Tulczyjew's Triplet with an Ehresmann connection I: Trivialization and Reduction
aa r X i v : . [ m a t h - ph ] J u l TULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATIONAND REDUCTION
OĞUL ESEN, MAHMUT KUDEYT, AND SERKAN SÜTLÜAbstract. The T ulczyjew’s triplet is a geometric framework that makes the Legendre transformation availableeven for the singular systems. In this work we present the trivialization, in both the horizontal and the verticalterms, and the reduction of the classical triplet under a symmetry group action in the presence of an Ehresmannconnection. We thus establish a geometric pathway for the Legendre transformations of singular dynamicalsystems after reduction.
Key words:
The Tulczyjew’s triplet; Hamiltonian reduction; Lagrangian reduction; Ehresmann connection.
MSC2010: C ontents1. Introduction 22. Preliminaries 42.1. (Special) Symplectic Manifolds 42.2. Tulczyjew’s Triplet 52.3. Group Action and Connection 82.4. Trivialization and Reduction of Tangent Bundle 102.5. Trivialization and Reduction of Cotangent Bundle 113. Trivializations and Reductions of Iterated Tangent Bundles 143.1. The Infinitesimal Group Action 143.2. Trivialization of TTQ 163.3. Reduction of TTQ 173.4. Trivialization of TT*Q 203.5. Reduction of TT*Q 224. Trivializations and Reductions of Cotangent Bundles 244.1. Trivialization of T*TQ 244.2. Reduction of T*TQ 254.3. Trivialization of T*T*Q 264.4. Reduction of T*T*Q 285. Trivialization and Reduction of Tulczyjew’s Triplet 295.1. Trivializations and Reductions of the Canonical Forms 295.2. Trivializations and Reductions of the Symplectomorphisms 305.3. Trivializations and Reductions of the Derivations 315.4. Trivializations and Reductions of the Tulczyjew’s Triplet 346. Horizontal-Vertical Decompositions of the Tulczyjew’s Triplet 366.1. Horizontal-Vertical Decomposition of the Second Order Tangent Bundles 366.2. Horizontal-Vertical Decomposition of the Second Order Cotangent Bundles 396.3. Horizontal-Vertical Decomposition of the Symplectomorphisms 427. Conclusion and Future Work 448. Acknowledgment 45References 45
1. Introduction
Euler-Lagrange equations governing the motion of a physical system, whose configuration space is afinite dimensional manifold, is determined by a Lagrangian function defined on the tangent bundle. Onthe other hand, Hamilton’s equation is generated by a Hamiltonian function defined on the cotangentbundle [1, 3, 15, 26, 44]. In order to find the (Legendre) transformation between the Lagrangian andthe Hamiltonian realizations of a system, it is significant to determine whether the Lagrangian (or theHamiltonian) function is (hyper)regular. For regular cases, it is immediate to perform the Legendretransformation as the fiber derivative of the generating function. On the other hand, for singular cases, toconstruct a relationship between these two formalisms is not so straightforward. Tulczyjew have proposed ageometric framework, called the Tulczyjew’s triplet, which makes the transformation possible even for thesingular and/or constrained systems, [63, 64, 65, 66, 67, 69, 70, 68].The Tulczyjew’s triplet is a construction of (the second order) iterated bundles of the configuration space Q connected by two symplectomorphisms, that is,(1.1) T ∗ TQ π T Q " " ❋❋❋❋❋❋❋❋ TT ∗ Q T π Q | | ①①①①①①①① Ω ♭ Q / / τ T ∗ Q ●●●●●●●● α Q o o T ∗ T ∗ Q π T ∗ Q { { ✈✈✈✈✈✈✈✈✈ TQ T ∗ Q In this theory, the dynamics of the system under consideration (whether it is in Lagrangian or Hamiltonianform) is formulated as a Lagrangian submanifold of the Tulczyjew symplectic space TT ∗ Q , that is, thetangent bundle of the momentum phase space. This is achieved by means of two special symplecticstructures [5, 62, 69] constituting the left and the right wings of the triplet. If the Lagrangian submanifoldsgenerated by a Hamiltonian function and a Lagrangian function coincide, the Legendre transformation thenis said to be established. For singular systems, one may need to refer to a Morse family in order to achieve thecoincidence of the Lagrangian submanifolds. We refer the reader to Subsections 2.1 and 2.2 for the detailsof this geometric construction, as well as the definitions of the projections and the symplectomorphismswith sharp mathematical rigor.The Tulczyjew’s triplet has been constructed for many physical systems and on several different geometrictheories. For instance, we refer to [18, 19, 31] for the triplet over a Lie group. In the case of the higher orderdynamics, we may cite [13], whereas in the case of the field theories we refer to [12, 16, 27, 28, 61], and forthe higher order field theories, [30]. On the other hand, for the graded bundles we cite [8, 32, 34, 33], andfor a discussion related to prolongations we refer to [40]. Finally, in [39], an extension of the Tulczyjew’striple to the level of Lie algebroids was proposed.Symmetry for a physical system, in its very classical sense, can be given by a Lie group, say G , action on theconfiguration space Q of a physical system [37, 38, 45, 50, 58, 57]. Moreover, with the group action beingfree and proper, the configuration space Q may be reduced to the orbit space G \ Q , along with a differentiablestructure. Besides, the tangent and the cotangent bundles can be reduced as well, by means of the liftedactions. Furthermore, if the existence of an Ehresmann connection is assumed, then both the tangent bundle ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 3 and the cotangent bundle can be decomposed into the sum of horizontal and vertical subbundles, that is,[9, 10, 47](1.2) TQ ≃ Q ⊕ T ( G \ Q ) × g , T ∗ Q ≃ Q ⊕ T ∗ ( G \ Q ) × g ∗ , where ⊕ stands for the Whitney sum over the base manifold G \ Q . On the other hand, we do remark theappearances of the Lie algebra g of the symmetry group, as well as its dual space g ∗ , in (1.2). Further,referring to these trivializations, one may arrive at the reductions of the tangent and cotangent bundles inthe forms(1.3) G \ TQ ≃ T ( G \ Q ) ⊕ ˜ g , G \ T ∗ Q ≃ T ∗ ( G \ Q ) ⊕ ˜ g ∗ , respectively. Here, ˜ g stands for the adjoint bundle, whereas ˜ g ∗ is the coadjoint bundle. We have reservedSubsections 2.3, 2.4 and 2.5 for the formal realizations of (1.2) and (1.3). These identifications are importantnot only for geometrical considerations, but also for the concrete physical applications; since they enable todecompose the dynamics into the vertical and the horizontal parts. Accordingly, the reduced dynamics maybe traced explicitly over the orbit spaces.We plan this work as the first in a series of papers where we address both the trivialization and the reductionof the classical Tulczyjew’s triplet in the presence of an Ehresmann connection. In more concrete terms,we shall construct the trivialization of the Tulczyjew’s triplet by substituting the decompositions in (1.2)of the tangent and the cotangent bundles into (1.1). To this end, of course, we shall need the lifts ofboth the connection and the momentum mappings to the level of the second order iterated bundles. Thisway, we shall further obtain the reduction of the Tulczyjew’s triplet simply by following the identificationsin (1.3), which in turn involves both the trivializations and the reductions of all the ingredients of thetriplet; namely, the second iterated bundles TTQ , T ∗ TQ , TT ∗ Q and T ∗ T ∗ Q , the symplectomorphisms Ω ♭ Q and α Q , all the projections, and finally the special symplectic structures (the wings). In addition, weshall express the horizontal-vertical decompositions of both the trivialized and the reduced Tulczyjew’striplets, inevitably of all ingredients, under the action of the group G . Our ultimate goal is to establish theLegendre transformation for singular Hamiltonian or/and Lagrangian systems admitting symmetries. Oncethe geometrical framework is set in the present paper, we shall focus on the Legendre transformations of thereduced Hamiltonian and the reduced Lagrangian dynamics, [20].Since one can find its roots in [59], the Lagrangian reduction theory [51, 52, 6] is rather more recentcompared to the Hamiltonian reduction theory. The reduced dynamics on the reduced tangent bundle G \ TQ is studied under the realm of the Lagrange–Poincaré equations. If, in particular, Q = G then theLagrange–Poincaré equations reduces to the Euler-Poincaré equations on the Lie algebra g . Accordingly,this paper may be considered as a generalization of [18, 19, 17], wherein the trivialization and reduction ofthe Tulczyjew’s triple were studied in the case Q = G . Referring the reader to [10] for further details on thereduced dynamics, especially the vertical and horizontal variations by means of an Ehresmann connection,we remak that the horizontal-vertical decomposition of the Tulczyjew’s triple we shall present in this noteprovides a proper setting for the Legendre transformations of the horizontal-vertical Lagrange-Poincaréequations studied in [9, 10, 47]. In other words, these works carry a motivational importance for the presentpaper. OĞUL ESEN, MAHMUT KUDEYT, AND SERKAN SÜTLÜ
In the Hamiltonian counterpart of the reduction, where we refer the reader to [53] for a brief history of thetheory, we shall discuss two approaches; the symplectic (or Marsden-Weinstein) reduction [46, 54], andthe Poisson reduction [49]. Since the Hamiltonian dynamics is defined on the cotangent bundle T ∗ Q , thereduction under the symmetry by a Lie group G is studied on the reduced cotangent bundle G \( T ∗ Q ) . Inthis case, the reduced dynamical equations are called the Hamilton-Poincaré equations [9, 47], generatedby a reduced Hamiltonian function. If, in particular, the configuration space coincides with the symmetrygroup, that is Q = G , then one arrives at the Lie-Poisson equations on the linear algebraic dual g ∗ of theLie algebra g , see for instance [50]. The variational formulations of the reduced Hamiltonian dynamics arepresented in [9].Let us remark that the Tulczyjew triplets over Lie algebroids [2, 7, 14, 29] have already been developed toaddress the problems similar to those considered in this present note, as the reduced tangent bundle G \ TQ is a (Atiyah) Lie algebroid. On the other hand, we also find it important to note that both the Hamilton-Poincaré and the Lagrange-Poincaré equations have already been recasted as Lagrangian submanifolds in[14]. Moreover, there are other recent works addressing the reduction of the Tulczyjew’s triplet undersymmetry, [4, 22, 23]. However, none of these strategies approaches to the Tulczyjew’s triplet from thepoint of view of an Ehresmann connection, the decompositions (1.2) it enables, and hence the reductions in(1.3). This, as depicted in [9, 10, 47], permits to analyze the dynamics over the orbit spaces concretely.The paper is planned over five main sections. In the following one, we review the classical Tulczyjew’striplet, the trivializations and the reductions of the first order bundles. In Section 3, we focus on thetrivializations and the reduction of the iterated tangent bundles TTQ and T ∗ TQ . Section 4 is for similardiscussion on the iterated cotangent bundles T ∗ TQ and T ∗ T ∗ Q . All these results are then merged in Section5 in order to arrive at the trivialization and the reduction of the Tulczyjew’s triplet. In Section 6, we showthe horizontal-vertical decompositions of the trivialized and the reduced Tulczyjew’s triplets.
2. Preliminaries (Special) Symplectic Manifolds.
A manifold P is said to be a symplectic manifold if it is equipped with a non-degenerate closed two-form Ω . In this case, Ω is called symplectic two-form. Cotangent bundles.
Consider a manifold Q , and its cotangent bundle T ∗ Q . We denote the canonicalcotangent projection by π Q mapping covectors in T ∗ Q to their initial points in Q . There exists a canonical(Liouville) one-form θ Q defined, on a vector field X over T ∗ Q as(2.1) θ Q ( X ) = h τ T ∗ Q ( X ) , T π Q ( X )i , where τ T ∗ Q is the tangent bundle projection from TT ∗ Q to T ∗ Q whereas T π Q is the tangent mapping ofthe cotangent bundle projection π Q . Minus of the exterior derivative of the canonical one-form θ Q , that is Ω Q : = − d θ Q , is the canonical symplectic two-form on T ∗ Q . In Darboux’ coordinates ( q i , p i ) on T ∗ Q , the ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 5 canonical one-form θ Q and the symplectic two-form Ω Q read(2.2) θ Q = p i dq i , Ω Q = dq i ∧ dp i , respectively. Special symplectic structures.
Let P be a symplectic manifold carrying an exact symplectic two-form Ω = − d θ where θ is being a potential one-form . Assume also that, P is the total space of a fibrebundle ( P , π, Q ) .Here, Q is base space and, π is projection. A special symplectic structure is a quintuple ( P , π, Q , θ, Θ ) where Θ is a fiber preserving symplectic diffeomorphism from P to the cotangent bundle T ∗ Q characterized by(2.3) h Θ ( x ) , π ∗ X ( q )i = h θ ( x ) , X ( x )i for a vector field X on P , for any point x in P where π ( x ) = q [43, 62, 67]. Here, on the left hand sidethere is the natural pairing between the cotangent space T ∗ q Q and the tangent space T q Q whereas on the righthand side there is the pairing between T ∗ x P and T x P . The two-tuple ( P , Ω ) is called underlying symplecticmanifold of the special symplectic structure. Here is a diagram exhibiting special symplectic structure:(2.4) P Θ / / π (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ T ∗ Q π Q ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤ Q Tulczyjew’s Triplet.
In this section, we are briefly summarizing the construction of the Tulczyjew’s triplet in its very classicalsense [66, 69, 70].
Derivations.
We introduce two derivations, namely i T and d T , mapping the exterior algebra Λ ( Q ) on amanifold Q to the exterior algebra Λ ( TQ ) on its tangent bundle TQ , [66, 67, 69, 70]. The derivation i T takesa k -form, say Ω k , on Q to a ( k − ) -form on TQ that is,(2.5) i T : Λ k ( Q ) −→ Λ k − ( TQ ) , i T Ω k (cid:0) X , . . . , X k − (cid:1) = Ω k (cid:0) τ T Q ( X ) , T τ Q ( X ) , . . . , T τ Q ( X k − ) (cid:1) for any collection of vector fields where X , . . . , X k − on TQ . Here, τ T Q is the tangent bundle projectionfrom
TTQ to TQ , and T τ Q is the tangent lift of τ Q . Referring to the derivation i T and the deRham exteriorderivative d , we define degree derivation(2.6) d T : Λ k ( Q ) −→ Λ k ( TQ ) , Ω k
7→ ( i T d + di T ) Ω k . Tulczyjew’s symplectic space.
The tangent bundle of a symplectic manifold ( P , Ω ) is also a symplecticmanifold determined by the two-tuple ( T P , d T Ω ) [66]. Here, the symplectic two-form d T Ω is the oneobtained by the application of the derivation d T in (2.6) to Ω . In particular, for the canonical symplecticmanifold ( T ∗ Q , Ω Q = − d θ Q ) , TT ∗ Q is a symplectic manifold equipped with the lifted symplectic two form OĞUL ESEN, MAHMUT KUDEYT, AND SERKAN SÜTLÜ d T Ω Q which admits two potential one-forms(2.7) ϑ = i T Ω Q , ϑ = − d T θ Q = − i T d θ Q − di T θ Q . If ( q i ) is a coordinate chart on Q then we employ the induces coordinates ( q i , Û q j ) on TQ . For the inducedcoordinates ( q i , p j , Û q k , Û p l ) on TT ∗ Q , the one-forms in (2.7) are computed to be [72](2.8) ϑ = i T Ω Q = Û q i dp i − Û p i dq i , ϑ = − d T θ Q = − Û p i dq i − p i d Û q i . The symplectic two-form on the bundle TT ∗ Q is(2.9) d T Ω Q = d ϑ = d ϑ = dq i ∧ d Û p i + d Û q i ∧ dp i . The canonical involution on
TTQ . The iterated tangent bundle
TTQ is a tangent bundle with the basemanifold TQ along with the tangent bundle projection τ T Q . It is possible to show that
TTQ can be writtenas a vector bundle over TQ apart from the canonical tangent bundle fibration. This fibration is achievedby the tangent mapping T τ Q of the projection τ Q . To manifest this, we plot here the double vector bundlestructure of the iterated tangent bundle TTQ in the following commutative diagram.(2.10)
TTQTQ TQQ τ T Q T τ Q τ Q τ Q Referring to this double bundle structure, we define a canonical involution κ Q by simply changing the orderof the fibrations in (2.10), see [70]. That is(2.11) κ Q : TTQ −→ TTQ , τ
T Q ◦ κ Q = T τ Q , T τ Q ◦ κ Q = τ T Q . We cite [35] for more general discussions on the double bundles.
Pairing between TT ∗ Q and TTQ . We now establish a pairing between an element Z in TT ∗ Q and anelement W in TTQ , see [70]. Recall that, there is a curve z ( t ) in T ∗ Q so that Z = Û z ( ) and, there is a curve v ( t ) in TQ so that W = Û v ( ) . In this framework, the pairing is defined as(2.12) h• , •i e : TT ∗ Q × TTQ −→ R , h Z , W i e = ddt h z ( t ) , v ( t )i (cid:12)(cid:12)(cid:12)(cid:12) t = , where the pairing on the right hand side is the one between T ∗ Q and TQ .In the induced coordinates ( q i , Û q j , q ′ k , Û q ′ l ) on the iterated tangent bundle TTQ , the fibration in (2.10) read τ T Q ( q i , Û q j , q ′ k , Û q ′ l ) = ( q i , Û q j ) , T τ Q ( q i , Û q j , q ′ k , Û q ′ l ) = ( q i , q ′ k ) , whereas the canonical involution κ Q in (2.11) is computed to be κ Q ( q i , Û q j , q ′ k , Û q ′ ℓ ) = ( q i , q ′ k , Û q j , Û q ′ ℓ ) . ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 7
Coordinate expression of the pairing (2.12) is as follows. Let us choose coordinates on TT ∗ Q as ( q i , p j , Û q k , Û p l ) ,and coordinates ( q i , Û q j , q ′ k , Û q ′ l ) on TTQ then, (cid:10) ( q i , p j , Û q k , Û p l ) , ( q i , Û q j , q ′ k , Û q ′ l ) (cid:11) e = p i Û q ′ i + q ′ i Û p i . Now we define two symplectomorphisms from TT ∗ Q . One is to T ∗ TQ and the other is to T ∗ T ∗ Q . Weassume that, being cotangent bundles, T ∗ TQ and T ∗ T ∗ Q equipped with the canonical symplectic forms Ω T Q = − d θ T Q and Ω T ∗ Q = − d θ T ∗ Q , respectively. Left Wing of the Tulczyjew’s Triplet.
