Tulczyjew's Triplet for Lie Groups III : Higher Order Dynamics and Reductions for Iterated Bundles
aa r X i v : . [ m a t h . S G ] F e b TULCZYJEW’S TRIPLET FOR LIE GROUPS III:HIGHER ORDER DYNAMICS AND REDUCTIONS FOR ITERATED BUNDLES
O ˘GUL ESEN, HASAN G ¨UMRAL, AND SERKAN S ¨UTL ¨U
Abstract.
Given a Lie group G , we elaborate the dynamics on T ∗ T ∗ G and T ∗ T G , which is givenby a Hamiltonian, as well as the dynamics on the Tulczyjew symplectic space
T T ∗ G , which may bedefined by a Lagrangian or a Hamiltonian function. As the trivializations we adapted respect the groupstructures of the iterated bundles, we exploit all possible subgroup reductions (Poisson, symplectic orboth) of higher order dynamics. MSC2010:
Key Words:
Euler-Poincar´e equations; Lie-Poisson equations, higher order dynamics on Lie groups.
Contents
1. Introduction 11.1. Trivializations 21.2. Content of the work 32. Geometry of iterated bundles 42.1. The first order tangent group
T G T ∗ G T ∗ T G T ∗ T ∗ G T T ∗ G
83. Dynamics on the First Order Bundles 93.1. Lagrangian dynamics on tangent group
T G T ∗ G T ∗ T G T ∗ T G by G T ∗ T G by g T ∗ T G by G s g T ∗ T ∗ G T ∗ T ∗ G by G T ∗ T ∗ G by g ∗ T ∗ T ∗ G by G s g ∗ T T ∗ G T T ∗ G T T ∗ G Introduction
The tangent and the cotangent bundles of a Lie group admit global trivializations, as well as theLie group structures, induced from the underlying Lie group itself. These structures may further becarried over the iterated bundles T ∗ T G , T T ∗ G , and T ∗ T ∗ G . These iterated bundles constitute the ulczyjew’s triplet, introduced for a geometric description of the Legendre transformation from theLagrangian description on T G to the Hamiltonian description on T ∗ G for a mechanical system having G as the configuration space. Such a system admits G as kinematical symmetries, and the reduction ofthe Lagrangian dynamics results in the Euler-Poincr´e equations on the Lie algebra g of G . Similarly,the reduction of the Hamiltonian dynamics to g ∗ is described by the Lie-Poisson equations.The present note is intended as a sequel to [15, 16]. In the first part [15], we gave a detailed descriptionof the possible trivializations of the iterated bundles T ∗ T G , T T ∗ G , and T ∗ T ∗ G , which are Lie groupisomorphisms. Moreover, we described the group structures up to the second iterated bundles, as wellas the canonical involutions on them. Having explicit descriptions of the cotangent and the Tulczyjewsymplectic structures, we performed the Marsden-Weinstein reduction by kinematical symmetries toobtain the reduced Tulczyjew triplet for the Legendre transformation from Euler-Poincar´e to Lie-Poisson equations. Then, in the second part [16], we studied the Lagrangian and the Hamiltoniandynamical equations at each stage of the Tulczyjew construction under the trivializations respectingthe Lie group structures. The dynamics we considered is defined either by a Lagrangian on T G , or bya Hamiltonian on T ∗ G , which, in the framework of Tulczyjew construction, corresponds to Lagrangiansubmanifolds of T ∗ T G or T ∗ T ∗ G , respectively. In other words, first order dynamics considered in [16]restricts to the fiber coordinates of the second iterated bundles.In this work, we aim to give a complete description of the higher order dynamics, and their reductionsby considering the Lagrangian and/or Hamiltonian functions on the second iterated bundles, taking fulladvantage of the trivializations at our disposal. Obviously, releasing the condition that the dynamicson iterated bundles are described by Lagrangian submanifolds opens up the possibility to obtain higherorder forms of Euler-Poincar´e and Lie-Poisson equations. The underlying structure will, indeed, offermore than this generalization.Immediate generalizations of the results of [15], [16], and the present work apply to fibered spacesadmitting local trivializations, or Ehresmann connections. A recent work [17] elaborates the parallelresults in the particular case of the principal G -bundles, and their associated vector bundles.1.1. Trivializations.
One observes that, the form of equations governing dynamics on Lie groups depends on the kindof trivializations adapted on iterated bundles [11, 12, 19, 33]. Additional terms in these equationsmay or may not appear depending on whether trivialization preserves semidirect product and groupstructures or not. If one preserves the group structures, canonical embeddings of factors involvingtrivialization defines subgroups of iterated bundles and reductions of dynamics with these subgroupsbecome possible.Based on exhaustive investigation of trivializations in our previous work [15], we shall present allreductions of dynamics on iterated bundles of a Lie group with the convenient trivialization of thefirst kind. In trivialization of the first kind, we identify tangent
T G and cotangent T ∗ G bundleswith their semidirect product trivializations G s g and G s g ∗ , respectively. Then, we trivialize theiterated bundles T ( G s g ) , T ( G s g ∗ ) , T ∗ ( G s g ) and T ∗ ( G s g ∗ ) by considering them as tangent andcotangent groups again. As an example, we obtain(1.1) T T ∗ G ≃ T ( G s g ∗ ) ≃ ( G s g ∗ ) s Lie ( G s g ∗ ) ≃ ( G s g ∗ ) s ( g s g ∗ ) or which, the trivialization maps preserve lifted group structures thereby making possible variousreductions of dynamics. On the other hand, in trivialization of the second kind, one distributesfunctors T and T ∗ to G s g and G s g ∗ , obtains products of first order bundles and then, trivializeseach factor involving the products. This results in, for example,(1.2) T T ∗ G ≃ T ( G s g ∗ ) → T G s T g ∗ ≃ ( G s g ) s ( g ∗ × g ∗ )for which distributions of functors mix up orders of fibrations, and do not preserve group structures[15]. Throughout this work we shall use trivialization of the first kind unless otherwise stated. Asubscript of g and g ∗ will show its position in the original trivialization of iterated bundle.1.2. Content of the work.
Here is a brief description of what we present in each section.Section 2. This section is intended as a reference section of the present work. Notations and conven-tions are fixed. Trivializations of all spaces
T G , T ∗ G , T ∗ T G , T T ∗ G , T ∗ T ∗ G , and their induced groupstructures are defined. Subgroups are listed. Subgroups with symplectic actions are identified. Thetrivialized form of the symplectic two-forms, as well as the associated one-forms and the invariantvector fields on the cotangent bundles and the Tulczyjew’s symplectic space T T ∗ G are given.Section 3. The dynamics on the first order (both tangent and cotangent) bundles are considered. Thefirst order Lagrangian and Hamiltonian dynamics on T G and T ∗ G are described by Euler-Lagrangeand Hamilton’s equations ddt δ ¯ Lδξ = T ∗ e R g δ ¯ Lδg − ad ∗ ξ δ ¯ Lδξ , (1.3) dgdt = T e R g (cid:18) δ ¯ Hδµ (cid:19) , dµdt = ad ∗ δ ¯ Hδµ µ − T ∗ e R g δ ¯ Hδg , (1.4)respectively. Reduction of (1.3) by G gives the Euler-Poincar´e equations. Poisson and Marsden-Weinstein reductions on T ∗ G are performed to obtain the Lie-Poisson equations.Section 4. Hamiltonian dynamics on T ∗ T G is given by the equations (cid:18) ddt − ad ∗ δHδµ (cid:19) (cid:0) ad ∗ ξ ν − µ (cid:1) = T ∗ e R g δHδg , dgdt = T e R g δHδµ equivalent to four component Hamilton’s equations. There are remarkable differences arising from theuse of different trivializations. Reductions by G , g and G s g are performed. Structures of the reducedspaces are studied in detail.Section 5. Hamiltonian dynamics on T ∗ T ∗ G is generated by the vector fields with components of theform dgdt = T e R g (cid:18) δHδν (cid:19) , dµdt = δHδξ + ad ∗ δHδν µ,dνdt = ad ∗ δHδµ µ + ad ∗ δHδν ν − T ∗ e R g (cid:18) δHδg (cid:19) − ad ∗ ξ δHδξ , dξdt = − δHδµ + [ ξ, δHδν ] . Reductions by G , g ∗ and G s g ∗ are performed. Structures of the reduced spaces are exhibited in details.The correspondence between the dynamics on T ∗ T ∗ G , and on T ∗ T G , is established by symplecticdiffeomorphisms and Poisson maps. ection 6. On T T ∗ G , there are both Lagrangian and Hamiltonian formalisms. If a function E on T T ∗ G is regarded as a Hamiltonian, then the Hamilton’s equations with the Tulczyjew symplecticstructure are ˙ g = T R g (cid:18) δEδν (cid:19) , ˙ µ = − δEδξ , ˙ ξ = δEδµ , ˙ ν = ad ∗ δEδν ν − T ∗ R g (cid:18) δEδg (cid:19) . Reduction by G results in reduced Tulczyjew triplet considered in [16] before. Reductions of Tulczyjewstructure by g , by a symplectic action of g ∗ that may be connected with a symplectic diffeomorphismfrom T T ∗ G to T ∗ T ∗ G , by G s g and by G s g ∗ are studied in detail.If the function E on T T ∗ G is regarded as a Lagrangian density, it then gives the Euler-Lagrangedynamics ddt (cid:18) δEδξ (cid:19) = T ∗ e R g (cid:18) δEδg (cid:19) − ad ∗ δEδµ µ + ad ∗ ξ (cid:18) δEδξ (cid:19) − ad ∗ δEδν νddt (cid:18) δEδν (cid:19) = δEδµ − ad ξ δEδν . Reductions of these equations by G , g ∗ and G s g ∗ are described. The latter gives the Euler-Poincar´eequations on g s g ∗ . 2. Geometry of iterated bundles
Let G be a Lie group, g = Lie ( G ) ≃ T e G be its Lie algebra, and g ∗ = Lie ∗ ( G ) be the dual of g . Weshall adapt the letters(2.1) g, h ∈ G, ξ, η, ζ ∈ g , µ, ν, λ ∈ g ∗ as elements of the spaces shown. For a tensor field which is either right or left invariant, we shalluse V g ∈ T g G , α g ∈ T ∗ g G , etc... We shall denote left and right multiplications on G by L g and R g , respectively. The right inner automorphism I g = L g − ◦ R g is a right representation of G on G satisfying I g ◦ I h = I hg . The right adjoint action Ad g = T e I g of G on g is defined as the tangent map of I g at the identity e ∈ G . The infinitesimal right adjoint representation ad ξ η is [ ξ, η ] and is defined asderivative of Ad g over the identity. A right invariant vector field X Gξ generated by ξ ∈ g is of the form X Gξ ( g ) = T e R g ξ . The identity [ ξ, η ] = [ X Gξ , X Gη ] JL defines the isomorphism between g and the space X R ( G ) of right invariant vector fields endowed with the Jacobi-Lie bracket. The coadjoint action Ad ∗ g of G on the dual g ∗ of the Lie algebra g is a right representation and is the linear algebraic dual of Ad g − , namely,(2.2) (cid:10) Ad ∗ g µ, ξ (cid:11) = (cid:10) µ, Ad g − ξ (cid:11) holds for all ξ ∈ g and µ ∈ g ∗ . The inverse element g − appears in the definition (2.2) in order tomake Ad ∗ g a right action. The infinitesimal coadjoint action ad ∗ ξ of g on g ∗ is the linear algebraic dualof ad ξ . Note that, the infinitesimal generator of the coadjoint action Ad ∗ g is minus the infinitesimalcoadjoint action ad ∗ ξ , that is, if g t ⊂ G is a curve passing through the identity in the direction of ξ ∈ g , then(2.3) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 Ad ∗ g t µ = − ad ∗ ξ µ. n the diagrams of this work, EL and EP will abbreviate Euler-Lagrange and Euler-Poincar´e equations,respectively, and PR, SR, LR, EPR and EPR will denote Poisson, symplectic, Lagrangian, and Euler-Poincar´e reductions, respectively.2.1. The first order tangent group
T G . The trivialization(2.4) tr T G : T G → G s g , V g ( g, T g R g − V g ) =: ( g, ξ ) , enables us to endow T G with the semi-direct product group structure on G s g given by(2.5) ( g, ξ (1) )(˜ g, ˜ ξ (1) ) = ( g ˜ g, ξ (1) + Ad g ˜ ξ (1) ) , for any ξ (1) , ˜ ξ (1) ∈ g = g . Accordingly, the Lie algebra of T G ∼ = G s g is the semi-direct sum Liealgebra g s g := g s g with the Lie bracket(2.6) [( ξ (2) , ξ (3) ) , ( ˜ ξ (2) , ˜ ξ (3) )] = ([ ξ (2) , ˜ ξ (2) ] , ad ξ (2) ˜ ξ (3) − ad ˜ ξ (2) ξ (3) )for any ξ (2) , ˜ ξ (2) ∈ g = g , and any ξ (3) , ˜ ξ (3) ∈ g = g . Here, indices on the Lie algebra g serve todistinguish the copies of g . For further details on tangent group see [21, 26, 33, 40, 41].2.2. The first order cotangent group T ∗ G . The cotangent bundle T ∗ G can also be endowed with a group structure borrowed from the semi-directproduct group G s g ∗ via the right trivialization(2.7) tr T ∗ G : T ∗ G → G s g ∗ , α g ( g, T ∗ e R g α g ) . The group operation on G s g ∗ is(2.8) ( g , µ )( g , µ ) := (cid:16) g g , µ + Ad ∗ g µ (cid:17) . On the other hand, we can use to pull back the canonical 1-form θ T ∗ G and the symplectic 2-form Ω T ∗ G on the cotangent bundle T ∗ G thereby decorating G s g ∗ with the structure of an exact symplecticmanifold symplectic 2-form Ω G s g ∗ and a potential 1-form θ ( λ,η ) G s g ∗ .A right invariant vector field X G s g ∗ ( ξ,ν ) on G s g ∗ corresponding to ( ξ, ν ) ∈ g s g ∗ at a point ( g, µ ) is givenby [18, App.B. (B.9)] X G s g ∗ ( ξ,ν ) ( g, µ ) = (cid:16) T e R g ξ, ν + ad ∗ ξ µ (cid:17) . Accordingly, the values of the canonical 1-form θ ( λ,η ) G s g ∗ and the symplectic 2-form Ω G s g ∗ on a rightinvariant vector field are [1, 3, 18, 29] D θ ( λ,η ) G s g ∗ , X G s g ∗ ( ξ,ν ) E ( g, µ ) = h λ, ξ i + h ν, η i , (2.9) D Ω G s g ∗ ; (cid:16) X G s g ∗ ( ξ ,ν ) , X G s g ∗ ( ξ ,ν ) (cid:17)E ( g, µ ) = h ν , ξ i − h ν , ξ i − h µ, [ ξ , ξ ] i . (2.10) Remark 2.1.
