Lagrangian reduction of nonholonomic discrete mechanical systems by stages
aa r X i v : . [ m a t h . DG ] S e p LAGRANGIAN REDUCTION OF NONHOLONOMIC DISCRETEMECHANICAL SYSTEMS BY STAGES
JAVIER FERN ´ANDEZ, CORA TORI, AND MARCELA ZUCCALLI
Abstract.
In this work we introduce a category
LDP d of discrete-time dy-namical systems, that we call discrete Lagrange–D’Alembert–Poincar´e sys-tems, and study some of its elementary properties. Examples of objectsof LDP d are nonholonomic discrete mechanical systems as well as their la-grangian reductions and, also, discrete Lagrange-Poincar´e systems. We alsointroduce a notion of symmetry group for objects of LDP d and a process ofreduction when symmetries are present. This reduction process extends thereduction process of discrete Lagrange–Poincar´e systems as well as the onedefined for nonholonomic discrete mechanical systems. In addition, we provethat, under some conditions, the two-stage reduction process (first by a closedand normal subgroup of the symmetry group and, then, by the residual sym-metry group) produces a system that is isomorphic in LDP d to the systemobtained by a one-stage reduction by the full symmetry group. Introduction
Mechanical systems are dynamical systems that are used to model a wide varietyof aspects of the real world, from the falling apple to the movement of astronomicalobjects, including machinery and billiards (see, for instance, [17] and [1]). One ofthe flavors of Mechanics —Lagrangian or Variational Mechanics— describes theevolution of a mechanical system using a variational principle defined in terms ofa function, the Lagrangian, L : T Q → R , where Q is the configuration manifold ofthe system. Nonholonomic mechanical systems, which describe systems containingrolling or sliding contact (such as wheels or skates), add constraints —in the formof a non-integrable subbundle D ⊂
T Q — to the variational principle (see, forinstance, [3] and [6]).
Numerical integrators and discrete mechanical systems.
As in many applications itis essential to predict the evolution of a mechanical system, the equations of motionthat can be derived from the corresponding variational principle must be solved.Solving these ordinary differential equations can be quite difficult in practice, sonumerical integrators are used to find approximate solutions to those equations.The standard methods for numerically approximating solutions of ODEs do notnecessarily preserve the structural characteristics of the solutions of the equations
Mathematics Subject Classification.
Primary: 37J15, 70G45; Secondary: 70G75.
Key words and phrases.
Geometric mechanics, discrete nonholonomic mechanical systems,symmetry and reduction.This research was partially supported by grants from Universidad Nacional de Cuyo ( of motion of mechanical systems (see [18]). Discrete mechanical systems were in-troduced as a way of modeling discrete-time analogues of mechanical systems; theevolution of a discrete mechanical system is also defined in terms of a variationalprinciple for the discrete Lagrangian L d : Q × Q → R ; this formalism is extended todeal with more general systems, including forced discrete systems as well as discretenonholonomic ones (see [24] and [9]). The equations of motion of discrete mechan-ical systems are algebraic equations whose solutions are numerical integrators forthe corresponding continuous system. In many cases, these integrators have verygood structural characteristics (especially when considering long-time evolution),that resemble those of the continuous system ([29] and [18]). Symmetries and symmetry reduction.
It is a natural idea to think that when amechanical or, more generally, a dynamical system has some degree of symmetry,it should be possible to gain some insight into its dynamics by studying some other“simplified system” obtained by eliminating or locking the symmetry. This processis usually known as the reduction of the given system and the resulting system isknown as the reduced system . In the case of Classical Mechanics, this idea seems togo back as far as the work of Lagrange. Over time, it has become a technique thathas been applied in both the Lagrangian and Hamiltonian formalisms, for uncon-strained systems as well as for holonomically and nonholonomically constrained ones(see among many other references, [3] [6], [2], [30, 31] [28], [25], [26], [4] and [23]).The reduction process has also been applied to discrete-time mechanical systemswith and without constraints (see, for instance, [21], [27], [20] and [13]).It is well known that, in most instances, the reduction of a mechanical systemis not a mechanical system but, rather, a more general dynamical system: thatis, while the dynamics of a mechanical system on Q is defined using a variationalprinciple for the Lagrangian, defined on T Q in the continuous case or on Q × Q inthe discrete case (and, maybe, other additional data), the dynamics of the reducedsystem is determined by a function that is usually not defined on a tangent bundleor a Cartesian product (of a manifold with itself). This can be problematic if oneexpects to analyze the reduced system with the same techniques as the originalone. That issue has usually been solved by passing from the family of mechanicalsystems to a larger class of dynamical systems, where there is a reduction processthat is closed within this larger class. Such is the case, for example, of mechanicalsystems on Lie algebroids and Lie groupoids (see [19] and [20]). In this paper wefollow this guiding principle, but choose the larger class following ideas adaptedfrom [7] and [14]. Reduction by stages.
Sometimes, it may be convenient to eliminate part of thesymmetric behavior of a mechanical system, while keeping some residual symmetrythat could be analyzed at a later point, if so desired. In this case a second reductionstep to eliminate the residual symmetry is possible. A natural question is, inthat case, whether the result of this two-stage reduction is equivalent to the fullreduction of all the symmetries at one time. The equivalence of the two-stage andone-stage reduction processes has been established in several cases. For instance, forLagrangian systems without constraints by H. Cendra, J. Marsden and T. Ratiuin [7], for Lagrangian systems with nonholonomic constraints by H. Cendra andV. D´ıaz in [5], for Hamiltonian systems by J. Marsden et al. in [26] and, forunconstrained discrete mechanical systems by the authors in [14].
ONHOLONOMIC DISCRETE LAGRANGIAN REDUCTION BY STAGES 3
Aims.
The main purpose of the present work is to establish the same equivalencedescribed in the previous paragraph for nonholonomically constrained discrete-timemechanical systems. In this respect, the paper is an extension of [14] to the con-strained setting that parallels [5] in the discrete-time context. We stress that, forthe present paper, whether a discrete-time system is related to a (continuous-time)mechanical system or not is irrelevant; in this respect, our analysis and results arecompletely independent of the discretization process chosen to produce the discretedynamical system in question, if one was used at all. As an aside, we also mentionthat, at the moment, the very interesting subject of geometric discretization of non-holonomic mechanical systems should be regarded as “work in progress”, withoutconclusive hard results on the quality of the numerical integrators obtained.
Constructions and results.
Except for a few special cases, the reduced systemobtained from a nonholonomic discrete mechanical system via the general reductionprocess defined in [13] is not a discrete mechanical system, constrained or not. So, aswe mentioned above, the first step is to construct a family of dynamical systems thatcontains all the systems of interest —nonholonomic discrete mechanical systemsas well as their reductions. The discrete Lagrange–D’Alembert–Poincar´e systems(DLDPSs) form such a family: one of these systems is determined by a fiber bundle φ : E → M , a function L d : E × M → R , the discrete Lagrangian , a nonholonomicinfinitesimal variation chaining map P (see Definition 3.3) as well as a regularsubmanifold D d ⊂ E × M , the kinematic constraints , and a subbundle D ⊂ p ∗ T E (where p : E × M → E is the projection), the variational constraints . All suchsystems are discrete-time dynamical systems whose trajectories are determined bya variational principle. Examples of DLDPSs include the nonholonomic discretemechanical systems (Example 3.9) and, when they are symmetric, their reductionsby the procedure defined in [13] (Section 3.2); also, the discrete Lagrange–Poincar´esystems considered in [14] are DLDPSs. A convenient notion of morphism betweenDLDPSs is introduced and a category LDP d is so defined. The category LP d ofdiscrete Lagrange–Poincar´e systems defined in [14] is a full subcategory of LDP d .Roughly speaking, a Lie group G is a symmetry group of a DLDPS if it acts onthe underlying fiber bundle in such a way that it preserves the different structures.When G is a symmetry group of a DLDPS M , we construct a new DLDPS M /G that we call the reduced system . In fact, the construction requires an additionalpiece of data: an affine discrete connection on a certain principal G -bundle; inter-estingly, we prove that the reduced systems obtained using different affine discreteconnections are always isomorphic in LDP d (Proposition 5.14). Also, the reductionmapping M → M /G is a morphism in LDP d ; Corollary 5.16 and Theorem 5.17prove that the reduction mapping determines a bijective correspondence betweenthe trajectories of M and those of M /G . It is important to notice that both thenotion of symmetry group and the reduction process extend the ones already inuse for nonholonomic discrete mechanical systems as well as for discrete Lagrange–Poincar´e systems.When G is a symmetry group of the DLDPS M and H ⊂ G is a closed andnormal subgroup, H is a symmetry group of M , so we can consider the reducedsystem M H := M /H using a discrete affine connection A Hd . Then, under a con-dition on A Hd , we prove that G/H is a symmetry group of M H , so that we can J. FERN´ANDEZ, C. TORI, AND M. ZUCCALLI consider a new reduced system M G/H := M H / ( G/H ). One of the main results ofthe paper, Theorem 6.6, is that M G/H is isomorphic in
LDP d to M /G . Plan for the paper.
Section 2 reviews the notion of affine discrete connection aswell as some basic results on principal bundles. Section 3 introduces the DLDPSsand their dynamics. Section 3.2 shows that both nonholonomic discrete mechanicalsystems as well as their reduction (in the sense of [13]) are examples of DLDPS and,also, that their dynamics as DLDPSs is the same as the “classical one”. Section 4introduces the category
LDP d whose objects are DLDPSs. Symmetries and areduction process in LDP d are analyzed in Section 5; in particular, in Section 5.4,we illustrate how these ideas can be applied by studying the discrete LL systems ona Lie group G . Finally, Section 6 establishes the equivalence between the two-stageand the single-stage reduction process, under appropriate conditions. Future work.
It would be very interesting to connect the analysis of this paper witha discretization process for continuous mechanical systems. This would allow, forinstance, the estimation of the error made when using a DLDPS as an approxima-tion of a (continuous) mechanical system. Indeed, a first step would be to tackle thissame problem with no constraints, that is, for discrete Lagrange–Poincar´e systems([14]). It should be noted that this error analysis is only known for unconstrainedsystems (see [29]) and forced mechanical systems (see [10] and [12]). Another av-enue for exploration would be the study of possible Poisson structures in DLDPSs:even though DLDPSs do not have a canonical Poisson structure, some of themdo (those coming from discrete mechanical systems, for instance) and it would beinteresting to see how those structures behave under the reduction process.
Notation.
Throughout the paper many spaces are Cartesian products. In generalwe denote the corresponding projections by p k : Q Nj =1 X j → X k and the obviousadaptations. Also, l X and r X will denote left and right smooth actions of a Liegroup on the manifold X . If G acts on the left on X we denote the correspondingquotient map by π X,G : X → X/G .2.
Revision of some discrete tools
In this section we review some basic notions and results about affine discreteconnections and smooth fiber bundles.2.1.
Affine discrete connections.
Let l Q : G × Q → Q be a smooth left actionof the Lie group G on the manifold Q . We consider several other actions of G ; forexample, we have the G actions l Q × Q and l Q × Q on Q × Q defined by l Q × Qg ( q , q ) :=( l Qg ( q ) , l Qg ( q )) and l Q × Q g ( q , q ) := ( q , l Qg ( q )). We also consider the left G -actionon itself given by l Gg ( g ′ ) := gg ′ g − . Definition 2.1.