Start with defining the vector fibration morphism(2.13) α Q : TT ∗ Q −→ T ∗ TQ , h α Q ( Z ) , W i = h Z , κ Q ( W )i e , where κ Q is the canonical involution defined in (2.11) whereas the pairing on the right hand side is the onegiven in (2.12). Here, the pairing of the left hand side is the canonical pairing between T ∗ TQ and TTQ . α Q is a symplectomorphism by satisfying α ∗ Q Ω T Q = d T Ω Q . Here, Ω T Q is the symplectic two-form T ∗ TQ , and d T Ω Q is the symplectic two-form on TT ∗ Q in (2.9). So that we arrive at a special symplectic structure(2.14) ( TT ∗ Q , T π Q , TQ , ϑ , α Q ) where ϑ is the one-form in (2.8), and T π Q is the tangent lift of the cotangent bundle projection π Q . Here,the underlying symplectic manifold for the special symplectic structure (2.14) is the symplectic manifold ( TT ∗ Q , d T Ω Q ) . We record this in the following diagram for future reference(2.15) T ∗ TQ TT ∗ QTQ π T Q α Q T π Q Right Wing of the Tulczyjew’s Triplet.
The nondegeneracy of the canonical symplectic two-form Ω Q leads to the existence of a (musical) diffeomorphism(2.16) Ω ♭ Q : TT ∗ Q −→ T ∗ T ∗ Q , Ω ♭ Q ( z ) = Ω Q ( z , •) . Further, Ω ♭ Q is a symplectomorphism by satisfying ( Ω ♭ Q ) ∗ Ω T ∗ Q = d T Ω Q . Here, Ω T ∗ Q is the symplectictwo-form T ∗ T ∗ Q . So that we arrive at the following special symplectic structure(2.17) ( TT ∗ Q , τ T ∗ Q , T ∗ Q , ϑ , Ω ♭ Q ) , over the symplectic manifold ( TT ∗ Q , d T Ω Q ) . Here, ϑ is the potential one-form in (2.7) and that τ T ∗ Q is thetangent bundle projection. The diagram is for the future reference.(2.18) TT ∗ Q T ∗ T ∗ QT ∗ Q τ T ∗ Q Ω ♭ Q π T ∗ Q Under the light of the discussions done in this subsection we can argue that there are two special symplecticstructures, given in (2.14) and (2.17), over the symplectic manifold ( TT ∗ Q , d T Ω Q ) . The Tulczyjew’s triplet OĞUL ESEN, MAHMUT KUDEYT, AND SERKAN SÜTLÜ is the following commutative diagram merging these two special symplectic structures in one picture that is[63, 66, 67, 70],(2.19) T ∗ TQ π T Q " " ❋❋❋❋❋❋❋❋ TT ∗ Q T π Q | | ①①①①①①①① Ω ♭ Q / / τ T ∗ Q ●●●●●●●● α Q o o T ∗ T ∗ Q π T ∗ Q { { ✈✈✈✈✈✈✈✈✈ TQ τ Q " " ❋❋❋❋❋❋❋❋❋ T ∗ Q π Q { { ✇✇✇✇✇✇✇✇✇ Q We finish this subsection by writing the local realizations of the symplectomorphisms in the triplet α Q ( q i , p j , Û q k , Û p l ) = ( q i , Û q k , Û p l , p j ) , Ω ♭ Q ( q i , p j , Û q k , Û p l ) = ( q i , p j , − Û p l , Û q k ) . (2.20)2.3. Group Action and Connection.
Let G be a Lie group, and Q be a smooth manifold. Assume that G is acting freely and properly on Q fromthe left(2.21) φ : G × Q −→ Q , ( g , q ) 7→ φ g ( q ) = φ q ( g ) = g · q , where we may regard φ g as a diffeomorphism on Q for a fixed g , and φ q as a differentiable mapping from G to Q for a fixed q . We denote the space of orbits in Q by ¯ Q : = G \ Q , then define a principal fiber bundle ( Q , π, ¯ Q ) with the total space Q , the base space ¯ Q . Here, the projection π maps a point q in Q to the orbit [ q ] passing through it that is, π : Q −→ ¯ Q , q
7→ [ q ] . By taking the tangent lift T π of the projection π , one arrives at a mapping from TQ to the tangent bundle T ¯ Q of the orbit space ¯ Q . The kernel of T π determines the vertical subbundle VQ of TQ . A vertical section ofthe tangent bundle is the one taking values only in VQ . An element ξ in the Lie algebra g of the symmetrygroup G generates a vertical vector field, called infinitesimal generator,(2.22) ξ Q ( q ) = T e φ q ( ξ ) = ddt exp t ξ · q (cid:12)(cid:12)(cid:12)(cid:12) t = , where exp is the exponential mapping from g to G . The map g X ( Q ) , taking a Lie algebra element ξ tothe associated infinitesimal generator ξ Q is a Lie algebra homomorphism. Connection.
In order to determine a complement, namely a horizontal subbundle, to the vertical subbundle
V Q in the tangent bundle TQ , one needs to define a connection. A connection, say A , is a Lie algebra valuedone-form on Q satisfying the following two conditions:(2.23) A ◦ ξ Q = ξ, A ◦ T φ g = Ad g ◦ A , where Ad is the adjoint representation of the group G on its Lie algebra g , [9, 10, 41, 56]. Defining aconnection A on Q is equivalent to defining a horizontal lift operator(2.24) h : X ( ¯ Q ) −→ X ( Q ) , u h ( u ) = u h ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 9 taking a vector field u on the quotient manifold ¯ Q to a vector field u h on Q satisfying(2.25) T q π ◦ u h ( q ) = u ◦ π ( q ) , for all q in Q . This can be stated in terms of a splitting of the following short exact sequence(2.26) 0 g TQ T ¯ Q T π h where the second right arrow stands for taking ξ in g to the infinitesimal generator ξ Q . So that, referringto a connection A , or equivalently defining a horizontal lift operator, we can decompose the tangent bundle TQ as a direct sum of VQ and a horizontal subbundle HQ defined to be the image space of horizontal liftoperator. Here, HQ can also equivalently be defined as the kernel of A . Such a decomposition manifeststwo projections, namely v and h , so that one can uniquely decompose a tangent vector v into the sum of avertical vector v ( v ) and a horizontal vector h ( v ) that is TQ = VQ ⊕ HQ , v = v ( v ) + h ( v ) . We can understand this decomposition as follows. If the initial point of a vector v is q that is τ Q ( v ) = q then, the vertical component v ( v ) of v equals to the value of an infinitesimal generator ξ Q ( q ) at q , whereasthe horizontal part h ( v ) can be written as the value u h ( q ) of the horizontal lift of a vector field u on ¯ Q . Associated bundle.
Assume once more a free and proper G action on a manifold Q as given in (2.21).Consider further that the same Lie group G has also a representation Φ on a vector space V . Define theproduct manifold Q × V , and an action of G on this manifold given by(2.27) G × ( Q × V ) −→ ( Q × V ) , g · ( q , v ) = ( g · q , Φ ( g , v )) . The space of orbits in Q × V is denoted by e V . We denote the equivalence class containing an element ( q , v ) of Q × V by [ q , v ] . Notice that, e V admits a fiber bundle structure, called associated bundle, over the basemanifold ¯ Q with projection(2.28) e V −→ ¯ Q , [ q , v ] 7→ [ q ] . Adjoint bundle.
A particular case of associated bundle (2.28) is important for the present work calledadjoint bundle. In this case, we choose the vector space V as the Lie algebra g and, accordingly, choose Φ as the adjoint action Ad of G on g in (2.27) that is(2.29) G × ( Q × g ) −→ Q × g , g · ( q , ξ ) = ( g · q , Ad g ξ ) . Thus, the adjoint bundle is the orbit space(2.30) ˜ g = {[ q , ξ ] : ( q , ξ ) ∈ Q × g } . So that, we arrive at a principal fiber bundle with typical fibers diffeomorphic to G , and the projectionmapping ˜ g −→ ¯ Q , [ q , ξ ] 7→ [ q ] . Coadjoint Bundle.
Recall the associated bundle structure given in (2.28), and choose, particularly, thevector space V as the linear algebraic dual g ∗ of g . In this case, the action Φ in (2.27) is determined as the coadjoint action Ad ∗ . So that we have G × ( Q × g ∗ ) −→ Q × g ∗ , g · ( q , µ ) = ( g · q , Ad ∗ g − µ ) . In this case the quotient space, called particularly as coadjoint bundle, turns out to be(2.31) ˜ g ∗ = {[ q , µ ] : q ∈ Q , µ ∈ g } with the projection ˜ g ∗ −→ ¯ Q , [ q , µ ] 7→ [ q ] . Trivialization and Reduction of Tangent Bundle.
Assume existence of a free and proper left action φ of a Lie group G on a manifold Q as given in (2.21).We may proceed by taking the tangent lift of the mappings φ g in (2.21) then arrive at an action of G on thetangent bundle TQ as follows(2.32) G × TQ −→ TQ , ( g , v ) 7→ g · v = T φ g ( v ) . This induces a fiber bundle structure for TQ with the base space is TQ : = G \ TQ . Trivialization of tangent bundle.
Consider the tangent bundle ( T ¯ Q , τ ¯ Q , ¯ Q ) of the quotient manifold ¯ Q , andaccordingly define the following Whitney product(2.33) Q ⊕ T ¯ Q = {( q , u ) ∈ Q × T ¯ Q : [ q ] = τ ¯ Q ( u )} , over the base manifold ¯ Q and, as always, ⊕ refers to × ¯ Q . In order to investigate the fibers of this bundlestructure we consider the following commutative diagram(2.34) Q ⊕ T ¯ Q QT ¯ Q ¯ Q pr pr πτ ¯ Q where pr i denoted the projection to the i -th factor. For the projection pr , a fiber at the point τ − Q [ q ] isdiffeomorphic to π − [ q ] . This determines a principal bundle structure pr with the fibers isomorphic to G .Later, we couple the total space Q ⊕ T ¯ Q of this fiber bundle with a Lie algebra g that is Q ⊕ T ¯ Q × g = (cid:8) ( q , u , ξ ) ∈ Q × T ¯ Q × g : [ q ] = τ ¯ Q ( u ) (cid:9) . This product space reads the following right trivialization of the tangent bundle λ T : TQ −→ Q ⊕ T ¯ Q × g , v (cid:0) τ Q ( v ) , T π ( v ) , A ( v ) (cid:1) ,λ − T : Q ⊕ T ¯ Q × g −→ TQ , ( q , u , ξ ) 7→ u hq + ξ Q ( q ) , (2.35)where u h is the horizontal lift of the vector u , and ξ Q is the infinitesimal generator, and A is the connectionone-form. ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 11
Reduction of tangent bundle.
We can transfer the group action (2.32) of G on TQ to its trivialization Q ⊕ T ¯ Q × g . See that, T ¯ Q is horizontal with respect to the action so that the Lie group G acts only on thefirst factor in Q ⊕ T ¯ Q . In accordance with this, from (2.32), we have that G × (cid:0) Q ⊕ T ¯ Q × g (cid:1) −→ Q ⊕ T ¯ Q × g , g · ( q , u , ξ ) = ( g · q , u , Ad g ξ ) , where Ad g is the adjoint action. The orbits space due to this action is G \ (cid:0) Q ⊕ T ¯ Q × g (cid:1) ≃ T ¯ Q ⊕ ˜ g = (cid:8) ( u , [ q , ξ ]) ∈ T ¯ Q × ˜ g : T π ( u ) = [ q ] (cid:9) . The reductions of the isomorphism λ T and the inverse mapping in (2.35) under the action of G are as follows T ¯ Q ⊕ ˜ g −→ TQ , ( u , [ q , ξ ]) 7→ [ u hq + ξ Q ( q )] , TQ −→ T ¯ Q ⊕ ˜ g , [ v ] 7→ ( T π ( v ) , [ τ Q ( v ) , A ( v )]) , (2.36)where we have once used ⊕ to denote the Whitney product over the base manifold ¯ Q . Collecting all thesediscussions on the trivialization and reduction of the tangent bundle, we have the following identificationsboth in the unreduced and reduced levels TQ Q ⊕ T ¯ Q × g TQ T ¯ Q ⊕ ˜ g G \ λ T G \ (2.36) where G \ denotes the reduction due to the symmetry group G . Here, λ T is the isomorphism in (2.35).2.5. Trivialization and Reduction of Cotangent Bundle.
Let ( P , Ω = − d θ ) be a symplectic manifold and assume that there is an action of a Lie group G on P . Wecall this action a symplectic action if it preserves the symplectic two-form. Further, it is called Hamiltonianaction if, additionally, there is an Ad ∗ -equivariant momentum map J Q from P to the dual of the Lie algebraof G that is(2.37) J Q : P −→ g ∗ , h J Q ( x ) , ξ i = h θ, ξ P i , where ξ P is the infinitesimal generator. Here, the pairing on left hand side is the one between the dual space g ∗ and the Lie algebra g whereas the pairing on right hand side is the one between T ∗ P and T P , [50].
Cotangent bundle reduction.
Consider a (left) group action φ of, say Lie group G , on a manifold Q as in(2.21). Lift the group action to the cotangent bundle T ∗ Q by means of the cotangent lift that is(2.38) G × T ∗ Q −→ T ∗ Q , g · z : = T ∗ φ g − ( z ) . Being a cotangent lift this action preserves the canonical symplectic structure on T ∗ Q [71]. For the presentcase, there is a nice characterization of the momentum mapping J Q : T ∗ Q −→ g ∗ , h J Q ( z ) , ξ i = h z , ξ Q ◦ π Q ( z )i , where the pairing on the left hand side is the one between g ∗ and g , whereas the pairing on the righthand side is the canonical pairing between T ∗ Q and TQ . Here, ξ Q is the infinitesimal generator definedin (2.22). It is possible to prove that such a momentum mapping satisfies equivariance property that is Ad ∗ g − ◦ J Q = J Q ◦ φ g .Being a smooth manifold, one can define the cotangent bundle T ∗ ¯ Q of the reduced space ¯ Q . Obeying ournotation policy, we denote the cotangent bundle projection by π ¯ Q : T ∗ ¯ Q ¯ Q . By assuming the existence ofa connection on Q , we take the dual of the short exact sequence in (2.26) in order to arrive at a decompositionof the cotangent bundle. This reads the following short exact sequence(2.39) 0 T ∗ ¯ Q T ∗ Q g ∗ , T ∗ π J Q h ∗ where T ∗ π is the cotangent lift of the projection π . Here, h ∗ is the linear algebraic dual of the horizontal liftoperator in (2.24) so that, it maps covectors in T ∗ Q to the cotangent bundle T ∗ ¯ Q of the reduced space ¯ Q . Trivialization of the cotangent bundle.
Define the following Whitney product Q ⊕ T ∗ ¯ Q = {( q , y ) ∈ Q × T ∗ ¯ Q : [ q ] = π ¯ Q ( y )} , over the base manifold ¯ Q . Two fibrations of this Whitney product is important for the present discussion.Before defining them properly, we first draw the following diagram to make it more easy to access thediscussion. Q ⊕ T ∗ ¯ QT ∗ ¯ Q ¯ Q pr τ ¯ Q The first fibration we are interested in is ( Q ⊕ T ∗ ¯ Q , pr , T ∗ ¯ Q ) with the fibration pr projecting to the secondcomponent. Other is the principal bundle containing Q ⊕ T ∗ ¯ Q as the total space with the Lie group G action.In this case, the base manifold is ¯ Q . Bringing the total space Q ⊕ T ∗ ¯ Q and the dual space g ∗ together andthen define Q ⊕ T ∗ ¯ Q × g ∗ = (cid:8) ( q , y , µ ) ∈ Q × T ∗ ¯ Q × g ∗ : [ q ] = π ¯ Q ( y ) (cid:9) . Later, we construct a trivialization of the cotangent bundle and the inverse as follows λ T ∗ : T ∗ Q −→ Q ⊕ T ∗ ¯ Q × g ∗ , z
7→ ( π Q ( z ) , h ∗ z , J Q ( z )) ,λ − T ∗ : Q ⊕ T ∗ ¯ Q × g ∗ −→ T ∗ Q , ( q , y , µ ) 7→ T ∗ q π ( y ) + A ∗ q µ, (2.40)where h ∗ is the linear algebraic dual of the horizontal lift operator in (2.24). Here, A ∗ q is the linear algebraicdual of the connection A q : T q Q g so that A ∗ q is from the dual space g ∗ to the covector space T ∗ q Q . Reduction of cotangent bundle.
By employing the trivialization (2.40) to the group action (2.38) on thecotangent level, we arrive at the induced action of G on the trivialization G × (cid:0) Q ⊕ T ∗ ¯ Q × g ∗ (cid:1) −→ Q ⊕ T ∗ ¯ Q × g ∗ , g · ( q , y , µ )) 7→ ( q · g , y , Ad ∗ g − µ ) , where Ad ∗ g − is the coadjoint action. Under this symmetry, the reduction of the isomorphism λ T ∗ in (2.40)can be written as(2.41) λ T ∗ : T ∗ Q −→ T ∗ ¯ Q ⊕ ˜ g ∗ , [ z ] 7→ ( h ∗ z , [ π Q ( z ) , J Q ( z )]) , ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 13 where ˜ g ∗ is the coadjoint bundle introduced in (coad-bundle). See also that in (2.41) we have used thefollowing identification G \ (cid:0) ( Q ⊕ T ∗ ¯ Q ) × g ∗ (cid:1) ≃ T ∗ ¯ Q ⊕ ˜ g ∗ . We present the commutative diagram involving trivializations and reduced bundles in one picture T ∗ Q ( Q ⊕ T ∗ ¯ Q ) × g ∗ T ∗ Q T ∗ ¯ Q ⊕ ˜ g ∗ G \ λ T ∗ G \ λ T ∗ where G \ denotes the reduction due to the symmetry group G . Here, λ T ∗ is the isomorphism in (2.40), and λ T ∗ is in (2.41). Tangent - cotangent duality.
