: iξP ( d(〈µ+ δ,AP 〉) + B ) = LξP (〈µ+ δ,AP 〉)− d(〈µ+ δ, ξ〉) + dδξ = 〈ξP (µ+ δ),AP 〉+ 〈µ+ δ, ad−ξ · AP 〉 − dδξ + dδξ = 〈ξP (δ)− ad∗ξµ− ad∗ξδ,AP 〉 = 0, where we used the fact that ξ ∈ gµ, or equivalently ad∗ξµ = iξΣδ. 5. Similar to cotangent bundle reduction, we use a ‘shift map’ to construct the required diffeomorphism. Recall that V πQ denotes the bundle of tangent vectors vertical to πQ : Q → Q/G. The subbundle V 0πQ of T ∗Q is defined as the annihilator of V πQ. Below we introduce a shift map φAµ between J−1 δ (µ) and V 0 Pπ Q = V 0πQ×QP and we show that it is equivariant w.r.t. to actions of Gµ obtained by restriction of ΨTPQ to J−1 δ (µ) and V 0 Pπ Q. This is sufficient for φAµ to project to a diffeomorphism [φAµ ] between the quotient spaces. This provides us with the desired diffeomorphism because the quotient of V 0 Pπ Q is well known: V 0 Pπ Q/Gµ = T ∗P/Gµ(Q/G). 12 B. Langerock, T. Mestdag and J. Vankerschaver Definition of the shift map. In the following we use a slight abuse of notations: if we write 〈µ + δ(p),AQ(q)〉 for some p ∈ P and q = ε(p), then this is the cotangent vector in T ∗qQ determined by vq 7→ 〈µ+ δ(p),AQ(q)(vq)〉. Let φAµ be the map φAµ : J−1 δ (µ)→ V 0 PπQ; (αq, p) 7→ ( αq − 〈µ+ δ(p),AQ(q)〉, p ) . It is well defined, i.e. φAµ (αq, p) ∈ V 0 Pπ Q for any (αq, p) in the level set J−1 δ (µ) since 〈φAµ (αq, p), (ξQ(q), p)〉 = 〈αq, ξQ(q)〉 − 〈µ+ δ(p),AQ(q)(ξQ(q))〉 = 〈Jδ(αq, p), ξ〉 − 〈µ, ξ〉 = 0. Equivariance of the shift map. For arbitrary g ∈ Gµ, φAµ ( Ψ T ∗PQ g (αq, p) ) = ( T ∗ΨQ g−1(αq)− 〈µ+ δ(pg),AQ(qg)〉, pg ) = ( T ∗ΨQ g−1 ( αq − 〈µ+ δ(pg),Adg−1 · AQ(q)〉 ) , pg ) = ( T ∗ΨQ g−1 ( αq − 〈µ+ δ(p),AQ(q)〉 ) , pg ) = Ψ T ∗PQ g ( φAµ (αq, p) ) . The projection of φAµ is thus well defined, and it is denoted by [φAµ ], i.e. [φAµ ] : J−1 δ (µ)/Gµ → V 0 Pπ Q/Gµ = T ∗P/Gµ(Q/G). � Definition 14. B̃ is the 2-form on P/Gµ obtained after reducing B + d(〈µ+ δ,AP 〉). In the following proposition we put a symplectic structure on the fibred product T ∗P/Gµ(Q/G) = T ∗(Q/G)×Q/G P/Gµ. The notations are similar to those on T ∗PQ. Definition 15. We define the projections 1) ε̃1 : T ∗P/Gµ(Q/G)→ T ∗(Q/G), 2) ε̃2 : T ∗P/Gµ(Q/G)→ P/Gµ. The canonical symplectic 2-form on T ∗(Q/G) is denoted by ωQ/G. Completely analogous to the construction on T ∗PQ, we introduce ε̃∗1ωQ/G + ε∗2B̃ as a 2-form on T ∗P/Gµ(Q/G). Figure 1. Cotangent bundle reduction (notations as in Theorem 2 with M = T ∗ PQ). Routh Reduction by Stages 13 Theorem 3 (Generalized magnetic cotangent bundle reduction). Let µ denote a regular value of the momentum map Jδ for the symplectic structure (T ∗PQ, ε ∗ 1ωQ+ε∗2B) with isotropy subgroup Gµ, and let AQ be a principal connection on Q → Q/G. Then the symplectically reduced manifold (Mµ, ωµ) is symplectomorphic to (T ∗P/Gµ(Q/G), ε̃∗1ωQ/G + ε̃∗2B̃)), with symplectomorphism [φAµ ]. 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. Proof. Let i0 denote the injection V 0 Pπ Q → T ∗PQ and π0 µ the projection V 0 Pπ Q → T ∗P/Gµ(Q/G). We know that [φAµ ] is a diffeomorphism, and therefore it only remains to show that the 2-form ε̃∗1ωQ/G + ε̃∗2B̃ is pull-backed to ωµ under this map (and therefore, implicitly ε̃∗1ωQ/G + ε̃∗2B̃ will be nondegenerate). We use the fact that ωµ is uniquely determined by i∗µωQ = π∗µωµ, with iµ : J−1 δ (µ) → T ∗PQ the natural inclusion and πµ : J−1 δ (µ) → Mµ the projection to the quotient space. Due to the uniqueness property, it is therefore sufficient to show that π∗µ ( [φAµ ]∗(ε̃∗1ωQ/G + ε̃∗2B̃) ) = i∗µ(ε∗1ωQ + ε∗2B). (2) The left-hand side of (2). Since [φAµ ]◦πµ = π0 µ◦φAµ the left-hand side of the above equation is (π0 µ ◦ φAµ )∗ ( ε̃∗1ωQ/G + ε̃∗2B̃ ) . The right-hand side of (2). We first show the equality of two 1-forms: i∗µ(ε∗1θQ) and ( φAµ )∗( i∗0 ( ε∗1θQ + ε∗2〈µ+ δ,AP 〉 )) . Let (αq, p) ∈ J−1 δ (µ), V(αq ,p) ∈ T (T ∗PQ) a tangent vector to J−1 δ (µ) and vq ∈ TqQ denotes the projection of V(αq ,p). Then, the first 1-form equals i∗µ(ε∗1θQ)(αq, p)(V(αq ,p)) = θQ(αq)(Tε1(V(αq ,p))) = 〈αq, vq〉. The second 1-form evaluated on this tangent vector gives (φAµ )∗ ( i∗0 ( ε∗1θQ + ε∗2〈µ+ δ,AP 〉 )) (αq, p)(V(αq ,p)) = θQ ( αq − 〈µ+ δ(p),AQ(q)〉 )( Tε1(T (φAµ )(V(αq ,p))) ) + 〈µ+ δ(p),AQ(q)(vq)〉 = 〈αq, vq〉. The two 1-forms are identical and after taking the exterior derivative and adding ε∗2B, we obtain that the right-hand side of (2) equals i∗µ(ε∗1ωQ + ε∗2B) = ( φAµ )∗( i∗0 ( ε∗1ωQ + ε∗2(d〈µ+ δ,AP 〉+ B) )) . Since (i) (ε̃2 ◦ π0 µ)∗B̃ = (ε2 ◦ i0)∗(d〈µ+ δ,AP 〉+ B), and (ii) (ε̃1 ◦ π0 µ)∗θQ/G = i∗0ε ∗ 1θQ, 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 re- duction. 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 ′ respec- tively. 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 ′. 14 B. Langerock, T. Mestdag and J. Vankerschaver Theorem 4. If f is an equivariant symplectic diffeomorphism M →M ′, such that J ′ = J ◦ f , then under symplectic reduction, the symplectic manifolds (Mµ,Ωµ) and (M ′µ,Ω ′ µ) are symplec- tically diffeomorphic under the map [fµ] : Mµ →M ′µ; [m]Gµ 7→ [f(m)]Gµ . Proof. This is a straightforward result. Since f is a diffeomorphism for which J ′ = J ◦ f , the restriction fµ of f to J−1(µ) determines a diffeomorphism from J−1(µ) to J ′−1(µ). The equivariance implies that fµ reduces to a diffeomorphism [fµ] from Mµ = J−1(µ)/Gµ to M ′µ = J ′−1(µ)/Gµ. It is our purpose to show that [fµ]∗Ω′µ = Ωµ, or since both πµ and π′µ are projec- tions, that π∗µΩµ = f∗µ(π′∗µ Ω′µ). The determining property for Ωµ is π∗µΩµ = i∗µΩ (and similarly for Ω′µ). From diagram chasing we have that i∗µΩ = f∗µ(i′∗µΩ′). Then π∗µΩµ = i∗µΩ = f∗µ(i′∗µΩ′) = f∗µ(π′∗µ Ω′µ) = π∗µ([fµ]∗Ω′µ), since π′µ ◦ fµ = [fµ] ◦ πµ by definition. This concludes the proof. � 5.2 Routh reduction for magnetic Lagrangian systems In Proposition 5 we have introduced the momentum map JL,δ = Jδ ◦ FL for a G-invariant hyperregular magnetic Lagrangian system (ε : P → Q,L,B), i.e. 