Let γ : Q → G be a smooth G -equivariant map with respectto l Q and l G , Γ := { ( q, l Qγ ( q ) ( q )) : q ∈ Q } and Hor ⊂ Q × Q be an l Q × Q -invariantsubmanifold containing Γ. We say that Hor defines an affine discrete connection A d on the principal G -bundle π Q,G : Q → Q/G if ( id Q × π Q,G ) | Hor : Hor → Q × ( Q/G )is an injective local diffeomorphism. We denote
Hor by Hor A d and we call γ thelevel of A d . As in this paper the only type of discrete connection that we consideris the affine, we will simply call them discrete connections . ONHOLONOMIC DISCRETE LAGRANGIAN REDUCTION BY STAGES 5
Given a discrete connection A d on π Q,G : Q → Q/G , the space U := l Q × Q G ( Hor A d ) = { ( q , l Qg ( q )) ∈ Q × Q : ( q , q ) ∈ Hor A d , g ∈ G } , is called the domain of A d . Proposition 2.2.
The space U is an open set in Q × Q .Proof. See point 1 of Proposition 2.4 in [15]. (cid:3)
Proposition 2.3.
Let A d be a discrete connection with level γ and domain U onthe principal G -bundle π Q,G : Q → Q/G . For each ( q , q ) ∈ U , there is a unique g ∈ G such that ( q , l Qg − ( q )) ∈ Hor A d .Proof. See Proposition 2.5 in [15]. (cid:3)
Definition 2.4.
Given a discrete connection A d with domain U on the principal G -bundle π Q,G : Q → Q/G , we define its discrete connection form A d : U ⊂ Q × Q → G by A d ( q , q ) := g, where g ∈ G is the element that appears in Proposition 2.3.In what follows we consider the open set U ′ := ( id × π Q,G )( Hor A d ) ⊂ Q × ( Q/G ). Definition 2.5.
Let A d be a discrete connection on the principal G -bundle π Q,G : Q → Q/G . The discrete horizontal lift h d : U ′ → Hor A d is the inverse map of theinjective local diffeomorphism ( id Q × π Q,G ) | Hor A d : Hor A d → U ′ . That is h q d ( r ) = h d ( q , r ) := ( q , q ) ⇔ ( q , q ) ∈ Hor A d and π Q,G ( q ) = r . In addition we define h q d := p ◦ h q d . Proposition 2.6.
Let A d be a discrete connection on the principal G -bundle π Q,G : Q → Q/G . Then,(1) the discrete connection form A d and the discrete horizontal lift h d aresmooth maps and,(2) if we consider the left G -actions on G and on Q × ( Q/G ) given by l G and l Q × ( Q/G ) g ( q , r ) := ( l Qg ( q ) , r ) , and the diagonal action l Q × Q on Q × Q then A d and h d are G -equivariant.(3) In general, for any g , g ∈ G , (2.1) A d ( l Qg ( q ) , l Qg ( q )) = g A d ( q , q ) g − for all ( q , q ) ∈ U . Proof.
All of the following references are from [15] and must be adapted to affinediscrete connections. Point 1 is Lemma 3.2 (smoothness of A d ) and Point 2 inTheorem 4.4 (smoothness of h d ). Point 3 is part of Theorem 3.4 while Point 2follows from point 3 just proved and point 2 in Theorem 4.4. (cid:3) Proposition 2.7.
Given a smooth function A : Q × Q → G such that (2.1) holds(with A instead of A d ), then Hor := { ( q , q ) ∈ Q × Q : A ( q , q ) = e } definesan affine discrete connection with level set γ ( q ) := A ( q, q ) − and whose discreteconnection -form is A .Proof. This proof is analogue to the proof of Proposition 4.12 in [13]. (cid:3)
J. FERN´ANDEZ, C. TORI, AND M. ZUCCALLI
Principal bundles.
Here we review a few basic notions and results on prin-cipal bundles. We refer to Section 9 of [14] and its references for additional details.
Definition 2.8.
Let G be a Lie group and ( E, M, φ, F ) a fiber bundle. We saythat G acts on the fiber bundle E if there are free left G -actions l E and l M on E and M respectively and a right G -action r F on F such that(1) l M induces a principal G -bundle structure π M,G : M → M/G ,(2) φ is a G -equivariant map for the given actions,(3) for every m ∈ M there is a trivializing chart ( U, Φ U ) of E such that U ⊂ M is G -invariant, m ∈ U and, when considering the left G -action l U × F on U × F given by l U × Fg ( m, f ) := ( l Mg ( m ) , r Fg − ( f )), the map Φ U is G -equivariant. Remark 2.9.
When a Lie group G acts on the fiber bundle ( E, M, φ, F ) and onthe manifold F ′ by a right action, it is possible to construct an associated bundle on M/G with total space ( E × F ′ ) /G and fiber F × F ′ . The special case when F ′ = G acting on itself by r g ( h ) := g − hg is known as the conjugate bundle and isdenoted by e G E . Proposition 2.10.
Let G be a Lie group that acts on the fiber bundle ( E, M, φ, F ) and A d be a discrete connection on the principal G -bundle π M,G : M → M/G . Wedefine e Φ A d : E × M → E × G × ( M/G ) and e Ψ A d : E × G × ( M/G ) → E × M by e Φ A d ( ǫ, m ) := ( ǫ, A d ( φ ( ǫ ) , m ) , π M,G ( m )) and e Ψ A d ( ǫ, w, r ) := ( ǫ, l Mw ( h φ ( ǫ ) d ( r ))) . Then, e Φ A d and e Ψ A d are smooth functions, inverses of each other. If we view E × M and E × G × ( M/G ) as fiber bundles over M via φ ◦ p , then e Φ A d and e Ψ A d arebundle maps (over the identity). In addition, if we consider the left G -actions l E × M and l E × M × ( M/G ) defined by l E × Mg ( ǫ, m ) := ( l Eg ( ǫ ) , l Mg ( m )) and l E × M × ( M/G ) g ( ǫ, w, r ) := ( l Eg ( ǫ ) , l Gg ( w ) , r ) , then e Φ A d and e Ψ A d are G -equivariant and they induce diffeomorphisms Φ A d : ( E × M ) /G → e G E × ( M/G ) and Ψ A d : e G E × ( M/G ) → ( E × M ) /G .Proof. This is Proposition 2.6 in [14] adapted to affine discrete connections. (cid:3)
Remark 2.11.
The discrete connection A d need not be defined on Q × Q but,rather, on the open subset U . This restricts the domain of e Ψ A d and e Φ A d to ap-propriate open sets, where the results of the Proposition 2.10 hold. We will ignorethis point and keep working as if A d were globally defined in order to avoid a moreinvolved notation.We have the commutative diagram(2.2) E × M e Φ A d ∼ / / π E × M,G (cid:15) (cid:15) Υ A d ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ ( E × G ) × ( M/G ) π E × G,G × id M/G (cid:15) (cid:15) ( E × M ) /G Φ A d ∼ / / e G E × ( M/G ) ONHOLONOMIC DISCRETE LAGRANGIAN REDUCTION BY STAGES 7 where Υ A d : E × M → e G E × ( M/G ) is defined as(2.3) Υ A d := Φ A d ◦ π E × M,G = ( π E × G,G × id M/G ) ◦ e Φ A d . Lemma 2.12.
Let G be a Lie group that acts on the fiber bundle ( E, M, φ, F ) and A d be a discrete connection on the principal G -bundle π M,G : M → M/G . Then, Υ A d : E × M → e G E × ( M/G ) defined by (2.3) is a principal G -bundle.Proof. This is Lemma 2.8 in [14] adapted to affine discrete connections. (cid:3)
All together, we have the following commutative diagram(2.4) E π M,G ◦ φ (cid:15) (cid:15) E × M p o o Υ A d ' ' PPPPPPPPPPPP e Φ A d / / ( E × G ) × ( M/G ) π E × G,G × id M/G (cid:15) (cid:15)
M/G e G E × ( M/G ) p M/G ◦ p o o Proposition 2.13.
Let ρ : X → Y be a principal G-bundle, Z ⊂ X a G -invariantregular submanifold, and S := ρ ( Z ) . Then S is a regular submanifold of Y.Proof. The statement can be proved locally, that is, it suffices to show that for each s ∈ S there is an open subset U ⊂ Y such that s ∈ U and ( S ∩ U ) ⊂ U is a regularsubmanifold.As ρ is a principal G -bundle, for each s ∈ S , there are an open subset U ⊂ Y such that s ∈ U and a diffeomorphism Φ U : ρ − ( U ) → U × G that is G -equivariant(for l U × Gg ( u, g ′ ) := ( u, gg ′ )) and that p ◦ Φ U = ρ | ρ − ( U ) .As Z ⊂ X is a regular submanifold and ρ − ( U ) is an open subset of X , Z ∩ ρ − ( U ) is a regular submanifold of ρ − ( U ). Then, as Φ U is a diffeomorphism, e Z := Φ U ( Z ∩ ρ − ( U )) is a regular submanifold of U × G . Furthermore, as Z is G -invariant and Φ U is G -equivariant, e Z is G -invariant.Let i : U → U × G be given by i ( u ) := ( u, e ), where e is the identity of G ; it iseasy to check that S ∩ U = i − ( e Z ). Then i is smooth and, furthermore, for each s ′ ∈ S ∩ U , di ( s ′ )( T s ′ U ) = T s ′ U ⊕ { } ⊂ T ( s ′ ,e ) ( U × G ). On the other hand, as( s ′ , e ) ∈ e Z and e Z is G -invariant, we have that { }⊕ T e G ⊂ T ( s ′ ,e ) e Z ⊂ T ( s ′ ,e ) ( U × G ).Then, T i ( s ′ ) e Z ⊕ di ( s ′ )( T s ′ U ) = T i ( s ′ ) ( U × G ) and i is transversal to e Z , so that S ∩ U = i − ( e Z ) is a regular submanifold of U (see Theorem 6.30 in [22]). (cid:3) Discrete Lagrange–D’Alembert–Poincar´e systems
In this section we introduce a type of discrete-time dynamical system that con-tains, among other examples, all nonholonomic discrete mechanical systems as wellas their reductions, as defined in [13].3.1.
Some definitions.
Given a fiber bundle φ : E → M we denote C ′ ( E ) := E × M , seen as a fiber bundle over M by φ ◦ p . We define the discrete secondorder manifold C ′′ ( E ) := ( E × M ) × p ,φ ◦ p ( E × M ) considered as a fiber bundleover M by e p := p | C ′′ ( E ) for the projection p : E × M × E × M → M . J. FERN´ANDEZ, C. TORI, AND M. ZUCCALLI
Remark 3.1.
Given a fiber bundle φ : E → M , the second order manifold e p : C ′′ ( E ) → M is isomorphic as a fiber bundle to the fiber bundle φ ◦ p : E × E × M → M with F E (( ǫ , m ) , ( ǫ , m )) := ( ǫ , ǫ , m ). Definition 3.2.
Given a fiber bundle φ : E → M , a discrete path in C ′ ( E ) is a set( ǫ · , m · ) = (( ǫ , m ) , . . . , ( ǫ N − , m N )) where (( ǫ k , m k +1 ) , ( ǫ k +1 , m k +2 )) ∈ C ′′ ( E ) for k = 0 , . . . , N − Definition 3.3.