The symplectic 2-form Ω G s g ∗ is not conserved under the group operation (2.8), assuch, G s g ∗ is not a symplectic Lie group as defined in [28].2.3. The cotangent group of tangent group T ∗ T G . .3.1. Trivialization.
The global trivialization of T ∗ T G ≃ T ∗ ( G s g ) can be achieved by trivializing T ∗ ( G s g ) into the semidirect product group G s g and the dual g ∗ × g ∗ of its Lie algebra g s g tr T ∗ ( G s g ) : T ∗ ( G s g ) → ( G s g ) s ( g ∗ × g ∗ ): ( α g , α ξ ) → (cid:0) g, ξ, T ∗ e R g ( α g ) + ad ∗ ξ α ξ , α ξ (cid:1) (2.11)which preserves the group multiplication rule( g, ξ, µ , µ ) ( h, η, ν , ν )= (cid:16) gh, ξ + Ad g η, µ + Ad ∗ g (cid:16) ν + ad ∗ Ad g − ξ ν (cid:17) , µ + Ad ∗ g ν (cid:17) = (cid:0) gh, ξ + Ad g η, µ + Ad ∗ g ν + ad ∗ ξ Ad ∗ g ν , µ + Ad ∗ g ν (cid:1) (2.12)on T ∗ T G and results in the following subgroups.
Proposition 2.2.
The canonical immersions of the following submanifolds G, g , g ∗ , g ∗ , G s g , G s g ∗ , G s g ∗ , g ∗ × g ∗ , g s ( g ∗ × g ∗ ) , ( G s g ) s g ∗ , G s ( g ∗ × g ∗ )(2.13) define subgroups of T ∗ T G and hence they act on T ∗ T G by actions induced from the multiplication inEq.(2.12).
Here, the group structure on G s g is the one given in (2.5) whereas the group structure on are in theform of Eq.(2.8) and, we obtain the multiplications( g, ξ, µ ) ( h, η, ν ) = ( gh, ξ + Ad g η, µ + Ad ∗ g ν )(2.14) ( g, µ , µ ) ( h, ν , ν ) = ( gh, µ + Ad ∗ g ν , µ + Ad ∗ g ν )(2.15) ( ξ, µ , µ ) ( η, ν , ν ) = ( ξ + η, µ + ν + ad ∗ ξ ν , µ + ν )(2.16)defining the group structures on ( G s g ) s g ∗ , G s ( g ∗ × g ∗ ) and g s ( g ∗ × g ∗ ), respectively.2.3.2. Symplectic Structure.
By requiring the trivialization tr T ∗ ( G s g ) be a symplectic map, we define acanonical one-form θ T ∗ T G and a symplectic two-form Ω T ∗ T G on the trivialized cotangent bundle T ∗ T G .To this end, we recall that a right invariant vector field X T ∗ T G ( η ,η ,ν ,ν ) on T ∗ T G is generated by anelement ( η , η , ν , ν ) in the Lie algebra ( g s g ) s ( g ∗ × g ∗ ) of T ∗ T G by means of the tangent lift ofright translation on T ∗ T G . At a point ( g, ξ, µ , µ ) in T ∗ T G , the value of such a right invariant vectorfield reads(2.17) X T ∗ T G ( η ,η ,ν ,ν ) ( g, ξ, µ , µ ) = (cid:0) T e R g η , η + ad η ξ, ν + ad ∗ η µ + ad ∗ η µ , ν + ad ∗ η µ (cid:1) and is an element of the fiber T ( g,ξ,µ ,µ ) ( T ∗ T G ). The values of canonical forms θ T ∗ T G and Ω T ∗ T G onright invariant vector fields can then be computed as h θ ( ν ,ν ,η ,η ) T ∗ T G ; X T ∗ T G ( ξ ,ξ ,µ ,µ ) i = h ν , ξ i + h ν , ξ i + h µ , η i + h µ , η i (2.18) D Ω T ∗ T G ; (cid:16) X T ∗ T G ( ξ ,ξ ,µ ,µ ) , X T ∗ T G ( η ,η ,ν ,ν ) (cid:17)E ( g, ξ, λ , λ ) = (cid:10) µ + ad ∗ ξ λ + ad ∗ ξ λ , η (cid:11) − h ν , ξ i + (cid:10) µ + ad ∗ ξ λ , η (cid:11) − h ν , ξ i . The musical isomorphism Ω ♭T ∗ T G , induced from the symplectic two-form Ω T ∗ T G , maps the tangentbundle T ( T ∗ T G ) to the cotangent bundle T ∗ ( T ∗ T G ). It takes the right invariant vector field in q.(2.17) to an element of the cotangent bundle T ∗ ( g,ξ,µ ,µ ) ( T ∗ T G ) with coordinatesΩ ♭T ∗ T G (cid:16) X T ∗ T G ( η ,η ,λ ,λ ) ( g, ξ, µ , µ ) (cid:17) = T ∗ ( g,ξ ) R ( g,ξ ) − ( λ , λ ) , − ( η , η )= (cid:16) T ∗ g R g − λ − ad ∗ Ad g − ξ λ , λ , − η , − η (cid:17) . (2.19) Remark 2.3.
The actions of the subgroups g ∗ and g ∗ are not symplectic, nor are any subgroup in thelist of Eq.(2.13) containing g ∗ and g ∗ . There remains only the action of the group G s g to performsymplectic reduction on T ∗ T G .2.4.
The cotangent group of cotangent group T ∗ T ∗ G . The global trivialization of the iteratedcotangent bundle can be achieved by semidirect product of the group G s g ∗ and the dual g ∗ × g ofits Lie algebra [15]. The trivialization map tr T ∗ T ∗ G : T ∗ ( G s g ∗ ) → ( G s g ∗ ) s ( g ∗ × g ): ( α g , α µ ) → (cid:16) g, µ, T ∗ e R g ( α g ) − ad ∗ α µ µ, α µ (cid:17) (2.20)implies on T ∗ T ∗ G , the group multiplication rule(2.21) ( g, µ , µ , ξ ) ( h, ν , ν , η ) = (cid:16) gh, µ + Ad ∗ g ν , µ + Ad ∗ g ν − ad ∗ Ad g η µ , ξ + Ad g η (cid:17) . Proposition 2.4.
Embeddings of following subspaces G, g ∗ , g ∗ , g , G s g ∗ , G s g ∗ , G s g , g ∗ s g ∗ , g ∗ × g , ( G s g ∗ ) s g ∗ , G s ( g ∗ × g ) , g ∗ s ( g ∗ × g )(2.22) define subgroups of T ∗ T ∗ G and hence they act on T ∗ T ∗ G by actions induced from the multiplicationin Eq.(2.21). The group structures on G s ( g ∗ × g ), G s g ∗ s g ∗ , g ∗ s ( g ∗ × g ) are (up to some reordering) given byEqs.(2.14), (2.15) and (2.16), respectively.2.4.1. Symplectic Structure on T ∗ T ∗ G . The canonical one-form and the symplectic two form on T ∗ T ∗ G can be mapped by tr T ∗ T ∗ G to T ∗ T ∗ G based on the fact that the trivialization map is a symplectic diffeo-morphism. Consider a right invariant vector field X T ∗ T ∗ G ( η ,ν ,ν ,η ) generated by an element ( η , ν , ν , η )in the Lie algebra ( g s g ∗ ) s ( g ∗ × g ) of T ∗ T ∗ G . At the point ( g, µ , µ , ξ ) , the right invariant vector(2.23) X T ∗ T ∗ G ( η ,ν ,ν ,η ) ( g, µ , µ , ξ ) = (cid:0) T R g η , ν + ad ∗ η µ , ν + ad ∗ η µ − ad ∗ ξ ν , η + ad η ξ (cid:1) is an element of T ( g,µ ,µ ,ξ ) ( T ∗ T ∗ G ) . The values of canonical forms θ T ∗ T ∗ G and Ω T ∗ T ∗ G at rightinvariant vector fields can now be evaluated to be h θ ( ν ,η ,η ,ν ) T ∗ T ∗ G , X T ∗ T ∗ G ( ξ ,µ ,µ ,ξ ) i = h ν , ξ i + h µ , η i + h µ , η i + h ν , ξ i (2.24) h Ω T ∗ T ∗ G ; (cid:16) X T ∗ T ∗ G ( ξ ,µ ,µ ,ξ ) , X T ∗ T ∗ G ( η ,ν ,ν ,η ) (cid:17) i ( g, λ , λ , ζ ) = (cid:10) µ + ad ∗ ξ λ − ad ∗ ζ µ , η (cid:11) − h µ , η i + (cid:10) − ν + ad ∗ ζ ν , ξ (cid:11) + h ν , ξ i . (2.25)The musical isomorphism Ω ♭T ∗ T ∗ G , induced from the symplectic two-form Ω T ∗ T ∗ G in Eq.(2.25), maps T ( T ∗ T ∗ G ) to T ∗ ( T ∗ T ∗ G ). At the point ( g, µ , µ , ξ ), Ω ♭T ∗ T ∗ G takes the vector in Eq.(2.23) to theelement Ω ♭T ∗ T ∗ G (cid:16) X T ∗ T ∗ G ( η ,ν ,ν ,η ) (cid:17) = (cid:16) T ∗ ( g,µ ) R ( g,µ ) − ( ν , η ) , − ( η , ν ) (cid:17) = (cid:16) T ∗ R g − ( ν ) − ad ∗ η Ad ∗ g − µ , η , − η , − ν (cid:17) (2.26) n T ∗ ( g,µ ,µ ,ξ ) ( T ∗ T ∗ G ) . Remark 2.5.
Actions of subgroups g ∗ and g , and hence any subgroup in the list (2.22) containing g ∗ and g , are not symplectic. Thus, there remains only the action of the group G s g ∗ to performsymplectic reduction on T ∗ T ∗ G .2.5. The tangent group of cotangent group
T T ∗ G . T T ∗ G ≃ T ( G s g ∗ ) can be trivialized as semidirect product of the group G s g ∗ and its Lie algebra g s g ∗ by tr T T ∗ G : T ( G s g ∗ ) → ( G s g ∗ ) s ( g s g ∗ ): ( V g , V µ ) → (cid:16) g, µ, T R g − V g , V µ − ad ∗ T R g − V g µ (cid:17) , (2.27)where ( V g , V µ ) ∈ T ( g,µ ) ( G s g ∗ ) [15]. The group multiplication on T T ∗ G is( g, µ , ξ, µ ) ( h, ν , η, ν )= (cid:16) gh, µ + Ad ∗ g ν , ξ + Ad g η, µ + Ad ∗ g ν − ad ∗ Ad g η µ (cid:17) (2.28)and embedded subgroups of T T ∗ G follow. Proposition 2.6.