Identifying the elements and their trivializations, let us consider a covector z ≃ ( q , y , µ ) in the cotangent bundle T ∗ Q and a vector v ≃ ( q , u , ξ ) in the tangent bundle TQ according tothe trivializations in (2.35) and (2.40), respectively. The duality is computed to be T ∗ Q × TQ −→ R , h( q , y , µ ) , ( q , u , ξ )i = h y , u i + h µ, ξ i , where the first pairing on the right hand side is between T ∗ ¯ Q and T ¯ Q whereas the second pairing on theright hand side is the one between g ∗ and g .
3. Trivializations and Reductions of Iterated Tangent Bundles
This is the main section where we present trivializations and reductions of iterated tangent bundles
TTQ and TT ∗ Q . We are first presenting the tangent lift of a group action.3.1. The Infinitesimal Group Action.
The tangent group
T G is isomorphic to the semi-direct product G × g T G −→ G × g , V g
7→ ( g , T L g − V g ) , where g is the Lie algebra of the Lie group G , and L g is the left translation on G . We refer [18, 21, 36, 42,48, 55, 60] for some further information about the tangent group.For a free and proper action φ of a Lie group G on the manifold Q as given in (2.21), consider the tangentmapping(3.1) T φ : T G × TQ −→ TQ , ( g , ξ ) · v T φ g ( v ) + ξ Q ( g · τ Q ( v )) , where T φ g is the tangent mapping of φ g whereas ξ Q is the infinitesimal generator of ξ on Q in (2.22). Themapping T φ determines a (free and proper [42]) action of the group T G on TQ by i ) ( e , ) · v = v , ii ) ( g , ξ ) · (cid:0) ( h , η ) · v (cid:1) = ( g , ξ )( h , η ) · v = ( g h , ξ + Ad g η ) · v , where the group structure on T G appears at the right hand side of the associativity condition. We denotethe orbit space in TQ by T G \ TQ . It is easy to verify that the quotient manifold T G \ TQ is diffeomorphic tothe tangent bundle T ¯ Q by means of the reduction by stages theorem [10]. In order to see this, we employthe trivialization of the tangent group T G ≃ G ⋉ g . In view of the fact that the Lie algebra g is a normalsubgroup, we decompose the reduction of TQ by T G in two steps; the reduction of the tangent bundle TQ by the normal subgroup g first, and then by the quotient group T G \ g ≃ G . The process may be summarizedas follows(3.2) T G \ TQ ≃ ( G × g )\ TQ ≃ G \ (cid:0) g \ TQ (cid:1) ≃ G \ (cid:0) g \( Q ⊕ T ¯ Q × g ) (cid:1) ≃ G \( Q ⊕ T ¯ Q ) ≃ T ¯ Q . We do note that we use the fact that the Lie algebra g acts on Q ⊕ T ¯ Q × g by simply addition to the thirdfactor in the fourth identification. The sixth identification is a manifestation of Q ⊕ T ¯ Q in (2.33) admitting aprincipal fiber bundle structure over the base T ¯ Q with the structure group G , as exhibited in Diagram 2.34.The identification in (3.2) reads that due to the tangent action (3.1), we arrive at the principal bundle ( TQ , T π, T ¯ Q ) where T π is the tangent mapping of the projection π . The associated vertical subbundle VTQ of the iterated tangent bundle
TTQ is defined as the kernel of the iterated tangent mapping TT π .Vertical vector fields are those obtained by the infinitesimal generators. Lie algebra of the tangent groupcan be considered to be the semi-direct product space g × g . For an element ( ζ, η ) in this Lie algebra, theinfinitesimal generator ( ζ, η ) T Q is computed to be(3.3) ( ζ, η ) T Q : TQ −→ TTQ , ( ζ, η ) T Q ( v ) = T ( e , ) ( T φ ) v ( ζ, η ) , ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 15 where T ( T φ ) v is the tangent lift of the mapping ( T φ ) v which induced from the tangent action (3.1) as follows ( T φ ) v : T G −→ TQ , ( T φ ) v ( g , ξ ) = ( g , ξ ) · v . The tangent connection.
In order to determine a complement to the vertical subbundle
VTQ of TTQ , oneneeds a to define a connection on ( TQ , T π, T ¯ Q ) . We now claim that the tangent mapping T A : TTQ −→ g × g of a connection A defined on ( Q , π, ¯ Q ) satisfies the requirements of being a connection. Let us remark thatthe Lie algebra in this case is the product space g × g , and the conditions in (2.23) become T A ◦ ( ζ, η ) T Q = ( ζ, η ) , T A ◦ T ( T φ ) ( g ,ξ ) = Ad ( g ,ξ ) ◦ T A , respectively. Here, Ad ( g ,ξ ) is the adjoint action of the tangent group T G on its Lie algebra g × g defined asAd ( g ,ξ ) : g × g −→ g × g , ( ζ, η ) 7→ ( Ad g ζ, Ad g ( η − [ ζ, ξ ])) , whereas ( T φ ) ( g ,ξ ) is the mapping induced from the tangent mapping (3.1) as follows ( T φ ) ( g ,ξ ) : TQ −→ TQ , ( T φ ) ( g ,ξ ) ( v ) = ( g , ξ ) · v . If A is a connection on Q , then we denote the corresponding horizontal lift operator by h , see (2.24). In thiscase, T A is a connection on TQ and the corresponding horizontal lift operator is computed to be the tangentlift of h that is,(3.4) T h : TT ¯ Q −→ TTQ . This reads the splitting of the following short exact sequence0 g × g TTQ TT ¯ Q . T ( T φ ) v TT π T A T h
Hence, we obtain a decomposition for the iterated bundle
TTQ as a direct sum of a vertical subbundle
VTQ and a horizontal bundle
HTQ . Here,
HTQ is the kernel of the connection
T A . By abusing the notation, weintroduce two projections, denoted by v and h , then we can decompose a vector W in TTQ into the sum ofa vertical vector v ( W ) and a horizontal vector h ( W ) , namely(3.5) TTQ = V ( TQ ) ⊕ H ( TQ ) , W = v ( W ) + h ( W ) . Realize that the vertical part v ( W ) is the value of an infinitesimal generator ( ζ, η ) T Q at v , whereas thehorizontal part h ( W ) is the value of a horizontal lift U T hv of a vector field U on T ¯ Q . Momentum map on T ∗ TQ . We shall introduce a mapping J T Q as(3.6) J T Q : T ∗ TQ −→ g ∗ × g ∗ , (cid:10) J T Q ( Υ ) , ( ζ, η ) (cid:11) = (cid:10) Υ , ( ζ, η ) T Q ◦ π T Q ( Υ ) (cid:11) for an element ( ζ, η ) in T g , where the pairing on the left hand side of (3.6) is the one between g ∗ × g ∗ and T g and the pairing on the right hand side of (3.6) is the canonical pairing between T ∗ TQ and TTQ , whereas ( ζ, η ) T Q is the infinitesimal generator of the vector ( ζ, η ) . In here, π T Q is the canonical projection from T ∗ TQ to TQ , see [11] for the mapping J T Q . Relationship between J
T Q and T J Q . Given the mapping J T Q (3.6) and the tangent map T J Q of themomentum map (2.37), there is a relationship between them given as(3.7) (cid:10) J T Q ◦ α Q ( Z ) , ( ζ, η ) (cid:11) = (cid:10) T J Q ( Z ) , ( η, ζ ) (cid:11) for elements ( ζ, η ) and ( η, ζ ) in T g and an element Z in TT ∗ Q by using the diagram TT ∗ Q g ∗ × g ∗ T ∗ TQ g ∗ × g ∗ T J Q α Q J T Q where α Q is the symplectomorphism in (2.13), whereas the mapping between bundles g ∗ × g ∗ and g ∗ × g ∗ is the swap operation interchanging the order of the elements of g ∗ × g ∗ . We also refer the reader to [11] forthe relation (3.7).3.2. Trivialization of TTQ.
It is possible to achieve a trivialization of the iterated tangent bundle
TTQ in many different ways. In thissection, our approach is to use the tangent mapping of the isomorphism λ T , given in (2.35), which resultswith a Whitney product over the base manifold T ¯ Q that is,(3.8) T λ T : TTQ −→ TQ × T ¯ Q TT ¯ Q × T g , W (cid:0) T τ Q ( W ) , TT π ( W ) , T A ( W ) (cid:1) , where TT π is the second tangent lift of the projection π : Q ¯ Q , whereas T A is the tangent lift of theconnection A in (2.23). See that, we have employed the following identity T ( Q ⊕ T ¯ Q × g ) ≃ TQ × T ¯ Q TT ¯ Q × T g , on the image space of (3.8). Recall now the inverse mapping λ − T in (2.35). The tangent mapping iscomputed to be T λ − T : TQ × T ¯ Q TT ¯ Q × T g −→ TTQ , ( v , U , ζ, η ) 7→ U T hv + T ( T φ ) v ( ζ, η ) , where the first term on the right hand side is the horizontal lift of a vector field U in T ¯ Q by means ofthe horizontal lift T h in (3.4), whereas T ( T φ ) v reads the image of the infinitesimal generator as in (3.3).Therefore we write the inverse mapping as T λ − T ( v , U , ζ, η ) = U T hv + ( ζ, η ) T Q ( v ) . Let us note also that, we can further trivialize the first term T τ Q ( W ) in the decomposition (3.8). As itbelongs to TQ , the substitution of the isomorphism (2.35) yields the presentation T τ Q ( W ) ≃ (cid:0) τ Q ◦ T τ Q ( W ) , T π ◦ T τ Q ( W ) , A ( T τ Q ( W )) (cid:1) ∈ Q ⊕ T ¯ Q × g of the vector T τ Q ( W ) . After scrutinizing each term, we arrive at a trivialization of TTQ as λ TT : TTQ −→ Q ⊕ TT ¯ Q × g × T g , W (cid:0) τ Q ◦ T τ Q ( W ) , TT π ( W ) , A ( T τ Q ( W )) , T A ( W ) (cid:1) , (3.9) ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 17 where the notation ⊕ is used to denote the Whitney product over the base manifold ¯ Q . Let us record hereabstractly the image space in (3.9) that is,(3.10) Q ⊕ TT ¯ Q × g × T g = (cid:8) ( q , U , ζ, ξ, η ) ∈ Q × TT ¯ Q × g × T g | [ q ] = τ ¯ Q ◦ τ T ¯ Q ( U ) (cid:9) . Trivialization of the tangent rhombic.
As recalled in Subsection 2.2 and pictured in Diagram 2.10, theiterated tangent bundle
TTQ admits a double vector bundle structure over TQ . Its trivialization (3.10) alsoadmits two different fibrations, this time, over the trivialized tangent bundle Q ⊕ T ¯ Q × g . To depict this,consider the trivializations of the projections τ T Q and T τ Q computed to be d τ T Q : Q ⊕ TT ¯ Q × g × T g −→ Q ⊕ T ¯ Q × g , ( q , U , ζ, ξ, η ) 7→ ( q , τ T ¯ Q ( U ) , ξ ) , d T τ Q : Q ⊕ TT ¯ Q × g × T g −→ Q ⊕ T ¯ Q × g , ( q , U , ζ, ξ, η ) 7→ ( q , T τ ¯ Q ( U ) , ζ ) , (3.11)respectively. We observe that, in the light of the trivializations λ T in (2.35) and λ TT in (3.9), the projectionssatisfy d τ T Q ◦ λ TT = λ T ◦ τ T Q , d T τ Q ◦ λ TT = λ T ◦ T τ Q . So that, in order to arrive at the trivialization of the tangent rhombic (2.10), we combine the projections in(3.11) then plot the following diagram where c τ Q is the projection to the first factor.(3.12) TTQ ≃ Q ⊕ TT ¯ Q × g × T g TQ ≃ Q ⊕ T ¯ Q × g Q ⊕ T ¯ Q × g ≃ TQQ š τ T Q š T τ Q c τ Q c τ Q Trivialization of the canonical involution.
In Subsection 2.2, as a manifestation of the double vectorbundle structure, the canonical involution κ Q has been introduced on TTQ in (2.11). So that, by changingthe order of the fibrations in the trivialized tangent rhombic (3.12), we define a canonical involution on thetrivialization Q ⊕ TT ¯ Q × g × T g of the iterated tangent bundle as follows(3.13) c κ Q : Q ⊕ TT ¯ Q × g × T g −→ Q ⊕ TT ¯ Q × g × T g , ( q , U , ζ, ξ, η ) 7→ ( q , κ ¯ Q ( U ) , ξ, ζ, η ) , where the mapping κ ¯ Q is the canonical involution on the bundle TT ¯ Q . Notice that, the involution c κ Q is alsocomputed by directly applying the trivialization λ TT in (3.9) to the canonical involution κ Q that is, c κ Q ◦ λ TT = λ TT ◦ κ Q . Reduction of TTQ.
We define an action of a Lie group G on the bundle TTQ (2.21) as follows(3.14) G × TTQ −→ TTQ , g · W = TT φ g ( W ) , where TT φ g stands for the tangent lift of the mapping T φ g so that we call it the iterated tangent lift of φ g .The action (3.14) induces an action on the trivialized bundle (3.10) as follows G × ( Q ⊕ TT ¯ Q × g × T g ) −→ Q ⊕ TT ¯ Q × g × T g , g · ( q , U , ζ, ξ, η ) 7→ ( g · q , U , Ad g ζ, Ad g ξ, Ad g η ) , (3.15)where Ad g is the adjoint action of the Lie algebra g .We shall next concentrate on the reduction procedure. However, let us first introduce an associated bundlefor which we shall consider the group action (2.27), where the vector space is taken to be the product space V = g × g × g whereas the action is G × ( Q × g × g × g ) −→ Q × g × g × g , g · ( q , ξ, η, ζ ) 7→ ( g · q , Ad g ξ, Ad g η, Ad g ζ ) . This determines an associated bundle with the total space(3.16) e V : = G \( Q × ˜ g × ˜ g × ˜ g ) ≃ ˜ g ⊕ ˜ g ⊕ ˜ g . The orbit space of the action (3.14) is
TTQ , which yields a projection from
TTQ to the orbit space
TTQ ,mapping a vector W to the orbit [ W ] passing through it. To see the reduced space, we offer the followingcomputation where we colour the base manifold by blue in order to exhibit the fiber bundle structure of TTQ . This helps to keep track of the base manifold in the computation as well. Accordingly,
TTQ ≃ T ( TQ )/ G ≃ T ( Q ⊕ T ¯ Q × g )/ G ≃ ( TQ × T ¯ Q TT ¯ Q × T g )/ G ≃ (cid:0) ( Q ⊕ T ¯ Q × g ) × T ¯ Q TT ¯ Q × g × g (cid:1) / G ≃ ( Q ⊕ TT ¯ Q × g × g × g )/ G ≃ TT ¯ Q ⊕ ( Q × g × g × g )/ G ≃ TT ¯ Q ⊕ ( ˜ g × ˜ g × ˜ g ) ≃ TT ¯ Q ⊕ e V , (3.17)where in the first and the third lines we have employed the identity (3.9), whereas at the last line we haveused (3.16). Hence, the reduction of the trivialized bundle under the action (3.15) is computed to be TTQ ≃ TT ¯ Q ⊕ e V = (cid:8) ( U , [ q , ζ, ξ, η ]) ∈ TT ¯ Q × e V : τ ¯ Q ◦ τ T ¯ Q ( U ) = [ q ] (cid:9) . Reduction of the tangent rhombic.
Two trivializations of the fibrations of
TTQ over TQ , namely d τ T Q and d T τ Q , have been obtained in (3.11), respectively. After the reduction (3.11), these projections reduce to τ T Q : TT ¯ Q ⊕ e V −→ T ¯ Q ⊕ ˜ g , ( U , [ q , ζ, ξ, η ]) 7→ ( τ T ¯ Q ( U ) , [ q , ξ ]) , T τ Q : TT ¯ Q ⊕ e V −→ T ¯ Q ⊕ ˜ g , ( U , [ q , ζ, ξ, η ]) 7→ ( T τ ¯ Q ( U ) , [ q , ζ ]) , respectively. Referring these reduced projections, we establish the reduction of the trivialized tangentrhombic (3.12) as follows. We note that, in this case, the reduced bundle TT ¯ Q × e V is the total space over ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 19 the reduced manifold T ¯ Q × ˜ g . That is,(3.18) TT ¯ Q ⊕ e VT ¯ Q ⊕ ˜ g T ¯ Q ⊕ ˜ g ¯ Q τ T Q T τ Q τ Q τ Q which we call reduced tangent rhombic. Here, τ Q is the tangent bundle projection acting on the first factorin T ¯ Q ⊕ ˜ g . Reduction of the canonical involution.
By changing the order of the fibration in the reduced rhombic(3.18), we define an involution κ Q for the reduced space TT ¯ Q ⊕ e V as κ Q : TT ¯ Q ⊕ e V −→ TT ¯ Q ⊕ e V , ( U , [ q , ζ, ξ, η ]) 7→ ( κ ¯ Q ( U ) , [ q , ξ, ζ, η ]) , where the mapping κ ¯ Q is the canonical involution for the bundle TT ¯ Q . The relationship between κ Q andthe trivialized involution c κ Q in (3.13) is as follows Q ⊕ TT ¯ Q × g × T g Q ⊕ TT ¯ Q × g × T g TT ¯ Q ⊕ e V TT ¯ Q ⊕ e V c κ Q G \ G \ κ Q where G \ stands for the reduction under the group action of G . Remark 3.1.