〈JL,δ(vq, p), ξ〉 = 〈FL(vq, p), (ξQ(q), p)〉 − δξ(p). 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)/Gµ and J−1 δ (µ)/Gµ (see also Theorem 4). The following diagram summarizes these previous observations. Figure 2. Diagram relating tangent and cotangent reduction (SR = symplectic reduction). We are now only three steps away from a description of Routh reduction for magnetic La- grangian systems. The final goal is to describe the symplectic reduced manifold (J−1 L,δ(µ)/Gµ,Ωµ) as a symplectic manifold associated with a new magnetic Lagrangian system, with Lagrangian say Lµ. 1. The first step is the construction of a diffeomorphism ∆µ : TP/Gµ(Q/G)→ J−1 L,δ(µ)/Gµ. This is crucial because the Hamiltonian dynamics determined by the Euler–Lagrange equations reduce to Hamiltonian dynamics on a manifold of the form TP/Gµ(Q/G). If we can characterize this reduced dynamics as the Euler–Lagrange equations of a magnetic Lagrangian system with configuration manifold P/Gµ → Q/G, we have developed a Routh reduction technique for mag- netic Lagrangian systems. This characterization consists of the two remaining steps mentioned before: Routh Reduction by Stages 15 2. We show that the composition [φAµ ] ◦ [(FL)µ] ◦ ∆µ equals the fibre derivative of some Lagrangian function L̃ on TP/Gµ(Q/G), and this implies that the symplectic 2-form ∆∗µΩµ on TP/Gµ(Q/G) is of the form FL̃∗(ε̃∗1ωQ/G + ε̃∗2B̃), which is the symplectic structure associated to the reduced magnetic Lagrangian system (P/Gµ → Q/G, L̃, B̃); 3. We show that the reduction of the energy EL in the symplectic reduction scheme is pre- cisely the energy EL̃ on TP/Gµ(Q/G). This guarantees that the Euler–Lagrange equations associated to the reduced Lagrangian L̃ are related to the Euler–Lagrange equations for L. The new reduced Lagrangian L̃ is what we call the Routhian (it is often alternatively denoted by R or Rµ). This is summarized in the diagram in Fig. 3. Figure 3. A symplectic view on Routh reduction. Step 1: the definition of ∆µ : TP/Gµ(Q/G) → J−1 L,δ(µ)/Gµ. First consider the map Πµ : J−1 L,δ(µ)/Gµ → TP/Gµ(Q/G) which is obtained from the Gµ-invariance of the map J−1 L,δ(µ)→ TP/Gµ(Q/G); (vq, p) 7→ (TπQ(vq), [p]Gµ). We will define the map ∆µ as the inverse of Πµ. In general however, Πµ 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 G- regular if the map JL,δ\|(vq ,p) : g → g∗; ξ 7→ JL,δ(vq + ξQ(q), p) is a diffeomorphism for all (vq, p) ∈ TPQ. Every mechanical G-invariant magnetic Lagrangian system is G-regular because JL,δ\|(vq ,p) is an affine map modeled on the metric on g obtained from the kinetic energy metric. Proposition 7. Πµ is a diffeomorphism if the Lagrangian is G-regular. Proof. We construct the inverse for Πµ. Choose an element (v[q]G , [p]Gµ) in TP/Gµ(Q/G) and fix a point (vq, p) ∈ TPQ that projects onto (v[q]G , [p]Gµ), i.e. TπQ(vq) = v[q] and p ∈ [p]Gµ . Due to the G-regularity of L, there exists a unique element ξ in g such that JL,δ(vq + ξQ(q), p) = µ. The orbit of the point (vq + ξQ(q), p) under the Gµ-action determines an element in J−1 L,δ(µ)/Gµ. We will show that this construction is independent of the chosen point (vq, p), and therefore defines a map ∆µ : TP/Gµ(Q/G)→ J−1 L,δ(µ)/Gµ which is the inverse to Πµ. To show that the construction is independent of the chosen point (vq, p) in TPQ, we choose any other point (v′q, p ′) in TPQ projecting onto (v[q]G , [p]Gµ). The point is of the form (v′q, p ′) = (ΨTQ g (vq + ηQ(q)), pg) for some g ∈ Gµ and η ∈ g. We now repeat the previous construction applied to (v′q, p ′): we consider a unique element ξ′ ∈ g for which JL,δ(Ψ TQ g (vq + ηQ(q)) + ξ′Q(qg), pg) = µ. Due to the equivariance of JL,δ, we conclude that µ = JL,δ(Ψ TQ g (vq + ηQ(q)) + ξ′Q(qg), pg) = Ad∗g · JL,δ(vq + (η + Adgξ ′)Q(q), p)− σδ(g−1). 16 B. Langerock, T. Mestdag and J. Vankerschaver Since g ∈ Gµ, we may conclude that JL,δ(vq+(η+Adgξ ′)Q(q), p) = µ and therefore η+Adgξ ′ = ξ. This implies that (v′q + ξ′Q(q), p′) = ΨTPQ g (vq + ξQ(q), p) for g ∈ Gµ such that p′ = pg. This concludes the proof. � The previous proposition guarantees that for a given G-regular Lagrangian and a bundle adapted coordinate chart (xr, gm) in Q → Q/G and (xr, gm, pa) in P → Q, the functions (xr, ẋr, gm, pa) 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 Lagrangian L̃ on TP/Gµ(Q/G) with the property that its Legendre transform FL̃ : TP/Gµ(Q/G)→ T ∗P/Gµ(Q/G) is precisely the reduced Legendre transform [(FL)µ], more specifically FL̃ = [φAµ ] ◦ [(FL)µ] ◦∆µ. As is already described in [10], the structure of the Lagrangian L̃ is completely determined by the cotangent bundle reduction scheme, and in particular by the shift map φAµ from J−1 δ (µ) → V 0 Pπ Q. To make this statement more precise, we remark that [φAµ ] ◦ [(FL)µ] is the quotient map of φAµ ◦ FL. Thus, if [(vq, p)]Gµ is arbitrary in J−1 L,δ(µ)/Gµ, then [φAµ ] ◦ [(FL)µ]([(vq, p)]Gµ) is the quotient of FL(vq, p)− ( 〈µ+ δ(p),AQ(q)〉, p ) . The latter is the fibre derivative of a new function on TPQ, namely L(vq, p)−〈µ+δ(p),AQ(q)(vq)〉. This function is clearly Gµ-invariant. Definition 17. The function L̃ is defined as the pull-back under ∆µ of the quotient map of the restriction to J−1 L,δ(µ) of the function (vq, p) 7→ L(vq, p)− 〈 µ+ δ(p),AQ(q)(vq) 〉 . Lemma 2. FL̃ = [φAµ ] ◦ [(FL)µ] ◦∆µ. Proof. Fix elements (v[q]G , [p]Gµ) ∈ TP/Gµ(Q/G) and fix a representative (vq, p) ∈ J−1 L,δ(µ) in the orbit ∆µ(v[q]G , [p]Gµ) ∈ J−1 L,δ(µ)/Gµ. By definition of the maps involved, we have( [φAµ ] ◦ [(FL)µ] ◦∆µ ) (v[q]G , [p]Gµ) = (π0 µ ◦ φAµ )(FL(vq, p)) = π0 µ ( FL(vq, p)− ( 〈µ+ δ(p),AQ(q)〉, p )) . Next we study the fibre derivative of the map L̃. Fix a point (w[q]G , [p]Gµ) and we compute 〈 FL̃(v[q]G , [p]Gµ), (w[q]G , [p]Gµ) 〉 = d du ∣∣∣∣ u=0 L̃(v[q]G + uw[q]G , [p]Gµ). We construct a curve u 7→ ζ(u) in J−1 L,δ(µ) that