The embeddings of the subspaces G, g ∗ , g , g ∗ , G s g ∗ , G s g , G s g ∗ , g ∗ s g ∗ , g s g ∗ , ( G s g ∗ ) s g ∗ , G s ( g s g ∗ ) , g ∗ s ( g s g ∗ )(2.29) of T T ∗ G define its subgroups. The group structures on G s g , G s g ∗ are defined by Eqs.(2.5) and (2.8),respectively. The group structures on ( G s g ∗ ) s g ∗ , G s ( g s g ∗ ) and g ∗ s ( g s g ∗ ) are defined (up tosome reordering) by Eqs.(2.15),(2.14) and (2.16), respectively. The group multiplications on g ∗ , g , g ∗ , g ∗ × g ∗ and g × g ∗ are vector additions. Tulczyjew symplectic strcuture on
T T ∗ G . T T ∗ G is central in Tulczyjew’s triplet and carriesa two-sided symplectic two-form. An element ( η , ν , η , ν ) in the semidirect product Lie algebra( g s g ∗ ) s ( g s g ∗ ) defines a right invariant vector field on T T ∗ G by the tangent lift of right translationin T T ∗ G . At a point ( g, µ , ξ, µ ), a right invariant vector is given by(2.30) X T T ∗ G ( η ,ν ,η ,ν ) ( g, µ , ξ, µ ) = (cid:0) T R g η , ν + ad ∗ η µ , η + ad η ξ, ν + ad ∗ η µ − ad ∗ ξ ν (cid:1) . The bundle T ( G s g ∗ ) carries Tulczyjew’s symplectic two-form Ω T ( G s g ∗ ) with two potential one-forms.The one-forms θ and θ are obtained by taking derivations of the symplectic two-form Ω G s g ∗ andthe canonical one-form θ G s g ∗ respectively in Eq.(2.9) [15]. By requiring the trivialization tr T T ∗ G inEq.(2.27) be a symplectic mapping, we obtain an exact symplectic structure Ω T T ∗ G with two potentialone-forms θ and θ taking the values D Ω T T ∗ G ; (cid:16) X T T ∗ G ( ξ ,ν ,ξ ,ν ) , X T T ∗ G ( ¯ ξ , ¯ ν , ¯ ξ , ¯ ν ) (cid:17)E ( g, µ, ξ, ν ) = (cid:10) ν , ¯ ξ (cid:11) − (cid:10) ν , ¯ ξ (cid:11) + h ¯ ν , ξ i− h ¯ ν , ξ i − (cid:10) ν, (cid:2) ξ , ¯ ξ (cid:3)(cid:11) + D ξ, ad ∗ ¯ ξ ν − ad ∗ ξ ¯ ν E , (2.31) D θ ( λ ,η ,λ ,η )1 , X T T ∗ G ( ξ ,ν ,ξ ,ν ) E = h λ , ξ i + h ν , η i + h λ , ξ i + h ν , η i , (2.32) D θ ( λ ,η ,λ ,η )2 , X T T ∗ G ( ξ ,ν ,ξ ,ν ) E = h µ, ξ i + h ν, ξ i + h µ, [ ξ, ξ ] i , (2.33)on right invariant vector fields of the form of Eq.(2.30). At a point ( g, µ, ξ, ν ) ∈ T T ∗ G , the musicalisomorphism Ω ♭T T ∗ G , induced from Ω T T ∗ G , maps the image of a right invariant vector field X T T ∗ G ( ξ ,ν ,ξ ,ν )8 o an element(2.34) Ω ♭T T ∗ G ( X T T ∗ G ( ξ ,ν ,ξ ,ν ) ) = (cid:16) T ∗ g R g − ν − ad ∗ ξ Ad ∗ g − µ, ξ , − ν , − ξ (cid:17) of T ∗ ( g,µ,ξ,ν ) ( T T ∗ G ) . Dynamics on the First Order Bundles
Lagrangian dynamics on tangent group
T G . Given a Lagrangian function L : T G → R , let ¯ L : G s g → R be the corresponding function determinedby ¯ L ◦ tr T G = L. The variation of the action integral of the latter is computed as(3.1) δ Z ba ¯ L ( ξ, g ) dt = Z ba (cid:28) δ ¯ Lδξ , δξ (cid:29) e + (cid:28) δ ¯ Lδg , δg (cid:29) g ! dt, applying the Hamilton’s principle to the variations of the group (base) component, and the reducedvariational principle(3.2) δξ = ˙ η + [ ξ, η ]to the variations of the Lie algebra (fiber) component. For the reduced variational principle we refer to[9, 16, 22, 25, 34] and for the Lagrangian dynamics on semidirect products to [4, 8, 23, 42, 43, 35, 32].For the following result see [5, 13, 12, 14, 16]. Proposition 3.1.
The trivialized Euler-Lagrange dynamics generated by a Lagrangian density ¯ L : G s g → R is given by (3.3) ddt δ ¯ Lδξ = T ∗ e R g δ ¯ Lδg − ad ∗ ξ δ ¯ Lδξ .
If, in addition, the Lagrangian density ¯ L : G s g → R is right invariant (namely it is independent ofthe group variable, that is ¯ L ( g, ξ ) = ℓ ( ξ )), then Eq.(3.3) reduces to the Euler-Poincar´e equations on g = ( G s g ) /G (3.4) ddt δlδξ = − ad ∗ ξ δlδξ . Along the motion, for any Lagrangian ¯ L = ¯ L ( g, ξ ), we compute that dLdt = (cid:10) δ ¯ Lδg , ˙ g (cid:11) + (cid:10) δ ¯ Lδξ , ˙ ξ (cid:11) = (cid:10) δ ¯ Lδg , T e R g ξ (cid:11) + (cid:10) δ ¯ Lδξ , ˙ ξ (cid:11) = (cid:10) T ∗ e R g δ ¯ Lδg , ξ (cid:11) + (cid:10) δ ¯ Lδξ , ˙ ξ (cid:11) = (cid:10) ddt δ ¯ Lδξ + ad ∗ ξ δ ¯ Lδξ , ξ (cid:11) + (cid:10) δ ¯ Lδξ , ˙ ξ (cid:11) = (cid:10) ddt δ ¯ Lδξ , ξ (cid:11) + (cid:10) δ ¯ Lδξ , ˙ ξ (cid:11) = ddt (cid:10) δ ¯ Lδξ , ξ (cid:11) (3.5)where, according to the trivialization (2.4), we have employed the identification ˙ g = T e R g ξ in the firstline whereas, we substitute the Euler-Lagrange equations (3.3) in the third line. The calculation (3.5)reads that the quantity (cid:10) δ ¯ L/δξ, ξ (cid:11) − L is a constant of the motion.3.2. Hamiltonian dynamics on cotangent group T ∗ G . iven ¯ H : G s g ∗ → R , one obtains the Hamilton’s equations(3.6) dgdt = T e R g (cid:18) δ ¯ Hδµ (cid:19) , dµdt = ad ∗ δ ¯ Hδµ µ − T ∗ e R g δ ¯ Hδg on the semidirect product G s g ∗ from the very definition(3.7) i G s g ∗ X ¯ H Ω G s g ∗ = − d ¯ H, where the right invariant vector field(3.8) X G s g ∗ ¯ H ( g, µ ) := (cid:18) T e R g δ ¯ Hδµ , ad ∗ δ ¯ Hδµ µ − T ∗ e R g δ ¯ Hδg (cid:19) is the Hamiltonian vector field associated to(3.9) (cid:18) δ ¯ Hδµ , − T ∗ e R g δ ¯ Hδg (cid:19) ∈ g s g ∗ . For further details of Hamiltonian dynamics on semi-direct products we refer the reader to [4, 11, 7,16, 24, 30, 32, 37, 33, 35, 31, 41].The canonical Poisson bracket of two functionals ¯
F , ¯ K : G s g ∗ → R . at a point ( g, µ ) ∈ G s g ∗ is givenby(3.10) (cid:8) ¯ F , ¯ K (cid:9) G s g ∗ ( g, µ ) = (cid:28) T ∗ e R g δ ¯ Fδg , δ ¯ Kδµ (cid:29) − (cid:28) T ∗ e R g δ ¯ Kδg , δ ¯ Fδµ (cid:29) + (cid:28) µ, (cid:20) δ ¯ Fδµ , δ ¯ Kδµ (cid:21)(cid:29) . Reduction of T ∗ G by G . The right action of G on G s g ∗ is(3.11) ( G s g ∗ ) × G → G s g ∗ : (( g, µ ) ; h ) → ( gh, µ )with the infinitesimal generator X G s g ∗ ( ξ, . If ¯ H, defined on G s g ∗ , is independent of g , it becomes rightinvariant under G . In this case, dropping the terms involving δ ¯ H /δg in Poisson bracket (3.10) is thePoisson reduction G s g ∗ → ( G s g ∗ ) /G ≃ g ∗ . When ¯ F and ¯ K are independent of the group variable g ∈ G , that is, when ¯ F = f ( µ ) and ¯ K = k ( µ ), we have the Lie-Poisson bracket(3.12) { f, k } g ∗ ( µ ) = (cid:28) µ, (cid:20) δfδµ , δkδµ (cid:21)(cid:29) from which the Lie-Poisson equations(3.13) ˙ µ = ad ∗ δhδµ µ. on the dual space g ∗ follows. The Lie-Poisson bracket given in Eq.(3.12) can also be obtained bypulling back the non-degenerate Poisson bracket in Eq.(3.10) with the embedding g ∗ → G s g ∗ .For the symplectic leaves of this Poisson structure [45], we apply Marsden-Weinstein symplectic re-duction theorem [38] to G s g ∗ with the action of G . The action (3.11) is symplectic and it inducesthe momentum mapping(3.14) J G s g ∗ : G s g ∗ −→ g ∗ : ( g, µ ) → µ which is also Poisson, and hence it projects trivialized Hamiltonian dynamics in (3.6) to the Lie-Poissondynamics in (3.13).The inverse image J − G s g ∗ ( µ ) ⊂ G s g ∗ of a regular value µ ∈ g ∗ consists of two-tuples ( g, µ ) for g ∈ G and fixed µ ∈ g ∗ . We may identify J − G s g ∗ ( µ ) with the group G . Let G µ be the isotropy group of the oadjoint action Ad ∗ , defined in (2.2), preserving the momenta µ . Then, we have the isomorphism(3.15) J − G s g ∗ ( µ ) . G µ ≃ G / G µ ≃ O µ identifying the equivalence class [ g ] of g in G/G µ with the coadjoint orbit(3.16) O µ = (cid:8) Ad ∗ g µ : g ∈ G (cid:9) . through the point µ in g ∗ [36]. We denote the reduced symplectic two-form on O µ by Ω /GG s g ∗ ( µ ) whichis the Kostant-Kirillov-Souriau two-form [25, 39, 36]. The value of Ω /GG s R g ∗ ( µ ) on two vector fields ad ∗ ξ µ , ad ∗ η µ in T µ O µ is(3.17) D Ω /GG s g ∗ ; (cid:0) ad ∗ ξ µ, ad ∗ η µ (cid:1)E = − D µ, [ ξ, η ] g E . Reduction of T ∗ G by G µ . The isotropy subgroup G µ acts on G s g ∗ as described by Eq.(3.11).Then, a Poisson and a symplectic reductions of dynamics are possible. The Poisson reduction of thesymplectic manifold G s g ∗ under the action of the isotropy group G µ results in( G s g ∗ ) /G µ ≃ O µ × g ∗ , with Poisson bracket(3.18) { H, K } O µ × g ∗ ( µ, ν ) = (cid:28) µ, (cid:20) δHδµ , δKδµ (cid:21)(cid:29) + (cid:28) ν, (cid:20) δHδµ , δKδν (cid:21) − (cid:20) δKδµ , δHδν (cid:21)(cid:29) which is not the direct product of Lie-Poisson structures on O µ and g ∗ .The coadjoint action of G s g on the dual g ∗ × g ∗ of its Lie algebra is(3.19) Ad ∗ ( g,ξ ) : g ∗ × g ∗ → g ∗ × g ∗ : ( µ, ν ) (cid:0) Ad ∗ g µ + ad ∗ ξ Ad ∗ g ν, Ad ∗ g ν (cid:1) . The symplectic reduction of G s g ∗ under the action of the isotropy subgroup G µ results in the coadjointorbit O ( µ,ν ) in g ∗ × g ∗ through the point ( µ, ν ) under the action in Eq.(3.19). The reduced symplectictwo-form Ω O ( µ,ν ) takes the value(3.20) D Ω O ( µ,ν ) ; ( η, ζ ) , (cid:0) ¯ η, ¯ ζ (cid:1)E ( µ, ν ) = h µ, [¯ η, η ] i + (cid:10) ν, [¯ η, ζ ] − (cid:2) η, ¯ ζ (cid:3)(cid:11) on two vectors ( η, ζ ) and (cid:0) ¯ η, ¯ ζ (cid:1) in T ( µ,ν ) O ( µ,ν ) .We summarize reductions of the symplectic space G s g ∗ in the following diagram.(3.21) g ∗ Poissonembedding (cid:15) (cid:15) O µ ? _ symplectic leaf o o symplecticembedding (cid:15) (cid:15) G s g ∗ PR by G ❱❱❱❱❱❱❱❱❱❱ j j ❱❱❱❱❱❱❱❱❱❱ SR by G ✐✐✐✐✐✐✐✐✐✐ ✐✐✐✐✐✐✐✐✐✐ PR by G µ ✐✐✐✐✐✐✐✐✐ t t ✐✐✐✐✐✐✐✐✐ SR by G µ ❯❯❯❯❯❯❯❯❯ * * ❯❯❯❯❯❯❯❯❯ O µ s g ∗ O ( µ,ν ) _? symplectic leaf o o Reductions of T ∗ G = G s g ∗ The Legendre Transformation. or a (hyper)regular Lagrangian ¯ L on T G = G s g , the Legendre transformation is G s g → G s g ∗ : ( g, ξ ) → (cid:0) g, δ ¯ Lδξ = µ (cid:1) which identifies δ ¯ L/δξ with the fiber variable µ of G s g ∗ . Define a Hamiltonian function(3.22) H ( g, µ ) = h µ, ξ i − L ( g, ξ )for which the Hamiltonian dynamics in Eq.(3.6) gives Euler-Lagrange equations (3.3). When ¯ L is inde-pendent of the group variable we have Euler-Poincare equations (3.4) and the Legendre transformation µ = δl/δξ maps these to Lie-Poisson equations (3.13) with the Hamiltonian function h ( µ ) = h µ, ξ i − l ( ξ ) . When the Lagrangian density is degenerate, the fiber derivative is not invertible hence a direct passagefrom the Lagrangian dynamics to Hamiltonian one is not possible. One possible way to define a generalLegendre trnasformation, including the degenerate cases, is possible in Tulczyjew’s approach [44]. Werefer [15, 16, 20] where the Tulczyjew’s triplet is constructed for Lie groups.4.