Let φ : E → M be a fiber bundle and D be a subbundle ofthe pullback bundle p ∗ ( T E ) ⊂ T ( C ′ ( E )). A nonholonomic infinitesimal variationchaining map (NIVCM) P on ( E, D ) is a homomorphism of vector bundles over e p ,according to the following commutative diagram D (cid:15) (cid:15) f p ∗ ( D ) o o (cid:15) (cid:15) P / / ker( dφ ) (cid:15) (cid:15) (cid:31) (cid:127) / / T E { { ✇✇✇✇✇✇✇✇✇✇ E × M C ′′ ( E ) g p o o f p / / E where e p (( ǫ , m ) , ( ǫ , m )) := ǫ and f p (( ǫ , m ) , ( ǫ , m )) := ( ǫ , m ). Remark 3.4.
The fiber of the bundle p ∗ ( T E ) on ( ǫ, m ) consists of vectors of theform ( δǫ, ∈ T ( ǫ,m ) ( C ′ ( E )). Definition 3.5.
Let φ : E → M be a fiber bundle, D ⊂ p ∗ T E be a subbundle, P be a NIVCM on ( E, D ) and ( ǫ · , m · ) = (( ǫ , m ) , . . . , ( ǫ N − , m N )) be a discrete pathin C ′ ( E ). An infinitesimal variation over ( ǫ · , m · ) is a tangent vector ( δǫ · , δm · ) =(( δǫ , δm ) , . . . , ( δǫ N − , δm N )) ∈ T ( ǫ · ,m · ) ( C ′ ( E ) N ) such that(3.1) δm k = dφ ( ǫ k )( δǫ k ) with k = 1 , . . . , N − . A nonholonomic infinitesimal variation over ( ǫ · , m · ) with fixed endpoints is an in-finitesimal variation ( δǫ · , δm · ) over ( ǫ · , m · ) such that δm N =0 ,δǫ N − = ^ δǫ N − ,δǫ k = f δǫ k + P (( ǫ k , m k +1 ) , ( ǫ k +1 , m k +2 ))( ^ δǫ k +1 , , if k = 1 , . . . , N − ,δǫ = P (( ǫ , m ) , ( ǫ , m ))( f δǫ , , (3.2)where ( f δǫ k , ∈ D ( ǫ k ,m k +1 ) is arbitrary for k = 1 , . . . , N − Definition 3.6.
Let φ : E → M be a fiber bundle. A discrete Lagrange–D’Alembert–Poincar´e system (DLDPS) over E is a collection M := ( E, L d , D d , D , P ) where L d : C ′ ( E ) → R is a smooth function, the discrete Lagrangian , D d ⊂ C ′ ( E ) is aregular submanifold, the kinematic constraints , D is a subbundle of p ∗ ( T E ), the variational constraints , and P is a NIVCM over ( E, D ). Definition 3.7.
Let M = ( E, L d , D d , D , P ) be a DLDPS. The discrete action of M is a function S d : C ′ ( E ) N → R defined by S d ( ǫ · , m · ) := P N − k =0 L d ( ǫ k , m k +1 ). A trajectory of M is a discrete path ( ǫ · , m · ) ∈ C ′ ( E ) N such that ( ǫ k , m k +1 ) ∈ D d forall k = 0 , . . . , N − dS d ( ǫ · , m · )( δǫ · , δm · ) = 0for all nonholonomic infinitesimal variations ( δǫ · , δm · ) on ( ǫ · , m · ) with fixed end-points. ONHOLONOMIC DISCRETE LAGRANGIAN REDUCTION BY STAGES 9
Given a DLDPS M = ( E, L d , D d , D , P ) we have the vector bundle ( f p ∗ ( D )) ∗ → C ′′ ( E ). Let ν d be the smooth section of this bundle defined by ν d (( ǫ , m ) , ( ǫ , m )) := D L d ( ǫ , m ) ◦ dp ( ǫ , m )+ D L d ( ǫ , m ) ◦ d ( φ ◦ p )( ǫ , m )+ D L d ( ǫ , m ) ◦ P (( ǫ , m ) , ( ǫ , m )) . (3.3)The next result characterizes the trajectories of a DLDPS in terms of its equationsof motion. Proposition 3.8.
Let M = ( E, L d , D d , D , P ) be a DLDPS and ( ǫ · , m · ) be a discretepath in C ′ ( E ) . Then ( ǫ · , m · ) is a trajectory of M if and only if ( ǫ k , m k +1 ) ∈ D d for all k = 0 , . . . , N − and ν d (( ǫ k − , m k ) , ( ǫ k , m k +1 )) = 0 for all k = 1 , . . . , N − , (3.4) where ν d is the section defined by (3.3) .Proof. Let ( δǫ · , δm · ) be a nonholonomic infinitesimal variation over ( ǫ · , m · ) withfixed endpoints. A straightforward but lengthy computation using Definition 3.5shows that dS d ( ǫ · , m · )( δǫ · , δm · ) = N − X k =1 (cid:0) D L d ( ǫ k , m k +1 ) ◦ dp ( ǫ k , m k +1 )+ D L d ( ǫ k − , m k ) ◦ P (( ǫ k − , m k ) , ( ǫ k , m k +1 ))+ D L d ( ǫ k − , m k ) ◦ d ( φ ◦ p )( ǫ k , m k +1 ) (cid:1) ( f δǫ k , . As the f δǫ k ∈ D ( ǫ k ,m k +1 ) are arbitrary, the result then follows by Definition 3.7. (cid:3) We refer to condition (3.4) as the equations of motion of the system.
Example 3.9.
We recall from [13] (Definition 3.1) that a discrete nonholonomicmechanical system is a collection (
Q, L d , D d , D nh ) where Q is a differentiable man-ifold, L d : Q × Q → R is a smooth function, D nh is a subbundle of T Q and D d is aregular submanifold of Q × Q . In an analogous way to what happens with discretemechanical systems and the discrete Lagrange–Poincar´e systems in [14] (Exam-ple 3.12), a discrete nonholonomic mechanical system can be seen as a discreteLagrange–D’Alembert–Poincar´e system with φ = id Q (so that C ′ ( E ) = Q × Q ), P = 0, the same D d as kinematic constraints and D := p ∗ ( D nh ) ⊂ p ∗ ( T Q ). In thiscase, a discrete path in C ′ ( E ) can be identified with path q · = ( q , . . . , q N ) ∈ Q N +1 and the equations of motion (3.4) become( q k , q k +1 ) ∈ D d for all k = 0 , . . . , N − D L d ( q k , q k +1 ) + D L d ( q k − , q k ) ∈ ( D nhq k ) ◦ for all k = 1 , . . . , N − k = 1 , . . . N −
1, which are the same equations of motion of the discrete non-holonomic mechanical system (
Q, L d , D d , D nh ) given by (6) in [8] or (3) in [13]. Remark 3.10.
Under appropriate regularity conditions on the discrete lagrangian L d and dimensional relation on the constraints spaces, the existence of trajectoriesof a DLDPS is guaranteed in a neighborhood of a given trajectory. Nonholonomic discrete mechanical systems with symmetry.
Symme-tries of a nonholonomic discrete mechanical system (Example 3.9) were consideredin [13]. Even more, a reduction process was developed there so that a new discrete-time dynamical system —called the reduced system — was constructed starting froma symmetric nonholonomic discrete mechanical system and whose dynamics cap-tured the essential features of that of the original system. Unfortunately, thatreduced system is not usually a nonholonomic discrete mechanical system. Thegoal of this section is to recall those constructions and results from [13] and provethat, indeed, the reduced system can be interpreted as a DLDPS whose trajectoriesin the sense of Definition 3.7 are the same as those of the reduced system (in thesense of [13]).Let l Q be a left G -action on Q such that π Q,G : Q → Q/G is a principal G -bundleand fix a discrete connection A d on this bundle. In this case, the commutativediagram (2.2) turns into(3.5) Q × Q e Φ A d ∼ / / π Q × Q,G (cid:15) (cid:15) Υ A d ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ ( Q × G ) × ( Q/G ) π Q × G,G × id Q/G (cid:15) (cid:15) ( Q × Q ) /G Φ A d ∼ / / e G × ( Q/G )where e G := ( Q × G ) /G with G acting on Q by l Q and on G by conjugation and(3.6) Υ A d ( q , q ) := ( π Q × G,G ( q , A d ( q , q )) , π Q,G ( q )) . A Lie group G is a symmetry group of the discrete nonholonomic mechanicalsystem ( Q, L d , D d , D nh ) if π Q,G : Q → Q/G is a principal bundle, L d and D d areinvariant by the diagonal action l Q × Q and D nh is invariant by the lifted action l T Q . By the G -invariance of L d , there is a well defined map ˇ L d : e G × ( Q/G ) → R such that ˇ L d ( v , r ) = L d ( q , q ) for any ( q , q ) ∈ Q × Q such that Υ A d ( q , q ) =( v , r ). The actions associated to L d and ˇ L d are S d ( q · ) := P k L d ( q k , q k +1 ) andˇ S d ( v · , r · ) := P k ˇ L d ( v k , r r +1 ). Also by G -invariance, being D d ⊂ Q × Q a regularsubmanifold, D d /G ⊂ ( Q × Q ) /G is a regular submanifold by Proposition 2.13; asΦ A d is a diffeomorphism, ˇ D d := Φ A d ( D d /G ) is a regular submanifold of e G × ( Q/G ).The next result of [13] relates the variational principle that describes the dynam-ics of (
Q, L d , D d , D nh ) with a variational principle for its reduced system definedon e G × ( Q/G ). Theorem 3.11.
Let G be a symmetry group of the discrete nonholonomic me-chanical system ( Q, L d , D d , D nh ) . Let A d be a discrete connection on the principal G -bundle π Q,G : Q → Q/G . Let q · be a discrete path in Q , r k := π Q,G ( q k ) , w k := A d ( q k , q k +1 ) and v k := π Q × G,G ( q k , w k ) be the corresponding discrete pathsin Q/G , G and e G . Then, the following statements are equivalent.(1) ( q k , q k +1 ) ∈ D d for all k and q · satisfies the criticality condition dS d ( q · )( δq · ) =0 for all fixed-endpoint variations δq · such that δq k ∈ D nhq k for all k .(2) ( v k , r k +1 ) ∈ ˇ D d for all k and d ˇ S d ( v · , r · )( δv · , δr · ) = 0 for all ( δv · , δr · ) suchthat (3.7) ( δv k , δr k +1 ) := d Υ A d ( q k , q k +1 )( δq k , δq k +1 ) ONHOLONOMIC DISCRETE LAGRANGIAN REDUCTION BY STAGES 11 for k = 0 , . . . , N − and where δq · is a fixed-endpoint variation on q · suchthat δq k ∈ D nhq k for all k . Remark 3.12.