projects onto the curve u 7→ ∆µ(v[q]G + uw[q]G , [p]Gµ) in J−1 L,δ(µ)/Gµ such that ζ(0) = (vq, p) and ζ̇(0) is vertical to the projection ρ2 ◦ iµ : J−1 L,δ(µ) → P (recall that ρ2 : TPQ → 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): (xr, ẋr, gm, pa) where the index r = 1, . . . ,dimQ/G, m = 1, . . . ,dimG and a = 1, . . . ,dimP − dimQ. Note that (xr, ẋr) is a coordinate chart in T (Q/G). In these coordi- nates, we let (vq, p) = (xr0, ẋ r 0, g m 0 , p a 0) and w[q]G = (xr0, w r 0). We define the curve ζ(u) to be the curve u 7→ (xr0, ẋ r 0 + uwi0, g m 0 , p a 0). Then the tangent vector to Tρ1(ζ̇(0)) is the vertical lift of some wq ∈ TqQ with TπQ(wq) = w[q]G . Routh Reduction by Stages 17 Finally, from the definition of L̃ and the fibre derivative FL̃ we obtain〈 FL̃(v[q]G , [p]Gµ), (w[q]G , [p]Gµ) 〉 = d du ∣∣∣∣ u=0 ( L− 〈 µ+ δ,AQ 〉) (ζ(u)) = 〈FL(vq, p)− (〈µ+ δ(p),AQ(q)〉, p), (wq, p)〉 = 〈 π0 µ ( FL(vq, p)− ( 〈µ+ δ(p),AQ(q)〉, p )) , (w[q]G , [p]Gµ) 〉 , since FL(vq, p)− (〈µ+ δ(p),AQ(q)〉, p) ∈ V 0 Pπ Q. 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 EL on TPQ to the energy of the Routhian L̃ on TP/Gµ(Q/G). In the following Lemma we again use the notations from Theorem 2 applied to the reduction of the symplectic structure on (TPQ,Ω L,B). Lemma 3. The energy EL̃ is the reduced Hamiltonian, i.e. it satisfies: (Πµ ◦ πµ)∗EL̃ = i∗µEL, with Πµ ◦ πµ : J−1 L,δ(µ)→ J−1 L,δ(µ)/Gµ → TP/Gµ(Q/G) and iµ : J−1 L,δ(µ)→ TPQ. Proof. Let (vq, p) ∈ J−1 L,δ(µ), such that (Πµ ◦ πµ)(vq, p) = (v[q]G , [p]Gµ). Then i∗µEL(vq, p) = 〈FL(vq, p), (vq, p)〉 − L(vq, p) = 〈(φAµ ◦ (FL)µ)(vq, p), (vq, p)〉 − ( L(vq, p)− 〈µ+ δ(p),AQ(q)(vq)〉 ) = 〈( [φAµ ] ◦ [(FL)µ] ◦∆µ ) (v[q]G , [p]Gµ), (v[q]G , [p]Gµ) 〉 − L̃(v[q]G , [p]Gµ). 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 (ε : P → Q,L,B) be a hy- perregular, G-invariant and G-regular magnetic Lagrangian system and let δ be a Bg-potential of the magnetic term B with 1-cocycle σδ. 1. Let µ ∈ g∗ be a regular value of the momentum map JL,δ and let Gµ be the isotropy subgroup of µ w.r.t. the affine action on g∗ with 1-cocycle −σδ, i.e. g ∈ Gµ if and only if µ = Ad∗gµ− σδ(g−1). 2. Fix a connection AQ on πQ : Q → Q/G and let AP be the corresponding connection on P → P/G. Compute the restriction of the Gµ-invariant function L(vq, p) − 〈µ + δ(p),AQ(q)(vq)〉 to J−1 L,δ(µ) and let L̃ be its quotient to TP/Gµ(Q/G) ∼= J−1 L,δ(µ)/Gµ. 3. Compute B̃ as the projection to P/Gµ of the 2-form B + d〈µ+ δ,AP 〉. 4. Consider the magnetic Lagrangian system: (ε̃µ : P/Gµ → Q/G, L̃, B̃). This reduced magnetic Lagrangian system is hyperregular and every solution p(t) ∈ P to the Euler–Lagrange equations for (ε, L,B) with momentum µ projects under P → P/Gµ to a solution of the Euler–Lagrange equations for (ε̃µ : P/Gµ → Q/G, L̃, B̃). Conversely, every solution to the Euler–Lagrange equations for (ε̃µ : P/Gµ → Q/G, L̃, B̃) is the projection of a solution to the Euler–Lagrange equations for (ε, L,B) with momentum µ. It is possible to say more about the structure of B̃ and its relation to the connection AQ. 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. 18 B. Langerock, T. Mestdag and J. Vankerschaver 5.3 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 ` on g and with it we associate a Lagrangian L on TG by right multiplication L(g, vg) = `(vgg −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. 1. The 1-cocycle σδ : G→ g∗ satisfies σδ(g) = δ(hg−1)−Ad∗g−1δ(h), for arbitrary h ∈ G. If we let h = e, then σδ(g) = δ(g−1)−Ad∗g−1δ(e) or equivalently, Ad∗gσδ(g) = Ad∗gδ(g −1)− δ(e). Similarly if we let h = g, then σδ(g) = δ(e) − Ad∗g−1δ(g). Since δ is determined up to a constant, we may assume without loss of generality that δ(e) = 0. 2. The associated 2-cocycle Σδ(ξ, η) = ξG(δη)− δ[ξ,η]. 3. We use the right identification of TG with G×g, i.e. (g, vg) is mapped to (g, vgg −1) ∈ G×g. The right action of G on TG 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)(vg) = g−1vg. In the right identification, the connection corresponds to the map (g, ξ) ∈ G× g 7→ Adg−1ξ ∈ g. 5. The momentum map JL,δ : TG→ g∗ equals JL,δ(g, ξg) = Ad∗gF`(ξ)−δ(g), for (g, ξ) ∈ G×g arbitrary. If L is G-regular then F` is invertible, i.e. there exists a function χ : g∗ → g such that F`(χ(ν)) = ν. 6. The affine action on g∗ is (g, µ) 7→ Ad∗gµ− σδ(g−1). 7. The isotropy group Gµ consists of group elements g such that µ+ δ(g) = Ad∗gµ. 8. The quotient G/Gµ (right coset space) can be identified with Õµ, i.e. [g]Gµ ∈ G/Gµ is mapped onto ν = Ad∗g−1µ− σδ(g) = Ad∗g−1(µ+ δ(g)). A tangent vector to G/Gµ at [g]Gµ which is the projection of ξg is mapped to a vector ν̇ = −ad∗ξν + iξΣδ in g∗. Lemma 4. The 2-form B + d(〈µ + δ,A〉) reduces to the Kirillov–Kostant–Souriau symplectic 2-form on Õµ ∼= G/Gµ. Proof. Let g be arbitrary, and let vg = ξg, wg = ηg be two tangent vectors in TgG with ξ, η ∈ g arbitrary. Note that vg = ξ′G(g) with ξ′ = Adg−1ξ, and similarly wg = η′G(g) with η′ = Adg−1η. Then B(g)(vg, wg) = B(g)(ξ′G(g), η′G(g)) = η′G(δξ′)(g). On the other hand d(〈µ+ δ,A〉)(g)(vg, wg) = ξ′G(〈µ+ δ, η′〉)(g)− η′G(〈µ+ δ, ξ′〉)(g)− 〈µ+ δ(g), [ξ′, η′]〉. Before continuing, we compute the equivariance of the 2-cocycle Σδ(ξ ′, η′): Σδ(ξ ′, η′) = − d ds 〈σδ(g−1 exp sξg), η′〉 = − d ds 〈σδ(g−1 exp sξ) + Ad∗exp−sξgσδ(g), η′〉 = − d ds 〈σδ(g−1) + Ad∗gσδ(exp sξ) + Ad∗gAd∗exp−sξσδ(g),Adg−1η〉 = Σδ(ξ, η) + 〈σδ(g), [ξ, η]〉. Summarizing, we have( B + d(〈µ+ δ,A〉) ) (g)(vg, wg) = ξ′G(〈µ+ δ, η′〉)(g)− 〈µ+ δ(g), [ξ′, η′]〉 Routh Reduction by Stages 19 = Σδ(ξ ′, η′) + δ[ξ′,η′](g)− 〈µ+ δ(g), [ξ′, η′]〉 = Σδ(ξ, η)− 〈Ad∗g−1µ− σδ(g), [ξ, η]〉. This 2-form is reducible to a 2-form on Õµ. If we use the isomorphism G/Gµ → Õµ introduced above, the 2-form reduces to B̃(ν)(ν̇, ν̇ ′) = 〈ν̇, η〉 with η ∈ g such that ν̇ ′ = −ad∗ην + iηΣδ. � We conclude by computing the Routhian L̃ as a function on Õµ. By definition, it equals the reduction to Õµ of L− 〈µ+ δ,A〉\|JL,δ=µ. In the right identification, the level set JL,δ(g, vg) = µ is precisely Ad∗g−1(µ + δ(g)) = F`(ξ), with ξg = vg. If we set ν = Ad∗g−1(µ + δ(g)) ∈ Õµ, the fixed momentum condition is ξ = χ(ν). The Routhian L̃(ν) becomes in the right identification L̃(ν) = `(χ(ν))− 〈ν, χ(ν)〉. By application of the chain rule, it easily follows that 〈dL̃(ν), ν̇ ′〉 = −〈ν̇ ′, χ(ν)〉. The reduced Euler–Lagrange equations are iν̇B̃(ν) = dL̃(ν) or ν̇ = −ad∗χ(ν)ν + iχ(ν)Σδ. (3) For later purpose, we remark that for a left action and ` originating from a left invariant Lagrangian, the reduced equations are ν̇ = ad∗χ(ν)ν−iχ(ν)Σδ (here the 1-cocycle satisfies σδ(g) = δ(g)). 