Hamiltonian Dynamics on T ∗ T G
For a Hamiltonian function(al) H on the symplectic manifold ( T ∗ T G, Ω T ∗ T G ) , the Hamilton’s equa-tions read(4.1) i X T ∗ TGH Ω T ∗ T G = − dH, where the right invariant Hamiltonian vector field X T ∗ T GH is generated by [2](4.2) (cid:18) δHδµ , δHδν , − T ∗ e R g (cid:18) δHδg (cid:19) − ad ∗ ξ (cid:18) δHδξ (cid:19) , − δHδξ (cid:19) ∈ ( g s g ) s ( g ∗ × g ∗ ) . Proposition 4.1.
Components of X T ∗ T GH are trivialized Hamilton’s equations on ( T ∗ T G, Ω T ∗ T G ) dgdt = T e R g δHδµ , (4.3) dξdt = δHδν − ad ξ δHδµ , (4.4) dµdt = − T ∗ e R g δHδg − ad ∗ ξ δHδξ + ad ∗ δHδµ µ + ad ∗ δHδν ν, (4.5) dνdt = − δHδξ + ad ∗ δHδµ ν. (4.6)From the equations (4.4) and (4.6), we single out δH/δν and δH/δξ , respectively. By substitutingthese into Eq.(4.5), we obtain the system(4.7) (cid:18) ddt − ad ∗ δHδµ (cid:19) (cid:0) ad ∗ ξ ν − µ (cid:1) = T ∗ e R g δHδg , dgdt = T e R g δHδµ equivalent to Eq.(4.3)-(4.6). Remark 4.2.
The Hamilton’s equations (4.3)-(4.6) have extra terms, compared to ones, for example,in [11, 19], coming from the choice of trivialization preserving group structure. The trivialization of[11] is of the second kind given by Eq.(1.2) whereas Eq.(4.3)-(4.6) results from trivializations of thefirst kind. Reference [6] studies geometric integrators of this Hamiltonian dynamics. .1. Reduction of T ∗ T G by G . We shall first perform Poisson reduction of Hamiltonian system on T ∗ T G under the action of G given by(( h, η, ν , ν ) ; g ) → ( hg, η, ν , ν )for a right invariant Hamiltonian H = H ( ξ, µ, ν ). Proposition 4.3.
The Poisson reduced manifold g s ( g ∗ × g ∗ ) carries the Poisson bracket { H, K } g s ( g ∗ × g ∗ ) ( ξ, µ, ν ) = (cid:28) δHδξ , δKδν (cid:29) − (cid:28) δKδξ , δHδν (cid:29) + (cid:28) µ, (cid:20) δHδµ , δKδµ (cid:21)(cid:29) − (cid:28) ad ∗ ξ δKδξ , δHδµ (cid:29) + (cid:28) ad ∗ ξ δHδξ , δKδµ (cid:29) + (cid:28) ν, (cid:20) δHδµ , δKδν (cid:21) − (cid:20) δKδµ , δHδν (cid:21)(cid:29) , (4.8) for two right invariant functionals H and K on T ∗ T G . Remark 4.4. T ∗ g = g × g ∗ carries a canonical Poisson bracket, and g ∗ carries Lie-Poisson bracket.The immersions g × g ∗ → g s ( g ∗ × g ∗ ) and g ∗ → g s ( g ∗ × g ∗ ) are Poisson maps. However, thePoisson structure described by Eq.(4.8) on g s ( g ∗ × g ∗ ) is not a direct product of these. In fact,direct product structure on g × ( g ∗ × g ∗ ) arises from the trivialization of for examples was done in[11] and [19]. In these cases the second line of (4.8) disappears. Proposition 4.5.
The Marsden-Weinstein symplectic reduction by the action of G on T ∗ T G withthe momentum mapping J GT ∗ T G : T ∗ T G → g ∗ : ( g, ξ, µ, ν ) → µ results in the reduced symplectic two-form Ω /GT ∗ T G on the reduced space O µ × g × g ∗ . The value of Ω /GT ∗ T G on two vectors ( η g ∗ ( µ ) , ζ, λ ) and (cid:0) ¯ η g ∗ ( µ ) , ¯ ζ, ¯ λ (cid:1) is (4.9) Ω /GT ∗ T G (cid:0) ( η g ∗ ( µ ) , ζ, λ ) , (cid:0) ¯ η g ∗ ( µ ) , ¯ ζ, ¯ λ (cid:1)(cid:1) = (cid:10) λ, ¯ ζ (cid:11) − (cid:10) ¯ λ, ζ (cid:11) − h µ, [ η, ¯ η ] i and the reduced Hamilton’s equations for a right invariant Hamiltonian H are dζdt = δHδλ , dλdt = − δHδζ , dµdt = ad ∗ δHδµ µ. Remark 4.6.
The reduced space O µ × g × g ∗ is a symplectic leaf [45] for the Poisson manifold g s ( g ∗ × g ∗ ) of Proposition 4.3 as well as for g × g ∗ × g ∗ with direct product Poisson structuredescribed in Remark 4.4 above.The symplectic two-form Ω /GT ∗ T G given in Eq.(4.9) on O µ × g × g ∗ is in a direct product form. Hencea reduction is possible by the additive action of g to the second factor in O µ × g × g ∗ . Proposition 4.7.
The momentum map of additive action of g on the symplectic manifold ( O µ × g × g ∗ , Ω /GT ∗ T G ) is J g O µ × g × g ∗ : O µ × g × g ∗ → g ∗ : ( µ, ξ, ν ) → ν and the symplectic reduction results in the orbit O µ with Kostant-Kirillov-Souriau two-form (3.17). Reduction of T ∗ T G by g . The vector space structure of g makes it an Abelian group, andaccording to the immersion in Eq.(2.13), g is an Abelian subgroup of T ∗ T G . It acts on the total space T ∗ T G by(4.10) (( h, η, µ, ν ) ; ξ ) → ( h, η + Ad h ξ, µ, ν ) . ince the action of G s g on its cotangent bundle T ∗ T G is symplectic, the subgroup g of G s g alsoacts on T ∗ T G symplectically. Following results describes Poisson and symplectic reductions of T ∗ T G by g assuming that functions K = K ( g, µ, ν ) and H = H ( g, µ, ν ) defined on G s ( g ∗ × g ∗ ) are rightinvariant under the above action of g . Proposition 4.8.
Poisson reduction of T ∗ T G by Abelian subgroup g gives the Poisson manifold G s ( g ∗ × g ∗ ) endowed with the Poisson bracket { H, K } G s ( g ∗ × g ∗ ) ( g, µ, ν ) = (cid:28) T ∗ e R g δHδg , δKδµ (cid:29) − (cid:28) T ∗ e R g δKδg , δHδµ (cid:29) + (cid:28) µ, (cid:20) δHδµ , δKδµ (cid:21)(cid:29) + (cid:28) ν, (cid:20) δHδµ , δKδν (cid:21) − (cid:20) δKδµ , δHδν (cid:21)(cid:29) . (4.11) Remark 4.9.
Recall that G × g ∗ is canonically symplectic with the Poisson bracket in Eq(3.10) andthe immersion G × g ∗ → G s ( g ∗ × g ∗ ) is a Poisson map. On the other hand, g ∗ is naturally Lie-Poissonand g ∗ → g s ( g ∗ × g ∗ ) is also a Poisson map. The Poisson bracket in Eq(4.11) is, however, not adirect product of these structures. Proposition 4.10.
The Marsden-Weinstein symplectic reduction by the action of g on T ∗ T G withthe momentum mapping J g T ∗ T G : T ∗ T G → g ∗ : ( g, ξ, µ, ν ) → ν results in the reduced symplectic space (cid:0) J g T ∗ T G (cid:1) − / g isomorphic to G s g ∗ and with the canonicalsymplectic two-from Ω G s g ∗ in Eq(2.10). It follows that the immersion G s g ∗ → G s ( g ∗ × g ∗ ) defines symplectic leaves of the Poisson manifold G s ( g ∗ × g ∗ ). The symplectic reduction of G s g ∗ under the action of G results in the total space O µ with Kostant-Kirillov-Souriau two-form (3.17). We arrive at the following proposition. Proposition 4.11.
Reductions by actions of g and G make the following diagram commutative (4.12) ( G s g ) s ( g ∗ × g ∗ ) SR by g ❦❦❦❦❦❦ u u ❦❦❦❦❦❦ SR by G at µ ❯❯❯❯❯❯ * * ❯❯❯❯❯❯ G s g ∗ SR by G at µ ❙❙❙❙❙❙❙❙ ) ) ❙❙❙❙❙❙❙❙ O µ × g × g ∗ SR by g ✐✐✐✐✐✐✐✐ t t ✐✐✐✐✐✐✐✐ O µ Symplectic Reductions of T ∗ T G
Note that, the symplectic reduction of T ∗ T G by the total action of the group G s g does not resultin O µ as reduced space. This is a matter of Hamiltonian reduction by stages theorem [30]. In thefollowing subsection, we will discuss the reduction of T ∗ T G under the action of G s g as well as theimplications of the Hamiltonian reduction by stages theorem for this case.4.3. Reduction of T ∗ T G by G s g . The Lie algebra of G s g is the space g s g endowed with thesemidirect product Lie algebra bracket(4.13) [( ξ , ξ ) , ( η , η )] g s g = ([ ξ , η ] , [ ξ , η ] − [ η , ξ ]) or ( ξ , ξ ) and ( η , η ) in g s g . Accordingly, the dual space g ∗ × g ∗ has the Lie-Poisson bracket { F, E } g ∗ × g ∗ ( µ, ν ) = (cid:28) µ, (cid:20) δFδµ , δEδµ (cid:21)(cid:29) + (cid:28) ν, (cid:20) δFδµ , δEδν (cid:21) − (cid:20) δEδµ , δFδν (cid:21)(cid:29) (4.14)for two functionals F and E on g ∗ × g ∗ . Proposition 4.12.
The Lie-Poisson structure on g ∗ × g ∗ is given by the bracket in Eq.(4.14) and theLie-Poisson equations for a function H ( µ, ν ) on g ∗ × g ∗ read (4.15) dµdt = ad ∗ δHδµ µ + ad ∗ δHδν ν, dνdt = ad ∗ δHδµ ν. Alternatively, the Lie-Poisson equations (4.15) can be obtained by Poisson reduction of T ∗ T G withthe action of G s g given by(4.16) (( h, η, µ, ν ) ; ( g, ξ )) (cid:0)(cid:0) hg, η + Ad h ξ, µ, ν (cid:1) and restricting the Hamiltonian function H to the fiber variables ( µ, ν ). In this case, the Lie-Poissondynamics of g and ξ remains the same but the dynamics governing µ and ν have the reduced formgiven by Eq.(4.15). This is a manifestation of the fact that the projections to the last two factors inthe trivialization (2.11) is a momentum map under the left Hamiltonian action of the group G s g toits trivialized cotangent bundle T ∗ T G . Yet another way is to reduce the bracket (4.8) on g s ( g ∗ × g ∗ )by assuming that functionals depend on elements of the dual spaces. That is, to consider the Abeliangroup action of g on g s ( g ∗ × g ∗ ) given by(( η, µ, ν ) ; ξ ) ( η + ξ, µ, ν )and then, apply Poisson reduction. Note finally that, the immersion g ∗ × g ∗ → g s ( g ∗ × g ∗ ) is aPoisson map.Application of the Marsden-Weinstein reduction to the symplectic manifold T ∗ T G results in thesymplectic leaves of Poisson structure on g ∗ × g ∗ . The action in Eq.(4.16) has the momentum mapping J G s g T ∗ T G : T ∗ T G → g ∗ × g ∗ : ( g, ξ, µ, ν ) → ( µ, ν ) . The pre-image (cid:16) J G s g T ∗ T G (cid:17) − ( µ, ν ) of an element ( µ, ν ) ∈ g ∗ × g ∗ is diffeomorphic to G s g . The isotropygroup ( G s g ) ( µ,ν ) of ( µ, ν ) consists of pairs ( g, ξ ) in G s g satisfying(4.17) Ad ∗ ( g,ξ ) ( µ, ν ) = (cid:0) Ad ∗ g (cid:0) µ + ad ∗ ξ ν (cid:1) , Ad ∗ g ν (cid:1) = ( µ, ν )which means that g ∈ G ν ∩ G µ and the representation of Ad g ξ on ν is null, that is ad ∗ Ad g ξ ν = 0 . Fromthe general theory, we deduce that the quotient space (cid:16) J G s g T ∗ T G (cid:17) − ( µ, ν ) (cid:30) ( G s g ) ( µ,ν ) ≃ O ( µ,ν ) is diffeomorphic to the coadjoint orbit O ( µ,ν ) in g ∗ × g ∗ through the point ( µ, ν ) under the action Ad ∗ ( g,ξ ) in Eq.(4.17), that is,(4.18) O ( µ,ν ) = n ( µ, ν ) ∈ g ∗ × g ∗ : Ad ∗ ( g,ξ ) ( µ, ν ) = ( µ, ν ) o . Proposition 4.13.