Theorem 3.11 is part of Theorem 5.11 in [13]; this last resultrequires the additional data of a connection on the principal bundle π Q,G : Q → Q/G to decompose the variations δq · in horizontal and vertical parts. We haveomitted this requirement and adapted the result accordingly.The reduced system associated to ( Q, L d , D d , D nh ) in Section 5 of [13] is thediscrete-time dynamical system on e G × ( Q/G ) whose trajectories are discrete pathsthat satisfy the variational principle of point 2 in Theorem 3.11. Next we constructa DLDPS that will, eventually, be equivalent to this reduced system. We definethe fiber bundle φ : E → M as p Q/G : e G → Q/G , where p Q/G ( π Q × G,G ( q, w )) := π Q,G ( q ). The reduced lagrangian ˇ L d : e G × ( Q/G ) → R is a smooth map on C ′ ( e G ) = e G × ( Q/G ). We already saw that ˇ D d is a regular submanifold of C ′ ( e G ) = e G × ( Q/G ).The space ˇ D := d Υ A d ( p ∗ ( D nh )) defines a subbundle of T ( C ′ ( e G )) (this is a specialcase of Lemma 5.8 in Section 5.2).In order to define the NIVCM ˇ P ∈ hom( f p ∗ ( ˇ D ) , ker( dp Q/G )) we consider theapplication Υ A d : Q × Q → e G × ( Q/G ) given by (3.6) and define(3.8) ˇ P (( v , r ) , ( v , r ))( f δv ,
0) := D ( p ◦ Υ A d )( q , q )( δq ) ∈ T v e G where ( q , q , q ) are such that ( v , r ) = Υ A d ( q , q ) and ( v , r ) = Υ A d ( q , q ) and δq ∈ D nhq satisfies that δv = D ( p ◦ Υ A d )( q , q )( δq ). That ˇ P is well definedfollows from Lemma 3.13.In this way, we associate a DLDPS M := ( E, ˇ L d , ˇ D d , ˇ D , ˇ P ) to the reduced systemand we will prove that the trajectories of both systems coincide. Lemma 3.13.
Let Q , D , A d and Υ A d be as before. Then, the following statementsare true.(1) For ( q , q ) ∈ Q × Q , d Υ A d | D nhq ×{ } ( q , q ) : ( D nhq × { } ) ⊂ T ( q ,q ) ( Q × Q ) → ˇ D Υ A d ( q ,q ) ⊂ p ∗ ( T e G ) Υ A d ( q ,q ) is an isomorphism of vector spaces.(2) For (( v , r ) , ( v , r )) ∈ C ′′ ( E ) and ( δv , ∈ ˇ D the map ˇ P given by (3.8) is well defined and it is linear in δv .(3) For (( v , r ) , ( v , r )) ∈ C ′′ ( E ) and ( δv , ∈ ˇ D we have dp Q/G ( v )( ˇ P (( v , r ) , ( v , r ))( δv , . Proof.
See point 2 in Lemma 5.1 for point 1 and Lemma 5.10 for points 2 and 3. (cid:3)
The following result of [14] proves that all discrete paths in C ′ ( E ) = C ′ ( e G ) arisefrom discrete paths in C ′ ( id Q ). Lemma 3.14.
Let ( v · , r · ) be a discrete path in C ′ ( E ) and q ∈ Q such that p Q/G ( v ) = π Q,G ( q ) . Then, there exists a unique discrete path in C ′ ( id Q : Q → Q ) such that Υ A d ( q k , q k +1 ) = ( v k , r k +1 ) for all k = 0 , . . . , N − . The following result compares the nonholonomic infinitesimal variations withfixed endpoints on the discrete path ( v · , r · ) in C ′ ( E ) = C ′ ( e G ) for the system M ,with the variations defined in point 2 of Theorem 3.11 on the same discrete path. Proposition 3.15.
Let ( v · , r · ) be a be discrete path in C ′ ( E ) such that ( v k , r k +1 ) =Υ A d ( q k , q k +1 ) for all k , where q · is a discrete path in C ′ ( id Q ) . Then, the followingstatements are true.(1) Given a fixed-endpoint variation δq · in D nh on q · , the infinitesimal varia-tion ( δv · , δr · ) defined in point 2 of Theorem 3.11 by (3.7) is a nonholonomicinfinitesimal variation on ( v · , r · ) with fixed endpoints (in the sense of Def-inition 3.5) for M .(2) Given a nonholonomic infinitesimal variation ( δv · , δr · ) on ( v · , r · ) with fixedendpoints (for M ), there exists a fixed-endpoint variation δq · in D nh on q · such that (3.7) is satisfied for all k .Proof. (1) Let δq · be a fixed-endpoint variation on q · in Q such that δq k ∈ D nhq k and let ( δv · , δr · ) be the variation defined by (3.7) in terms of δq · . Let( g δv k ,
0) := d Υ A d ( q k , q k +1 )( δq k , ∈ d Υ A d ( p ∗ ( D nhq k )) = ˇ D ( v k ,r k +1 ) for k =0 , . . . , N − δv · , δr · ) is a nonholonomic infinitesimal variationon ( v · , r · ) with fixed endpoints. Recall thatΥ A d ( q k , q k +1 ) =( π Q × G,G ( q k , A d ( q k , q k +1 )) , π Q,G ( q k +1 ))=(( p ◦ Υ A d )( q k , q k +1 ) , π Q,G ( q k +1 )) , and given that δq · is a fixed-endpoint variation, we notice that( δv N − , δr N ) = d Υ A d ( q N − , q N )( δq N − , δq N )= d Υ A d ( q N − , q N )( δq N − ,
0) = ( ^ δv N − , , and δv = d ( p ◦ Υ A d )( q , q )( δq , δq ) = d ( p ◦ Υ A d )( q , q )(0 , δq )= D ( p ◦ Υ A d )( q , q )( δq ) = ˇ P (( v , r ) , ( v , r ))( f δv , . Also, taking into account that ^ δv k +1 = D ( p ◦ Υ A d )( q k , q k +1 )( δq k +1 ),for k = 1 , . . . , N − δv k , δr k +1 ) = d Υ A d ( q k , q k +1 )( δq k , δq k +1 )= d Υ A d ( q k , q k +1 )( δq k ,
0) + d Υ A d ( q k , q k +1 )(0 , δq k +1 )=( g δv k ,
0) + D Υ A d ( q k , q k +1 )( δq k +1 )=( g δv k ,
0) + ( D ( p ◦ Υ A d )( q k , q k +1 )( δq k +1 ) ,D ( p ◦ Υ A d )( q k , q k +1 )( δq k +1 ))=( g δv k ,
0) + ( ˇ P (( v k , r k +1 ) , ( v k +1 , r k +2 ))( ^ δv k +1 , , δr k +1 ) . Thus, ( δv · , δr · ) satisfies conditions (3.2). By construcion of the discretepath ( v · , r · ), r k = p Q/G ( v k ) for all k and since p Q/G ◦ p ◦ Υ A d = π Q,G ◦ p ,then dp Q/G ( p ◦ Υ A d )( q k ,q k +1 ) ( d ( p ◦ Υ A d ) ( q k ,q k +1 ) ( δq k , δq k +1 )) = dπ Q,G ( q k )( δq k ) dp Q/G ( v k )( δv k ) = δr k (3.9) so that δr k = dp Q/G ( v k )( δv k ), where δv k is given by the condition (3.7).Hence ( δv · , δr · ) satisfies condition (3.1), hence part 1 is true. ONHOLONOMIC DISCRETE LAGRANGIAN REDUCTION BY STAGES 13 (2) We consider ( δv · , δr · ) that satisfies (3.1) and (3.2) for some vectors ( g δv k , ∈ ˇ D ( v k ,r k +1 ) with k = 1 , . . . , N −
1. Let δq := 0 ∈ D nhq and δq N := 0 ∈ D nhq N and, for each k = 1 , . . . , N −
1, using point 1 of Lemma 3.13, let δq k ∈ D nhq k such that d Υ A d ( q k , q k +1 )( δq k ,
0) = ( g δv k , r k = φ ( v k ) = p Q/G ( v k ) for all k . Also, using (3.9), we have δr k = dφ ( v k )( δv k ) = dp Q/G ( v k )( δv k )= dp Q/G ( v k )( g δv k + ˇ P (( v k , r k +1 ) , ( v k +1 , r k +2 ))( ^ δv k +1 , dp Q/G ( v k )( D ( p ◦ Υ A d )( q k , q k +1 )( δq k ) + D ( p ◦ Υ A d )( q k , q k +1 )( δq k +1 ))= dp Q/G ( v k )( d ( p ◦ Υ A d )( q k , q k +1 )( δq k , δq k +1 ))= d ( p Q/G ◦ p ◦ Υ A d )( q k , q k +1 )( δq k , δq k +1 ) = dπ Q,G ( q k )( δq k ) , and δv k = g δv k + ˇ P (( v k , r k +1 ) , ( v k +1 , r k +2 ))( ^ δv k +1 , d ( p ◦ Υ A d )( q k , q k +1 )( δq k ,
0) + D ( p ◦ Υ A d )( q k , q k +1 )( δq k +1 )= d ( p ◦ Υ A d )( q k , q k +1 )( δq k ,
0) + d ( p ◦ Υ A d )( q k , q k +1 )(0 , δq k +1 )= d ( p ◦ Υ A d )( q k , q k +1 )( δq k , δq k +1 ) . Finally, putting all together( δv k , δr k +1 ) =( d ( p ◦ Υ A d )( q k , q k +1 )( δq k , δq k +1 ) , dπ Q,G ( q k +1 )( δq k +1 ))= d Υ A d ( q k , q k +1 )( δq k , δq k +1 ) , and we have verified that (3.7) is satisfied for all k . Hence, part 2 is true. (cid:3) Corollary 3.16.
A discrete path ( v · , r · ) is a trajectory of M if and only if it is atrajectory of the reduced system according to part 2 of Theorem 3.11.Proof. The equivalence between the two descriptions of a trajectory follows imme-diately by the correspondence of the infinitesimal variations established in Propo-sition 3.15. (cid:3) Categorical formulation
Definition 4.1.
We define the category of discrete Lagrange–D’Alembert–Poincar´esystems
LDP d as the category whose objects are DLDPSs. Given M , M ′ ∈ ob LDP d with M = ( E, L d , D d , D , P ) and M ′ = ( E ′ , L ′ d , D ′ d , D ′ , P ′ ) a map Υ : C ′ ( E ) → C ′ ( E ′ ) is a morphism in mor LDP d ( M , M ′ ) if(1) Υ is a surjective submersion,(2) D ( p ◦ Υ) = 0,(3) As maps from C ′′ ( E ) in M ′ (4.1) p C ′ ( E ′ ) ,M ′ ◦ Υ ◦ p C ′′ ( E ) ,C ′ ( E )1 = φ ′ ◦ p C ′ ( E ′ ) ,E ′ ◦ Υ ◦ p C ′′ ( E ) ,C ′ ( E )2 where p A,Bj : A → B are the maps induced by the canonical projections ofthe Cartesian product onto its factors,(4) L d = L ′ d ◦ Υ, (5) D ′ d = Υ( D d ),(6) D ′ = d Υ( D ),(7) For all ((( ǫ , m ) , ( ǫ , m )) , ( δǫ , ∈ f p ∗ ( D ), P ′ (Υ (2) (( ǫ , m ) , ( ǫ , m )))( d Υ( ǫ , m )( δǫ , d ( p ◦ Υ)( ǫ , m )( P (( ǫ , m ) , ( ǫ , m ))( δǫ , , dφ ( ǫ )( δǫ ))(4.2) where Υ × Υ defines a map Υ (2) : C ′′ ( E ) → C ′′ ( E ′ ).We recall the statement of Lemma 4.3 of [14]. Lemma 4.2.