6 Routh reduction by stages In reduction by stages, we study the reduction of a G-invariant system (symplectic or La- grangian) 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. Definition 18. 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 Gµ is the isotropy subgroup of µ under the Ad∗-action of G on g∗; Gν 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 Kν is the isotropy of ν w.r.t. to standard coadjoint action of K on K∗. These groups satisfy Gν ∩K = Kν and Kν is normal in Gν . 4. gν and Kν denote the Lie algebras of Gν and Kν respectively. Ḡν denotes the quotient group Gν/Kν and its Lie algebra equals ḡν = gν/Kν . 5. The projections onto the quotient groups are denoted by r : G→ Ḡ = G/K and rν : Gν → Ḡν , and on the level of the Lie algebra: r′ : g→ ḡ = g/K and r′ν : gν → ḡν . The inclusion map Gν → G induces a map kν : gν → g, with its dual k∗ν : g∗ → g∗ν . 6. ρ denotes an element in ḡ∗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 = ν̄, there exists an element k ∈ Kν and g ∈ (Gν)ν̄ such that Ad∗kgµ ′ = µ. The stages hypothesis is a condition on a chosen momentum value and 20 B. Langerock, T. Mestdag and J. Vankerschaver 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. 6.1 Symplectic reduction by stages Theorem 6 (Symplectic reduction by stages [12]). Let (M,ω) be a symplectic manifold with a canonical G-action ΨM with an equivariant momentum map JG. 1. Fix a regular value µ ∈ g∗ of the momentum map and perform symplectic reduction to obtain the symplectic manifold (Mµ, ωµ). 2. The restriction of the action ΨM to K is canonical and the map JK = i∗ ◦ JG : M → K∗ determines an equivariant momentum map for this induced action. Fix a regular value ν of JK and perform symplectic reduction to obtain the symplectic manifold (Mν , ων). 3. The level set J−1 K (ν) is Gν-invariant. 4. The group Ḡν = Gν/Kν acts on Mν by projecting the restricted action of Gν on J−1 K (ν). This induced action ΨMν is free, proper and canonical on (Mν , ων). Assume that Kν is connected. 5. Fix an element ν̄ in g∗ν such that the restriction of ν̄\|Kν equals ν\|Kν . There is a well-defined momentum map JḠν : Mν → ḡ∗ν for the induced action ΨMν . This momentum map is determined from JG and ν̄ and has a non-equivariance cocycle: JḠν satisfies (r′ν)∗ ◦ JḠν ◦ πν = k∗ν ◦ JG ◦ iν − ν̄. 6. Let ρ ∈ ḡ∗ν be a regular value for the momentum map JḠν and let (Ḡν)ρ be the isotropy subgroup of ρ w.r.t. the affine action of Ḡν on ḡ∗ν . Perform symplectic reduction to obtain the symplectic manifold ((Mν)ρ, (ων)ρ) with (Mν)ρ = J−1 Ḡν (ρ)/(Ḡν)ρ. If ρ is chosen such that (r′ν)∗ρ = µ\|gν − ν̄, then there exists a symplectic diffeomorphism F : (Mµ, ωµ)→ ((Mν)ρ, (ων)ρ). For our purpose it is also important to understand the reduction of a G-invariant Hamil- tonian 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 jµ of J−1 G (µ) in J−1 K (ν). Recall that we use πν for the projection J−1 K (ν) → Mν . It was shown in [12] that the image of πν ◦ jµ : J−1 G (µ) → Mν is contained in J−1 Ḡν (ρ). Moreover, this map is equivariant w.r.t. the action of Gµ on J−1 G (µ) and (Ḡν)ρ on J−1 Ḡν (ρ) (this makes sense, since Gµ projects to a subset of (Ḡν)ρ). The quotient of πν ◦ jµ is the symplectic diffeomorphism F mentioned in the previous theorem (see also Fig. 4). Let h be a G-invariant Hamiltonian on M and let hµ be the function on Mµ obtained from π∗µhµ = i∗µH. On the other hand we let hν be the function satisfying π∗νhν = i∗νh. This function is Ḡν-invariant: hν([m]Kν [g]Kν ) = h(iν(mg)) = h(iν(m)g), with m ∈ J−1 K (ν) and g ∈ Gν arbitrary. Note that (πν ◦ jµ)∗hν = i∗µh. The Hamiltonian hν is a Ḡν-invariant function on (Mν , ων). Applying the second symplectic reduction to this manifold, we obtain a new reduced Hamiltonian (hν)ρ on (Mν)ρ. Proposition 8. F ∗((hν)ρ) = hµ. Routh Reduction by Stages 21 Figure 4. Commuting diagram relating the different reduced symplectic manifolds. Proof. We rely on the commuting diagram in Fig. 4: π∗µ(F ∗((hν)ρ)) = (πν ◦ jµ)∗hν = i∗µ(h). This uniquely characterizes F ∗((hν)ρ) as the function hµ. � 6.2 Routh reduction by stages Routh reduction by stages is symplectic reduction by stages applied to the symplectic struc- ture 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 re- duction 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 vq ∈ TQ and any subspace K′ < g with injection i′ : K′ → g, the mapping i′∗ ◦ JL\|vq ◦ i′ : K′ → K′∗ defined by ξ 7→ i′∗ ( JL(vq + (i′(ξ))Q(q)) ) is invertible. 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 JL, the map i∗ ◦ JL = i∗ ◦ (ψTQ)∗ ◦ 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 JL and A0 a G-connection on Q. Let (Q/Gµ → Q/G,L0,B0) 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∗ ◦ JL for the K-action and a K- connection A1 on Q. Assume that A1 is G-equivariant w.r.t. the action of G on Q and K∗. Consider the magnetic Lagrangian system (Q/Kν → Q/K,L1,B1) obtained by performing Routh reduction with respect to K. 3. Ḡν acts on Q/Kν and Q/K by projecting the induced action of Gν on Q. These induced actions are free and proper. 