In this remark, we present the reduction of
TTQ under the tangent group action
T G . First wedefine the product space TQ × T g and an action of T G on this space given by
T G × (cid:0) TQ × T g (cid:1) −→ TQ × T g , ( g , ξ ) · ( v , η, ζ ) 7→ (( g , ξ ) · v , Ad ( g ,ξ ) ( η, ζ )) , where Ad ( g ,ξ ) is the adjoint action of the group T G on its Lie algebra g × g . Notice that this induces anassociated bundle with the total space being G : = T G \ (cid:0) TQ × T g (cid:1) −→ T ¯ Q : = T G \ TQ , [ v , η, ζ ] T G
7→ [ v ] T G , where the subscripts under the bracelets refer to the orbit spaces under the tangent group action. Let usdefine an action of T G on the trivialized bundle TQ × T ¯ Q TT ¯ Q × T g as T G × ( TQ × T ¯ Q TT ¯ Q × T g ) −→ TQ × T ¯ Q TT ¯ Q × T g , ( g , ξ ) · ( v , U , ζ, η ) 7→ (cid:0) ( g , ξ ) · v , U , Ad ( g ,ξ ) ( ζ, η ) (cid:1) , where Ad ( g ,ξ ) is the adjoint action of T G on g × g . Here, ( g , ξ ) · v is the action in (3.1). Hence, the reductionof the mapping T λ T in (3.8) can be done under the action of T G as follows
T G \ TTQ −→ T G \( TQ × T ¯ Q TT ¯ Q × T g ) ≃ TT ¯ Q × T ¯ Q G , [ W ] T G
7→ ( TT π ( W ) , [ T τ Q ( W ) , T A ( W )] T G ) , (3.19)where we employed × T ¯ Q to denote the Whitney product over the base manifold T ¯ Q . Thus, the image spacecan be viewed as TT ¯ Q × T ¯ Q G = n ( U , [ v , ( η, ζ )] T G ) : TT π ( U ) = [ v ] T G o . The inverse mapping of (3.19) is computed to be TT ¯ Q × T ¯ Q G −→ T G \ TTQ , ( U , [ v , ( η, ζ )] T G ) 7→ (cid:2) U Thv + ( η, ζ ) T Q ( v ) (cid:3) T G . In view of all these discussions on the trivialized and the reduced iterated bundle, we have the followingdiagram containing both reduced and unreduced levels
TTQ TQ × T ¯ Q TT ¯ Q × T g T G \ TTQ TT ¯ Q × T ¯ Q G T λ T T G \ T G \ (3.19) where T G \ denotes the reduction under the symmetry of the group T G , and T λ T is the isomorphism in(3.8).3.4. Trivialization of TT*Q.
We employ the tangent mapping of the isomorphism λ T ∗ in (2.40) to attain a trivialization of the tangentbundle TT ∗ Q that is,(3.20) T λ T ∗ : TT ∗ Q −→ TQ × T ¯ Q TT ∗ ¯ Q × T g ∗ , Z (cid:0) T π Q ( Z ) , T h ∗ ( Z ) , T J Q ( Z ) (cid:1) , where T π Q is the tangent mapping of the cotangent bundle projection π Q , whereas T h ∗ is the tangentmapping of the linear algebraic dual h ∗ of the horizontal lift operator in (2.24), and T J Q is tangent mappingof the momentum mapping in (2.37). In (3.20), the notation × T ¯ Q is used for the Whitney product over thebase manifold T ¯ Q . Notice additionally that we have used the identity T ( Q ⊕ T ∗ ¯ Q × g ∗ ) ≃ TQ × T ¯ Q TT ∗ ¯ Q × T g ∗ on the image space of (3.20). Consider now the inverse mapping λ − T ∗ in (2.40), then its tangent mapping iscomputed to be T λ − T ∗ : TQ × T ¯ Q TT ∗ ¯ Q × T g ∗ −→ TT ∗ Q , ( v , Y , µ, ν ) 7→ TT ∗ π ( Y ) + T A ∗ ( µ, ν ) , where TT ∗ π is the tangent mapping of the cotangent lifted mapping T ∗ π in (2.39), whereas T A ∗ q is thetangent mapping of the linear algebraic dual A ∗ q of the connection (2.23).Let us remark that the first term T π Q ( Z ) in the factorization (3.20) is an element of the tangent bundle TQ .As such, we can apply the trivialization λ T in (2.35) to this term and arrive at its trivialization(3.21) T π Q ( Z ) ≃ (cid:0) τ Q ◦ T π Q ( Z ) , T π ◦ T π Q ( Z ) , A ◦ T π Q ( Z ) (cid:1) ∈ Q ⊕ T ¯ Q × g . By merging the tangent mapping in (3.20) and the trivialization in (3.21), we arrive at a trivialization of TT ∗ Q as follows λ TT ∗ : TT ∗ Q −→ Q ⊕ TT ∗ ¯ Q × g × T g ∗ , Z (cid:0) τ Q ◦ T π Q ( Z ) , T h ∗ ( Z ) , A ( T π Q ( Z )) , T J Q ( Z ) (cid:1) , (3.22) ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 21 where the symbol ⊕ stands for the Whitney product over the base manifold ¯ Q . For future reference, let usrecord here the image space in the most abstract way Q ⊕ TT ∗ ¯ Q × g × T g ∗ = {( q , Y , ζ, µ, ν ) ∈ Q × TT ∗ ¯ Q × g × T g ∗ | [ q ] = π ¯ Q ◦ T π ¯ Q ( Y )} , where we use the product space of two times of the dual space g ∗ isomorphic to T g ∗ . Trivializations of the fibrations of TT ∗ Q . The diamond shape subdiagram in the classical Tulczyjew’striplet (2.19) illustrates the bundle structure of TT ∗ Q over both the tangent bundle TQ and the cotangentbundle T ∗ Q , along with the fibrations T π Q and τ T ∗ Q respectively. Let us now present the trivialization ofthese projections one by one under the Lie group action. Accordingly, we offer the commutative diagram(3.23) TT ∗ Q Q ⊕ TT ∗ ¯ Q × g × T g ∗ TQ Q ⊕ T ¯ Q × g λ TT ∗ T π Q š T π Q λ T to realize the relation between the trivialization λ TT ∗ of (3.22) and the trivialization λ T of (2.35). Accord-ingly, obeying the commutation of Diagram 3.23, we compute the trivialization of T π Q as(3.24) d T π Q : Q ⊕ TT ∗ ¯ Q × g × T g ∗ −→ Q ⊕ T ¯ Q × g , (cid:0) q , Y , ζ, µ, ν (cid:1) (cid:0) q , T π ¯ Q ( Y ) , ζ (cid:1) , where T π ¯ Q is the tangent mapping of the canonical cotangent bundle projection π ¯ Q : T ∗ ¯ Q ¯ Q . Noticethat, the relationship between T π Q and T π ¯ Q is T π ¯ Q ◦ T h ∗ = T π ◦ T π Q , where h ∗ is the dual of the horizontal lift. Next, we compute the trivialization of the tangent bundleprojection τ T ∗ Q according to the commutative diagram TT ∗ Q Q ⊕ TT ∗ ¯ Q × g × T g ∗ TQ Q ⊕ T ∗ ¯ Q × g ∗ λ TT ∗ τ T ∗ Q › τ T ∗ Q λ T ∗ which reads the following explicit realization(3.25) š τ T ∗ Q : Q ⊕ TT ∗ ¯ Q × g × T g ∗ −→ Q ⊕ T ∗ ¯ Q × g ∗ , ( q , Y , ζ, µ, ν ) 7→ ( q , τ T ∗ ¯ Q ( Y ) , µ ) . Here, τ T ∗ ¯ Q is a tangent bundle projection satisfying τ T ∗ ¯ Q ◦ T h ∗ = h ∗ ◦ τ T ∗ Q . We are now ready to present the trivialization of the diamond subdiagram, that is, two fibrations of TT ∗ Q ,inside the Tulczyjew’s triplet (2.19). To this end, we draw the trivialized diamond as(3.26) TT ∗ Q ≃ Q ⊕ TT ∗ ¯ Q × g × T g ∗ TQ ≃ Q ⊕ T ¯ Q × g Q ⊕ T ∗ ¯ Q × g ∗ ≃ T ∗ QQ š T π Q › τ T ∗ Q c τ Q d π Q collecting the fibrations d T π Q in (3.23) and š τ T ∗ Q in (3.23), while both c τ Q and c π Q are the projections ontothe first factor. Trivialization of the pairing between TT ∗ Q and TTQ . Even though there is no canonical pairing between
TTQ and TT ∗ Q , in the equation (2.12), we have presented a (tilde)-pairing h• , •i e between these spaces.The pairing is established by properly lifting the pairing between T ∗ Q and TQ . In view of the isomorphisms λ TT in (3.9) and λ TT ∗ in (3.22), we now calculate a pairing between trivialized bundles. Accordingly, wefirst fix their base manifolds by considering the fibrations d T τ Q in (3.11) and d T π Q in (3.24). This reads thatthe first and the third components in the trivializations λ TT and λ TT ∗ must be the same, say q and ζ . Assuch, for any Z in TT ∗ Q and for any W in TT ∗ Q , we may define a (hat-)pairing(3.27) h• , •i b : (cid:0) Q ⊕ TT ∗ ¯ Q × g × T g ∗ (cid:1) × ( Q ⊕ TT ¯ Q × g × T g ) −→ R for the trivializations as follows h λ TT ∗ ( Z ) , λ TT ( W )i b : = h Z , W i e . Letting T g ∗ ≃ g ∗ × g ∗ and T g ≃ g × g , the hat-pairing is explicitly computed to be h( q , Y , ζ, µ, ν ) ; ( q , U , ζ, ξ, η )i b = h Y , U i e + dds (cid:12)(cid:12) s = h µ + s ν, ξ + s η i = h Y , U i e + h ν, ξ i + h µ, η i , where the first pairing on the last line is the tilde pairing h• , •i e between TT ∗ ¯ Q and TT ¯ Q , whereas the secondand the third terms are the dualities between the Lie algebra and the dual space.3.5. Reduction of TT*Q.
In this part, we examine a trivialization of the reduced tangent bundle of the cotangent TT ∗ Q , [73]. First,we lift the group action of G in (2.21) to the tangent bundle TT ∗ Q as follows(3.28) G × TT ∗ Q −→ TT ∗ Q , g · Z : = TT ∗ φ g − ( Z ) , where the mapping TT ∗ φ g − is the tangent lift of the cotangent lifted action T ∗ φ g − in (2.38). The space oforbits via the action (3.28) is denoted by TT ∗ Q , further we can define a projection from TT ∗ Q to its spaceof orbits TT ∗ Q , mapping a vector Z to its equivalence class [ Z ] . By employing the trivialization (3.22) ofthe tangent bundle TT ∗ Q we arrive at the following (trivialized) action G × ( Q ⊕ TT ∗ ¯ Q × g × T g ∗ ) −→ Q ⊕ TT ∗ ¯ Q × g × T g ∗ , g · ( q , Y , ζ, µ, ν ) 7→ (cid:0) g · q , Y , Ad g ζ, Ad ∗ g − µ, Ad ∗ g − ν (cid:1) . (3.29)Here, g · q is the action of G on Q , Ad g is the adjoint action, and Ad ∗ g − is the coadjoint action.We shall first present an associated bundle. In this case, we consider the vector space in (2.27) as the productspace V = g × T g ∗ of the Lie algebra g and the tangent space of its dual g ∗ . Hence, after taking T g ∗ ≃ g ∗ × g ∗ we can define an action of G on Q × ( g × T g ∗ ) as G × ( Q × g × T g ∗ ) −→ Q × g × T g ∗ , g · ( q , ζ, µ, ν ) 7→ (cid:0) g · q , Ad g ζ, Ad ∗ g − µ, Ad ∗ g − ν (cid:1) . The orbit space is the Whitney product ˜ g ⊕ ˜ g ∗ ⊕ ˜ g ∗ on the reduced manifold ¯ Q . ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 23
In order to comprehend the reduction of the the orbit space TT ∗ Q explicitly, we give the calculation TT ∗ Q ≃ T ( T ∗ Q )/ G ≃ T ( Q ⊕ T ∗ ¯ Q × g ∗ )/ G ≃ ( TQ × T ¯ Q TT ∗ ¯ Q × T g ∗ )/ G ≃ (cid:0) ( Q ⊕ T ¯ Q × g ) × T ¯ Q TT ∗ ¯ Q × T g ∗ (cid:1) / G ≃ (cid:0) (( Q ⊕ T ¯ Q ) × T ¯ Q TT ∗ ¯ Q (cid:1) × g × g ∗ × g ∗ )/ G ≃ ( Q ⊕ TT ∗ ¯ Q × g × g ∗ × g ∗ )/ G ≃ TT ∗ ¯ Q ⊕ ( Q × g × g ∗ × g ∗ )/ G ≃ TT ∗ ¯ Q ⊕ ( ˜ g × ˜ g ∗ × ˜ g ∗ ) ≃ TT ∗ ¯ Q ⊕ e V , by colouring the base manifold T ∗ Q by red in order to keep track of it in the computation. In the first andthird lines, we have used the identity (3.22). In the last line of the calculation we see exactly the manifoldin (3.31), and(3.30) e V : = G \( Q × g × g ∗ × g ∗ ) ≃ ˜ g ⊕ ˜ g ∗ ⊕ ˜ g ∗ , where the sub-index 1 defines the placement of the Lie algebra g in the order of spaces. We shall adopt thisnotation in the sequel. As a result, the reduction with respect to the action (3.29) is calculated as(3.31) TT ∗ Q ≃ TT ∗ ¯ Q ⊕ e V = (cid:8) ( Y , [ q , ζ, µ, ν ]) ∈ TT ∗ ¯ Q × e V : π ¯ Q ◦ τ T ∗ ¯ Q ( Y ) = [ q ] (cid:9) . Reductions of fibrations of TT ∗ Q . As discussed previously, TT ∗ Q admits bundle structures both on TQ and T ∗ Q by means of the fibrations T π Q and τ T ∗ Q , respectively. In (3.24) and (3.25), we have presented thetrivializations of these mappings. On the other hand, implementing the reductions to the total and the basespaces, we arrive at the reduced projections T π Q : TT ∗ ¯ Q ⊕ e V −→ T ¯ Q × ˜ g , ( Y , [ q , ζ, µ, ν ]) 7→ ( T π ¯ Q ( Y ) , [ q , ζ ]) , (3.32) τ T ∗ Q : TT ∗ ¯ Q ⊕ e V −→ T ∗ ¯ Q × ˜ g ∗ , ( Y , [ q , ζ, µ, ν ]) 7→ ( τ T ∗ ¯ Q ( Y ) , [ q , µ ]) , (3.33)respectively. Next, referring to the trivialized diamond (3.26), we draw the following commutative diagram,which we call the reduced diamond, TT ∗ ¯ Q ⊕ ( ˜ g × ˜ g ∗ × ˜ g ∗ ) T ¯ Q ⊕ ˜ g T ∗ ¯ Q ⊕ ˜ g ∗ ¯ Q T π Q τ T ∗ Q τ Q π Q where τ Q and π Q are the projections of T ¯ Q ⊕ ˜ g and T ∗ ¯ Q ⊕ ˜ g ∗ onto the base manifold ¯ Q .
4. Trivializations and Reductions of Cotangent Bundles
Trivialization of T*TQ.
In Subsection 3.2, we have established a trivialization of
TTQ as the product space Q ⊕ TT ¯ Q × g × T g bymeans of the trivialization mapping (3.9). The iterated tangent bundle TTQ is a vector bundle over TQ withthe tangent bundle projection τ T Q . The dual of this fibration reads the bundle structure of the cotangentbundle T ∗ TQ over TQ . In order to arrive at a trivialization of the cotangent bundle T ∗ TQ , we refer thisdualization as follows. The trivialization Q ⊕ TT ¯ Q × g × T g of TTQ determines a vector bundle over thetrivialization Q ⊕ T ¯ Q × g of TQ . Here, the projection is defined to be d τ T Q in (3.11). For any element ( q , u , η ) on the base manifold, we compute the linear algebraic dual of each fiber ( T u T ¯ Q × g × T η g ) ∗ = T ∗ u T ¯ Q × g ∗ × T ∗ η g . Collecting all these dual spaces, let us propose the following bundle structure ( Q ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g , d π T Q , Q ⊕ T ¯ Q × g ) , where the total space is(4.1) Q ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g = {( q , K , µ, ξ, ν ) : [ q ] = τ ¯ Q ◦ π T ¯ Q ( K )} , whereas the projection is d π T Q : Q ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g −→ Q ⊕ T ¯ Q × g , ( q , K , µ, ξ, ν ) 7→ ( q , π T ¯ Q ( K ) , ξ ) . (4.2)To convince ourselves for the duality once more, let us exhibit the pairing between Q ⊕ TT ¯ Q × g × T g and Q ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g as h• , •i : (cid:0) Q ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g (cid:1) × (cid:0) Q ⊕ TT ¯ Q × g × T g (cid:1) −→ R , (cid:10) ( q , K , µ, ξ, ν ) , ( q , U , ζ, ξ, η ) (cid:11) = h K , U i + h µ, ζ i + h ν, η i , (4.3)where the first pairing in the right hand side is the one between bundles T ∗ T ¯ Q and TT ¯ Q , whereas each ofthe second and third pairings are between the Lie algebra g and its dual g ∗ . Then we indicate that this dualbundle is convenient as a trivialization of the bundle T ∗ TQ .Let us calculate the inverse mapping of the trivialization λ TT in (3.9) as follows. The inverse λ − TT of λ TT isdefined by λ − TT : Q ⊕ TT ¯ Q × g × T g −→ TTQ , ( q , U , ζ, ξ, η ) 7→ U Thv + T ( e , ) ( T φ ) v ( ζ, η ) (4.4)for the vector v = ( τ T ¯ Q ( U )) hq + ξ Q ( q ) in TQ , where T h is the tangent mapping (3.4) of the horizontal liftoperator (2.24) and ξ Q is the infinitesimal generator of the vector ξ in the Lie algebra g , whereas T ( T φ ) v isthe tangent mapping of the induced action ( T φ ) v by the action (3.1). In here, ( e , ) is the identity elementof the tangent group T G . In order to display a trivialization for the bundle T ∗ TQ , we need to calculate thedual mapping of λ − TT in (4.4) as follows. ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 25
Consider an arbitrary covector Υ in T ∗ TQ . We project Υ down to the base manifold TQ and arrive at avector π T Q ( Υ ) = v = ( τ T ¯ Q ( U )) hq + ξ Q ( q ) , where the trivialization of TQ in (2.35) has been employed to arrive the sum of a vertical vector ξ Q ( q ) andthe horizontal lift of τ T ¯ Q ( U ) . Let us also denote the image of the vector v under the connection (2.23) as ξ that is A ( π T Q ( Υ )) = ξ . Under the light of these notations, for any ( q , U , ζ, ξ, η ) , we compute the dualmapping as (cid:10) ( λ − TT ) ∗ ( Υ ) , ( q , U , ζ, ξ, η ) (cid:11) = (cid:10) Υ , λ − TT ( q , U , ζ, ξ, η ) (cid:11) = (cid:10) Υ , U Thv + T ( e , ) ( T φ ) v ( ζ, η ) (cid:11) = (cid:10) Υ , U Thv (cid:11) + (cid:10) Υ , T ( e , ) ( T φ ) v ( ζ, η ) (cid:11) = h T ∗ h ( Υ ) , U i + D T ∗( e , ) ( T φ ) v ( Υ ) , ( ζ, η ) E , (4.5)where T ∗ h is the dual mapping of (3.4). The cotangent mapping T ∗( e , ) ( T φ ) v reads a pair of dual elementsdenoted by(4.6) T ∗( e , ) ( T φ ) v : T ∗ v TQ −→ g ∗ × g ∗ , T ∗( e , ) ( T φ ) v ( Υ ) = ( µ, ν ) . By merging the calculation (4.5) and the mapping (4.6), we compute the dual mapping ( λ − TT ) ∗ . We denotethis dual mapping by λ T ∗ T : = ( λ − TT ) ∗ since we consider it a trivialization of T ∗ TQ that is, λ T ∗ T : T ∗ TQ −→ Q ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g , Υ
7→ ( τ Q ◦ π T Q ( Υ ) , T ∗ h ( Υ ) , µ, ξ, ν ) . (4.7)We remark here that the projections d π T Q and π T Q , and trivializations λ T in (2.35) and λ T ∗ T in (4.7) commutethe diagram T ∗ TQ Q ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g TQ Q ⊕ T ¯ Q × g π T Q λ T ∗ T š π T Q λ T that is, d π T Q ◦ λ T ∗ T = λ T ◦ π T Q . Reduction of T*TQ.