The symplectic reduction of T ∗ T G results in the coadjoint orbit O ( µ,ν ) in g ∗ × g ∗ through the point ( µ, ν ) . The reduced symplectic two-form Ω G s g \ T ∗ T G (denoted simply by Ω O ( µ,ν ) ) takesthe value (4.19) D Ω O ( µ,ν ) ; ( η, ζ ) , (cid:0) ¯ η, ¯ ζ (cid:1)E ( µ, ν ) = h µ, [¯ η, η ] i + (cid:10) ν, [¯ η, ζ ] − (cid:2) η, ¯ ζ (cid:3)(cid:11) n two vectors ( η, ζ ) and (cid:0) ¯ η, ¯ ζ (cid:1) in T ( µ,ν ) O ( µ,ν ) . This reduction can also be achieved by stages as described in [23, 30]. That is, first trivialize dy-namics by the action of Lie algebra g on T ∗ T G which results in the Poisson structure on the product G s ( g ∗ × g ∗ ) given by Eq.(4.11). Symplectic leaves of this Poisson structure are spaces diffeomorphicto G s g ∗ with symplectic two-form given in Eq.(2.10). The isotropy group G µ of an element µ ∈ g ∗ acts on G s g ∗ by the same way as assigned in Eq.(3.11), that is,(4.20) ( G s g ∗ ) × G µ → G s g ∗ : (( h, λ ) ; g ) → ( hg, λ ) . Then, the Hamiltonian reduction by stages theorem states that, the symplectic reduction of G s g ∗ under the action of G µ will result in O ( µ,ν ) as the reduced space endowed with the symplectic two-formΩ O ( µ,ν ) in Eq.(4.19). Following diagram summarizes the Hamiltonian reduction by stages theorem forthe case of T ∗ T G under consideration(4.21) ( G s g ) s ( g ∗ × g ∗ ) SR by g at µ ❣❣❣❣❣❣❣❣❣ s s ❣❣❣❣❣❣❣❣❣ SR by G × g at ( µ,ν ) (cid:15) (cid:15) G s g ∗ G µ at ν ❲❲❲❲❲❲❲❲❲❲❲ + + ❲❲❲❲❲❲❲❲❲❲❲ O ( µ,ν ) Hamiltonian reduction by stages for T ∗ T G
There exists a momentum mapping J G µ G s g ∗ from G s g ∗ to the dual space g ∗ µ of the isotropy subalgebra g µ of G µ . Isotropy subgroup G µ,ν of the coadjoint action is G µ,ν = n g ∈ G µ : Ad ∗ g − ν = ν o . The quotient symplectic space (cid:16) J G µ G s g ∗ (cid:17) − ( ν ) (cid:30) G µ,ν ≃ O ( µ,ν ) is diffeomorphic to the coadjoint orbit O ( µ,ν ) defined in (4.18).It is also possible to establish the Poisson reduction of the symplectic manifold G s g ∗ under the actionof the isotropy group G µ . This results in G µ \ ( G s g ∗ ) ≃ O µ × g ∗ with the Poisson bracket(4.22) { H, K } O µ × g ∗ ( µ, ν ) = (cid:28) µ, (cid:20) δHδµ , δKδµ (cid:21)(cid:29) + (cid:28) ν, (cid:20) δHδµ , δKδν (cid:21) − (cid:20) δKδµ , δHδν (cid:21)(cid:29) . We note again, that, the Poisson structure on O µ × g ∗ is not a direct product of the Lie-Poissonstructures on O µ and g ∗ . Following diagram illustrates various reductions of T ∗ T G under the actionsof G , g and G s g . Diagram (3.21), describing reductions of G s g ∗ , can be attached to the lower right orner of this to have a complete picture of reductions.(4.23) g s ( g ∗ × g ∗ ) O µ × g × g ∗ ? _ symplectic leaf o o g ∗ × g ∗ Poissonembedding (cid:15) (cid:15)
Poissonembedding O O ( G s g ) s ( g ∗ × g ∗ ) PR by G ◆◆◆◆◆◆◆◆◆◆◆ g g ◆◆◆◆◆◆◆◆◆◆◆ SR by G ♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣ PR by g ♣♣♣♣♣♣♣♣♣♣♣ w w ♣♣♣♣♣♣♣♣♣♣♣ SR by g ◆◆◆◆◆◆◆◆◆◆◆ ' ' ◆◆◆◆◆◆◆◆◆◆◆ SR by G s g / / PR by G s g o o O ( µ,ν ) symplecticembedding (cid:15) (cid:15) symplecticembedding O O symplecticleaf { { G s ( g ∗ × g ∗ ) G s g ∗ _? symplectic leaf o o Reductions of T ∗ T G Hamiltonian Dynamics on T ∗ T ∗ G Proposition 5.1.
A Hamiltonian function H on T ∗ T ∗ G determines the Hamilton’s equations i X T ∗ T ∗ GH Ω T ∗ T ∗ G = − dH by uniquely defining Hamiltonian vector field X T ∗ T ∗ GH . The Hamiltonian vector field is a right invariantvector field generated by a 4-tuple Lie algebra element (cid:18) δHδν , δHδξ , ad ∗ δHδµ µ − T ∗ e R g (cid:18) δHδg (cid:19) , − δHδµ (cid:19) in ( g s g ∗ ) s ( g ∗ × g ) . At the point ( g, µ, ν, ξ ) , the Hamilton’s equations are dgdt = T e R g (cid:18) δHδν (cid:19) , (5.1) dµdt = δHδξ + ad ∗ δHδν µ (5.2) dνdt = ad ∗ δHδµ µ + ad ∗ δHδν ν − T ∗ e R g (cid:18) δHδg (cid:19) − ad ∗ ξ δHδξ (5.3) dξdt = − δHδµ + [ δHδν , ξ ] . (5.4)5.1. Reduction of T ∗ T ∗ G by G . It follows from Eq.(2.21) that the right action of G on T ∗ T ∗ G is(5.5) (( h, ν, λ , ξ ) ; g ) → ( hg, ν, λ , ξ )with the infinitesimal generator X T ∗ T ∗ G ( η, , , being a right invariant vector field as in Eq.(2.23) generatedby ( η, , ,
0) for η ∈ g . roposition 5.2. Poisson reduction of T ∗ T ∗ G under the action of G results in g ∗ s ( g ∗ × g ) endowedwith the Poisson bracket { H, K } g ∗ s ( g ∗ × g ) ( µ, ν, ξ ) = (cid:28) δHδµ , δKδξ (cid:29) − (cid:28) δKδµ , δHδξ (cid:29) + (cid:28) ν, (cid:20) δHδν , δKδν (cid:21)(cid:29) + (cid:28) ξ, ad ∗ δHδν δKδξ − ad ∗ δKδν δHδξ (cid:29) + (cid:28) µ, (cid:20) δHδµ , δKδν (cid:21) − (cid:20) δKδµ , δHδν (cid:21)(cid:29) , (5.6) and symplectic reduction gives O µ × g × g ∗ with the symplectic two-form defined by (5.7) Ω /GT ∗ T ∗ G (cid:0) ( η g ∗ ( µ ) , λ, ζ ) , (cid:0) ¯ η g ∗ ( µ ) , ¯ λ, ¯ ζ (cid:1)(cid:1) = (cid:10) ζ, ¯ λ (cid:11) − (cid:10) ¯ ζ, λ (cid:11) − h µ, [ η, ¯ η ] i on two elements ( η g ∗ ( µ ) , λ, ζ ) and (cid:0) ¯ η g ∗ ( µ ) , ¯ λ, ¯ ζ (cid:1) of T µ O µ × g × g ∗ . Recall that, in previous section, the Poisson and symplectic reductions of T ∗ T G result in reducedspaces g s ( g ∗ × g ∗ ) and O µ × g × g ∗ , respectively. The reduced Poisson bracket on g s ( g ∗ × g ∗ ) isgiven by Eq.(4.8) and the reduced symplectic two-form Ω /GT ∗ T G on O µ × g × g is in Eq.(4.9). We havethe following proposition from [15] relating the reductions of cotangent bundles T ∗ T ∗ G and T ∗ T G .We refer to [27] for a detailed study on the canonical maps between semidirect products.5.2.
Reduction of T ∗ T ∗ G by g ∗ . The action of g ∗ on T ∗ T ∗ G , given by(5.8) (( g, µ, ν, ξ ) ; λ ) → (cid:0) g, µ + Ad ∗ g λ, ν, ξ (cid:1) generated by X T ∗ T ∗ G (0 ,λ, , = (cid:0) , λ, − ad ∗ ξ λ, (cid:1) . As the action of G s g ∗ on its cotangent bundle T ∗ T ∗ G issymplectic, and g ∗ is a subgroup. The action in Eq(5.8) is symplectic hence we can perform a Poissonand a symplectic reductions of T ∗ T ∗ G . Proposition 5.3.
The Poisson reduction of T ∗ T ∗ G with the action of g ∗ results in G s ( g ∗ × g ) endowed with the bracket { H, K } G s ( g ∗ × g ) ( g, ν, ξ ) = (cid:28) T ∗ e R g δHδg , δKδν (cid:29) − (cid:28) T ∗ e R g δKδg , δHδν (cid:29) + (cid:28) ξ, ad ∗ δHδν δKδξ − ad ∗ δKδν δHδξ (cid:29) + (cid:28) ν, (cid:20) δHδν , δKδν (cid:21)(cid:29) . (5.9) The application of Marsden-Weinstein symplectic reduction with the action of g ∗ on T ∗ T ∗ G havingthe momentum mapping J g ∗ T ∗ T ∗ G : T ∗ T ∗ G → g : ( g, µ, ν, ξ ) → ξ results in the reduced symplectic space (cid:16) J g ∗ T ∗ T ∗ G (cid:17) − ( ξ ) / g ∗ isomorphic to G s g ∗ with the canonicalsymplectic two-form Ω G s g ∗ in Eq.(2.10). Reduction of T ∗ T ∗ G by G s g ∗ . The Lie algebra of the group G s g ∗ is the space g s g ∗ carryingthe bracket(5.10) [( ξ, µ ) , ( η, ν )] g s g ∗ = (cid:0) [ ξ, η ] , ad ∗ ξ ν − ad ∗ η µ (cid:1) . The dual space g ∗ × g carries the Lie-Poisson bracket(5.11) { F, E } g ∗ × g ( ν, ξ ) = (cid:28) ν, (cid:20) δFδν , δEδν (cid:21)(cid:29) + (cid:28) ξ, ad ∗ δFδν δEδξ − ad ∗ δFδν δEδξ (cid:29) , that follows from the Lie algebra bracket in Eq.(5.10). roposition 5.4. The Lie-Poisson bracket, in Eq.(5.11), on g ∗ × g defines the Hamiltonian vectorfield X g ∗ × g E by { F, E } g ∗ × g = − D dF, X g ∗ × g E E whose components are the Lie-Poisson equations (5.12) dνdt = ad ∗ δHδν ν − ad ∗ ξ δHδξ , dξdt = [ δHδν , ξ ] . Although, these equations result from Eq.(5.11), it is possible to obtain them starting from the Hamil-ton’s equations (5.1)-(5.4) on T ∗ T ∗ G and applying a Poisson reduction with the action of G s g ∗ givenby T ∗ T ∗ G × ( G s g ∗ ) → T ∗ T ∗ G : (( h, ν, λ, ξ ); ( g, µ )) ( hg, ν + Ad ∗ h µ, λ, ξ ) . (5.13)In other words, choosing the Hamiltonian function H in Eqs.(5.1)-(5.4) depending on fiber variablesonly, that is H = H ( ν, ξ ), Eq.(5.12) follows.To reduce the Hamilton’s equations (5.1)-(5.4) on T ∗ T ∗ G symplectically, we first compute the mo-mentum mapping J G ξ G s g ∗ : T ∗ T ∗ G → g ∗ × g : ( g, µ, ν, ξ ) → ( ν, ξ ) , associated with the action of G s g ∗ in Eq.(5.13) and the quotient space(5.14) (cid:16) J G ξ G s g ∗ (cid:17) − ( ν, ξ ) (cid:30) G ( ν,ξ ) ≃ O ( ν,ξ ) . Here, G ( ν,ξ ) is the isotropy subgroup of G s g ∗ consisting of elements preserved under the coadjointaction G s g ∗ on the dual space g ∗ × g of its Lie algebraAd ∗ : ( G s g ∗ ) × ( g ∗ × g ) → g ∗ × g : (( g, µ ) , ( ν, ξ )) → (cid:16) Ad ∗ g ν − ad ∗ Ad g ξ µ, Ad g ξ (cid:17) (5.15)and, the space O ( ν,ξ ) is the coadjoint orbit passing through the point ( ν, ξ ) under this coadjoint action. Proposition 5.5.