4. Assume that Kν is connected. Fix an element ν̄ ∈ g∗ν such that ν̄\|Kν = ν\|Kν . Then the magnetic Lagrangian system (Q/Kν → Q/K,L1,B1) is Ḡν-invariant, Ḡν-regular and 22 B. Langerock, T. Mestdag and J. Vankerschaver admits a B1ḡν-potential δ1 entirely determined by the choice of ν̄. The potential satisfies, for arbitrary q ∈ Q (r′ν)∗ ( δ1([q]Kν ) ) = −(ψQ ◦ kν)∗ ( 〈ν,A1(q)〉 ) + ν. Let J1 denote the momentum map associated with δ1. 5. Fix a regular value ρ ∈ ḡ∗ν for the momentum map J1 and let (Ḡν)ρ be the isotropy sub- group of ρ w.r.t. the affine action of Ḡν on ḡν . Fix a Ḡν-connection on Q/K. Consider the magnetic Lagrangian system ((Q/Kν)/(Ḡν)ρ → (Q/K)/Ḡν), L2,B2) obtained by per- forming Routh reduction with respected to Ḡν . If ρ is chosen such that (r′ν)∗ρ = µ\|gν − ν̄, then every solution γ(t) ∈ Q/Gµ to the Euler– Lagrange equations for (Q/Gµ → Q/G,L0,B0) is mapped to a solution in (Q/Kν)/(Ḡν)ρ to the Euler–Lagrange equations for ((Q/Kν)/(Ḡν)ρ → (Q/K)/Ḡν , L2,B2). Conversely, a solution in (Q/Kν)/(Ḡν)ρ to the Euler–Lagrange equations for ((Q/Kν)/(Ḡν)ρ → (Q/K)/Ḡν , L2,B2) is the projection of a solution in Q/Gµ to the Euler–Lagrange equations for (Q/Gµ → Q/G,L0,B0). Proof. 1 and 2 are obtained by applying Routh reduction. 3 follows from [12, p. 152]: we know that the quotient groups Ḡν = Gν/Kν acts in a free and proper way on the quotient space Q/Kν . The group Ḡν is a subgroup of Ḡ and acts freely and properly and Q/K. We now show 4. Ḡν-Invariance of the Routh reduced system (Q/Kν → Q/K,L1,B1). Lemma 5. If the connection A1 is chosen such that it is equivariant w.r.t. the action of the full group G, i.e. if (ΨQ g )∗A1 = Adg−1A1, then the magnetic Lagrangian system (Q/Kν → Q/K,L1,B1) is Ḡν-invariant and δ1 is a B1ḡν- potential. Proof. We first show that L1 is Ḡν-invariant. For that purpose, we choose an arbitrary ḡ ∈ Ḡν and let g ∈ Gν be a representative. Similar we choose a point (v[q]K , [q]Kν ) ∈ TQ/Kν (Q/K) such that it is the projection of vq ∈ (i ◦ JL)−1(ν) ⊂ TQ. By definition of the quotient action on TQ/Kν (Q/K), the action of ḡ on an element (v[q]K , [q]Kν ) equals the projection of vqg. We now check the invariance of L1 at an arbitrary point in TQ/Kν (Q/K): L1 ( Ψ TQ/Kν (Q/K) ḡ ( v[q]K , [q]Kν )) = L(ΨTQ g (vq))− 〈ν,A1(qg)(TΨQ g (vq))〉 = L(vq)− 〈Ad∗g−1ν,A1(q)(vq)〉 = L1 ( v[q]K , [q]Kν ) . Next, we check the Ḡν-invariance of B1. Recall that B1 is the projection to Q/Kν of the 2-form d〈ν,A1〉 on Q. We first consider the equivariance of this 2-form under Gν . Let g ∈ Gν be arbitrary, then (ΨQ g )∗(d〈ν,A1〉) = d〈Ad∗g−1ν,A1〉 = d〈ν,A1〉. We thus obtain Gν-invariance for d〈ν,A1〉, and we may conclude that (Ψ Q/Kν ḡ )∗B1 = B1 holds on Q/Kν . The third and final step is the definition of the B1ḡν-potential. We consider an element ξ̄ = [ξ]Kν ∈ ḡν = gν/Kν and let ξ ∈ gν be a representative. Then, by definition of B1, the 1-form iξ̄Q/Kν B1 is the projection to Q/Kν of the 1-form iξQd(〈ν,A1〉) on Q (i.e. ξQ projects to ξ̄Q/Kν ). Again we concentrate on the 1-form on Q: iξQd〈ν,A 1〉 = LξQ(〈ν,A1〉)− d ( iξQ〈ν,A 1〉 ) . Routh Reduction by Stages 23 Since 〈ν,A1〉 is Gν-invariant, we conclude that iξQd〈ν,A1〉 = −d ( iξQ〈ν,A1〉 ) . The exact 1-from on the right gives a strong hint of the structure of the ḡν-potential. Assume now that we fixed an element ν ∈ g∗ν such that ν\|Kν = ν\|Kν . The function δ on Q, defined by −δξ(q) = 〈ν,A1(q)(ξQ(q))〉 − 〈ν, ξ〉 is our candidate for the B1ḡν-potential. This statement makes sense provided that δξ projects to a function on Q/Kν and that it only depends on the equivalence class ξ̄ = ξ + Kν of ξ ∈ gν . The latter is a straightforward consequence of the fact that A1 is a principal K-connection. The Kν-invariance is more involved, and we rely on a result in [12]. For any k in Kν , we have −δξ(qk) = 〈ν,A1(qk)(ξQ(qk))〉 − 〈ν, ξ〉 = 〈ν,A1(q)((Adkξ)Q(q))〉 − 〈ν, ξ〉. Therefore δξ is constant on the orbits of Kν in Q if 〈ν,A1(q)((ξ − Adkξ)Q(q))〉 vanishes for all k. To show this we introduce a function f on Kν given by f(k) = 〈ν,A1(q)((ξ−Adkξ)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(k1k2) = f(k1) + f(k2) for arbitrary k1,2 ∈ Kν , we may conclude that f = 0 (since Kν is assumed connected). The first condition f(e) = 0 is trivial. To check the second condition: let κ ∈ Kν be arbitrary, then df \|e(κ) = 〈ν,A1(q)(−adκξ)Q(q))〉 = −〈ν, adκξ〉 = −〈ad∗κ, ξ〉 = 0. Above, we have used the fact that Kν is normal in Gν and that, as a consequence, the Lie bracket [κ, ξ] is in Kν . Therefore the contraction of the corresponding fundamental vector field with A1 is precisely [κ, ξ]. Next, we check the third condition and compute f(k1k2). Given the identity ξ −Adk1Adk2ξ = ξ −Adk1ξ + Adk1(ξ −Adk2ξ) and the fact that k1 ∈ Kν , f(k1k2) = 〈ν,A1(q)((ξ −Adk1k2ξ)Q(q))〉 = 〈ν,A1(q)((ξ −Adk1ξ)Q(q))〉+ 〈ν,A1(q)(Adk1(ξ −Adk2ξ)Q(q))〉 = f(k1) + 〈Ad∗k1ν,A 1(q)((ξ −Adk2ξ)Q(q))〉 = f(k1) + f(k2). This completes the proof: the g∗ν-valued function δ is shown to be projectable to a ḡ∗ν-valued function on Q/Kν . This is the sought-after potential δ1: for arbitrary q ∈ Q, we have (r′ν)∗ ( δ1([q]Kν ) ) = −(ψQ ◦ kν)∗ ( 〈ν,A1(q)〉 ) + ν. � Symplectic structure of (Q/Kν → Q/K,L1,B1) 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 Mν with the symplectic structure on TQ/Kν (Q/K) induced by the magnetic Lagrangian system (Q/Kν → Q/K,L1,B1). Then: 1. The action ΨTQ/Kν (Q/K) of Ḡν on TQ/Kν (Q/K) is precisely the induced action on the first reduced space Mν in symplectic reduction by stages. 2. For a chosen ν̄ ∈ g∗ν , the momentum map J1 : TQ/Kν (Q/K)→ ḡ∗ν associated with the mag- netic Lagrangian system (Q/Kν → Q/K,L1,B1) corresponds to the induced momentum map JḠν from symplectic reduction by stages. 24 B. Langerock, T. Mestdag and J. Vankerschaver Proof. 