In this part, we present the reduction of T ∗ TQ under the Lie group G action by referring to the isomorphism λ T ∗ T in (4.7). First of all, we note the action G × T ∗ TQ −→ T ∗ TQ , g · Υ : = T ∗ T φ g − ( Υ ) , (4.8)where the mapping T ∗ T φ g − is the cotangent lift of the tangent lifted action in (2.32). The orbit space via theaction (4.8) is denoted by T ∗ TQ , then we can define a orbit projection from T ∗ TQ to its orbit space T ∗ TQ which is mapping a vector Υ to its orbit [ Υ ] passing through it. The action (4.8) generates an action of G on the trivialized bundle (4.1) as follows G × (cid:0) Q ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g (cid:1) −→ Q ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g , g · ( q , K , µ, ξ, ν ) = ( g · q , K , Ad ∗ g − µ, Ad g − ξ, Ad ∗ g − ν ) , where Ad g is the adjoint action of the Lie algebra g and Ad ∗ g is the coadjoint action of its dual Lie algebra g ∗ . Let us consider the group action (2.27) where the vector space is considered to be the product space V = g ∗ × g × g ∗ (sub-index 2 defines the placement of the Lie algebra g in the product space) which is theproduct space of the dual Lie algebra g ∗ and the cotangent space of the Lie algebra g isomorphic to g × g ∗ .Thus we can define an action of G on Q × ( g ∗ × g × g ∗ ) . The reduction reads an associated bundle(4.9) e V : = G \( Q × g ∗ × g × g ∗ ) ≃ ˜ g ∗ ⊕ ˜ g ⊕ ˜ g ∗ . Hence, the reduction of the trivialized bundle under the action (4.8) is computed to be the Whitney product(4.10) T ∗ TQ ≃ T ∗ T ¯ Q ⊕ e V = {( K , [ q , µ, ξ, ν ]) ∈ T ∗ T ¯ Q × e V : [ q ] = τ ¯ Q ◦ π T ¯ Q ( K )} . By reducing the trivialized cotangent projection d π T Q in (4.2), we obtain the reduced projection(4.11) π T Q : T ∗ T ¯ Q ⊕ e V −→ T ¯ Q ⊕ ˜ g , ( K , [ q , µ, ξ, ν ]) 7→ ( π T ¯ Q ( K ) , [ q , ξ ]) . The pairing between T ∗ TQ and TTQ . In previous sections, we have found the pairing (4.3) betweentrivializations of T ∗ TQ and TTQ . By employing the reductions, we get(4.12) h• , •i : ( T ∗ T ¯ Q ⊕ e V ) × ( TT ¯ Q ⊕ e V ) −→ R computed to be (cid:10) ( K , [ q , µ, ξ, ν ]) , ( U , [ q , ζ, ξ, η ]) (cid:11) = h K , U i + h µ, ζ i + h ν, η i for an element ( K , [ q , µ, ξ, ν ]) in T ∗ T ¯ Q ⊕ e V and an element ( U , [ q , ζ, ξ, η ]) in TT ¯ Q ⊕ e V , where the firstpairing on the right hand side is between T ∗ T ¯ Q and TT ¯ Q , whereas the second and the third pairings arebetween the Lie algebra g and its dual.4.3. Trivialization of T*T*Q.
In Subsection 3.4, we have established a trivialization of the tangent bundle TT ∗ Q as the product space Q ⊕ TT ∗ ¯ Q × g × T g ∗ referring to the mapping (3.22). The bundle TT ∗ Q is a vector bundle over T ∗ Q with the projection τ T ∗ Q . The dual of this fibration is the cotangent bundle T ∗ T ∗ Q over T ∗ Q . In order toarrive at a trivialization of the cotangent bundle T ∗ T ∗ Q , we refer this dualization as follows. The product Q ⊕ TT ∗ ¯ Q × g × T g ∗ is a vector bundle over the base manifold Q ⊕ T ∗ ¯ Q × g ∗ with the bundle projection š τ T ∗ Q in (3.25). For an arbitrary base element ( q , y , µ ) , we take the dual of the each fiber as ( T y T ∗ ¯ Q × g × T µ g ∗ ) ∗ = T ∗ y T ∗ ¯ Q × g ∗ × T ∗ µ g ∗ . Collection of the dual spaces determines the dual bundle(4.13) ( Q ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ , š π T ∗ Q , Q ⊕ T ∗ ¯ Q × g ∗ ) . Here, the total space is given by Q ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ = {( q , L , γ, µ, ξ ) : [ q ] = π ¯ Q ◦ π T ∗ ¯ q ( L )} , ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 27 whereas the bundle projection š π T ∗ Q is defined through(4.14) š π T ∗ Q : Q ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ −→ Q ⊕ T ∗ ¯ Q × g ∗ , ( q , L , γ, µ, ξ ) 7→ ( q , π T ∗ ¯ Q ( L ) , µ ) . We remark that π T ∗ ¯ Q is the cotangent bundle projection satisfying π T ∗ ¯ Q ◦ T ∗ T ∗ π = h ∗ ◦ π T ∗ Q . The pairing between the trivializations of the bundles T ∗ T ∗ Q and TT ∗ Q is h• , •i : ( Q ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ ) × ( Q ⊕ TT ∗ ¯ Q × g × T g ∗ ) −→ R , (cid:10) ( q , L , γ, µ, ξ ) , ( q , Y , ζ, µ, ν ) (cid:11) = h L , Y i + h γ, ζ i + h ν, ξ i , (4.15)where the first pairing is between bundles TT ∗ ¯ Q and T ∗ T ∗ ¯ Q , whereas each of the second and the thirdpairings are between the Lie algebra g and the dual space g ∗ .Next, we show that the dual bundle (4.13) is suitable as a trivialization of the bundle T ∗ T ∗ Q . We computethe inverse of the trivialization λ TT ∗ in (3.22) as follows λ − TT ∗ : Q ⊕ TT ∗ ¯ Q × g × T g ∗ −→ TT ∗ Q , ( q , Y , ζ, µ, ν ) 7→ T z T ∗ π ( Y ) + T A ∗ ( ζ, ν ) , where z = T ∗ π ( τ T ∗ ¯ Q ( Y )) hq + A ∗ ( µ ) in T ∗ Q , whereas T A ∗ is the tangent mapping of the linear algebraic dual A ∗ of the connection A in (2.23).In order to express a trivialization for the bundle T ∗ T ∗ Q , let us calculate the dual map of λ − TT ∗ . Consider acovector Ξ in T ∗ T ∗ Q , its projection is π T ∗ Q ( Ξ ) = z and under the momentum mapping we have J Q ( z ) = µ .For an element ( q , Y , ζ, µ, ν ) in the trivialization of the bundle TT ∗ Q we compute (cid:10) ( λ − TT ∗ ) ∗ ( Ξ ) , ( q , Y , ζ, µ, ν ) (cid:11) = (cid:10) Ξ , λ − TT ∗ ( q , Y , ζ, µ, ν ) (cid:11) = h Ξ , T z T ∗ π ( Y ) + T A ∗ ( ζ, ν )i = h Ξ , T z T ∗ π ( Y )i + h Ξ , T A ∗ ( ζ, ν )i = (cid:10) T ∗ z T ∗ π ( Ξ ) , Y (cid:11) + h T ∗ A ∗ ( Ξ ) , ( ζ, ν )i . (4.16)See that the image of the mapping T ∗ A ∗ can be denoted by(4.17) T ∗ A ∗ : T ∗ T ∗ Q −→ g ∗ × g , Ξ T ∗ A ∗ ( Ξ ) = ( γ, ξ ) . Referring to these notations, by taking T ∗ g ∗ ≃ g ∗ × g , and by combining (4.16) and (4.17), we compute thedual map ( λ − TT ∗ ) ∗ and introduce it as a trivialization of the bundle T ∗ T ∗ Q that is, λ T ∗ T ∗ : T ∗ T ∗ Q −→ Q ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ , Ξ
7→ ( π Q ◦ π T ∗ Q ( Ξ ) , T ∗ T ∗ π ( Ξ ) , γ, µ, ξ ) . (4.18)The cotangent bundle projection π T ∗ Q , its trivialization š π T ∗ Q in (4.14), and the trivializations λ T ∗ in (2.40)and λ T ∗ T ∗ in (4.18) commute the diagram T ∗ T ∗ Q Q ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ T ∗ Q Q ⊕ T ∗ ¯ Q × g ∗ π T ∗ Q λ T ∗ T ∗ › π T ∗ Q λ T ∗ by satisfying the relation š π T ∗ Q ◦ λ T ∗ T ∗ = λ T ∗ ◦ π T ∗ Q . Reduction of T*T*Q.
In this subsection, we present a trivialization of the reduced iterated cotangent bundle T ∗ T ∗ Q in the realmof the isomorphism λ T ∗ T ∗ in (4.18). We first start with the action of G on T ∗ T ∗ Q (4.19) G × T ∗ T ∗ Q −→ T ∗ T ∗ Q , g · Ξ : = T ∗ T ∗ φ g − ( Ξ ) , where the mapping T ∗ T ∗ φ g − is the cotangent lift of the action T ∗ φ g − in (2.38). The orbit space of theaction (4.19) is denoted by T ∗ T ∗ Q , then there is a projection from T ∗ T ∗ Q to its orbit space T ∗ T ∗ Q , which ismapping a vector Ξ to its orbit [ Ξ ] generated by the action (4.19).Referring to λ T ∗ T ∗ in (4.18), the induced action of G on the trivialization of T ∗ T ∗ Q is G × ( Q ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ ) −→ ( Q ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ ) , g · ( q , L , γ, µ, ξ ) : = ( g · q , L , Ad ∗ g − γ, Ad ∗ g − µ, Ad g − ξ ) . Consider the product space V = g ∗ × g ∗ × g and the group action G on Q × V . This determines an associatedbundle with the total space(4.20) e V : = G \( Q × g ∗ × g ∗ × g ) ≃ ˜ g ∗ ⊕ ˜ g ∗ ⊕ ˜ g . In this notation, the reduced bundle due to the action (4.19) is calculated to be [73](4.21) T ∗ T ∗ Q ≃ T ∗ T ∗ ¯ Q ⊕ e V = {( L , [ q , γ, µ, ξ ]) : [ q ] = π Q ◦ π T ∗ Q ( L )} . Remarking the projection (4.14), we arrive at the reduced mapping(4.22) π T ∗ Q : T ∗ T ∗ ¯ Q ⊕ e V −→ T ∗ ¯ Q ⊕ ˜ g ∗ , ( L , [ q , γ, µ, ξ ]) 7→ ( π T ∗ ¯ Q ( L ) , [ q , µ ]) . Pairing between T ∗ T ∗ Q and TT ∗ Q . In (4.15), we have showed the pairing (4.15) between trivializationsof bundles T ∗ T ∗ Q and TT ∗ Q . For the reduced spaces, we have h• , •i : ( T ∗ T ∗ ¯ Q ⊕ e V ) × ( TT ∗ ¯ Q ⊕ e V ) −→ R , (cid:10) ( L , [ q , γ, µ, ξ ]) , ( Y , [ q , ζ, µ, ν ]) (cid:11) = h L , Y i + h γ, ζ i + h ν, ξ i , (4.23)where the first pairing is between TT ∗ ¯ Q and T ∗ T ∗ ¯ Q , whereas the second and the third pairings are betweenthe Lie algebra g and its dual g ∗ . ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 29
5. Trivialization and Reduction of Tulczyjew’s Triplet
In this section, our main goal is to construct trivialized and reduced Tulczyjew’s triplets using the result inSections 3 and 4.5.1.
Trivializations and Reductions of the Canonical Forms.
Recall the canonical one-form θ Q in (2.1), and the canonical symplectic two-form Ω Q . In this subsectionwe first write these forms on the trivialized cotangent bundle Q ⊕ T ∗ ¯ Q × g ∗ in (2.40), see also [73]. Thenwe obtain their reduced forms on the reduced cotangent bundle T ∗ ¯ Q ⊕ ˜ g ∗ in (2.41). Trivialization and reduction of the canonical one-form.
Being a one-form, θ Q is a section of the cotangentbundle projection so that it is a map T ∗ Q T ∗ T ∗ Q . By employing the trivialization λ T ∗ in (2.40) to thebase manifold T ∗ Q , and the trivialization (4.18) to the total manifold T ∗ T ∗ Q we arrive at the trivialization c θ Q of the canonical one-form as follows(5.1) c θ Q : Q ⊕ T ∗ ¯ Q × g ∗ −→ Q ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ , ( q , y , µ ) 7→ ( q , θ ¯ Q ( y ) , µ, µ, ) , where θ ¯ Q is the canonical one-form on the cotangent bundle of the reduced manifold T ∗ ¯ Q . To prove this,let us refer the definition (2.1). Then by using the mapping š τ T ∗ Q in (3.25), and d T π Q in (3.24), we compute D c θ Q ( q , y , µ ) , ( q , Y , ζ, µ, ν ) E = Dš τ T ∗ Q ( q , Y , ζ, µ, ν ) , d T π Q ( q , Y , ζ, µ, ν ) E = (cid:10) ( q , τ T ∗ ¯ Q ( Y ) , µ ) , ( q , T π ¯ Q ( Y ) , ζ ) (cid:11) = (cid:10) τ T ∗ ¯ Q ( Y ) , T π ¯ Q ( Y ) (cid:11) + h µ, ζ i = (cid:10) θ ¯ Q ( y ) , Y (cid:11) + h µ, ζ i (5.2)for an arbitrary vector ( q , Y , ζ, µ, ν ) in Q ⊕ TT ∗ ¯ Q × g × T g ∗ . The first pairing on the third line is between T ∗ ¯ Q and T ¯ Q whereas the second pairing is between g ∗ and g . A direct observation proves (5.1).By employing the reduction (2.41) of T ∗ Q and the reduction (4.21) of T ∗ T ∗ , the reduced one-form θ Q iscomputed by θ Q : T ∗ ¯ Q ⊕ ˜ g ∗ −→ T ∗ T ∗ ¯ Q ⊕ ˜ V , ( y , [ q , µ ]) 7→ ( θ ¯ Q ( y ) , [ q , µ, µ, ]) , where ˜ V = ˜ g ∗ ⊕ ˜ g ∗ ⊕ ˜ g is the abbreviation stated in (4.20). Trivialization and reduction of the canonical symplectic two-form.
The trivialization of the symplectictwo-form Ω Q on T ∗ Q is computed to be [73] c Ω Q (( q , Y , ζ, µ, ν ) , ( q , ´ Y , ´ ζ, µ, ´ ν )) = Ω ¯ Q ( Y , ´ Y ) + h ´ ν, ζ i − (cid:10) ν, ´ ζ (cid:11) + (cid:10) µ, [ ζ, ´ ζ ] (cid:11) − π ∗ ¯ Q B µ ( Y , ´ Y ) (5.3)for two arbitrary vectors ( q , Y , ζ, µ, ν ) and ( q , ´ Y , ´ ζ, µ, ´ ν ) in trivialized bundle Q ⊕ TT ∗ ¯ Q × g × T g ∗ . Here, Ω ¯ Q is the canonical symplectic two-form on T ∗ ¯ Q whereas B µ is a two-form on T ∗ ¯ Q , defined by means of thecurvature B of the connection A in (2.23), such that(5.4) B µ ( Y , ´ Y ) = (cid:10) µ, B ( Y , ´ Y ) (cid:11) . For two vectors ( Y , [ q , ζ, µ, ν ]) and ( ´ Y , [ q , ´ ζ, µ, ´ ν ]) in the bundle TT ∗ ¯ Q ⊕ ˜ V , the reduced form Ω Q is Ω Q (( Y , [ q , ζ, µ, ν ]) , ( ´ Y , [ q , ´ ζ, µ, ´ ν ])) = Ω ¯ Q ( Y , ´ Y ) + h[ q , ´ ν ] , [ q , ζ ]i − (cid:10) [ q , ν ] , [ q , ´ ζ ] (cid:11) + (cid:10) [ q , µ ] , (cid:2) q , [ ζ, ´ ζ ] (cid:3) (cid:11) − π ∗ ¯ Q B [ q ,µ ] ( Y , ´ Y ) , (5.5)where Ω ¯ Q is the canonical symplectic form on reduced manifold ¯ Q , whereas B [ q ,µ ] is the two-form definedas(5.6) B [ q ,µ ] ( T π ¯ Q ( Y ) , T π ¯ Q ( ´ Y )) = (cid:10) [ q , µ ] , B ( T π ¯ Q ( Y ) , T π ¯ Q ( ´ Y )) (cid:11) for the curvature B . In here, T π ¯ Q is the projection from the bundle TT ∗ ¯ Q to the bundle T ¯ Q . Notice that, [ q , ζ ] , [ q , ´ ζ ] and [ q , [ ζ, ´ ζ ]] belong to the adjoint bundle ˜ g whereas [ q , µ ] , [ q , ν ] and [ q , ´ ν ] are in ˜ g ∗ .5.2. Trivializations and Reductions of the Symplectomorphisms.