Let Υ ∈ mor LDP d ( M , M ′ ) , (( ǫ , m ) , ( ǫ , m )) ∈ C ′′ ( E ) and also ( ǫ ′ , m ′ ) := Υ( ǫ , m ) . Then, if δǫ ∈ T ǫ E , D ( p ◦ Υ)( ǫ , m )( dφ ( ǫ )( δǫ )) = dφ ′ ( ǫ ′ )( D ( p ◦ Υ)( ǫ , m )( δǫ )) . Proposition 4.3.
LDP d is a category with the standard composition of functionsand identity mappings.Proof. This proof is analogous to the proof of Proposistion 4.4 of [14]. (cid:3)
Remark 4.4.
Any discrete Lagrange–Poincar´e system [14, Definition 3.4] can beseen as a Lagrange–D’Alembert–Poincar´e system “without constraints”, that is,with D d := C ′ ( E ) and D := p ∗ T E . Also, with this interpretation any morphism ofLagrange–Poincar´e systems is a morphism of Lagrange–D’Alembert–Poincar´e sys-tems. It is immediate to check that the category of Lagrange–Poincar´e systems [14,Definition 4.1] is a full subcategory of
LDP d . Lemma 4.5.
Let Υ ′ ∈ mor LDP d ( M , M ′ ) and Υ ′′ ∈ mor LDP d ( M , M ′′ ) where M = ( E, L d , D d , D , P ) , M ′ = ( E ′ , L ′ d , D ′ d , D ′ , P ′ ) and M ′′ = ( E ′′ , L ′′ d , D ′′ d , D ′′ , P ′′ ) .If F : C ′ ( E ′ ) → C ′ ( E ′′ ) is a smooth map such that the diagram C ′ ( E ) Υ ′ z z ✉✉✉✉✉✉✉✉✉ Υ ′′ $ $ ■■■■■■■■■ C ′ ( E ′ ) F / / C ′ ( E ′′ ) is commutative, then F ∈ mor LDP d ( M ′ , M ′′ ) . Also, if F is a diffeomorphism, then F is an isomorphism in LDP d .Proof. The proof that F satisfies the points 1 to 4 and 7 in Definition 4.1 andthe last assertion of the statement is the same as in the proof of Lemma 4.5in [14]. We want to prove that F satisfies points 5 and 6 of Definition 4.1. SinceΥ ′ ∈ mor LDP d ( M , M ′ ) and Υ ′′ ∈ mor LDP d ( M , M ′′ ) and the previous diagram iscommutative, we have that F ( D ′ d ) = F (Υ ′ ( D d )) = ( F ◦ Υ ′ )( D d ) = Υ ′′ ( D d ) = D ′′ d , and, then, point 5 in Definition 4.1 is satisfied. Similarly, as Υ ′ and Υ ′′ are mor-phisms in LDP d , using the commutativity of the diagram, we have D ′′ = d Υ ′′ ( D ) = d ( F ◦ Υ ′ )( D ) = dF ( d Υ ′ ( D )) = dF ( D ′ ) . This proves that point 6 in Definition 4.1 is satisfied. (cid:3)
ONHOLONOMIC DISCRETE LAGRANGIAN REDUCTION BY STAGES 15
Theorem 4.6.
Given Υ ∈ mor LDP d ( M , M ′ ) with M = ( E, L d , D d , D , P ) and M ′ = ( E ′ , L ′ d , D ′ d , D ′ , P ′ ) , let ( ǫ · , m · ) = (( ǫ , m ) , . . . , ( ǫ N − , m N )) be a discretepath in C ′ ( E ) and define ( ǫ ′ k , m ′ k +1 ) := Υ( ǫ k , m k +1 ) for k = 0 , . . . , N − .(1) If ( ǫ · , m · ) is a trajectory of M , then ( ǫ ′· , m ′· ) is a trajectory of M ′ .(2) If D d = Υ − ( D ′ d ) and ( ǫ ′· , m ′· ) is a trajectory of M ′ , then ( ǫ · , m · ) is atrajectory of M .Proof. By hypothesis, ( ǫ · , m · ) is a discrete path in C ′ ( E ). It follows from its def-inition, the fact that Υ is a morphism and (4.1) that ( ǫ ′· , m ′· ) is a discrete path in C ′ ( E ).Assume that ( δǫ · , δm · ) is an infinitesimal variation in M over ( ǫ · , m · ) and that( δǫ ′· , δm ′· ) is an infinitesimal variation in M ′ over ( ǫ ′· , m ′· ) satisfying(4.3) d Υ( ǫ k , m k +1 )( δǫ k , δm k +1 ) = ( δǫ ′ k , δm ′ k +1 ) for k = 0 , . . . , N − . Then, using the chain rule and that L d = L ′ d ◦ Υ, we see that(4.4) dS d ( ǫ · , m · )( δǫ · , δm · ) = dS ′ d ( ǫ ′· , m ′· )( δǫ ′· , δm ′· ) . In order to prove point 1, we assume that ( ǫ · , m · ) is a trajectory of M . Then( ǫ k , m k +1 ) ∈ D d for k = 0 , . . . , N −
1. Then,(4.5) ( ǫ ′ k , m ′ k +1 ) = Υ( ǫ k , m k +1 ) ∈ Υ( D d ) = D ′ d for k = 0 , . . . , N − . Let ( δǫ ′· , δm ′· ) be an infinitesimal variation with fixed endpoints in M ′ over thediscrete path ( ǫ ′· , m ′· ). That is, there are ( f δǫ ′ k , ∈ D ′ ( ǫ ′ k ,m ′ k +1 ) for k = 1 , . . . , N − δǫ ′ k and f δǫ ′ k instead of δǫ k and f δǫ k .By morphism’s property 6 applied to Υ, there exist ( f δǫ k , ∈ D ( ǫ k ,m k +1 ) suchthat d Υ( ǫ k , m k +1 )( f δǫ k ,
0) = ( f δǫ ′ k ,
0) for k = 1 , . . . , N −
1; we fix one such vector foreach k . Next apply (3.1) and (3.2) to define an infinitesimal variation ( δǫ · , δm · ) on( ǫ · , m · ) with fixed endpoints based on the f δǫ · constructed above.Direct computations using the morphism properties of Υ show that condition (4.3)holds for the ( δǫ · , δm · ) and ( δǫ ′· , δm ′· ) variations. Then, using (4.4), dS ′ d ( ǫ ′· , m ′· )( δǫ ′· , δm ′· ) = dS d ( ǫ · , m · )( δǫ · , δm · ) = 0 , where the last equality holds because ( δǫ · , δm · ) is an infinitesimal variation withfixed endpoints in M over ( ǫ · , m · ), that is a trajectory of M . Finally, as ( δǫ ′· , δm ′· )was an arbitrary infinitesimal variation with fixed endpoints in M ′ over the path( ǫ ′· , m ′· ), and we have (4.5), we conclude that ( ǫ ′· , m ′· ) is a trajectory of M ′ . Thisproves point 1.In order to prove point 2, assume that ( ǫ ′· , m ′· ) is a trajectory of M ′ . Then,as D d = Υ − ( D ′ d ) and ( ǫ ′ k , m ′ k +1 ) ∈ D ′ d for k = 0 , . . . , N −
1, we have thatΥ( ǫ k , m k +1 ) = ( ǫ ′ k , m ′ k +1 ) ∈ D ′ d , so that ( ǫ k , m k +1 ) ∈ Υ − ( D ′ d ) = D d for k =0 , . . . , N − ǫ · , m · )satisfies the criticality condition in M , so that it is a trajectory of M , thus provingpoint 2. (cid:3) The following result, whose proof is immediate, is useful when working withconcrete DLDPSs.
Lemma 4.7.
Let φ : E → M and φ ′ : E ′ → M ′ be two fiber bundles and ( F, f ) be afiber bundle isomorphism from E to E ′ . For any M = ( E, L d , D d , D , P ) ∈ ob LDP d ,let L ′ d := L d ◦ ( F × f ) − , D ′ d := ( F × f )( D d ) , D ′ := d ( F × f )( D ) and P ′ so that,for all ((( ǫ ′ , m ′ ) , ( ǫ ′ , m ′ )) , ( δǫ ′ , ∈ f p ∗ ( D ′ ) , we have P ′ (( ǫ ′ , m ′ ) , ( ǫ ′ , m ′ ))( δǫ ′ , dF ( F − ( ǫ ′ ))(( P (( F − ( ǫ ′ ) , f − ( m ′ )) , ( F − ( ǫ ′ ) , f − ( m ′ )))( dF − ( ǫ ′ )( δǫ ′ ) , . Then, M ′ := ( E ′ , L ′ d , D ′ d , D ′ , P ′ ) ∈ ob LDP d and F × f ∈ hom LDP d ( M , M ′ ) is anisomorphism in LDP d . In particular, the corresponding sections ν d and ν ′ d definedby (3.3) satisfy ( ^ F × f (2) ) ∗ ( ν ′ d ) = ν d or, explicitly, ν d (( ǫ , m ) , ( ǫ , m ))( δǫ ,
0) = ν ′ d (( F ( ǫ ) , f ( m )) , ( F ( ǫ ) , f ( m )))( dF ( ǫ )( δǫ ) , . Reduction of discrete Lagrange–D’Alembert–Poincar´e systems
In this section we introduce the notion of symmetry group of a DLDPS and areduction procedure to associate a “reduced” DLDPS system to a symmetric one.In addition, we prove that the reduction procedure is a morphism in
LDP d andcompare the dynamics of the reduced system to that of the original one.5.1. Discrete Lagrange–D’Alembert–Poincar´e systems with symmetry.
Let G be a Lie group that acts on the fiber bunble ( E, M, φ, F ) as in Definition 2.8.We consider the G -actions on C ′ ( E ) and C ′′ ( E ) given by(5.1) l C ′ ( E ) g ( ǫ , m ) := ( l Eg ( ǫ ) , l Mg ( m )) , (5.2) l C ′′ ( E ) g (( ǫ , m ) , ( ǫ , m )) := ( l C ′ ( E ) g ( ǫ , m ) , l C ′ ( E ) g ( ǫ , m )) . Also, we consider the G -actions on ker( dφ ) ⊂ T E and on f p ∗ T ( C ′ ( E )) ⊂ T C ′′ ( E )given by(5.3) l T Eg ( ǫ , δǫ ) := dl Eg ( ǫ )( δǫ )(5.4) l T ( C ′ ( E )) g ( ǫ , m )( δǫ , δm ) := ( dl Eg ( ǫ )( δǫ ) , dl Mg ( m )( δm )) l g p ∗ T ( C ′ ( E )) g (( ǫ , m ) , ( ǫ , m ) , ( δǫ , δm )) :=( l C ′′ ( E ) g (( ǫ , m ) , ( ǫ , m )) , dl C ′ ( E ) g ( ǫ , m )( δǫ , δm )) . (5.5) Lemma 5.1.