1. The momentum map for the K action is precisely JK := i∗K ◦JL, with JL : TQ→ g∗. By definition, the induced action of Ḡν on J−1 K (ν)/Kν is obtained by projecting the action of Gν on J−1 K (ν). If we take into account that we realize the quotient manifold J−1 K (ν)/Kν as TQ/Kν (Q/K), the induced action on TQ/Kν (Q/K) is obtained by projection of the action of Gν on TQ under the projection TQ→ TQ/Kν (Q/K). This is precisely the action we have introduced above. 2. The induced momentum map JḠν is defined in the following way (we consider it directly as a function on TQ/Kν (Q/K) instead of on Mν):〈 JḠν ( v[q]K , [q]Kν ) , ξ̄ 〉 = 〈JL(vq), kν(ξ)〉 − 〈ν̄, ξ〉 = 〈FL(vq), ξQ〉 − 〈ν̄, ξ〉, (4) where ξ ∈ gν is arbitrary and projects to ξ̄ ∈ ḡν , vq projects to (v[q]K , [q]Kν ). By definition of the momentum map of the magnetic Lagrangian system (Q/Kν → Q/K,L1,B1), we have〈 J1 ( v[q]K , [q]Kν ) , ξ̄ 〉 = 〈 FL1 ( v[q]K , [q]Kν ) , ( (ξ̄)Q/K , [p]Kν )〉 − (δ1)ξ̄([p]Kν ). (5) We now show that the right-hand side of (5) equals the right-hand side of (4). We therefore use the definition of FL1 and δ1 as being the projection of maps upstairs:〈 FL1 ( v[q]K , [q]Kν ) , ( (ξ̄)Q/K , [q]Kν )〉 = 〈FL(vq), ξQ(q)〉 − 〈ν,A1(q)(ξQ(q))〉, (δ1)ξ̄([q]Kν ) = δξ(p) = −〈ν,A1(q)(ξQ(q))〉+ 〈ν, ξ〉. � Finally, before we can reapply Routh reduction for the second stage, we need to check that L1 is Ḡν-regular. Lemma 7. The magnetic Lagrangian system (Q/Kν → Q/K,L1,B1) is Ḡν-regular. Proof. We have to show that, for any (v[q]K , [q]Kν ) the map ḡν → ḡ∗ν ; ξ̄ 7→ J1(v[q]K + ξ̄Q/K([q]K), [q]Kν ) is invertible. Let vq determine a tangent vector in J−1 K (ν) representing (v[q]K , [q]Kν ). Let η denote an arbitrary element in ḡ∗ν . Due to the assumed G-hyperregularity, there is a unique ξ ∈ gν such that k∗ν(JL(vq + ξQ(q))) = (r′ν)∗η + ν. The projection ξ̄ = r′ν(ξ) of ξ defines the inverse element for η, since it is such that k∗ν(JL(vq + ξQ(q)))− ν̄ = (r′ν)∗J1(v[q]K + ξ̄Q/K) = (r′ν)∗η. � Symplectic and Routh reduction by stages. Summarizing the above lemmas, we con- clude that the magnetic Lagrangian system (Q/Kν → Q/K,L1,B1) is amenable to Routh re- duction and that the symplectic structure and momentum map associated to this Lagrangian system correspond to the symplectic structure and momentum map encountered in symplec- tic reduction by stages. If ρ is chosen such that the compatibility relation (r′ν)∗ρ = µ\|gν − ν̄ holds, then from symplectic reduction by stages we have that the symplectic structures asso- ciated to (Q/Gµ → Q/G,L0,B0) and ((Q/Kν)/(Ḡν)ρ → (Q/K)/Ḡν , L2,B2) are symplectically diffeomorphic by means of the symplectic diffeomorphism F introduced earlier. From Proposi- tion 8 it follows that F ∗EL2 = EL0 and therefore the corresponding Hamiltonian vector fields are F -related. We define a map τ : Q/Gµ → (Q/Kν)/(Ḡν)ρ as τ([q]Gµ) = [ [q]Kν ] (Ḡν)ρ . The map is well-defined since Gµ is a subgroup of Gν and since rν(Gµ) ⊂ (Ḡν)ρ. Lemma 8. The symplectic diffeomorphism F is fibred over τ . Routh Reduction by Stages 25 Figure 5. Fibration of the symplectic diffeomorphism F . Proof. We recall the definition of the map F : fix an element (v[q]G , [q]Gµ) and let vq ∈ J−1 L (µ) be a representative. The point F (v[q]G , [q]Gµ) is obtained by taking the consecutive quotients of vq. In particular, the component of the final quotient in the configuration space (Q/Kν)/(Ḡν) of the magnetic Lagrangian system, is precisely the image τ . � Since F is a diffeomorphism, τ is onto. The Hamiltonian vector field on TQ/Gµ(Q/G) and T(Q/Kν)/(Ḡν)ρ((Q/K)/Ḡν) are F -related. Their integral curves project onto solutions of the Euler–Lagrange equations. This concludes the proof of Theorem 7. � 7 Examples 7.1 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)× S1, 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 L = 1 2 m(ẋ2 + ẏ2) + 1 2 I1θ̇ 2 + 1 2 I2(θ̇ + ψ̇)2 − V (ψ). The Euler–Lagrange equations of the system are, in normal form, ẍ = 0, ÿ = 0, θ̈ = 1 I1 V ′, ψ̈ = −I1 + I2 I1I2 V ′. 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 cos θ − sin θ x sin θ cos θ y 0 0 1  . (6) The identity of the group is (x = 0, y = 0, θ = 0) and the multiplication is given by (x1, y1, θ1) ∗ (x2, y2, θ2) = (x2 cos θ1 − y2 sin θ1 + x1, x2 sin θ1 + y2 cos θ1 + y1, θ1 + θ2). The matrices e1 = 0 0 1 0 0 0 0 0 0  , e2 = 0 0 0 0 0 1 0 0 0  , e3 = 0 −1 0 1 0 0 0 0 0  , 26 B. Langerock, T. Mestdag and J. Vankerschaver form a basis for the Lie algebra, for which [e1, e2] = 0, [e1, e3] = e2 and [e2, e3] = −e1. A corre- sponding basis of fundamental vector fields on Q is ẽ1 = ∂ ∂x , ẽ2 = ∂ ∂y , ẽ3 = −y ∂ ∂x + x ∂ ∂y + ∂ ∂θ , and a basis of invariant vector fields is ê1 = cos θ ∂ ∂x + sin θ ∂ ∂y , ê2 = − sin θ ∂ ∂x + cos θ ∂ ∂y , ê3 = ∂ ∂θ . 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)× S1 → Q/SE(2) = S1, which locally takes the form (dx+ ydθ)e1 + (dy − xdθ)e2 + dθe3. The momentum map JL is given by: JL = mẋe1 +mẏe2 + ( m(xẏ − yẋ) + I1θ̇ + I2(θ̇ + ψ̇) ) e3. 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 R2. Full reduction. Let µ = µ1e 1 + µ2e 2 + µ3e 3 be a generic element in g∗. An element ξ = ξ1e1 + ξ2e2 + ξ3e3 of the isotropy algebra gµ satisfies ξ3µ2 = 0, ξ3µ1 = 0, ξ1µ2 − ξ2µ1 = 0. So if we suppose that µ1 and µ2 do not both vanish – we will set µ1 = 1 from now on – then a typical element of gµ is ξ = ξ1(e1 + µ2e2). Since gµ is 1-dimensional, Gµ is of course Abelian. A convenient way to describe the manifold P/Gµ = SE(2)/Gµ locally is by considering a coor- dinate transformation (x′, y′, θ′) in the group coordinates such that the vector field associated to an element in gµ becomes a coordinate vector field: in the new coordinates, we should have ∂x′ = ∂x + µ2∂y. This is obtained by the following transformation x′ = x, y′ = y − µ2x, θ′ = θ. Then clearly (y′, θ) is a coordinate chart on SE(2)/Gµ. And simultaneously, we have that (y′, θ, ψ) is a coordinate chart on the reduced configuration manifold P/Gµ = (SE(2)×S1)/Gµ, and the fibration P/Gµ → Q/G = S1 is locally represented by (y′, θ, ψ) 7→ (ψ). For