In this subsection, we compute trivializations and reductions of the symplectomorphisms α Q in (2.13) and Ω ♭ Q in (2.16). Trivialization and Reduction of α Q . Recall the pairing h• , •i b in (3.27), and the trivialized involution c κ Q in (3.13). The trivialization c α Q is explicitly computed to be (cid:10) c α Q ( q , Y , ξ, µ, ν ) , ( q , U , ζ, ξ, η ) (cid:11) = (cid:10) ( q , Y , ξ, µ, ν ) , c κ Q ( q , U , ζ, ξ, η ) (cid:11) b = (cid:10) ( q , Y , ξ, µ, ν ) , ( q , κ ¯ Q ( U ) , ξ, ζ, η ) (cid:11) b = (cid:10) Y , κ ¯ Q ( U ) (cid:11) ˜ + h ν, ζ i + h µ, η i = (cid:10) α ¯ Q ( Y ) , U (cid:11) + h ν, ζ i + h µ, η i , (5.7)for vectors ( q , U , ζ, ξ, η ) in Q ⊕ TT ¯ Q × g × T g and ( q , Y , ξ, µ, ν ) in Q ⊕ TT ∗ ¯ Q × g × T g ∗ . That is we have c α Q : Q ⊕ TT ∗ ¯ Q × g × T g ∗ −→ Q ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g , ( q , Y , ξ, µ, ν ) 7→ ( q , α ¯ Q ( Y ) , ν, ξ, µ ) . (5.8)Here, the projection mapping α ¯ Q satisfies the property α ¯ Q ◦ T h ∗ = T ∗ h ◦ α Q , where T ∗ h is the cotangent lift of the lifted horizontal operator T h in (3.4), whereas
T h ∗ is the tangent mapof the dual mapping of horizontal lift operator (2.24).We arrive at the reduced mapping(5.9) α Q : TT ∗ ¯ Q ⊕ e V −→ T ∗ T ¯ Q ⊕ e V , ( Y , [ q , ξ, µ, ν ]) 7→ ( α ¯ Q ( Y ) , [ q , ν, ξ, µ ]) , where α ¯ Q is the symplectomorphism from the bundle TT ∗ ¯ Q to the bundle T ∗ T ¯ Q . Here, e V = ˜ g ⊕ ˜ g ∗ ⊕ ˜ g ∗ in(3.30), and e V = ˜ g ∗ ⊕ ˜ g ⊕ ˜ g ∗ in (4.9). ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 31
Trivialization and Reduction of Ω ♭ Q . Recall the trivialized symplectic two-form c Ω Q exhibited in (5.3).Referring to this mapping and for vectors ( q , Y , ζ, µ, ν ) , ( q , ´ Y , ´ ζ, µ, ´ ν ) in the trivialization of TT ∗ Q we compute D c Ω Q ♭ ( q , Y , ζ, µ, ν ) , ( q , ´ Y , ´ ζ, µ, ´ ν ) E = D c Ω Q ; ( q , Y , ζ, µ, ν ) , ( q , ´ Y , ´ ζ, µ, ´ ν ) E = Ω ¯ Q ( Y , ´ Y ) + h ´ ν, ζ i − (cid:10) ν, ´ ζ (cid:11) + (cid:10) µ, [ ζ, ´ ζ ] (cid:11) − B µ ( T π ¯ Q ( Y ) , T π ¯ Q ( ´ Y )) = h Ω ♭ ¯ Q ( Y ) − π ∗ ¯ Q B ♭µ ( Y ) , ´ Y i + h ´ ν, ζ i + h ad ∗ ζ µ − ν, ´ ζ i . (5.10)Thus we have that c Ω Q ♭ : Q ⊕ TT ∗ ¯ Q × g × T g ∗ −→ Q ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ , ( q , Y , ζ, µ, ν ) 7→ (cid:0) q , Ω ♭ ¯ Q ( Y ) − π ∗ ¯ Q B ♭µ ( Y ) , ad ∗ ζ µ − ν, µ, ζ (cid:1) . (5.11)Under the symmetry, we obtain the reduced mapping Ω ♭ Q : TT ∗ ¯ Q ⊕ e V −→ T ∗ T ∗ ¯ Q ⊕ e V , ( Y , [ q , ζ, µ, ν ]) 7→ (cid:0) Ω ♭ ¯ Q ( Y ) − π ∗ ¯ Q B ♭µ ( Y ) , [ q , ad ∗ ζ µ − ν, µ, ζ ] (cid:1) , (5.12)where e V = ˜ g ∗ ⊕ ˜ g ∗ ⊕ ˜ g in (4.9).5.3. Trivializations and Reductions of the Derivations.
Recall the derivations i T in (2.5) and d T in (2.6). We have used them to obtain one-forms ϑ in (2.7) and ϑ in (2.7), respectively. In this part, we transfer these derivations to trivialized and reduced second iteratedbundle. For this, we trivialize the third iterated bundle TTT ∗ Q as TTT ∗ Q −→ Q ⊕ TTT ∗ ¯ Q × g × T g × TT g ∗ , Û Z
7→ ( τ Q ◦ T τ Q ◦ TT π Q ( Û Z ) , TT h ∗ ( Û Z ) , A ( T τ Q ◦ TT π Q ( Û Z )) , T A ( TT π Q ( Û Z )) , TT J Q ( Û Z )) . Here, the image space is(5.13) Q ⊕ TTT ∗ ¯ Q × g × T g × TT g ∗ = {( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) : [ q ] = π ¯ Q ◦ T π ¯ Q ◦ TT π ¯ Q ( Û Y )} admitting the projections › τ TT ∗ Q and › T τ T ∗ Q by › τ TT ∗ Q : Q ⊕ TTT ∗ ¯ Q × g × T g × TT g ∗ −→ Q ⊕ TT ∗ ¯ Q × g × T g ∗ , ( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) 7→ ( q , τ TT ∗ ¯ Q ( Û Y ) , ζ, µ, ν ) , › T τ T ∗ Q : Q ⊕ TTT ∗ ¯ Q × g × T g × TT g ∗ −→ Q ⊕ TT ∗ ¯ Q × g × T g ∗ , ( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) 7→ ( q , T τ T ∗ ¯ Q ( Û Y ) , ξ, µ, w ) , where τ TT ∗ ¯ Q is the canonical projection from TTT ∗ ¯ Q to TT ∗ ¯ Q whereas T τ T ∗ ¯ Q is the tangent mapping ofthe canonical projection τ T ∗ ¯ Q from TT ∗ ¯ Q to T ∗ ¯ Q . Trivialization of the derivation i T . The trivialization b i T of the derivation i T in (2.5) is defined as b i T : Λ k ( Q ) −→ Λ k − (cid:0) Q ⊕ T ¯ Q × g (cid:1) , b i T Ω k ( X , · · · , X k − ) = Ω k ( d τ T Q ( X ) , d T τ Q ( X ) , · · · , d T τ Q ( X k − )) (5.14)for any collection of vector fields X , . . . , X k − on the trivialized tangent bundle Q ⊕ T ¯ Q × g . Recall that theone-form ϑ in (2.7). Let us construct a trivialized version of ϑ by applying the derivation b i T in (5.14) to the trivialized symplectic form c Ω Q in (5.3) as follows b ϑ ( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) = b i T c Ω Q ( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) = c Ω Q (cid:0)› τ TT ∗ Q ( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) , › T τ T ∗ Q ( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) (cid:1) = c Ω Q (cid:0) (cid:16) q , τ TT ∗ ¯ Q ( Û Y ) , ζ, µ, ν (cid:17) , (cid:16) q , T τ T ∗ ¯ Q ( Û Y ) , ξ, µ, w (cid:17) (cid:1) = Ω ¯ Q ( τ TT ∗ ¯ Q ( Û Y ) , T τ T ∗ ¯ Q ( Û Y )) + h w , ζ i − h ν, ξ i + h µ, [ ζ, ξ ]i− B µ ( T π ¯ Q ( τ TT ∗ ¯ Q ( Û Y )) , T π ¯ Q ( T τ T ∗ ¯ Q ( Û Y ))) = i T Ω ¯ Q ( Û Y ) + h w , ζ i − h ν, ξ i + h µ, [ ζ, ξ ]i − π ∗ ¯ Q B µ ( τ TT ∗ ¯ Q ( Û Y ) , T τ T ∗ ¯ Q ( Û Y )) = i T Ω ¯ Q ( Û Y ) + h w , ζ i − h ν, ξ i + h µ, [ ζ, ξ ]i − i T ( π ∗ ¯ Q B µ )( Û Y ) (5.15)for a vector ( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) in Q ⊕ TTT ∗ ¯ Q × g × T g × TT g ∗ , where i T Ω ¯ Q is a one-form generated bythe derivative i T in (2.5), whereas Ω ¯ Q is the canonical symplectic form on T ∗ ¯ Q . In here, B µ is the inducedcurvature in (5.4) and π ∗ ¯ Q is pull back of the canonical projection π ¯ Q from T ∗ ¯ Q to ¯ Q . Trivialization of the derivation d T . The trivialization b d T of the derivation d T in (2.6) is defined as c d T : Λ k ( Q ) −→ Λ k ( Q ⊕ T ¯ Q × g ) , c d T = d b i T + b i T d , where d is the exterior derivative on Q , whereas b i T is the trivialized derivation given in (5.14). A trivialization b ϑ of the derivation ϑ in (2.7) is defined as(5.16) b ϑ = − c d T c θ Q for a vector ( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) in Q ⊕ TTT ∗ ¯ Q × g × T g × TT g ∗ , where c θ Q is the trivialized one-form in(5.1).Replacing the manifold Q with trivialized bundle Q ⊕ T ∗ ¯ Q × g ∗ , let us find an one-form c d T c θ Q by − b ϑ ( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) = c d T c θ Q ( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) = ( d b i T + b i T d ) c θ Q ( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) = d b i T c θ Q ( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) + b i T d c θ Q ( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) = d ( b i T ( θ ¯ Q + µ )( q , y , µ ))( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) − b i T (− d c θ Q )( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) = d ( i T θ ¯ Q ( y )( τ TT ∗ ¯ Q ( Û Y )) + µ ( ζ ))( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) − b i T c Ω Q ( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) = d θ ¯ Q ( τ TT ∗ ¯ Q ( Û Y ) , T τ T ∗ ¯ Q ( Û Y )) + h µ, η i + h w , ζ i− ( i T Ω ¯ Q ( Û Y ) + h w , ζ i − h ν, ξ i + h µ, [ ζ, ξ ]i − i T ( π ∗ ¯ Q B µ )( Û Y )) = di T θ ¯ Q ( Û Y ) − i T Ω ¯ Q ( Û Y ) + h µ, η i + h ν, ξ i − h µ, [ ζ, ξ ]i + i T ( π ∗ ¯ Q B µ )( Û Y ) = di T θ ¯ Q ( Û Y ) + i T d θ ¯ Q ( Û Y ) + h µ, η i + h ν, ξ i − h µ, [ ζ, ξ ]i + i T ( π ∗ ¯ Q B µ )( Û Y ) = d T θ ¯ Q ( Û Y ) + h µ, η i + h ν, ξ i − h µ, [ ζ, ξ ]i + i T ( π ∗ ¯ Q B µ )( Û Y ) (5.17)for a vector ( q , Û Y , ζ, ξ, η, µ, ν, w , Û ν ) in Q ⊕ TTT ∗ ¯ Q × g × T g × TT g ∗ , where d T θ ¯ Q is a one-form generated bythe derivative d T in (2.6), whereas θ ¯ Q is the canonical one-form on T ∗ ¯ Q . In here, B µ is in (5.4), and π ∗ ¯ Q isthe pull back of the canonical projection π ¯ Q from T ∗ ¯ Q to ¯ Q . ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 33
Symplectic form on the trivialization of TT ∗ Q . In this part, we display the symplectic form on thetrivialization of the bundle TT ∗ Q by employing one forms b ϑ in (5.15) and b ϑ in (5.16). Actually, thissymplectic form denoted by c d T c Ω Q can be calculated either of exterior derivative of b ϑ or b ϑ . We shallcalculate this symplectic form c d T c Ω Q by using each one-forms and we realize that this two-form can begenerated by this one-forms, separately. In order to observe this fact, let us calculate exterior derivative ofone-forms b ϑ and b ϑ as follows: d b ϑ ( X , ´ X )( q , Y , ζ, µ, ν ) = d ( i T Ω ¯ Q ( Û Y ) − h w , ζ i + h ν, ξ i − h µ, [ ζ, ξ ]i − i T ( π ∗ ¯ Q B µ )( Û Y ))( ´ X ( q , Y , ζ, µ, ν )) = di T Ω ¯ Q ( Û Y , Û Y ) + h w , [ ζ, ζ ]i − h ν, [ ξ , ξ ]i + h µ, [[ ζ, ξ ] , [ ζ, ξ ]]i − di T ( π ∗ ¯ Q B µ )( Û Y , Û Y ) = di T Ω ¯ Q ( Û Y , Û Y ) + i T d Ω ¯ Q ( Û Y , Û Y ) − h ν, [ ξ , ξ ]i + h µ, [[ ζ, ξ ] , [ ζ, ξ ]]i − di T ( π ∗ ¯ Q B µ )( Û Y , Û Y ) = d T Ω ¯ Q ( Û Y , Û Y ) − h ν, [ ξ , ξ ]i + h µ, [ ζ, [ ξ , ξ ]]i − di T ( π ∗ ¯ Q B µ )( Û Y , Û Y ) − i T d ( π ∗ ¯ Q B µ )( Û Y , Û Y ) = d T Ω ¯ Q ( Û Y , Û Y ) + D − ν + ad ∗ ζ µ, [ ξ , ξ ] E − d T ( π ∗ ¯ Q B µ )( Û Y , Û Y ) . for vectors X ( q , Y , ζ, µ, ν ) = ( q , Û Y , ζ, ξ , η, µ, ν, w , Û ν ) , ´ X ( q , Y , ζ, µ, ν ) = ( q , Û Y , ζ, ξ , η, µ, ν, w , Û ν ) , (5.18)in the trivialization of the bundle TTT ∗ Q , where we use the identities(5.19) i T d Ω ¯ Q = , d Ω ¯ Q = − d θ ¯ Q = , i T d ( π ∗ ¯ Q B µ ) = . For the second one-form b ϑ , we have d b ϑ ( X , ´ X )( q , Y , ζ, µ, ν ) = d (− d T θ ¯ Q ( Û Y ) − h µ, η i − h ν, ξ i + h µ, [ ζ, ξ ]i − i T ( π ∗ ¯ Q B µ )( Û Y ))( X ( q , Y , ζ, µ, ν )) = − dd T θ ¯ Q ( Û Y )( Û Y ) − h µ, [ η, η ]i − h ν, [ ξ , ξ ]i + h µ, [[ ζ, ξ ] , [ ζ, ξ ]]i − di T ( π ∗ ¯ Q B µ )( Û Y , Û Y ) = − d T d θ ¯ Q ( Û Y )( Û Y ) − h ν, [ ξ , ξ ]i + h µ, [[ ζ, ξ ] , [ ζ, ξ ]]i − di T ( π ∗ ¯ Q B µ )( Û Y , Û Y )− i T d ( π ∗ ¯ Q B µ )( Û Y , Û Y ) = d T Ω ¯ Q ( Û Y )( Û Y ) − h ν, [ ξ , ξ ]i + h µ, [[ ζ, ξ ] , [ ζ, ξ ]]i − d T ( π ∗ ¯ Q B µ )( Û Y , Û Y ) = d T Ω ¯ Q ( Û Y )( Û Y ) − h ν, [ ξ , ξ ]i + h µ, [ ζ, [ ξ , ξ ]]i − d T ( π ∗ ¯ Q B µ )( Û Y , Û Y ) = d T Ω ¯ Q ( Û Y )( Û Y ) + D − ν + ad ∗ ζ µ, [ ξ , ξ ] E − d T ( π ∗ ¯ Q B µ )( Û Y , Û Y ) , where we employ the identities dd T = d T d , i T d ( π ∗ ¯ Q B µ ) = . Then, the trivialized symplectic form c d T c Ω Q of d T Ω Q in (2.9) is equal to c d T c Ω Q = d b ϑ = d b ϑ . Reductions of the potential one-forms on the reduced bundle of TT ∗ Q . Since horizontal componentsare invariant under the group action (2.21), they remain the same. Vertical terms, that is ξ ∈ g and µ ∈ g ∗ ,will be replaced by ¯ ξ = [ q , ξ ] ∈ ˜ g and ¯ µ = [ q , µ ] ∈ ˜ g ∗ , respectively. We compute the reduced one-form(5.20) ϑ = i T Ω ¯ Q ( Û Y ) + (cid:10) ¯ w , ¯ ζ (cid:11) − (cid:10) ¯ ν, ¯ ξ (cid:11) + (cid:10) ¯ µ, [ ¯ ζ, ¯ ξ ] (cid:11) − i T ( π ∗ ¯ Q B ¯ µ )( Û Y ) , where i T Ω ¯ Q is the image of the canonical symplectic form on T ∗ ¯ Q under the derivative i T in (2.5), whereas¯ w = [ q , w ] , ¯ ν = [ q , ν ] , ¯ µ = [ q , µ ] are elements of the coadjoint bundle ˜ g ∗ , and ¯ ξ = [ q , ξ ] and ¯ ζ = [ q , ζ ] areelements of the adjoint bundle ˜ g . Similarly, the reduction of the form b ϑ in (5.16) is computed to be(5.21) ϑ = − d T θ ¯ Q ( Û Y ) − h ¯ µ, ¯ η i − (cid:10) ¯ ν, ¯ ξ (cid:11) + (cid:10) ¯ µ, [ ¯ ζ, ¯ ξ ] (cid:11) − i T ( π ∗ ¯ Q B ¯ µ )( Û Y ) , where d T θ ¯ Q is a one-form generated by the derivative d T in (2.6), whereas θ ¯ Q is the canonical one-form on T ∗ ¯ Q .5.4. Trivializations and Reductions of the Tulczyjew’s Triplet.
We now ready to perform the trivializations and the reductions of two special symplectic structures, givenin (2.14) and (2.17), admitted by the symplectic manifold ( TT ∗ Q , d T Ω Q ) . Both for the trivializations andreductions of the structures, we simply substitute all the bundles, mappings and differential forms by theirtrivialized and reduced versions, respectively. Trivialization of the Left Wing.