Let G be a Lie group acting on the fiber bundle φ : E → M and A d a discrete connection on the principal G -bundle π M,G : M → M/G . We define Υ (2) A d : C ′′ ( E ) → C ′′ ( e G E ) as the restriction of Υ A d × Υ A d : C ′ ( E ) × C ′ ( E ) → C ′ ( e G E ) × C ′ ( e G E ) to the corresponding spaces, where Υ A d is defined by (2.3) . Then,(1) Υ (2) A d is well defined.(2) d Υ A d ( ǫ , m ) | ( p ∗ T E ) ( ǫ ,m : ( p ∗ T E ) ( ǫ ,m ) → ( p ∗ T ( e G E )) Υ A d ( ǫ ,m ) is anisomorphism of vector spaces for every ( ǫ , m ) ∈ C ′ ( E ) .(3) Υ (2) A d : C ′′ ( E ) → C ′′ ( e G E ) is a principal G -bundle with structure group G .In particular, C ′′ ( E ) /G ≃ C ′′ ( e G E ) . ONHOLONOMIC DISCRETE LAGRANGIAN REDUCTION BY STAGES 17 (4) For each (( v , r ) , ( v , r )) ∈ C ′′ ( e G E ) and ( ǫ , m ) ∈ C ′ ( E ) such that Υ A d ( ǫ , m ) = ( v , r ) , there is a unique pair ( ǫ , m ) ∈ C ′ ( E ) such that (( ǫ , m ) , ( ǫ , m )) ∈ C ′′ ( E ) and Υ (2) A d (( ǫ , m ) , ( ǫ , m )) = (( v , r ) , ( v , r )) .Proof. This result is almost identical to Lemma 5.1 in [14], the only difference beingthat, here, we are using affine discrete connections instead of discrete connections.It is easy to see that the proof of Lemma 5.1 remains valid for affine discreteconnections. (cid:3)
Proposition 5.2.
Let G be a Lie group acting on the fiber bundle φ : E → M and A d be a discrete connection on the principal G -bundle π M,G : M → M/G . Givena discrete path ( v · , r · ) = (( v , r ) , . . . , ( v N − , r N )) in C ′ ( e G E ) and ( e ǫ , e m ) ∈ C ′ ( E ) such that Υ A d ( e ǫ , e m ) = ( v , r ) , there is a unique discrete path ( ǫ · , m · ) ∈ C ′ ( E ) such that ( ǫ , m ) = ( e ǫ , f m ) and Υ A d ( ǫ k , m k +1 ) = ( v k , r k +1 ) for all k .Proof. This is Proposition 5.2 in [14] except for using affine discrete connectionsinstead of discrete connections, which doesn’t alter the proof. (cid:3)
Definition 5.3.
Let M = ( E, L d , D d , D , P ) ∈ ob LDP d . A Lie group G is a sym-metry group of M if(1) G acts on φ : E → M (Definition 2.8),(2) L d is G -invariant by the action l C ′ ( E ) (5.1),(3) D d is G -invariant by the action l C ′ ( E ) (5.1),(4) D is G -invariant by the lifted action l T E (5.3),(5) P is G -equivariant for the actions l g p ∗ T ( C ′ ( E )) (5.5) and l T E (5.3).
Remark 5.4.
In the context of Example 3.9, if G is a symmetry group of thenonholonomic discrete mechanical system ( Q, L d , D d , D nh ) in the sense of [13], thenit is a symmetry group of ( Q, L d , D d , D , ∈ ob LDP d in the sense of Definition 5.3. Lemma 5.5.
Let M = ( E, L d , D d , D , P ) ∈ ob LDP d and G be a Lie group. Then,for g ∈ G , if Υ := l C ′ ( E ) g and M ′ = M ,(1) D d is G -invariant by the diagonal action (5.1) if and only if point 5 inDefinition 4.1 is satisfied for Υ ,(2) D is G -invariant by the lifted action (5.4) if and only if point 6 in Defini-tion 4.1 is satisfied for Υ ,(3) Point 5 in Definition 5.3 is equivalent to point 7 in Definition 4.1 for Υ .Proof. Points 1 and 2 are directly satisfied by the definitions of l C ′ ( E ) g and l T ( C ′ ( E )) g .To prove point 3 we start by noting that P ( l C ′′ ( E ) g (( ǫ , m ) , ( ǫ , m )))( dl C ′ ( E ) g ( ǫ , m )( δǫ , P ( l g p ∗ T ( C ′ ( E )) g (( ǫ , m ) , ( ǫ , m )) , ( δǫ , d ( p ◦ l C ′ ( E ) g )( ǫ , m )( P (( ǫ , m ) , ( ǫ , m ))( δǫ , , dφ ( ǫ )( δǫ ))= dl Eg ( ǫ )( P (( ǫ , m ) , ( ǫ , m ))( δǫ , . Then, by point 7 in Definition 4.1 for Υ := l C ′ ( E ) g and M ′ = M we have that thefirst members of the previous identities are the same, and by point 5 of Definition 5.3the last members of the previous identities are the same, proving the equivalenceof the conditions. (cid:3) Proposition 5.6.
Let M = ( E, L d , D d , D , P ) ∈ ob LDP d and G be a Lie group.Then, G is a symmetry group of M if and only if G acts on the fiber bundle φ : E → M and l C ′ ( E ) g ∈ mor LDP d ( M , M ) for all g ∈ G .Proof. Assume that G is a symmetry group of M . Then, by definition, G acts onthe fiber bundle φ : E → M . We have to prove that l C ′ ( E ) g ∈ mor LDP d ( M , M ) forall g ∈ G .Proving that l C ′ ( E ) g satisfies conditions 1 to 4 of Definition 4.1 is analogous towhat was done in the proof of the Proposition 5.6 in [14]. Lemma 5.5 provesthat l C ′ ( E ) g satisfies the remaining conditions of the Definition 4.1. Thus, l C ′ ( E ) g ∈ mor LDP d ( M , M ).Conversely, if G acts on the fiber bundle φ : E → M and l C ′ ( E ) g ∈ mor LDP d ( M , M ),the first condition of Definition 5.3 is satisfied and the remaining conditions followfrom morphism’s properties and Lemma 5.5. (cid:3) Reduced discrete Lagrange–D’Alembert–Poincar´e system.
Let G be asymmetry group of M = ( E, L d , D d , D , P ) ∈ ob LDP d . Since G acts on ( E, M, φ, F )the conjugate bundle ( e G E , M/G, p M/G , F × G ) is a fiber bundle (see Section 9in [14]).Let A d be a discrete connection on the principal G -bundle π M,G : M → M/G and Υ A d : E × M → e G E × ( M/G ) be the map defined by (2.3) that is a principalbundle with structure group G by Lemma 2.12.We define ˇ L d : e G E × ( M/G ) → R by ˇ L d ( v , r ) := L d ( ǫ , m ) for any ( ǫ , m ) ∈ Υ − A d ( v , r ) that, by the G -invariance of L d , is well defined. Hence, ˇ L d ◦ Υ A d = L d . Lemma 5.7.
The space ˇ D d := Υ A d ( D d ) is a regular submanifold of e G E × ( M/G ) .Also, D d = Υ − A d ( ˇ D d ) .Proof. Since Υ A d : E × M → e G E × ( M/G ) is a principal G -bundle and D d is a G -invariant regular submanifold of E × M , by Proposition 2.13, ˇ D d is a regularsubmanifold of e G E × ( M/G ). Also, as D d is G -invariant and Υ A d is a principal G -bundle, we have D d = Υ − A d (Υ A d ( D d )) = Υ − A d ( ˇ D d ). (cid:3) Lemma 5.8.
The space ˇ D := d Υ A d ( D ) is a subbunble of p ∗ ( T ( e G E )) where p : C ′ ( e G E ) → e G E is the projection onto the first factor. In addition, rank( ˇ D ) =rank( D ) .Proof. Given ( v, r ) ∈ C ′ ( e G E ) and ( ǫ, m ) ∈ C ′ ( E ) such that ( v, r ) = Υ A d ( ǫ, m ), wewant to prove that the subspace d Υ A d ( v, r )( D ( ǫ,m ) ) ⊂ T ( v,r ) C ′ ( e G E ) is independentof the particular choice of ( ǫ, m ) in Υ − A d ( v, r ). ONHOLONOMIC DISCRETE LAGRANGIAN REDUCTION BY STAGES 19
If ( v, r ) = Υ A d (¯ ǫ, ¯ m ) then, since Υ A d : C ′ ( E ) → C ′ ( e G E ) is a G -principal bundle,there is g ∈ G such that (¯ ǫ, ¯ m ) = l C ′ ( E ) g ( ǫ, m ). Since D is G -invariant we have that D (¯ ǫ, ¯ m ) = D l C ′ ( E ) g ( ǫ,m ) = dl C ′ ( E ) g ( ǫ, m )( D ( ǫ,m ) ) , and, as Υ A d is G -invariant, d Υ A d (¯ ǫ, ¯ m )( D (¯ ǫ, ¯ m ) ) = d Υ A d ( l C ′ ( E ) g ( ǫ, m ))( dl C ′ ( E ) g ( D ( ǫ,m ) ))= d (Υ A d ◦ l C ′ ( E ) g )( ǫ, m )( D ( ǫ,m ) )= d Υ A d ( ǫ, m )( D ( ǫ,m ) ) . Then d Υ A d ( ǫ, m )( D ( ǫ,m ) ) is a vector subspace of T ( v,r ) ( C ′ ( e G E )) for each ( v, r ) ∈ C ′ ( e G E ) and is independent of the particular ( ǫ, m ) chosen in Υ − A d ( v, r ); we call itˇ D ( v,r ) . This construction gives a fiberwise vector structure to ˇ D . That rank( ˇ D ) =rank( D ) follows immediately from point 2 in Lemma 5.1.Now we need to check that for every ( v, r ) ∈ C ′ ( e G E ) there exist smooth sec-tions defined in an open neighborhood of ( v, r ) that generate ˇ D ( v ′ ,r ′ ) for all ( v ′ , r ′ )in that neighborhood. To do this, notice that given ( v, r ) ∈ C ′ ( e G E ), for any( ǫ, m ) ∈ Υ − A d ( v, r ), as D is a subbundle of T ( C ′ ( E )), there is an open neighbor-hood U ⊂ C ′ ( E ) of ( ǫ, m ) and d = dim( D ( ǫ,m ) ) smooth local sections σ , . . . , σ d : U → T ( C ′ ( E )) such that { σ ( ǫ ′ , m ′ ) , . . . , σ d ( ǫ ′ , m ′ ) } is a basis of D ( ǫ ′ ,m ′ ) for each( ǫ ′ , m ′ ) ∈ U (Lemma 10.32 in [22]). Then, Υ A d ( U ) is an open neighborhood of( v, r ) (because Υ A d , being a principal bundle map, is an open map; see Lemma21.1 [22]). In addition, as Υ A d is a principal bundle, there is an open neighbor-hood V ⊂ Υ A d ( U ) of ( v, r ) and a smooth section Σ : V → U of Υ A d . Define η j := d Υ A d ◦ σ j ◦ Σ, j = 1 , . . . , d , which are smooth sections over V of T C ′ ( e G E )such that, for each ( v ′ , r ′ ) ∈ V , { η ( v ′ , r ′ ) , . . . , η d ( v ′ , r ′ ) } generates ˇ D ( v ′ ,r ′ ) . (cid:3) Remark 5.9.