a more systematic treatment on appropriate coordinate changes, we refer to [6]. We now compute the Routhian L0 and the 2-form B0. Following [19], a convenient way to compute the (unreduced) Routhian for mechanical Lagrangians is by using 2(L0 + V ) =( − pxẋ− pyẏ − θ̇pθ + ψ̇pψ ) J−1 L (µ) , where pi is the momentum in the ith coordinate. We have: 2(L0 + V ) = ( −m(ẋ2 + ẏ2)− I1θ̇ 2 − I2(θ̇ + ψ̇)θ̇ + I2(θ̇ + ψ̇)ψ̇ ) J−1 L (µ) = ( −m(ẋ2 + ẏ2)− (I1 + I2)θ̇2 + I2ψ̇ 2 ) J−1 L (µ) = − 1 m (1 + µ2 2)− ( µ3 − (xµ2 − y)− I2ψ̇ )2 I1 + I2 + I2ψ̇ 2 = I1I2 I1 + I2 ψ̇2 + 2I2 µ3 − (xµ2 − y) I1 + I2 ψ̇ − (µ3 − (xµ2 − y))2 I1 + I2 . Routh Reduction by Stages 27 In the last step we have left out some constant terms. The reduced Lagrangian is then ob- tained by taking the quotient w.r.t. the action of Gµ. This is done by applying the coordinate transformation introduced above. We get: L0 = 1 2 I1I2 I1 + I2 ψ̇2 + I2 µ3 + y′ I1 + I2 ψ̇ − ( V (ψ) + 1 2 (µ3 + y′)2 I1 + I2 ) , which is clearly independent of x′. The 2-form B0 is obtained by reducing the 2-form d ((dx+ ydθ) + µ2(dy − xdθ) + µ3dθ) = d(y − µ2x) ∧ dθ. Using the coordinate change we get B0 = dy′ ∧ dθ. In this example the Routhian L0 depends on the velocity corresponding to the coordinate ψ on S1, but is independent of the velocities corresponding to the two remaining coordinates (y′, θ). With the above, the reduced Euler–Lagrange equations take the form ẏ′ = 0, θ̇ = 1 I1 + I2 (y′ + µ3 − I2ψ̇), ψ̈ = −I1 + I2 I1I2 V ′ − 1 I1 ẏ′. 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 R2 of translations in the x and y direction. Let us denote the symmetry group by K = R2 and study the quotient spaces. We will use the same notations as before: the Lie algebra elements e1, e2 denote a basis for the subalgebra K of K in g. The momentum map for this action is now JK = i∗ ◦ JL = mẋe1 + mẏe2. We choose ν to be the projection of the momentum µ we had used in the full reduction: let ν = e1 +µ2e 2 ∈ g∗. Since K is Abelian, Kν = K and the quotient space is SE(2)× S1/R2 = S1 × S1. If we choose A1 = dxe1 + dye2 to be the trivial connection, we simply get B1 = 0. The Routhian L1 can now be obtained from 2(L1 + V ) = ( −pxẋ− pyẏ + pθθ̇ + pψψ̇ ) J−1 L (ν) = ( −mẋ2 −mẏ2 + I1θ̇ 2 + I2(θ̇ + ψ̇)2 ) J−1 L (ν) = I1θ̇ 2 + I2(θ̇ + ψ̇)2, where we ignored again some constant terms. The Routh reduced system is now a standard Lagrangian system on S1 × S1 with Lagrangian L1 = 1 2I1θ̇ 2 + 1 2I2(θ̇ + ψ̇)2 − V (ψ) (see the paragraphs on Abelian Routh reduction). Its equations of motion are θ̈ = 1 I1 V ′, ψ̈ = −I1 + I2 I1I2 V ′. For this example there is actually no second stage: the group Ḡν = Gν/Kν is the trivial one {e}, and the vector space ḡν = gν/Kν 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 µ3. We now show that the two ways of reducing the system are equivalent. Equivalence between direct reduction and reduction by stages. Let us compute the diffeomorphism F for this example. Here, it is a map J−1 L (µ)/Gµ → J−1 K (ν)/K that is obtained by projection of the inclusion map J−1 L (µ)→ J−1 K (ν). The latter equals, in coordinates (x, y, θ, ψ, ψ̇) 7→ ( x, y, θ, ψ, θ̇ = 1 I1 + I2 (µ3 − (xµ2 − y)− I2ψ̇), ψ̇ ) . 28 B. Langerock, T. Mestdag and J. Vankerschaver The map is reducible, and after taking the quotient it becomes F : J−1 L (µ)/Gµ → J−1 K (ν)/K; (y′, θ, ψ, ψ̇) 7→ ( θ, ψ, θ̇ = 1 I1 + I2 (µ3 + y′ − I2ψ̇), ψ̇ ) . This diffeomorphism maps the Gµ-reduced system on the Kν-reduced system, as is obvious from the respective equations of motion. 7.2 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 [ px py ] = Γ [ −vy vx ] , where [ px py ] = M [ vx vy ] . (7) 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 R3, equipped with the multi- plication (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 R2 con- structed by means of the cocycle B((x, y), (x′, y′)) = 1 2(xy′ − yx′). We now introduce the infinitesimal cocycle C : R2 × R2 → R, given by C((vx, vy), (v ′ x, v ′ y)) = vxv ′ y − vyv′x. For the relation between B and C, see for instance [12]. The Lie algebra h of H can then be identified with R3 with the bracket [(vx, vy, v), (v′x, v ′ y, v ′)] = (0, 0, C((vx, vy), (v ′ x, v ′ y))) = (0, 0, vxv ′ y − vyv′x). The dual Lie algebra h∗ can again be identified with R3, with coordinates (px, py, p) and duality pairing 〈(px, py, p), (vx, vy, v)〉 = pxvx + pyvy + pv. We now induce the following quadratic Lagrangian ` on h: `(vx, vy, v) = 1 2 [vx, vy]M [ vx vy ] + v2 2 , Routh Reduction by Stages 29 and we define L on H by left extension: L(g, ġ) = `(g−1ġ), or explicitly L(x, y, s; ẋ, ẏ, ṡ) = 1 2 [ẋ, ẏ]M [ ẋ ẏ ] + 1 2 ( ṡ− 1 2 (xẏ − yẋ) )2 . (8) The Euler–Poincaré equations obtained from ` are given by d dt pxpy p  = p −vyvx 0  . 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 ∼= R2 we consider the connection one-form given at the identity by A(e)(vx, vy, 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 JL : TH → R for the R-action on the tangent bundle TH is given by JL(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) with ṡ = Γ + (xẏ − yẋ)/2. The isotropy subgroup RΓ = R acts on this level set by translations in the s-direction, so that the reduced velocity space is given by J−1(Γ)/RΓ = TR2. 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 TR2: L1(x, y; ẋ, ẏ) = 1 2 [ẋ, ẏ]M [ ẋ ẏ ] , (9) up to constant terms. Second reduction. We now perform reduction with respect to the remaining symmetry group, H/R ∼= R2, using the results from Section 5.3. We have a left invariant magnetic Lagrangian system on the group R2, with Lagrangian (9) and magnetic form BΓ = −Γdx ∧ dy. The potential δ : R2 → R2 corresponding to the latter is given by δ(x, y) = Γ [ −y x ] , and the momentum map is therefore J2(x, y, ẋ, ẏ) = M(ẋ, ẏ)T − δ(x, y). The non-equivariance 2-cocycle of the momentum map is Σδ = BΓ. The affine action of R2 on itself is given by (x, y) · (px, py) = (px − Γy, py + Γx). If we fix a momentum value (λ, µ) ∈ R2, the isotropy group R2 (λ,µ) of the affine action consists of just the zero element, and consequentially the twice-reduced space J−1 2 (λ, µ)/R2 (λ,µ) is nothing but R2. 