Recall the special symplectic structure ( TT ∗ Q , T π Q , TQ , ϑ , α Q ) in(2.14) with its picture in (2.15). Notice that, there are three manifolds TT ∗ Q , T ∗ T ∗ Q and TQ in thisspecial symplectic structure. We substitute the trivializations of these spaces in (3.22), (4.18) and (2.35),respectively. Further, we use the trivialized mappings c α Q in (5.8), d T π Q in (3.24) and d π T Q in (4.2). As a finalstep, we replace the one-form ϑ in (2.7) by its trivialized version b ϑ in (5.16). Notice that the trivializedsymplectomorphism c α Q still preserves symplectic structures. So that we have the following quintable(5.22) ( Q × TT ∗ ¯ Q × g × T g ∗ , d T π Q , Q ⊕ T ¯ Q × g , b ϑ , c α Q ) and the diagram(5.23) Q × T ∗ T ¯ Q × g ∗ × T ∗ g Q ⊕ TT ∗ ¯ Q × g × T g ∗ Q ⊕ T ¯ Q × g š π T Q d α Q š T π Q Reduction of the left wing.
We refer the special symplectic structure (2.14) once more and, this time, wesubstitute the reduced spaces TT ∗ ¯ Q ⊕ ˜ V in (3.31), T ∗ T ¯ Q ⊕ ˜ V in (4.10), and T ¯ Q ⊕ ˜ g in (2.36). Additionally,we recall reduced mappings α Q in (5.9), T π Q in (3.32) and π T Q in (4.11). For final step, we replace theone- form ϑ in (2.7) by its reduced version ϑ in (5.21). So that we have(5.24) ( TT ∗ ¯ Q ⊕ ˜ V , T π Q , T ¯ Q ⊕ ˜ g , ϑ , α Q ) and the diagram(5.25) T ∗ T ¯ Q ⊕ ˜ V TT ∗ ¯ Q ⊕ ˜ V T ¯ Q ⊕ ˜ g π T Q α Q T π Q ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 35
Trivialization of the right wing.
At first, recall the special symplectic stucture ( TT ∗ Q , τ T ∗ Q , T ∗ Q , ϑ , Ω ♭ Q ) in (2.17). Similar to what we have done to the left wing, we substitute the trivializations of the manifolds TT ∗ Q , T ∗ T ∗ Q , and T ∗ Q referring to (3.22), (4.18) and (2.40), respectively. Later, we recall the mappings c Ω Q ♭ in (5.11), š τ T ∗ Q in (3.25) and š π T ∗ Q in (4.14). As the final step, we replace the one-form ϑ by itstrivialized version b ϑ in (5.15). So that we arrive at(5.26) (cid:16) Q ⊕ TT ∗ ¯ Q × g × T g ∗ , š τ T ∗ Q , Q ⊕ T ∗ ¯ Q × g ∗ , b ϑ , c Ω Q ♭ (cid:17) and(5.27) Q ⊕ TT ∗ ¯ Q × g × T g ∗ Q ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ Q ⊕ T ∗ ¯ Q × g ∗ › τ T ∗ Q d Ω Q ♭ › π T ∗ Q being the trivialization of (2.18). Reduction of the right wing.
To obtain the reduction of the special symplectic structures (2.17), we referthe reduced versions of the mappings and manifolds. That is, we recall the reductions of TT ∗ Q , T ∗ T ∗ Q and T ∗ Q given in (3.31), (4.21), and (2.41), respectively. By considering the section ϑ in (5.20), we have(5.28) (cid:16) TT ∗ ¯ Q ⊕ ˜ V , τ T ∗ Q , T ∗ ¯ Q ⊕ ˜ g ∗ , ϑ , Ω Q ♭ (cid:17) and the diagram(5.29) TT ∗ ¯ Q ⊕ ˜ V T ∗ T ∗ ¯ Q ⊕ ˜ V T ∗ ¯ Q ⊕ ˜ g ∗ τ T ∗ Q Ω ♭ Q π T ∗ Q Trivialized Tulczyjew’s triplet.
We construct trivialized Tulczyjew’s triplet by combining trivializedspecial symplectic structures (5.22) and (5.26). That is to draw the diagrams (5.23) and (5.27) in one picture(5.30) T ∗ TQQ ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g TT ∗ QQ ⊕ TT ∗ ¯ Q × g × T g ∗ T ∗ T ∗ QQ ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ TQQ ⊕ T ¯ Q × g T ∗ QQ ⊕ T ∗ ¯ Q × g ∗ Q š π T Q › τ T ∗ Q š T π Q d α Q d Ω Q ♭ › π T ∗ Q pr pr where c α Q in (5.8) is the trivialization of symplectomorphism α Q , whereas c Ω Q ♭ in (5.11) is the trivializationof symplectomorphism Ω ♭ Q . Reduced Tulczyjew’s triplet.
We construct reduced trivialized Tulczyjew’s triplet by merging reducedspecial symplectic structures (5.24) and (5.28) that is(5.31) T ∗ TQT ∗ T ¯ Q ⊕ ˜ V TT ∗ QTT ∗ ¯ Q ⊕ ˜ V T ∗ T ∗ QT ∗ T ∗ ¯ Q ⊕ ˜ V TQT ¯ Q ⊕ ˜ g T ∗ QT ∗ ¯ Q ⊕ ˜ g ∗ ¯ Q π T Q τ T ∗ Q T π Q α Q Ω Q ♭ π T ∗ Q τ Q π Q where we use abbreviations ˜ V = ˜ g ⊕ ˜ g ∗ ⊕ ˜ g ∗ , ˜ V = ˜ g ∗ ⊕ ˜ g ⊕ ˜ g ∗ and ˜ V = ˜ g ∗ ⊕ ˜ g ∗ ⊕ ˜ g , whereas α Q and Ω Q ♭ are reduced symplectomorphism given in (5.8) and (5.12), respectively.
6. Horizontal-Vertical Decompositions of the Tulczyjew’s Triplet
In this part, we exploit horizontal-vertical decompositions of both the trivialized and reduced Tulczyjew’striplets obtained in the previous subsection.6.1.
Horizontal-Vertical Decomposition of the Second Order Tangent Bundles.
Referring to a connection A in (2.23), we start by recording the following decompositions for future reference TQ = HQ × Q VQ , HQ ≃ Q ⊕ T ¯ Q , VQ ≃ Q × g , T ∗ Q = H ∗ Q × Q V ∗ Q , H ∗ Q ≃ Q ⊕ T ∗ ¯ Q , V ∗ Q ≃ Q × g ∗ , (6.1)where HQ and H ∗ Q are horizontal bundles of the tangent bundle TQ and the cotangent bundle T ∗ Q ,respectively, whereas VQ and V ∗ Q are vertical bundles of TQ and T ∗ Q , respectively as well. Here, × Q denotes the Whitney sum over Q whereas ⊕ denotes the Whitney sum over ¯ Q .We note the horizontal-vertical decomposition of the reduced first order bundles as TQ = HQ ⊕ VQ , HQ = T ¯ Q , VQ = ˜ g , T ∗ Q = H ∗ Q ⊕ V ∗ Q , H ∗ Q = T ∗ ¯ Q , V ∗ Q = ˜ g ∗ , (6.2)where ˜ g is the adjoint bundle whereas ˜ g ∗ is the coadjoint bundle. ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 37
Horizontal-vertical decomposition of
TTQ . We have that(6.3)
TTQ = T HQ × T Q
TV Q , where the tangent bundles on the right hand side are T HQ = T ( Q ⊕ T ¯ Q ) = TQ × T ¯ Q TT ¯ Q ≃ ( Q ⊕ T ¯ Q × g ) × T ¯ Q TT ¯ Q ≃ Q ⊕ TT ¯ Q × g , TVQ = T ( Q × g ) = TQ × T g ≃ Q ⊕ T ¯ Q × g × T g , (6.4)where × T ¯ Q is the Whitney sum on T ¯ Q . We present the following commutative diagram to show theprojections and decompositions in one look T HQQ ⊕ TT ¯ Q × g TTQQ ⊕ TT ¯ Q × g × T g TV QQ ⊕ T ¯ Q × g × T g HQQ ⊕ T ¯ Q TQQ ⊕ T ¯ Q × g VQQ × g τ H Q T h T T v T š τ T Q τ V Q h T v T where we employ the following mappings h T : Q ⊕ T ¯ Q × g −→ Q ⊕ T ¯ Q , ( q , u , ξ ) 7→ ( q , u ) , v T : Q ⊕ T ¯ Q × g −→ Q ⊕ g , ( q , u , ξ ) 7→ ( q , ξ ) , T h T : Q ⊕ TT ¯ Q × g × T g −→ Q ⊕ TT ¯ Q × g , ( q , U , ζ, ξ, η ) 7→ ( q , U , ζ ) , T v T : Q ⊕ TT ¯ Q × g × T g −→ Q ⊕ T ¯ Q × g × T g , ( q , U , ζ, ξ, η ) 7→ ( q , T τ ¯ Q ( U ) , ζ, ξ, η ) ,τ HQ : Q ⊕ TT ¯ Q × g −→ Q ⊕ T ¯ Q , ( q , U , ζ ) 7→ ( q , τ T ¯ Q ( U )) ,τ V Q : Q ⊕ T ¯ Q × g × T g −→ Q × g , ( q , T τ ¯ Q ( U ) , ζ, ξ, η ) 7→ ( q , ξ ) . (6.5)Recall the horizontal-vertical decomposition of the iterated tangent bundle (6.3) given as the Whitey sumof T HQ and
TVQ in (6.4). Then we have the reduced bundles
T HQ ≃ Q ⊕ TT ¯ Q × g = TT ¯ Q ⊕ Q × g = TT ¯ Q ⊕ ˜ g , TV Q ≃ Q ⊕ T ¯ Q × g × T g ≃ T ¯ Q ⊕ Q × g × g × g = T ¯ Q ⊕ ( ˜ g × ˜ g × ˜ g ) , (6.6)where ˜ g is the adjoint bundle. So that the horizontal-vertical decomposition of TTQ is written as(6.7)
TTQ = T HQ × T ¯ Q ⊕ ˜ g TVQ , where × T ¯ Q ⊕ ˜ g is the Whitney sum over TQ = T ¯ Q ⊕ ˜ g . Here are the projections τ HQ : T HQ −→ HQ , ( U , [ q , ξ ]) 7→ τ T ¯ Q U ,τ V Q : TVQ −→ V Q , ( u , [ q , ζ, ξ, η ]) 7→ [ q , ξ ] . (6.8) Horizontal-vertical decomposition of TT ∗ Q . In this case,(6.9) TT ∗ Q = T H ∗ Q × T ∗ Q TV ∗ Q , where × T ∗ Q denotes the Whitney sum over T ∗ Q . Here, the tangent bundles on the right hand side are T H ∗ Q = T ( Q ⊕ T ∗ ¯ Q ) = TQ × T ¯ Q TT ∗ ¯ Q ≃ ( Q ⊕ T ¯ Q × g ) × T ¯ Q TT ∗ ¯ Q ≃ Q ⊕ TT ∗ ¯ Q × g , TV ∗ Q = T ( Q × g ∗ ) = TQ × T g ∗ ≃ Q ⊕ T ¯ Q × g × T g ∗ . (6.10)Here is a diagram summarizing the situation T H ∗ QQ ⊕ TT ∗ ¯ Q × g TT ∗ QQ ⊕ TT ∗ ¯ Q × g × T g ∗ TV ∗ QQ ⊕ T ¯ Q × g × T g ∗ H ∗ QQ ⊕ T ∗ ¯ Q T ∗ QQ ⊕ T ∗ ¯ Q × g ∗ V ∗ QQ × g ∗ τ H ∗ Q T h T ∗ T v T ∗ › τ T ∗ Q τ V ∗ Q h T ∗ v T ∗ along with the mappings h T ∗ : Q ⊕ T ∗ ¯ Q × g ∗ −→ Q ⊕ T ∗ ¯ Q , ( q , y , µ ) 7→ ( q , y ) , v T ∗ : Q ⊕ T ∗ ¯ Q × g ∗ −→ Q ⊕ g ∗ , ( q , y , µ ) 7→ ( q , µ ) , T h T ∗ : Q ⊕ TT ∗ ¯ Q × g × T g ∗ −→ Q ⊕ TT ∗ ¯ Q × g , ( q , Y , ζ, µ, ν ) 7→ ( q , Y , ζ ) , T v T ∗ : Q ⊕ TT ∗ ¯ Q × g × T g ∗ −→ Q ⊕ T ¯ Q × g × T g ∗ , ( q , Y , ζ, µ, ν ) 7→ ( q , T π ¯ Q ( Y ) , ζ, µ, ν ) ,τ H ∗ Q : Q ⊕ TT ∗ ¯ Q × g −→ Q ⊕ T ∗ ¯ Q , ( q , Y , ζ ) 7→ ( q , τ T ∗ ¯ Q ( Y )) ,τ V ∗ Q : Q ⊕ T ¯ Q × g × T g ∗ −→ Q × g ∗ , ( q , T π ¯ Q ( Y ) , ζ, µ, ν ) 7→ ( q , µ ) . (6.11)The bundle TT ∗ Q has a projection to the tangent bundle TQ as well. Accordingly, horizontal-verticaldecompositions of these spaces read the following picture T H ∗ QQ ⊕ TT ∗ ¯ Q × g TT ∗ QQ ⊕ TT ∗ ¯ Q × g × T g ∗ TV ∗ QQ ⊕ T ¯ Q × g × T g ∗ HQQ ⊕ T ¯ Q TQQ ⊕ T ¯ Q × g VQQ × g T π H Q T h T ∗ T v T ∗ š T π Q T π V Q h T v T where we have referred h T and v T in (6.5), and T h T ∗ and T v T ∗ in (6.11) whereas T π HQ : Q ⊕ TT ∗ ¯ Q × g −→ Q ⊕ T ¯ Q , ( q , Y , ζ ) 7→ ( q , T π ¯ Q ( Y )) , T π V Q : Q ⊕ T ¯ Q × g × T g ∗ −→ Q × g , ( q , T π ¯ Q ( Y ) , ζ, µ, ν ) 7→ ( q , ζ ) , (6.12)where d T π Q is the trivialized projection given in (3.24), whereas T π ¯ Q is the canonical projection from TT ∗ ¯ Q to T ¯ Q . Observe that the image d T π Q ( q , Y , ζ, µ, ν ) under d T π Q is ( q , T π ¯ Q ( Y ) , ζ ) .The reduction of the decomposition of the bundle TT ∗ Q in (6.9) is(6.13) TT ∗ Q = T H ∗ Q T ∗ ¯ Q ⊕ ˜ g ∗ TV ∗ Q ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 39 where, in the light of the trivializations (6.10), we have that
T H ∗ Q ≃ Q ⊕ TT ∗ ¯ Q × g ≃ TT ∗ ¯ Q × Q × g ≃ TT ∗ ¯ Q × ˜ g , TV ∗ Q ≃ Q ⊕ T ¯ Q × g × T g ∗ ≃ T ¯ Q ⊕ Q × g × g ∗ × g ∗ ≃ T ¯ Q ⊕ ˜ V , (6.14)where ˜ V is identified with the product space ( ˜ g × ˜ g ∗ × ˜ g ∗ ) . The projections are τ H ∗ Q : T H ∗ Q −→ H ∗ Q , ( Y , [ q , ζ ]) 7→ τ T ∗ ¯ Q Y ,τ V ∗ Q : TV ∗ Q −→ V ∗ Q , ( y , [ q , ζ, µ, ν ]) 7→ [ q , µ ] . (6.15)6.2. Horizontal-Vertical Decomposition of the Second Order Cotangent Bundles.
In the previous section we have presented the tangent and cotangent bundles of horizontal-vertical decom-positions. In the present one, we compute the cotangent bundles by replacing each tangent space with itsdual. This approach is the one that we have pursued in Section 4 as well.