As, by point 2 of Lemma 5.1, d Υ A d ( ǫ , m ) : ( p ∗ T E ) ( ǫ ,m ) → ( p ∗ T ( e G E )) Υ A d ( ǫ ,m ) is an isomorphism, d Υ A d ( ǫ , m ) | D ( ǫ ,m is an isomorphismfrom D ( ǫ ,m ) onto ˇ D Υ A d ( ǫ ,m ) .As, by point 3 of Lemma 5.1 Υ (2) A d is a principal G -bundle, given (( v , r ) , ( v , r )) ∈ C ′′ ( e G E ), there are (( ǫ , m ) , ( ǫ , m )) ∈ C ′′ ( E ) such that Υ (2) A d (( ǫ , m ) , ( ǫ , m )) =(( v , r ) , ( v , r )). We fix one element in the G -orbit formed by those elements.Using Remark 5.9, given ((( v , r ) , ( v , r )) , ( δv , ∈ f p ∗ ( ˇ D ) there is a unique((( ǫ , m ) , ( ǫ , m )) , ( δǫ , ∈ f p ∗ ( D ) such that ( δv ,
0) = d Υ A d ( ǫ , m )( δǫ , P (( v , r ) , ( v , r ))( δv ,
0) := D ( p ◦ Υ A d )( ǫ , m )( P (( ǫ , m ) , ( ǫ , m ))( δǫ , D ( p ◦ Υ A d )( ǫ , m )( dφ ( ǫ )( δǫ ))= d ( p ◦ Υ A d )( ǫ , m )( P (( ǫ , m ) , ( ǫ , m ))( δǫ , , dφ ( ǫ )( δǫ )) . (5.6) Lemma 5.10.
Under the previous conditions, the map ˇ P defined by (5.6) is a welldefined element of hom( f p ∗ ( ˇ D ) , ker( dp M/G )) . Proof.
The proof is similar to the proof of the Lemma 5.10 of [14] with f p ∗ ( ˇ D )instead of p ∗ ( T e G E ) and taking into account the G -invariance of D . (cid:3) Definition 5.11.
Let G be a symmetry group of M = ( E, L d , D d , D , P ) ∈ ob LDP d and A d be a discrete connection on the principal bundle π M,G : M → M/G .The system ( e G E , ˇ L d , ˇ D d , ˇ D , ˇ P ) ∈ ob LDP d defined previously is called the reduceddiscrete Lagrange–D’Alembert–Poincar´e system obtained as the reduction of M bythe symmetry group G using the discrete connection A d . We denote this systemby M /G or M / ( G, A d ). Example 5.12.
Given a discrete nonholonomic mechanical system (
Q, L d , D d , D nh )let M := ( Q, L d , D d , D ,
0) be the discrete Lagrange–D’Alembert–Poincar´e systemconstructed in Example 3.9. Let G be a symmetry group of ( Q, L d , D d , D nh ). Asnoted in Remark 5.4, G is a symmetry group of M . Let A d be a discrete con-nection on the principal G -bundle π Q,G : Q → Q/G . The system M / ( G, A d ) is( e G E , ˇ L d , ˇ D d , ˇ D , ˇ P ) where the fiber bundle φ : e G E → M/G is p Q/G : e G → Q/G , thelagrangian is determined by ˇ L d ◦ Υ A d = L d and, by (5.6),ˇ P (( v k − , r k ) , ( v k , r k +1 ))( δv k ,
0) = D ( p ◦ Υ A d )( q k − , q k )( δq k ) , where we have ( v k − , r k ) = Υ A d ( q k − , q k ), ( v k , r k +1 ) = Υ A d ( q k , q k +1 ) and ( δv k ,
0) = d Υ A d ( q k , q k +1 )( δq k , δq k ∈ D nhq k . This DLDPS coincides with the one ob-tained in Section 3.2 as associated to the reduction of ( Q, L d , D d , D nh ) modulo G in the sense of [13]. Thus, the reduction process of DLDPSs extends the reductionconstruction of discrete nonholonomic mechanical systems introduced in [13]. Proposition 5.13.
Let G be a symmetry group of M = ( E, L d , D d , D , P ) ∈ ob LDP d and A d a discrete connection on the principal G -bundle π M,G : M → M/G .Then, Υ A d ∈ mor LDP d ( M , M / ( G, A d )) , where Υ A d is the map defined by (2.3) .Proof. The proof that Υ A d satisfies the conditions 1 to 4 and 7 of Definition 4.1 isanalogous to the proof of Proposition 5.13 in [14], with f p ∗ ( D ) instead of p ∗ T E .Since the constraint spaces of the system M / ( G, A d ) have been defined as ˇ D d :=Υ A d ( D d ) and ˇ D := d Υ A d ( D ) conditions 5 and 6 of Definition 4.1 hold. (cid:3) The following result proves that given a discrete Lagrange–D’Alembert–Poincar´esystem with symmetry, the reduced systems obtained using different discrete con-nections are all isomorphic in
LDP d . Proposition 5.14.
Let G be a symmetry group of M = ( E, L d , D d , D , P ) ∈ ob LDP d and A d , A d be two discrete connections on the principal G -bundle π M,G : M → M/G . Then, the reduced systems M / ( G, A d ) and M / ( G, A d ) are isomorphicin LDP d .Proof. The proof is analogous to the proof of Proposition 5.14 in [14], using Lem-mas 2.12 and 4.5 and Proposition 5.13. (cid:3)
Dynamics of the reduced discrete Lagrange–D’Alembert–Poincar´esystem.
In this section we consider the dynamics of the reduced system defined inSection 5.2.
ONHOLONOMIC DISCRETE LAGRANGIAN REDUCTION BY STAGES 21
Theorem 5.15.
Let G be a symmetry group of M = ( E, L d , D d , D , P ) , A d be adiscrete connection on the principal G -bundle π M,G : M → M/G and M / ( G, A d ) =( e G E , ˇ L d , ˇ D d , ˇ D , ˇ P ) ∈ ob LDP d be the corresponding reduced DLDPS. Assume that ( ǫ · , m · ) = (( ǫ , m ) , . . . , ( ǫ N − , m N )) is a discrete path in C ′ ( E ) , and ( v · , r · ) =(( v , r ) , . . . , ( v N − , r N )) is a discrete path in C ′ ( e G E ) such that Υ A d ( ǫ k , m k +1 ) =( v k , r k +1 ) for k = 0 , . . . , N − . Then, ( ǫ · , m · ) is a trajectory of M if and only if ( v · , r · ) is a trajectory of M / ( G, A d ) .Proof. By Proposition 5.13, Υ A d ∈ mor LDP d ( M , M / ( G, A d )) and, by Lemma 5.7, D d = Υ − A d ( ˇ D d ). Then the result follows from Theorem 4.6. (cid:3) Corollary 5.16.
Let G be a symmetry group of M = ( E, L d , D d , D , P ) ∈ ob LDP d ,and A d be a discrete connection on the principal G -bundle π M,G : M → M/G . Forthe discrete path ( ǫ · , m · ) in C ′ ( E ) we define a discrete path ( v · , r · ) in C ′ ( e G E ) as ( v k , r k +1 ) := Υ A d ( ǫ k , m k +1 ) for k = 0 , . . . , N − . Then, the following statementsare equivalent.(1) ( ǫ · , m · ) is a traejctory of the system M .(2) Condition (3.4) is satisfied for ν d := ν M d defined by (3.3) for M .(3) ( v · , r · ) is a trajectory of the system M / ( G, A d ) .(4) Condition (3.4) is satisfied for ν d := ν M / ( G, A d ) d and D d := ˇ D d being thoseof M / ( G, A d ) .Proof. The equivalence 1 ⇔ ⇔ M / ( G, A d ). Theequivalence 1 ⇔ (cid:3) Theorem 5.17.
Let G be a symmetry group of M = ( E, L d , D d , D , P ) ∈ ob LDP d and A d be a discrete connection on the principal G -bundle π M,G : M → M/G .Let ( v · , r · ) be a trajectory of the system M / ( G, A d ) and ( e ǫ , e m ) ∈ D d such that Υ A d ( e ǫ , e m ) = ( v , r ) . Then, there exists a unique trajectory ( ǫ · , m · ) of M suchthat ( ǫ , m ) = ( e ǫ , e m ) and Υ A d ( ǫ k , m k +1 ) = ( v k , r k +1 ) for all k .Proof. By Proposition 5.2, the discrete path ( v · , r · ) lifts to a unique discrete path( ǫ · , m · ) in C ′ ( E ) starting at ( e ǫ , e m ). Then, ( ǫ , m ) = ( e ǫ , e m ) and ( v k , r k +1 ) =Υ A d ( ǫ k , m k +1 ) for all k . As ( v · , r · ) is a trajectory of M / ( G, A d ), by Theorem 5.15,( ǫ · , m · ) is a trajectory of M . (cid:3) Remark 5.18.
Theorem 5.17 states that all trajectories of a reduced discreteLagrange–D’Alembert–Poincar´e system M / ( G, A d ) come from trajectories of theoriginal system M . A direct description of the reconstruction process in terms ofthe lifting of discrete paths is given in Remark 5.18 of [14]. In that respect, itshould be kept in mind that, as D d = Υ − A d ( ˇ D d ) and the trajectories of M / ( G, A d )are in ˇ D d , the lifted discrete paths are in D d automatically.Next we study the relationship between the equation of motion of a symmetricDLDPS and that of its reduction. Before we can state the result, we recall thepullback construction for sections of vector bundles. If ρ j : V j → X j for j = 1 , F : V → V is a morphism of vector bundles over f : X → X , there is a pullback map F ∗ : Γ( X , V ∗ ) → Γ( X , V ∗ ) determined by F ∗ ( α )( x )( v ) := α ( f ( x ))( F ( v )), for α ∈ Γ( X , V ∗ ), x ∈ X and v ∈ ( V ) x .Let G be a symmetry group of M = ( E, L d , D d , D , P ) ∈ ob LDP d and A d be a discrete connection on the principal G -bundle π M,G : M → M/G . Let M / ( G, A d ) be the corresponding reduced system. We have the vector bundles f p ∗ ( D ) → C ′′ ( E ) and f p ∗ ( ˇ D ) → C ′′ ( e G E ). Then, as d Υ (2) A d : T C ′′ ( E ) → T C ′′ ( e G E )is a morphism of vector bundles (over Υ (2) A d : C ′′ ( E ) → C ′′ ( e G E )) that restrictsto a morphism d Υ (2) A d | g p ∗ ( D ) : f p ∗ ( D ) → f p ∗ ( ˇ D ), we have the pullback map( d Υ (2) A d | g p ∗ ( D ) ) ∗ : Γ( C ′′ ( e G E ) , f p ∗ ( ˇ D ) ∗ ) → Γ( C ′′ ( E ) , f p ∗ ( D ) ∗ ). The following re-sult relates the equation of motion of M , ν M d ∈ Γ( C ′′ ( E ) , f p ∗ ( D ) ∗ ), to the one of M / ( G, A d ), ν M /Gd ∈ Γ( C ′′ ( e G E ) , f p ∗ ( ˇ D ) ∗ ). Lemma 5.19.