The symplectic reduction of T ∗ T ∗ G results in the coadjoint orbit O ( ν,ξ ) in g ∗ × g through the point ( ν, ξ ) . The reduced symplectic two-form Ω / ( G s g ∗ ) T ∗ T ∗ G (denoted simply by Ω O ( ν,ξ ) ) takesthe value (5.16) D Ω O ( ν,ξ ) ; ( λ, η ) , (cid:0) ¯ λ, ¯ η (cid:1)E ( ν, ξ ) = h ν, [¯ η, η ] i + (cid:10) ξ, ad ∗ η ¯ λ − ad ∗ ¯ η λ ] (cid:11) on two vectors ( λ, η ) and (cid:0) ¯ λ, ¯ η (cid:1) in T ( ν,ξ ) O ( ν,ξ ) . Alternatively, this reduction can be performed in two steps by applying the Hamiltonian reduction bystages theorem [30]. The first step consists of the symplectic reduction of T ∗ T ∗ G with the action of g ∗ which has already been established in previous subsection and resulted in the reduced symplecticspace (cid:16) J g ∗ T ∗ T ∗ G (cid:17) − ( ξ ) / g ∗ , isomorphic to G s g ∗ , with the canonical symplectic two-from Ω G s g ∗ inEq.(2.10). For the second step, we recall the adjoint group action Ad g − of G on g and define theisotropy subgroup(5.17) G ξ = (cid:8) g ∈ G : Ad g − ξ = ξ (cid:9) for an element ξ ∈ g under the adjoint action. The Lie algebra g ξ of G ξ consists of vectors η ∈ g satisfying [ η, ξ ] = 0. The isotropy subgroup G ξ acts on G s g ∗ by the same way as described in q.(3.11). This action is Hamiltonian and has the momentum mapping J G ξ G s g ∗ : G s g ∗ → g ∗ ξ , where g ∗ ξ is the dual space of g ξ . The quotient space (cid:16) J G ξ G s g ∗ (cid:17) − ( ν ) (cid:30) G ξ,ν ≃ O ( ν,ξ ) is diffeomorphic to the coadjoint orbit O ( ν,ξ ) in Eq.(5.14).(5.18) g ∗ s ( g ∗ × g ) O µ × g ∗ × g ? _ symplectic leaf o o g ∗ × g Poissonembedding (cid:15) (cid:15)
Poissonembedding O O ( G s g ∗ ) s ( g ∗ × g ) PR by G
PPPPPPPPPPPP g g PPPPPPPPPPPP
SR by G ♥♥♥♥♥♥♥♥♥♥♥♥ ♥♥♥♥♥♥♥♥♥♥♥♥ PR by g ∗ ♥♥♥♥♥♥♥♥♥♥♥♥ w w ♥♥♥♥♥♥♥♥♥♥♥♥ SR by g ∗ PPPPPPPPPPPP ' ' PPPPPPPPPPPP
SR by G s g ∗ / / PR by G s g ∗ o o O ( µ,ξ ) symplecticembedding (cid:15) (cid:15) symplecticembedding O O symplecticleaf z z G s ( g ∗ × g ) G s g ∗ _? symplectic leaf o o Reduction of T ∗ T ∗ G = ( G s g ∗ ) s ( g ∗ × g ) Hamiltonian and Lagrangian Dynamics on Tulczyjew Symplectic Space
T T ∗ G Hamiltonian Dynamics on
T T ∗ G .Proposition 6.1. Given a Hamiltonian function E on T T ∗ G , the Hamilton’s equation i X TT ∗ GE Ω T T ∗ G = − dE defines a Hamiltonian right invariant vector field X T T ∗ GE generated by the element (cid:18) δEδν , − (cid:18) δEδξ + ad ∗ δEδν µ (cid:19) , − δEδµ + ad ξ δEδν , − (cid:18) T ∗ R g δEδg + ad ∗ ξ δEδξ + ad ∗ ξ ad ∗ δEδν µ (cid:19)(cid:19) of the Lie algebra ( g s g ∗ ) s ( g s g ∗ ) . Components of X T T ∗ GE define the Hamilton’s equations (6.1) ˙ g = T R g (cid:18) δEδν (cid:19) , ˙ µ = − δEδξ , ˙ ξ = δEδµ , ˙ ν = ad ∗ δEδν ν − T ∗ R g (cid:18) δEδg (cid:19) in the adapted trivialization of T T ∗ G .6.1.1. Reduction of
T T ∗ G by G . Proposition 6.2.
The Poisson reduction of
T T ∗ G under the action of G results in the total space g ∗ s ( g s g ∗ ) endowed with the Poisson bracket (6.2) { E, F } g ∗ s ( g s g ∗ ) ( µ, ξ, ν ) = (cid:28) δFδξ , δEδµ (cid:29) − (cid:28) δEδξ , δFδµ (cid:29) + (cid:28) ν, (cid:20) δEδν , δFδν (cid:21)(cid:29) . emark 6.3. Here, the Poisson bracket on g ∗ s ( g s g ∗ ) is the direct product of canonical Poissonbracket on g ∗ × g and Lie-Poisson bracket on g ∗ whereas in Eq.(4.8) we obtained a Poisson bracket,on the isomorphic space g s ( g ∗ × g ∗ ), which is not in the form of a direct product.The action of G is Hamiltonian with the momentum mapping(6.3) J GT T ∗ G : T T ∗ G → g ∗ : ( g, µ, ξ, ν ) → ν + ad ∗ ξ µ. The quotient space of the preimage J − T T ∗ G ( λ ) of an element λ ∈ g ∗ under the action of isotropysubgroup G λ is J − T T ∗ G ( λ ) (cid:14) G λ ≃ O λ × g ∗ × g .Pushing forward a right invariant vector field X T T ∗ G ( η,υ,ζ, ˜ υ ) in the form of Eq.(2.30) by the symplecticprojection T T ∗ G → O λ × g ∗ × g , we obtain the vector field(6.4) X O λ × g ∗ × g ( η,υ,ζ ) (cid:16) Ad ∗ g − λ, µ, ξ (cid:17) = (cid:16) ad ∗ η ◦ Ad ∗ g − λ, υ + ad ∗ η µ, ζ + [ ξ, η ] (cid:17) on the quotient space O λ × g ∗ × g . We refer to [16] for the proof of the following proposition. Proposition 6.4.
The reduced Tulczyjew’s space O λ × g ∗ × g has an exact symplectic two-form Ω O λ × g ∗ × g with two potential one-forms χ and χ whose values on vector fields of the form of Eq.(2.30)at the point (cid:16) Ad ∗ g − λ, µ, ξ (cid:17) are (cid:28) Ω O λ × g ∗ × g , (cid:18) X O λ × g ∗ × g ( η,υ,ζ ) , X O λ × g ∗ × g ( ¯ η, ¯ υ, ¯ ζ ) (cid:19)(cid:29) = (cid:10) υ, ¯ ζ (cid:11) − h ¯ υ, ζ i − h λ, [ η, ¯ η ] i , (6.5) D χ , X O λ × g ∗ × g ( η,υ,ζ ) E (cid:16) Ad ∗ g − λ, µ, ξ (cid:17) = h λ, η i − h υ, ξ i , (6.6) D χ , X O λ × g ∗ × g ( η,υ,ζ ) E (cid:16) Ad ∗ g − λ, µ, ξ (cid:17) = h λ, η i + h µ, ζ i . (6.7)The potential one-forms θ and θ of Eq.(2.32) and Eq.(2.33) for Tulczyjew symplectic structure on T T ∗ G and the one-forms χ and χ of reduced Tulczyjew space are related by the equations D θ , X T T ∗ G ( η,υ,ζ, ˜ υ ) E ( g, µ, ξ, ν ) = D χ , X O λ × g ∗ × g ( η,υ,ζ ) E (cid:16) Ad ∗ g − λ, µ, ξ (cid:17) , D θ , X T T ∗ G ( η,υ,ζ, ˜ υ ) E ( g, µ, ξ, ν ) = D χ , X O λ × g ∗ × g ( η,υ,ζ ) E (cid:16) Ad ∗ g − λ, µ, ξ (cid:17) . Reduction of
T T ∗ G by g . Proposition 6.5.
The action of g on T T ∗ G is given, for η ∈ g , by (6.8) ϕ η : T T ∗ G → T T ∗ G : (( g, µ, ξ, ν ) ; η ) → ( g, µ, ξ + η, ν ) and it is symplectic.Proof. Push forward of a vector field X T T ∗ G ( ξ ,ν ,ξ ,ν ) in the form of Eq.(2.30) by the transformation ϕ η is also a right invariant vector field( ϕ η ) ∗ X T T ∗ G ( ξ ,ν ,ξ ,ν ) = X T T ∗ G ( ξ ,ν ,ξ − [ η,ξ ] ,ν + ad ∗ η ν ) . By direct calculation, one establishes the identity(6.9) ϕ ∗ η Ω T T ∗ G ( X, Y ) ( g, µ, ξ, ν ) = Ω
T T ∗ G (cid:0) ( ϕ η ) ∗ X, ( ϕ η ) ∗ Y (cid:1) ( g, µ, ξ + η, ν )which gives the desired result. In Eq.(6.9) X and Y are right invariant vector fields as in Eq.(2.30)and Ω T T ∗ G is the symplectic two-form given in Eq.(2.31). (cid:3) roposition 6.6. The Poisson reduction of
T T ∗ G under the action in Eq.(6.8) of g results in ( G s g ∗ ) s g ∗ endowed with the bracket { E, F } ( G s g ∗ ) s g ∗ ( g, µ, ν ) = (cid:28) T ∗ e R g δEδg , δFδν (cid:29) − (cid:28) T ∗ e R g δFδg , δEδν (cid:29) + (cid:28) ν, (cid:20) δEδν , δFδν (cid:21)(cid:29) . Remark 6.7.
The Poisson bracket { E, F } ( G s g ∗ ) s g ∗ is independent of functions with respect to µ ,that is, it does not involve δE/δµ and δF/δµ . Its structure resembles the canonical Poisson bracket inEq.(3.10) on G s g ∗ . We recall that, on ( G s g ∗ ) s g ∗ there is another Poisson bracket given in Eq.(4.11)that involves δE/δµ, δF/δµ , δE/δν and δF/δν . This latter comes from reduction of T ∗ T G by g .The infinitesimal generator X T T ∗ G (0 , ,ξ , of the action in Eq.(6.8) corresponds to the element ξ ∈ g andis a right invariant vector field. Since the action is Hamiltonian, and the symplectic two-form is exact,we can derive the associated momentum map J g T T ∗ G from the equation (cid:10) J g T T ∗ G ( g, µ, ξ, ν ) , ξ (cid:11) = D θ , X T T ∗ G (0 , ,ξ , E = h µ, ξ i , where θ is the potential one-form of Tulczyjew in Eq.(2.33) satisfying dθ = Ω T T ∗ G . We find that(6.10) J g T T ∗ G : T T ∗ G → Lie ∗ ( g ) = g ∗ : ( g, µ, ξ, ν ) → µ is the projection to the second entry in T T ∗ G . The preimage of an element µ ∈ g ∗ by J g T T ∗ G is thespace G s ( g s g ∗ ). Following proposition describes the symplectic reduction of T T ∗ G with the actionof g . Proposition 6.8.
The symplectic reduction of
T T ∗ G under the action of g given by Eq.(6.8) givesthe reduced space (cid:0) J g T T ∗ G (cid:1) − ( µ ) . g ≃ G s g ∗ with the canonical symplectic two-from Ω G s g ∗ as in Eq.(2.10). Remark 6.9.
Existence of the symplectic action of g on T T ∗ G is directly related to the existenceof symplectic diffeomorphism(6.11) ¯ σ G : T T ∗ G → T ∗ T G : ( g, µ, ξ, ν ) → (cid:0) g, ξ, ν + ad ∗ ξ µ, µ (cid:1) in Tulczyjew triplet described in [15].6.1.3. Reduction of
T T ∗ G by g ∗ . Induced from the group operation on
T T ∗ G , there are two canonicalactions of g ∗ on T T ∗ Gψ : g ∗ × T T ∗ G → T T ∗ G, φ : g ∗ × T T ∗ G → T T ∗ G described by ψ λ ( g, µ, ξ, ν ) = ( g, µ + λ, ξ, ν ) , (6.12) φ λ ( g, µ, ξ, ν ) = ( g, µ, ξ, ν + λ ) . (6.13) Proposition 6.10. ψ is a symplectic action whereas φ is not. roof. Pushing forward of a vector field X T T ∗ G ( ξ ,ν ,ξ ,ν ) in the form of Eq.(2.30) by transformations ψ λ and φ λ results in right invariant vector fields( ψ λ ) ∗ X T T ∗ G ( ξ ,ν ,ξ ,ν ) = X T T ∗ G (cid:16) ξ ,ν − ad ∗ ξ λ,ξ ,ν − ad ∗ ξ ad ∗ ξ λ (cid:17) , ( φ λ ) ∗ X T T ∗ G ( ξ ,ν ,ξ ,ν ) = X T T ∗ G (cid:16) ξ ,ν ,ξ ,ν − ad ∗ ξ λ (cid:17) . If Ω
T T ∗ G is the symplectic two-form on T T ∗ G given in Eq.(2.31), direct calculations show that theidentity(6.14) ψ λ ∗ Ω T T ∗ G ( X, Y ) ( g, µ, ξ, ν ) = Ω
T T ∗ G (( ψ λ ) ∗ X, ( ψ λ ) ∗ Y ) ( g, µ + λ, ξ, ν )holds for all vector fields X and Y , and λ ∈ g ∗ whereas(6.15) φ λ ∗ Ω T T ∗ G ( X, Y ) ( g, µ, ξ, ν ) = Ω
T T ∗ G (( φ λ ) ∗ X, ( φ λ ) ∗ Y ) ( g, µ, ξ, ν + λ )does not necessarily hold. Hence, ψ λ is a symplectic action but not φ λ . (cid:3) Proposition 6.11.