The bundle T ∗ HQ . The bundle ( T HQ , τ HQ , HQ ) is a vector bundle over the base HQ . According to (6.4),for any element ( q , u ) on the base manifold HQ ≃ Q ⊕ T ¯ Q , a fiber is isomorphic to T u T ¯ Q × g . In order tohave the dual of the bundle τ HQ , we first calculate the linear algebraic dual of each fiber(6.16) ( T u T ¯ Q × g ) ∗ = T ∗ u T ¯ Q × g ∗ . Then combining all the dual spaces, propose the cotangent(6.17) ( T ∗ HQ , π HQ , HQ ) , with the trivialization(6.18) T ∗ HQ ≃ Q ⊕ T ∗ T ¯ Q × g ∗ = {( q , K , µ ) : u = π T ¯ Q K } . and the projection(6.19) π HQ : T ∗ HQ ≃ Q ⊕ T ∗ T ¯ Q × g ∗ −→ HQ ≃ Q ⊕ T ¯ Q , ( q , K , µ ) 7→ ( q , π T ¯ Q ( K )) . The bundle T ∗ V Q . Similar to the previous case, the trivialized vertical bundle
TVQ in (6.4) defines avector bundle over the bundle
V Q ≃ Q × g . Here, for any element ( q , ξ ) on the base manifold VQ , a fiber isisomorphic to T [ q ] ¯ Q × g × T ξ g . So that in order to find the trivialization of the cotangent bundle T ∗ VQ wesimply take the linear algebraic dual of each fiber as(6.20) ( T [ q ] ¯ Q × g × T ξ g ) ∗ = T ∗[ q ] ¯ Q × g ∗ × T ∗ ξ g , where [ q ] is the equivalence class of the element q in Q generated by the action (2.21). Accumulating allthese dual spaces, we shall propose the following bundle structure(6.21) ( T ∗ V Q , π
V Q , VQ ) , where the total space T ∗ V Q has the following trivialization(6.22) T ∗ V Q : = Q ⊕ T ∗ ¯ Q × g ∗ × T ∗ g : = {( q , k , µ, ξ, ν ) : [ q ] = π ¯ Q ( k )} , whereas the projection is(6.23) π VQ : T ∗ V Q ≃ Q ⊕ T ∗ ¯ Q × g ∗ × T ∗ g −→ VQ ≃ Q × g , ( q , k , µ, ξ, ν ) 7→ ( q , ξ ) . Horizontal-vertical decomposition of T ∗ TQ . We sum the discussion by recording the following horizontal-vertical decomposition(6.24) T ∗ TQ = T ∗ VQ × T ∗ Q T ∗ HQ . For this decomposition, to see the horizontal π HQ (6.19) and the vertical π VQ (6.23) parts of the trivializedmapping d π T Q in (4.2), we consider the following diagram T ∗ HQQ ⊕ T ∗ T ¯ Q × g ∗ T ∗ TQQ ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g T ∗ VQQ ⊕ T ∗ ¯ Q × g ∗ × T ∗ g HQQ ⊕ T ¯ Q TQQ ⊕ T ¯ Q × g V QQ × g π H Q h T ∗ T v T ∗ T š π T Q π V Q h T v T where we employ mappings h T ∗ T : Q ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g −→ Q ⊕ T ∗ T ¯ Q × g ∗ , ( q , K , µ, ξ, ν ) 7→ ( q , K , µ ) , v T ∗ T : Q ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g −→ Q ⊕ T ∗ ¯ Q × g ∗ × T ∗ g , ( q , K , µ, ξ, ν ) 7→ ( q , τ T ∗ ¯ Q ◦ α − Q ( K ) , µ, ξ, ν ) . (6.25)Here, π HQ is in (6.19) and π V Q is in (6.23) are the horizontal and the vertical parts of (4.2).By reducing the bundle T ∗ HQ in (6.18) and the bundle T ∗ VQ in (6.22) we arrive at T ∗ HQ ≃ Q ⊕ T ∗ T ¯ Q × g ∗ ≃ T ∗ T ¯ Q × Q × g ∗ = T ∗ T ¯ Q ⊕ ˜ g ∗ , T ∗ VQ ≃ Q ⊕ T ∗ ¯ Q × g ∗ × T ∗ g ≃ T ∗ ¯ Q ⊕ Q × g ∗ × g × g ∗ = T ∗ ¯ Q ⊕ ˜ V , (6.26)where ˜ g is the adjoint bundle in (2.30), whereas ˜ g ∗ is the coadjoint bundle in (2.31). In here, ˜ V is identifiedwith the product space ( ˜ g ∗ × ˜ g × ˜ g ∗ ) . Notice the following projections π HQ : T ∗ HQ −→ HQ , ( K , [ q , µ ]) 7→ π T ¯ Q K ,π V Q : T ∗ VQ −→ VQ , ( k , [ q , µ, ξ, ν ]) 7→ [ q , ξ ] , (6.27) The bundle T ∗ H ∗ Q . We have obtained the trivialization of the tangent bundle
T H ∗ Q in (6.10). For arbitraryelement ( q , y ) in H ∗ Q ≃ Q ⊕ T ¯ Q , the fiber is T y T ∗ ¯ Q × g . By taking the dual of each fiber(6.28) ( T y T ∗ ¯ Q × g ) ∗ = T ∗ y T ∗ ¯ Q × g ∗ and then by collecting all the dual spaces, we arrive at the trivialization of the total of(6.29) ( T ∗ H ∗ Q , π H ∗ Q , H ∗ Q ) , ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 41 given by(6.30) T ∗ H ∗ Q : = Q ⊕ T ∗ T ∗ ¯ Q × g ∗ = {( q , L , µ ) : y = π T ∗ ¯ Q ( L )} . In here, the projection π H ∗ Q is the horizontal part of (4.14) that is(6.31) π H ∗ Q : T ∗ H ∗ Q ≃ Q ⊕ T ∗ T ∗ ¯ Q × g ∗ −→ H ∗ Q ≃ Q ⊕ T ∗ ¯ Q , ( q , L , µ ) 7→ ( q , π T ∗ ¯ Q ( L )) . The bundle T ∗ V ∗ Q . Recall the tangent bundle TV ∗ Q in (6.10) over the base manifold V ∗ Q ≃ Q × g ∗ . For ( q , µ ) in V ∗ Q , the fiber is T [ q ] ¯ Q × g × T µ g ∗ . whose dual is computed to be(6.32) ( T [ q ] ¯ Q × g × T µ g ∗ ) ∗ = T ∗[ q ] ¯ Q × g ∗ × T ∗ µ g ∗ . Gathering all these dual spaces together, we arrive at the following bundle structure(6.33) ( T ∗ V ∗ Q , π V ∗ Q , V ∗ Q ) , where the total space of the bundle is displayed(6.34) T ∗ V ∗ Q : = Q ⊕ T ∗ ¯ Q × g ∗ × T ∗ g ∗ = {( q , l , γ, µ, ξ ) : [ q ] = π ¯ Q ( l )} , whereas the projection π V ∗ Q is the vertical part of (4.14) defined by(6.35) π V ∗ Q : T ∗ V ∗ Q −→ V ∗ Q , ( q , l , γ, µ, ξ ) 7→ ( q , µ ) . Horizontal-vertical decomposition of T ∗ T ∗ Q . according to the previous definitions of T ∗ H ∗ Q in (6.30)and T ∗ V ∗ Q in (6.34) we have that(6.36) T ∗ T ∗ Q = T ∗ V ∗ Q × T ∗ Q T ∗ H ∗ Q . along with the following diagram T ∗ H ∗ QQ ⊕ T ∗ T ∗ ¯ Q × g ∗ T ∗ T ∗ QQ ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ T ∗ V ∗ QQ ⊕ T ∗ ¯ Q × g ∗ × T ∗ g ∗ H ∗ QQ ⊕ T ∗ ¯ Q T ∗ QQ ⊕ T ∗ ¯ Q × g ∗ V ∗ QQ × g ∗ π H ∗ Q h T ∗ T ∗ v T ∗ T ∗ › π T ∗ Q π V ∗ Q h T ∗ v T ∗ where we employ mappings h T ∗ T ∗ : Q ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ −→ Q ⊕ T ∗ T ∗ ¯ Q × g ∗ , ( q , L , γ, µ, ξ ) 7→ ( q , L , µ ) , v T ∗ T ∗ : Q ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ −→ Q ⊕ T ∗ ¯ Q × g ∗ × T ∗ g ∗ , ( q , L , γ, µ, ξ ) 7→ ( q , π T ∗ ¯ Q ( L ) , γ, µ, ξ ) . (6.37)The reduction of the decomposition in (6.36) is(6.38) T ∗ T ∗ Q = T ∗ V ∗ Q × T ∗ ¯ Q ⊕ ˜ g ∗ T ∗ H ∗ Q . where × T ∗ ¯ Q ⊕ ˜ g ∗ is the Whitney sum over T ∗ ¯ Q ⊕ ˜ g ∗ . Here the reduced spaces are T ∗ H ∗ Q = T ∗ H ∗ Q / G = Q ⊕ T ∗ T ∗ ¯ Q × g ∗ ≃ T ∗ T ∗ ¯ Q ⊕ Q × g ∗ ≃ T ∗ T ∗ ¯ Q ⊕ ˜ g ∗ , T ∗ V ∗ Q = T ∗ V ∗ Q / G = Q ⊕ T ∗ ¯ Q × g ∗ × T ∗ g ∗ ≃ T ∗ ¯ Q ⊕ Q × g ∗ × g ∗ × g ≃ T ∗ ¯ Q ⊕ ˜ V , (6.39)where ˜ g ∗ is the coadjoint bundle, and ˜ V is identified with the product space ˜ g ∗ ⊕ ˜ g ∗ ⊕ ˜ g . The projections arecomputed to be π H ∗ Q : T ∗ H ∗ Q −→ H ∗ Q , ( L , [ q , µ ]) 7→ π T ∗ ¯ Q ( L ) π V ∗ Q : T ∗ V ∗ Q −→ V ∗ Q , ( l , [ q , γ, µ, ξ ]) 7→ [ q , µ ] . (6.40)6.3. Horizontal-Vertical Decomposition of the Symplectomorphisms.
By referring to all of done in this section, we now present horizontal-vertical decompositions of thesymplectomorphisms α Q and Ω ♭ Q accommodated in the Tulczyjew’s triplet. Horizontal-vertical decomposition of α Q . Recall the symplectomorphism α Q from TT ∗ Q to T ∗ TQ , andits trivialization c α Q in (5.8). Then referring to the horizontal-vertical decomposition of TT ∗ Q in (6.9), andthe horizontal-vertical decomposition of T ∗ TQ in (6.24), we plot the following diagram(6.41) T H ∗ QQ ⊕ TT ∗ ¯ Q × g TT ∗ QQ ⊕ TT ∗ ¯ Q × g × T g ∗ TV ∗ QQ ⊕ T ¯ Q × g × T g ∗ T ∗ HQQ ⊕ T ∗ T ¯ Q × g ∗ T ∗ TQQ ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g T ∗ V QQ ⊕ T ∗ ¯ Q × g ∗ × T ∗ g α H T h T ∗ T v T ∗ d α Q α V h T ∗ T v T ∗ T where we have employed the mappings T h T ∗ and T v T ∗ in (6.11), and the mappings h T ∗ T and v T ∗ T in (6.25).Assuming the commutativity of this diagram, we have the following decomposition(6.42) α H ( q , Y , ζ ) = ( q , α ¯ Q ( Y ) , ν ) , α V ( q , T π ¯ Q ( Y ) , ζ, µ, ν ) = ( q , τ T ∗ ¯ Q ( Y ) , ν, ξ, µ ) . Reductions of α H and α V are(6.43) α H ( Y , [ q , ζ ]) = ( α ¯ Q ( Y ) , [ q , ν ]) , α V ( T π ¯ Q ( Y ) , [ q , ζ, µ, ν ]) = ( τ T ∗ ¯ Q ( Y ) , [ q , ν, ξ, µ ]) . ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 43
Horizontal-vertical decomposition of c Ω Q ♭ . Recall now the trivialized symplectomorphism c Ω Q ♭ in (5.11)by using the following diagram, we plot(6.44) T H ∗ QQ ⊕ TT ∗ ¯ Q × g TT ∗ QQ ⊕ TT ∗ ¯ Q × g × T g ∗ TV ∗ QQ ⊕ T ¯ Q × g × T g ∗ T ∗ H ∗ QQ ⊕ T ∗ T ∗ ¯ Q × g ∗ T ∗ T ∗ QQ ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ T ∗ V ∗ QQ ⊕ T ∗ ¯ Q × g ∗ × T ∗ g ∗ Ω ♭ H T h T ∗ T v T ∗ d Ω Q ♭ Ω ♭ V h T ∗ T ∗ v T ∗ T ∗ where we have referred the mappings h T ∗ T ∗ and v T ∗ T ∗ in (6.37) and the mappings T h T ∗ and T v T ∗ in (6.11).So that we have the following decomposition Ω ♭ H ( q , Y , ζ ) = ( q , Ω ♭ ¯ Q ( Y ) − π ∗ ¯ Q B ♭µ ( Y ) , µ ) , Ω ♭ V ( q , T π ¯ Q ( Y ) , ζ, µ, ν ) = ( q , π T ∗ ¯ Q ( Ω ♭ ¯ Q ( Y ) − π ∗ ¯ Q B ♭µ ( Y )) , ad ∗ ζ µ − ν, µ, ζ ) . (6.45)Reductions of Ω ♭ H and Ω ♭ V are Ω ♭ H ( Y , [ q , ζ ]) = ( Ω ♭ ¯ Q ( Y ) − π ∗ ¯ Q B ♭µ ( Y ) , [ q , µ ]) , Ω ♭ V ( T π ¯ Q ( Y ) , [ q , ζ, µ, ν ]) = ( π T ∗ ¯ Q ( Ω ♭ ¯ Q ( Y ) − π ∗ ¯ Q B ♭µ ( Y )) , [ q , ad ∗ ζ µ − ν, µ, ζ ]) . (6.46)By merging the horizontal-vertical diagrams (6.41) and (6.44), the horizontal-vertical decompositions ofthe trivialized Tulczyjew’s triplet in (5.30) is(6.47) T ∗ VQQ ⊕ T ∗ ¯ Q × g ∗ × T ∗ g TV ∗ QQ ⊕ T ¯ Q × g × T g ∗ T ∗ V ∗ QQ ⊕ T ∗ ¯ Q × g ∗ × T ∗ g ∗ T ∗ TQQ ⊕ T ∗ T ¯ Q × g ∗ × T ∗ g TT ∗ QQ ⊕ TT ∗ ¯ Q × g × T g ∗ T ∗ T ∗ QQ ⊕ T ∗ T ∗ ¯ Q × g ∗ × T ∗ g ∗ T ∗ HQQ ⊕ T ∗ T ¯ Q × g ∗ T H ∗ QQ ⊕ TT ∗ ¯ Q × g T ∗ H ∗ QQ ⊕ T ∗ T ∗ ¯ Q × g ∗ Ω ♭ V α V v T ∗ T h T ∗ T d Ω Q ♭ d α Q T v T ∗ T h T ∗ v T ∗ T ∗ h T ∗ T ∗ Ω ♭ H α H where we use the mappings α V and α H given in (6.42), Ω ♭ V and Ω ♭ H given in (6.45), whereas we employ themaappings v T ∗ T and h T ∗ T given in (6.25), v T ∗ T ∗ and h T ∗ T ∗ given in (6.37), T v T ∗ and T h T ∗ given in (6.11).In here, c α Q in (5.8) and c Ω Q ♭ in (5.11) are trivialized symplectomorphisms. By reducing each term of (5.30) under the group action G , the horizontal-vertical decompositions of thereduced Tulczyjew’s triplet in (5.31) is(6.48) T ∗ VQT ∗ ¯ Q ⊕ ˜ V TV ∗ QT ∗ ¯ Q ⊕ ˜ V T ∗ V ∗ QT ∗ ¯ Q ⊕ ˜ V T ∗ TQT ∗ T ¯ Q ⊕ ˜ V TT ∗ QTT ∗ ¯ Q ⊕ ˜ V T ∗ T ∗ QT ∗ T ∗ ¯ Q ⊕ ˜ V T ∗ HQT ∗ T ¯ Q ⊕ ˜ g ∗ T H ∗ QTT ∗ ¯ Q ⊕ ˜ g T ∗ H ∗ QT ∗ T ∗ ¯ Q ⊕ ˜ g ∗ Ω ♭ V α V v T ∗ T h T ∗ T Ω ♭ Q α Q T v T ∗ T h T ∗ v T ∗ T ∗ h T ∗ T ∗ Ω ♭ H α H where we emlpoy the mappings α V and α H given in (6.43), Ω ♭ V and Ω ♭ H given in (6.46). Here, the reducedversions of T h T ∗ and T v T ∗ in (6.11), h T ∗ T and v T ∗ T in (6.25), h T ∗ T ∗ and v T ∗ T ∗ in (6.37) are T h T ∗ : TT ∗ ¯ Q ⊕ ˜ V −→ TT ∗ ¯ Q ⊕ ˜ g , ( Y , [ q , ζ, µ, ν ]) 7→ ( Y , [ q , ζ ]) , T v T ∗ : TT ∗ ¯ Q ⊕ ˜ V −→ T ¯ Q ⊕ ˜ V , ( Y , [ q , ζ, µ, ν ]) 7→ ( T π ¯ Q ( Y ) , [ q , ζ, µ, ν ]) , h T ∗ T : T ∗ T ¯ Q ⊕ ˜ V −→ T ∗ T ¯ Q ⊕ ˜ g ∗ , ( K , [ q , µ, ξ, ν ]) 7→ ( K , [ q , µ ]) , v T ∗ T : T ∗ T ¯ Q ⊕ ˜ V −→ T ∗ ¯ Q ⊕ T ∗ T ¯ Q ⊕ ˜ V , ( K , [ q , µ, ξ, ν ]) 7→ ( τ T ∗ ¯ Q ◦ α − Q ( K ) , [ q , µ, ξ, ν ]) , h T ∗ T ∗ : T ∗ T ∗ ¯ Q ⊕ ˜ V −→ T ∗ T ∗ ¯ Q ⊕ ˜ g ∗ , ( L , [ q , γ, µ, ξ ]) 7→ ( L , [ q , µ ]) , v T ∗ T ∗ : T ∗ T ∗ ¯ Q ⊕ ˜ V −→ T ∗ ¯ Q ⊕ ˜ V , ( L , [ q , γ, µ, ξ ]) 7→ ( π T ∗ ¯ Q ( L ) , [ q , γ, µ, ξ ]) , (6.49)respectively.
7. Conclusion and Future Work
In this note, we have presented both the trivialization and the reduction of the Tulczyjew’s triplet inthe presence of a proper Ehresmann connection. More precisely, we obtained the trivializations and thereductions of the iterated tangent bundles in Section 3, as well as the trivializations and the reductions of theiterated cotangent bundles in Section 4. The trivializations and the reductions of the rest of the ingredientsof the Tulczyjew’s triplet; such as the symplectic forms, the symplectomorphisms, and the projectionshave been established in Section 5 to obtain the trivialized Tulczyjew’s triplet in (5.30) and the reducedTulczyjew’s triplet in (5.31). Finally, the horizontal-vertical decompositions of these triplets have beengiven in (6.47) and (6.48), respectively. This will be the starting point of the upcoming paper [20] where weshall show that the Legendre transformation of (possibly singular) horizontal-vertical Lagrange-Poincarésystems introduced in [9, 10, 47].
ULCZYJEW’S TRIPLET WITH AN EHRESMANN CONNECTION I: TRIVIALIZATION AND REDUCTION 45
This present paper establishes the necessary geometric framework, while the next one will implement thisgeometry to the Hamilton-Poincaré and the Lagrange-Poincaré equations. Accordingly, we shall discussthe trivialization and the reductions of Morse families, generating functions, as well as the Lagrangiansubmanifolds under symmetry. As a result, it will be possible to consider the Legendre transformation evenfor the singular Lagrangian/Hamiltonian dynamics under symmetries.The higher order versions of the Lagrange–Poincaré and the Hamilton–Poincaré reductions are presented in[25], see also [24]. As an independent but a related research, we find it interesting to study trivialized andreduced Tulczyjew’s triplets for the higher order reduced dynamics as well.
8. Acknowledgment
Authors are extremely grateful Prof. Hasan Gümral for his guidance and for sharing his expertise generouslyboth on the field of Geometric Mechanics and, in particular, on the Tulczyjew’s triplets. MK gratefullyacknowledge Gebze Technical University and the mathematics department, and especially the chairmanProf. Mansur İsmailov (İsgenderoğlu), for providing an excellent research environment.This work is supported by TUBITAK (the Scientific and Technological Research Council of Turkey) with apost-doctoral grant number 2218.
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