30 B. Langerock, T. Mestdag and J. Vankerschaver The reduced Euler–Lagrange equations (3) in the case of a left action, assume the following form d dt [ px py ] = −i(vx,vy)Σδ = Γ [ −vy vx ] , with (px, py) T = M(vx, vy) T , and these are nothing but the equations (7). Acknowledgements BL is an honorary postdoctoral researcher at the Department of Mathematics of Ghent Uni- versity and associate academic staff at the Department of Mathematics of K.U.Leuven. BL is sponsored by a Research Programme of the Research Foundation – Flanders (FWO). Part of this work was supported by the Sint-Lucas department of Architecture, K.U.Leuven Association. TM is a Postdoctoral Fellow of the Research Foundation – Flanders (FWO). JV is a postdoc at the Department of Mathematics of UC San Diego, partially supported by NSF CAREER award DMS-1010687 and NSF FRG grant DMS-1065972, and is on leave from a Postdoctoral Fellow- ship of the Research Foundation–Flanders. This work is part of the irses project geomech (nr. 246981) within the 7th European Community Framework Programme. We are indebted to F. Cantrijn, M. Crampin and E. Garćıa-Toraño Andres for many useful discussions. We thank one of the referees for pointing out reference [16] on the reduction hypothesis. References [1] Abraham R., Marsden J.E., Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. [2] Adamec L., A route to Routh – the classical setting, J. Nonlinear Math. Phys. 18 (2011), 87–107. [3] Arnold V.I., Kozlov V.V., Neishtadt A.I., Mathematical aspects of classical and celestial mechanics, Springer- Verlag, Berlin, 1997. [4] Cendra H., Marsden J.E., Ratiu T.S., Lagrangian reduction by stages, Mem. Amer. Math. Soc. 152 (2001), no. 722. [5] Cortés J., de León M., Marrero J.C., Mart́ın de Diego D., Mart́ınez E., A survey of Lagrangian mecha- nics and control on Lie algebroids and groupoids, Int. J. Geom. Methods Mod. Phys. 3 (2006), 509–558, math-ph/0511009. [6] Crampin M., Mestdag T., Routh’s procedure for non-Abelian symmetry groups, J. Math. Phys. 49 (2008), 032901, 28 pages, arXiv:0802.0528. [7] Echeverŕıa-Enŕıquez A., Muñoz-Lecanda M.C., Román-Roy N., Reduction of presymplectic manifolds with symmetry, Rev. Math. Phys. 11 (1999), 1209–1247, math-ph/9911008. [8] Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. I, Interscience Publishers, New York – London, 1963. Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. II, Interscience Publishers, New York – London, 1969. [9] Lamb H., Hydrodynamics, Reprint of the 1932 6th ed., Cambridge Mathematical Library, Cambridge Uni- versity Press, Cambridge, 1993. [10] Langerock B., Cantrijn F., Vankerschaver J., Routhian reduction for quasi-invariant Lagrangians, J. Math. Phys. 51 (2010), 022902, 20 pages, arXiv:0912.0863. [11] Langerock B., Castrillón Lopéz M., Routh reduction for singular Lagrangians, Int. J. Geom. Methods Mod. Phys. 7 (2010), 1451–1489, arXiv:1007.0325. [12] Marsden J.E., Misio lek G., Ortega J.P., Perlmutter M., Ratiu T.S., Hamiltonian reduction by stages, Lecture Notes in Mathematics, Vol. 1913, Springer, Berlin, 2007. [13] Marsden J.E., Montgomery R., Ratiu T.S., Reduction, symmetry, and phases in mechanics, Mem. Amer. Math. Soc. 88 (1990), no. 436. http://dx.doi.org/10.1142/S1402925111001180 http://dx.doi.org/10.1142/S0219887806001211 http://arxiv.org/abs/math-ph/0511009 http://dx.doi.org/10.1063/1.2885077 http://arxiv.org/abs/0802.0528 http://dx.doi.org/10.1142/S0129055X99000386 http://arxiv.org/abs/math-ph/9911008 http://dx.doi.org/10.1063/1.3277181 http://dx.doi.org/10.1063/1.3277181 http://arxiv.org/abs/0912.0863 http://dx.doi.org/10.1142/S0219887810004907 http://dx.doi.org/10.1142/S0219887810004907 http://arxiv.org/abs/1007.0325 Routh Reduction by Stages 31 [14] Marsden J.E., Ratiu T.S., Scheurle J., Reduction theory and the Lagrange–Routh equations, J. Math. Phys. 41 (2000), 3379–3429. [15] Mestdag T., Crampin M., Invariant Lagrangians, mechanical connections and the Lagrange–Poincaré equa- tions, J. Phys. A: Math. Theor. 41 (2008), 344015, 20 pages, arXiv:0802.0146. [16] Mikityuk I.V., Stepin A.M., A sufficient condition for stepwise reduction: proof and applications, Sb. Math. 199 (2008), 663–671. [17] Milne-Thomson L., Theoretical hydrodynamics, 5th ed., MacMillan, London, 1968. [18] Montgomery R., A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, Vol. 91, American Mathematical Society, Providence, RI, 2002. [19] Pars L.A., A treatise on analytical dynamics, Heinemann Educational Books Ltd., London, 1965. [20] Vankerschaver J., Kanso E., Marsden J.E., The dynamics of a rigid body in potential flow with circulation, Regul. Chaotic Dyn. 15 (2010), 606–629, arXiv:1003.0080. http://dx.doi.org/10.1063/1.533317 http://dx.doi.org/10.1088/1751-8113/41/34/344015 http://arxiv.org/abs/0802.0146 http://dx.doi.org/10.1070/SM2008v199n05ABEH003938 http://dx.doi.org/10.1134/S1560354710040143 http://arxiv.org/abs/1003.0080 1 Introduction 2 Preliminaries on Routh reduction 3 Magnetic Lagrangian systems 4 Magnetic Lagrangian systems with symmetry 4.1 Symplectic reduction: a brief introduction 4.2 Invariant magnetic Lagrangian systems and momentum maps 5 Routh reduction for magnetic Lagrangian systems 5.1 Magnetic cotangent bundle reduction 5.2 Routh reduction for magnetic Lagrangian systems 5.3 Reduction of magnetic Lagrangian systems on Lie groups 6 Routh reduction by stages 6.1 Symplectic reduction by stages 6.2 Routh reduction by stages 7 Examples 7.1 Elroy's Beanie 7.2 Rigid bodies on the Heisenberg group References

Routh Reduction by Stages

Репозитарії

Схожі ресурси