Poisson reduction of
T T ∗ G under the action ψ of g ∗ results in G s ( g s g ∗ ) endowed with the bracket (6.16) { E, F } G s ( g × g ∗ ) ( g, ξ, ν ) = (cid:28) T ∗ e R g δFδg , δEδν (cid:29) − (cid:28) T ∗ e R g δEδg , δFδν (cid:29) + (cid:28) ν, (cid:20) δEδν , δFδν (cid:21)(cid:29) . Remark 6.12.
The Poisson bracket { E, F } G s ( g s g ∗ ) is independent of derivatives of functions withrespect to ξ and it resembles to the canonical Poisson bracket in Eq.(3.10) on G s g ∗ . On the otherhand, the space G s ( g ∗ × g ), which is isomorphic to G s ( g s g ∗ ), has the Poisson bracket in Eq.(5.9)involving derivatives with respect to both of ξ and ν . This latter is obtained from T ∗ T G via reductionby g ∗ .The infinitesimal generator X T T ∗ G (0 ,ν , , of the action are defined by ν ∈ Lie ( g ∗ ). We compute theassociated momentum map from the equation D J g ∗ T T ∗ G ( g, µ, ξ, ν ) , ν E = D θ , X T T ∗ G (0 ,ν , , E = − h ν , ξ i , where θ is the Tulczyjew potential one-form in Eq.(2.32). We find that(6.17) J g ∗ T T ∗ G : T T ∗ G → Lie ∗ ( g ∗ ) ≃ g : ( g, µ, ξ, ν ) → − ξ is minus the projection to third factor in T T ∗ G . The preimage of an element ξ ∈ g is the space G s ( g ∗ × g ∗ ). Proposition 6.13.
The symplectic reduction of
T T ∗ G under the action of g ∗ defined in Eq.(6.8)results in the reduced space (cid:16) J g ∗ T T ∗ G (cid:17) − ( ξ ) (cid:30) g ∗ ≃ G s ( g ∗ × g ∗ )/ g ∗ ≃ G s g ∗ with the canonical symplectic two-from Ω G s g ∗ as given in Eq.(2.10). Remark 6.14.
The existence of symplectic action of g ∗ on T T ∗ G can be traced back to existence ofthe symplectic diffeomorphism(6.18) Ω ♭G s g ∗ : T T ∗ G → T ∗ T ∗ G : ( g, µ, ξ, ν ) → (cid:0) g, µ, ν + ad ∗ ξ µ, − ξ (cid:1) described in [15]. n the following proposition, we discuss the actions ψ and φ of g ∗ on T T ∗ G in Eqs(6.12) and (6.13)from a different point of view. Proposition 6.15.
The mappings
Emb : G s g ∗ ֒ → T T ∗ G : ( g, µ ) → ( g, µ, , Emb : G s g ∗ ֒ → T T ∗ G : ( g, ν ) → ( g, , , ν )(6.19) define a Lagrangian and a symplectic, respectively, embeddings of G s g ∗ into T T ∗ G .Proof. The first embedding is Lagrangian because it is the zero section of the fibration
T T ∗ G → G s g ∗ .The second one is symplectic because the pull-back of Ω T T ∗ G to G s g ∗ by Emb results in thesymplectic two-form Ω G s g ∗ in Eq.(2.9). On the image of Emb , the Hamilton’s equations (6.1) reduceto the trivialized Hamilton’s equations (3.6) on G s g ∗ . Consequently, the embedding g ∗ → T T ∗ G isa Poisson map. When E = h ( ν ) the Hamilton’s equations (6.1) reduce to the Lie-Poisson equations(3.13). (cid:3) Reduction of
T T ∗ G by G s g . The action ϑ : T T ∗ G × ( G s g ) → T T ∗ G, (6.20) (( g, µ, ξ, ν ) ; ( h, η )) ϑ ( h,η ) ( g, µ, ξ, ν ) := (cid:16) gh, µ, ξ + Ad g η, ν − ad ∗ Ad g η µ (cid:17) (6.21)of G s g on T T ∗ G can be described as a composition ϑ ( h,η ) = ϑ ( h, ◦ ϑ ( e,Ad g η ) , where ϑ ( h, and ϑ ( e,Ad g η ) can be identified with the actions of G and g on T T ∗ G , respectively. Sinceboth of these are symplectic, the action ϑ of G s g on T T ∗ G is symplectic. Proposition 6.16.
The Poisson reduction of
T T ∗ G under the action of G s g in Eq.(6.21) resultsin g ∗ × g ∗ endowed with the bracket (6.22) { E, F } g ∗ × g ∗ ( µ, ν ) = (cid:28) ν, (cid:20) δEδν , δFδν (cid:21)(cid:29) . Remark 6.17.
Although the Poisson bracket (6.22) structurally resembles the Lie-Poisson bracket on g ∗ , it is not a Lie-Poisson bracket on g ∗ × g ∗ considered as dual of Lie algebra g s g of the group G s g .We refer to the Poisson bracket in Eq.(4.14) for the Lie-Poisson structure on Lie ∗ ( G s g ) = g ∗ × g ∗ .Right invariant vector field generating the action ϑ is associted to two tuples ( ξ , ξ ) in the Lie algebraof G s g , and is given by(6.23) X T T ∗ G ( ξ , ,ξ , ( g, µ, ξ, ν ) = (cid:0) T R g ξ , ad ∗ ξ µ, ξ + [ ξ, ξ ] , ad ∗ ξ ν (cid:1) . The momentum map for this Hamiltonian action is defined by the equation D J G s g T T ∗ G ( g, µ, ξ, ν ) , ( ξ , ξ ) E = D θ , X T T ∗ G ( ξ , ,ξ , E = h µ, ξ i + (cid:10) ν + ad ∗ ξ µ, ξ (cid:11) , where θ , in Eq.(2.33), is the Tulczyjew potential one-form on T T ∗ G . We find J G s g T T ∗ G : T T ∗ G → Lie ∗ ( G s g ) = g ∗ × g ∗ : ( g, µ, ξ, ν ) = (cid:0) ν + ad ∗ ξ µ, µ (cid:1) . Note that, we have the following relation J G s g T T ∗ G ( g, µ, ξ, ν ) = (cid:0) J GT T ∗ G ( g, µ, ξ, ν ) , J g T T ∗ G ( g, µ, ξ, ν ) (cid:1) or momentum maps in Eqs.(6.3) and (6.10) for the actions of G and g on T T ∗ G . The preimage ofa fixed element ( λ, µ ) ∈ g ∗ × g ∗ is (cid:16) J G s g T T ∗ G (cid:17) − ( λ, µ ) = (cid:8) ( g, µ, ξ, ν ) : ν = λ − ad ∗ ξ µ (cid:9) which we may identify with the semidirect product G s g . We recall the coadjoint action Ad ∗ ( g,ξ ) , inEq.(4.17), of the group G s g on the dual g ∗ × g ∗ of its Lie algebra. The isotropy subgroup ( G s g ) ( λ,µ ) of this coadjoint action is( G s g ) ( λ,µ ) = n ( g, ξ ) ∈ G s g : Ad ∗ ( g,ξ ) ( λ, µ ) = ( λ, µ ) o and acts on the preimage (cid:16) J G s g T T ∗ G (cid:17) − ( λ, µ ). A generic quotient space (cid:16) J G s g T T ∗ G (cid:17) − ( λ, µ ) (cid:30) ( G s g ) ( λ,µ ) ≃ G s g / ( G s g ) ( λ,µ ) ≃ O ( λ,µ ) is a coadjoint orbit in g ∗ × g ∗ through the point ( λ, µ ) under the coadjoint action Ad ∗ ( g,ξ ) in Eq.(4.17). Proposition 6.18.
The symplectic reduction of
T T ∗ G under the action of G s g given in Eq.(6.21)results in the coadjoint orbit O ( λ,µ ) in g ∗ × g ∗ through the point ( λ, µ ) under the coadjoint action Ad ∗ ( g,ξ ) in Eq.(4.17) as the total space and the symplectic two-from Ω O ( λ,µ ) in Eq.(4.19). It is also possible to obtain the symplectic space O ( λ,µ ) in two steps. Recall the symplectic reduction of T T ∗ G under the action of g at µ ∈ g ∗ which results in G s g ∗ with the canonical symplectic two-fromΩ G s g ∗ . Then, consider the action of isotropy subgroup G µ on G s g ∗ and apply symplectic reductionwhich results in (cid:16) O ( λ,µ ) , Ω O ( λ,µ ) (cid:17) . Following is the diagram summarizing this two stage reduction of T T ∗ G .(6.24) ( G s g ∗ ) s ( g s g ∗ ) SR by g at µ ✐✐✐✐✐✐✐✐ t t ✐✐✐✐✐✐✐✐ SR by G s g at ( λ,µ ) (cid:15) (cid:15) G s g ∗ G µ at λ ❯❯❯❯❯❯❯❯❯ * * ❯❯❯❯❯❯❯❯❯ O ( λ,µ ) Reductions of
T T ∗ G by G s g Reduction of
T T ∗ G by G s g ∗ . The action α : T T ∗ G × ( G s g ∗ ) → T T ∗ G : (( g, µ, ξ, ν ) ; ( h, λ )) α ( h,λ ) ( g, µ, ξ, ν )of G s g ∗ on T T ∗ G is given by(6.25) α ( h,λ ) ( g, µ, ξ, ν ) = ( gh, µ + Ad ∗ g λ, ξ, ν ) . As in the case of the action of G s g , it can also be described by composition of two actions α ( h,λ ) = α ( h, ◦ α ( e,Ad ∗ g λ ) , where, α ( h, and α ( e,Ad ∗ g λ ) can be identified with the actions of G and g ∗ on T T ∗ G , respectively. Sinceboth of them are symplectic, α is also symplectic. roposition 6.19. Poisson reduction of
T T ∗ G under the action (6.25) of G s g ∗ results in g s g ∗ endowed with the bracket (6.26) { F, H } g s g ∗ ( ξ, ν ) = (cid:28) ν, (cid:20) δEδν , δFδν (cid:21)(cid:29) . Remark 6.20.
Regarding g ∗ × g as dual of the Lie algebra g s g ∗ of G s g ∗ , we obtained the Lie-Poisson bracket in Eq.(5.11). Although g ∗ × g and g × g ∗ are isomorphic as vector spaces, (5.11) isdifferent from the Poisson bracket in Eq.(6.26) as manifestation of group structure carried by adaptedtrivialization.Infinitesimal generator of α is associated to the two tuple ( ξ , ν ) in the Lie algebra g s g ∗ of G s g ∗ and is in the form(6.27) X T T ∗ G ( ξ ,ν , , ( g, µ, ξ, ν ) = (cid:0) T R g ξ , ν + ad ∗ ξ µ, [ ξ, ξ ] , ad ∗ ξ ν − ad ∗ ξ ν (cid:1) . The momentum mapping J G s g ∗ T T ∗ G is defined by the equation D J G s g ∗ T T ∗ G ( g, µ, ξ, ν ) , ( ξ , ν ) E = D θ , X T T ∗ G ( ξ ,ν , , E = − h ν , ξ i + (cid:10) ν + ad ∗ ξ µ, ξ (cid:11) , where θ is the potential one-form given by Eq.(2.32). We obtain J G s g ∗ T T ∗ G : T T ∗ G → Lie ∗ ( G s g ∗ ) = g ∗ × g : ( g, µ, ξ, ν ) → (cid:0) ν + ad ∗ ξ µ, − ξ (cid:1) which can be decomposed as J G s g ∗ T T ∗ G ( g, µ, ξ, ν ) = (cid:16) J GT T ∗ G ( g, µ, ξ, ν ) , J g ∗ T T ∗ G ( g, µ, ξ, ν ) (cid:17) where J GT T ∗ G and J g ∗ T T ∗ G are momentum mappings in Eqs.(6.3) and (6.17) for the actions of G and g ∗ on T T ∗ G , respectively. The preimage of an element ( λ, ξ ) ∈ g ∗ × g is (cid:16) J G s g ∗ T T ∗ G (cid:17) − ( λ, ξ ) = (cid:8) ( g, µ, − ξ, ν ) : ν = λ + ad ∗ ξ µ (cid:9) which can be identified with the space G s g ∗ . The isotropy subgroup of coadjoint action of G s g on g ∗ × g is ( G s g ∗ ) ( λ,ξ ) = n ( g, µ ) ∈ G s g : Ad ∗ ( g,µ ) ( λ, ξ ) = ( λ, ξ ) o , where the coadjoint action is given by Eq.(5.15). Isotropy subgroup acts on preimage of ( λ, ξ ) andresults in the coadjoint orbit through the point ( λ, ξ ) ∈ g ∗ × g (cid:16) J G s g ∗ T T ∗ G (cid:17) − ( λ, ξ ) (cid:30) ( G s g ∗ ) ( λ,ξ ) ≃ G s g ∗ / ( G s g ) ( λ,ξ ) ≃ O ( λ,ξ ) . Proposition 6.21.
Symplectic reduction of
T T ∗ G under the action of G s g ∗ given by Eq.(6.25) resultsin the coadjoint orbit O ( λ,ξ ) and the symplectic two-from Ω O ( λ,ξ ) in Eq.(5.16). Similar to the reduction of
T T ∗ G by G s g , we may perform symplectic reduction of T T ∗ G withaction of G s g ∗ by two stages. Recall symplectic reduction of T T ∗ G with action of g ∗ at ξ ∈ g whichresults in G s g ∗ and the canonical symplectic two-from Ω G s g ∗ . Then, consider the action of isotropysubgroup G ξ , defined in Eq.(5.17), on G s g ∗ and apply symplectic reduction. This gives O ( λ,ξ ) and he symplectic two-form Ω O ( λ,ξ ) . Following diagram shows this two stage reduction of T T ∗ G (6.28) ( G s g ∗ ) s ( g s g ∗ ) SR by g ∗ at ξ ✐✐✐✐✐✐✐ t t ✐✐✐✐✐✐✐ SR by G s g ∗ at ( λ,ξ ) (cid:15) (cid:15) G s g ∗ G ξ at λ ❯❯❯❯❯❯❯❯❯ * * ❯❯❯❯❯❯❯❯❯ O ( λ,ξ ) Reduction of
T T ∗ G by G s g ∗ We summarize diagrammatically all possible reductions of Hamiltonian dynamics on the Tulczyjewsymplectic space
T T ∗ G .(6.29) ( G s g ∗ ) s g ∗ G s g ∗ (cid:31) (cid:127) symplecticleaf / / _? symplecticleaf o o G s ( g s g ∗ ) g ∗ s g ∗ Poissonembedding O O ( G s g ∗ ) s ( g s g ∗ ) PR by g g g PPPPPPPPPPPPPPPPPPPPPPPPPP
PR by g ∗ ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ SR by g (cid:19) (cid:19) SR by g ∗ (cid:11) (cid:11) PR by G s g o o PR by G s g ∗ / / SR by G s g w w ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ PR by G s g ∗ ' ' PPPPPPPPPPPPPPPPPPPPPPPPPPP
SR by g S S SR by g ∗ K K g s g ∗ Poissonembedding O O O ( µ,ν ) (cid:31) ? symplecticleaf O O G s g ∗ G µ o o SR by G ξ / / O ( µ,ξ ) (cid:31) ? symplecticleaf O O Hamiltonian reductions of
T T ∗ Q = ( G s g ∗ ) s ( g s g ∗ )6.2. Lagrangian Dynamics on
T T ∗ G . As it is a tangent bundle, we can study Lagrangian dynamics on
T T ∗ G ≃ ( G s g ∗ ) s ( g s g ∗ ). Wedefine variation of the base element ( g, µ ) ∈ G s g ∗ by tangent lift of right translation of the Lie algebraelement ( η, λ ) ∈ g s g ∗ , that is δ ( g, µ ) = T ( e, R ( g,µ ) ( η, λ ) = (cid:0) T e R g η, λ + ad ∗ η µ (cid:1) . To obtain the reduced variational principle δ ( ξ, ν ) on the Lie algebra g s g ∗ we compute δ ( ξ, ν ) = ddt ( η, λ ) + [( ξ, ν ) , ( η, λ )] g s g ∗ = ddt ( η, λ ) + (cid:0) [ ξ, η ] , ad ∗ η ν − ad ∗ ξ λ (cid:1) = (cid:16) ˙ η + [ ξ, η ] , ˙ λ + ad ∗ η ν − ad ∗ ξ λ (cid:17) , (6.30)for any ( η, λ ) ∈ g s g ∗ . Assuming δ ( ξ, ν ) = ( δξ, δν ) and δ ( g, µ ) = ( δg, δµ ), we have the set of variations(6.31) δg = T e R g η, δµ = λ + ad ∗ η µ, δξ = ˙ η + [ ξ, η ], δν = ˙ λ + ad ∗ η ν − ad ∗ ξ λ or an arbitrary choice of ( η, λ ) ∈ g s g ∗ . Note that, these variations are the image of the right invariantvector field X T T ∗ G ( η,λ, ˙ η, ˙ λ ) generated by (cid:16) η, λ, ˙ η, ˙ λ (cid:17) . Proposition 6.22.
For a given Lagrangian E on T T ∗ G , extremals of action integral are defined bythe trivialized Euler-Lagrange equations ddt (cid:18) δEδξ (cid:19) = T ∗ e R g (cid:18) δEδg (cid:19) − ad ∗ δEδµ µ + ad ∗ ξ (cid:18) δEδξ (cid:19) − ad ∗ δEδν νddt (cid:18) δEδν (cid:19) = δEδµ + ad ξ δEδν (6.32) obtained by the variational principles in Eq.(6.31).Proof. Let us begin with the observation that for any ( g, ξ ) ∈ G s g , the variation δ ( g, ξ ) = ( δg, δξ ) at( η, ˙ η ) ∈ g s g may be given by( δg, δξ ) = X T G ( η, ˙ η ) ( g, ξ ) = ( T R g η, ˙ η + [ η, ξ ]) . Accordingly, given a Lagrangian L : T G ∼ = G s g → R , the action integral δ Z ba L ( g, ξ ) dt = Z ba (cid:18)D δ L δg , δg E + D δ L δξ , δξ E(cid:19) dt = Z ba (cid:18)D δ L δg , T R g η E + D δ L δξ , ˙ η E + D δ L δξ , [ η, ξ ] E(cid:19) dt = D δ L δξ , η E(cid:12)(cid:12)(cid:12)(cid:12) ba + Z ba (cid:18)D T ∗ R g δ L δg , η E + D − ddt δ L δξ , η E + D ad ∗ ξ δ L δξ , η E(cid:19) dt leads to the (trivialized) Euler-Lagrange equations ddt δ L δξ = T ∗ R g δ L δg + ad ∗ ξ δ L δξ . Accordingly, the (trivialized) Euler-Lagrange equations on
T T ∗ G are given by ddt δEδ ( ξ, ν ) = T ∗ R ( g,µ ) δEδ ( g, µ ) + ad ∗ ( ξ,ν ) δEδ ( ξ, ν ) , where the first summand on the right hand side is T ∗ R ( g,µ ) δEδ ( g, µ ) = (cid:16) T ∗ R g δEδg − ad ∗ δEδµ µ, δEδµ (cid:17) , while the second summand beingad ∗ ( ξ,ν ) δEδ ( ξ, ν ) = (cid:16) ad ∗ ξ δEδξ − ad ∗ δEδν ν, δEδν + ad ξ δEδν (cid:17) . (cid:3) Proposition 6.23.
Given a Lagrangian E = E ( g, µ, ξ, ν ) on T T ∗ G , the quantity (6.33) D δEδξ , ξ E + D δEδν , ν E − E is constant. roof. Let us begin with dEdt = D ∂E∂g , ˙ g E + D ∂E∂µ , ˙ µ E + D ∂E∂ξ , ˙ ξ E + D ∂E∂ν , ˙ ν E = D ∂E∂g , T R g ξ E + D ∂E∂µ , ν + ad ∗ ξ µ E + D ∂E∂ξ , ˙ ξ E + D ∂E∂ν , ˙ ν E = D T ∗ R g (cid:18) ∂E∂g (cid:19) − ad ∗ ∂E∂µ µ, ξ E + D ∂E∂µ , ν E + D ∂E∂ξ , ˙ ξ E + D ∂E∂ν , ˙ ν E . Next, substituting the (trivialized) Euler-Lagrange equations (6.32), we obtain dEdt = D ddt (cid:18) ∂E∂ξ (cid:19) − ad ∗ ξ ∂E∂ξ + ad ∗ ∂E∂ν ν, ξ E + D ν, ddt (cid:18) ∂E∂ν (cid:19) − ad ξ ∂E∂ν E + D ∂E∂ξ , ˙ ξ E + D ∂E∂ν , ˙ ν E = D ddt (cid:18) ∂E∂ξ (cid:19) , ξ E + D ∂E∂ξ , ˙ ξ E + D ˙ ν, ∂E∂ν E + D ν, ddt (cid:18) ∂E∂ν (cid:19) E + D ad ∗ ∂E∂ν ν, ξ E − D ad ∗ ξ ∂E∂ξ , ξ E − D ν, ad ξ ∂E∂ν E = ddt (cid:18)D ∂E∂ξ , ξ E + D ν, ∂E∂ν E(cid:19) , from which we result follows. (cid:3) Reductions on
T T ∗ G . When the Lagrangian density E in the trivialized Euler-Lagrange equa-tions (6.32) is independent of the group variable g ∈ G , we arrive at Euler-Lagrange equations (6.38)on g ∗ s ( g s g ∗ ). In addition, if the Lagrangian E depends only on fiber coordinates E = E ( ξ, ν ) , wehave the Euler-Poincar´e equations (6.34). Proposition 6.24.
The Euler-Poincar´e equations on the Lie algebra g s g ∗ are (6.34) ddt (cid:18) δEδξ (cid:19) = ad ∗ ξ (cid:18) δEδξ (cid:19) − ad ∗ δEδν ν, ddt (cid:18) δEδν (cid:19) = − ad ξ δEδν . If, moreover, E = E ( ξ ) , the Euler-Poincar´e equations (3.4) on g arise. This procedure is calledreduction by stages [10, 23].Alternatively, the Lagrangian density E in trivialized Euler-Lagrange equations (6.32) can be in-dependent of µ ∈ g ∗ , that is, E can be invariant under the action of g ∗ on T T ∗ G . In this case,we have Euler-Lagrange equations (6.36) on G s ( g s g ∗ ). When E = E ( g, ξ ), we have trivializedEuler-Lagrange equations (6.37) on G s g . The following diagram summarizes this discussion. G s g ∗ ) s ( g s g ∗ )EL in (6.32) L.R. by G ③③③③③③③③ | | ③③③③③③③③ L.R. by g ∗ ❉❉❉❉❉❉❉❉ " " ❉❉❉❉❉❉❉❉ EPR by G s g ∗ (cid:15) (cid:15) g ∗ s ( g s g ∗ )EL in (6.38) ( g s g ∗ )EP in (6.34) G s ( g s g ∗ )EL in (6.36) g EP in (3.4) ?(cid:31) canonicalimmersion O O P0 canonicalimmersion ❉❉❉❉❉❉ b b ❉❉❉❉❉❉ .(cid:14) canonicalimmersion ③③③③③③ < < ③③③③③③ G s g EL in (6.37) ?(cid:31) canonicalimmersion O O (6.35) Lagrangian reductions on
T T ∗ G ddt (cid:18) δEδξ (cid:19) = T ∗ e R g (cid:18) δEδg (cid:19) + ad ∗ ξ (cid:18) δEδξ (cid:19) − ad ∗ δEδν νddt (cid:18) δEδν (cid:19) = − ad ξ δEδν (6.36) ddt (cid:18) δEδξ (cid:19) = T ∗ e R g (cid:18) δEδg (cid:19) + ad ∗ ξ (cid:18) δEδξ (cid:19) (6.37) ddt (cid:18) δEδξ (cid:19) = − ad ∗ δEδµ µ + ad ∗ ξ (cid:18) δEδξ (cid:19) − ad ∗ δEδν νddt (cid:18) δEδν (cid:19) = δEδµ − ad ξ δEδν (6.38) 7. Summary, Discussions and Prospectives
We write Hamilton’s equations on the cotangent bundles T ∗ T G and T ∗ T ∗ G . Symplectic and Poissonreductions of T ∗ T G are performed under actions of G, g and G s g as shown in diagram (4.23). T ∗ T ∗ G is also reduced by actions of G, g ∗ and G s g ∗ c.f. diagram (5.18).On the Tulczyjew’s symplectic space T T ∗ G = ( G s g ∗ ) s ( g s g ∗ ), we obtain both Hamilton’s andEuler-Lagrange equations. Hamilton’s equations are reduced by symplectic and Poisson actions of G, g ∗ , g , G s g and G s g ∗ . These reductions are summarized in diagram (6.29). As it is a tangentbundle, Lagrangian reductions are performed with actions of G , g ∗ , G s g ∗ and G s ( g ∗ × g ) and, areshown in diagram (6.35).Hamiltonian reductions of the Tulczyjew’s symplectic space T T ∗ G can be generalized to symplecticreduction of tangent bundle of a symplectic manifold with lifted symplectic structure. This may bea first step towards reduction of special symplectic structures and reduction of Tulczyjew’s triplet or arbitrary configuration manifold Q . In order to obtain this more general picture for trivializationand reduction of Tulczyjew triplet, we plan to pursue a new project where the reduction is appliedto Lagrangian dynamics on T Q and Hamiltonian dynamics on T ∗ Q for an arbitrary manifold Q . Inthis case, the reduced Lagrangian dynamics on the orbit space T Q/G is called Lagrange-Poincar´eequations. If, particularly, Q = G then the Lagrange-Poincar´e equations turns out to be Euler-Poincar´e equations on g . Similarly, the Hamiltonian dynamics on T ∗ Q/G is called Hamilton-Poincar´eequations and, reduce to Lie-Poisson equations on g ∗ for the case of Q = G . In the first paper [17]of that series, we have already presented the trivialization and reduction of Tulczyjew triplet for anarbitrary manifold under the presence of an Ehresmann connection. References [1] R. Abraham and J. E. Marsden.
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J. Differential Geom. , 18(3):523–557, 1983. epartment of Mathematics, Gebze Technical University, 41400 Gebze-Kocaeli, Turkey Email address : [email protected] Department of Mathematics, Yeditepe University, 34755 Atas¸ehir-˙Istanbul, Turkey
Email address : [email protected] Department of Mathematics, Is¸ık University, 34980 S¸ile-˙Istanbul, Turkey
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