With the previous notation, ν M d = ( d Υ (2) A d | g p ∗ ( D ) ) ∗ ( ν M /Gd ) .Proof. For any µ := (( ǫ , m ) , ( ǫ , m )) ∈ C ′′ ( E ) and ( δǫ , ∈ ( f p ∗ ( D )) µ , we have( d Υ (2) A d | g p ∗ ( D ) ) ∗ ( ν M /Gd )( µ )( δǫ ,
0) = ν M /Gd (Υ (2) A d ( µ ))( d Υ A d ( ǫ , m )( δǫ , . Then, a direct computation shows that ν M /Gd (Υ (2) A d ( µ ))( d Υ A d ( ǫ , m )( δǫ , ν M d ( µ )( δǫ , , proving the statement. (cid:3) Let M = ( E, L d , D d , D , P ) ∈ ob LDP d and G be a symmetry group of M . Con-sider the vertical bundle ker( T π
E,G ) ⊂ T E over E and let V := p ∗ ker( T π
E,G ),where p : C ′ ( E ) → E is the projection map. Assume that S := D ∩ V is a vec-tor bundle over C ′ ( E ) and that there is another vector bundle H → C ′ ( E ) suchthat D = S ⊕ H ; the vector bundle H may be constructed using a (continuous)connection on the principal bundle π E,G : E → E/G . Then, as the section ofmotion ν M d takes values in f p ∗ ( D ) ∗ ≃ f p ∗ ( S ∗ ) ⊕ f p ∗ ( H ∗ ) we can decompose itas ν M d = (( ν M d ) S , ( ν M d ) H ). Clearly, for any µ ∈ C ′′ ( E ), the condition ν M d ( µ ) = 0is equivalent to ( ν M d ) S ( µ ) = 0 and ( ν M d ) H ( µ ) = 0 . The last two conditions are usually called the vertical and horizontal equations ofmotion.
Definition 5.20.
Let M = ( E, L d , D d , D , P ) ∈ ob LDP d and G be a symmetrygroup of M . We define the discrete nonholonomic momentum map J d : C ′ ( E ) → ( g D ) ∗ by J d ( ǫ , m )( ξ ) := − D L d ( ǫ , m )( ξ E ( ǫ ))for ( ǫ , m ) ∈ C ′ ( E ) and (( ǫ , m ) , ξ ) ∈ g D := { (( ǫ , m ) , ξ ) ∈ C ′ ( E ) × g :( ξ E ( ǫ ) , ∈ D ( ǫ ,m ) } , where g := Lie ( G ); we assume that g D → C ′ ( E ) is asmooth vector bundle. We also define, for each ξ ∈ Γ( g D ), ( J d ) ξ : C ′ ( E ) → R by( J d ) ξ ( ǫ , m ) := J d ( ǫ , m )( ξ ( ǫ , m )). ONHOLONOMIC DISCRETE LAGRANGIAN REDUCTION BY STAGES 23
In the context of Definition 5.20 we have that L d ( l Eg ( ǫ ) , l Mg ( m )) = L d ( ǫ , m )for all g ∈ G and ( ǫ , m ) ∈ C ′ ( E ). Then, for each (( ǫ , m ) , ξ ) ∈ C ′ ( E ) × g ,0 = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 L d ( l E exp( tξ ) ( ǫ ) , l M exp( tξ ) ( m ))= D L d ( ǫ , m )( ξ E ( ǫ )) + D L d ( ǫ , m )( ξ M ( m )) , thus, for (( ǫ , m ) , ξ ) ∈ C ′ ( E ) × g D ,(5.7) J d ( ǫ , m )( ξ ) = − D L d ( ǫ , m )( ξ E ( ǫ )) = D L d ( ǫ , m )( ξ M ( m )) . By Proposition 3.8, if ( ǫ · , m · ) is a trajectory of M then (3.4) holds. Let ξ ∈ Γ( g D ), so that, for each k = 1 , . . . , N − ξ ( ǫ k , m k +1 ) ∈ g D ( ǫ k ,m k +1 ) ; then, for eachsuch k , we can evaluate the second condition in (3.4) at ( ξ ( ǫ k , m k +1 )) E ( ǫ k ) , ∈S ( ǫ k ,m k +1 ) ⊂ D ( ǫ k ,m k +1 ) to obtain − D L d ( ǫ k ,m k +1 )(( ξ ( ǫ k , m k +1 )) E ( ǫ k ))= D L d ( ǫ k − , m k ) ◦ dφ ( ǫ k )( ξ ( ǫ k , m k +1 )) E ( ǫ k )) | {z } =( ξ ( ǫ k ,m k +1 )) M ( m k ) + D L d ( ǫ k − , m k ) ◦ P (( ǫ k − , m k ) , ( ǫ k , m k +1 ))( ξ ( ǫ k , m k +1 ) E ( ǫ k ) , . Plugging this last result into (5.7), we see that( J d ) ξ ( ǫ k , m k +1 ) = ( J d ) ξ ( ǫ k − , m k )+ D L d ( ǫ k − , m k ) ◦ P (( ǫ k − , m k ) , ( ǫ k , m k +1 ))( ξ ( ǫ k , m k +1 ) E ( ǫ k ) , k = 1 , . . . , N −
1. The following result completes the previous discussion.
Proposition 5.21.
Let M = ( E, L d , D d , D , P ) ∈ ob LDP d and G be a symmetrygroup of M . Also, let ( ǫ · , m · ) be a discrete path in C ′ ( E ) . The following statementsare equivalent.(1) ( ǫ · , m · ) satisfies ν S d (( ǫ k − , m k ) , ( ǫ k , m k +1 )) = 0 for all k = 1 , . . . N − .(2) For all sections ξ ∈ Γ( g D ) and k = 1 , . . . N − , equation (5.8) is satisfied.Proof. The previous argument leading to (5.8) shows that 1 ⇒ k = 1 , . . . , N − ǫ k , m k +1 ) , s ) ∈S ( ǫ k ,m k +1 ) , there is ξ ∈ g such that s = ξ E ( ǫ k ). Then, as we are assuming that g D is a smooth vector bundle, there is ξ ∈ Γ( g D ) such that ξ ( ǫ k , m k +1 ) = ξ . Usingthis particular ξ , equation (5.8) leads to ν S d (( ǫ k − , m k ) , ( ǫ k , m k +1 ))( s ) = 0 and,eventually, to the validity of point 1. (cid:3) The equation (5.8) is known as the discrete nonholonomic momentum evolutionequation and has been considered, for instance, in [8] and [13].5.4.
Discrete LL systems on Lie groups.
In this section we review the notionsof discrete and continuous LL system on a Lie group G and then show how, inthe discrete case, LL systems are examples of DLDPSs. We also show that theirreduced and “momentum description” on Lie ( G ) ∗ are also examples of DLDPSs ina natural way and find their equations of motion, which agree with the ones thatappear in the literature. As a concrete case, we explore the discrete Suslov system. An LL system on the Lie group G is a nonholonomic mechanical system ( G, L, D )for whom G , acting on itself by left multiplication, is a symmetry group. Such asystem can be described alternatively as a reduced system on g := Lie ( G ) withreduced lagrangian ℓ and constraint subspace d ⊂ g . Yet another description, usinga (reduced) Legendre transform, is as a dynamical system on g ∗ satisfying theEuler–Poincar´e–Suslov equations (see, for instance [3]). Example 5.22.
A well known example of this type of system, due to G. Suslov [32],is a model for a rigid body, with a fixed point and constrained so that one of thecomponents of its angular velocity relative to the body frame vanishes. Explicitly,the configuration space is the Lie group G := SO (3), with Lagrangian L ( g, ˙ g ) := h I dL g − ( g )( ˙ g ) , dL g − ( g )( ˙ g ) i , where L g is the left multiplication by g map in G ,so that dL g − ( g )( ˙ g ) ∈ T e SO (3) = so (3) ≃ R (with the Lie algebra operationgiven by the vector product × ), I is the inertia tensor of the body and h· , ·i is thecanonical inner product of R . The nonholonomic constraint D is determined bythe subspace d := { ω ∈ R : ω = 0 } , requiring that D g := dL g ( e )( d ) ⊂ T g G for all g ∈ G . The dynamics of this nonholonomic system is completely determined by theEuler–Poincar´e–Suslov equations in so (3) ∗ that, in terms of the angular momentum M := I ω are ( ˙ M = M × ( I − M ) + λe ,M ∈ d ∗ := I ( d ) . A discrete analogue of the LL systems has been considered by Yu. Fedorov andD. Zenkov in [11] and, also, by R. McLachlan and M. Perlmutter in [27]; the purposeof this section is to show that all the discrete systems that have been considered(reduced, non-reduced and on g ∗ ) can be seen as DLDPS. A discrete LL system onthe Lie group G is a discrete nonholonomic system ( G, L d , D d , D nh ) for whom G ,acting on itself by left multiplication, is a symmetry group. As seen in Example 3.9,such a discrete nonholonomic system can naturally be seen as a DLDPS M LL =( E LL , L LLd , D LLd , D LL , P LL ) where the fiber bundle E LL → M LL is id G : G → G (with G acting by left multiplication on both G s), so that C ′ ( id G ) = G × G while L LLd = L d , D LLd = D d , P LL = 0 and D LL = p ∗ ( D nh ) for p : G × G → G theprojection onto the first factor. Example 5.23.
The discrete version of the Suslov system has been extensivelystudied in [11] and [16] as a reduced discrete mechanical system on SO (3) andas a discrete dynamical system obeying the discrete Euler–Lagrange–Suslov equa-tions. Here we follow the notation of [16] . This nonholonomic discrete me-chanical system is defined on the space G := SO (3), with discrete Lagrangian L d ( g , g ) := − Tr( g J g t ), where J := ( I + I − I ) 0 − I ( I + I − I ) − I − I − I
23 12 ( I + I − I ) is the mass tensor associated to the rigid body’s inertia tensor I (which can beassumed to have component I = 0). The infinitesimal variations are determined We consider here only the system whose discrete Lagrangian originates in ℓ (1 ,ǫ ) d with ǫ = 1.Still, the analysis remains valid for arbitrary ǫ and, also, for ℓ ( ∞ ,ǫ ) d . ONHOLONOMIC DISCRETE LAGRANGIAN REDUCTION BY STAGES 25 by the subspace d := − ω ω ω − ω − ω ω ∈ R × : ω = 0 ⊂ so (3)—that corresponds to the subspace d ⊂ R defined in Example 5.22 under theisomorphism R ≃ so (3)— through D nhg := dL g (1)( d ) = { gω ∈ R × : ω ∈ d } for any g ∈ G . In order to define the discrete dynamical constraints, we recallthe Cayley transform
Cay : so (3) → G defined by Cay( ω ) := (1 + ω )(1 − ω ) − ;then we define S d := Cay( d ). An explicit parametrization of S d is given in (25)of [16], but it won’t be necessary for our purposes. It suffices to say that W ∈ G is in S d if and only if its rotation axis is orthogonal to the e axis in R andthe angle of rotation is in ( − π, π ). Then, the discrete dynamical constraint is D d := { ( g , g ) ∈ G × G : g − g ∈ S d } . All together, ( G, L d , D d , D nh ) is a discretenonholonomic mechanical system in the sense of Example 3.9. Hence, as seen in thesame Example, M LL := ( E LL , L LLd , D LLd , D LL , P LL ) is a DLDPS, where the fiberbundle E LL → M LL is id G : G → G , L LLd := L d , D LLd := D d , D LL := p ∗ ( D nh ) and P LL = 0. We conclude that the discrete Suslov system can be seen as a discreteLagrange–D’Alembert–Poincar´e system in a natural way.Back in the setting of an arbitrary discrete LL system, we are given that G is asymmetry group of the nonholonomic discrete mechanical system ( G, L d , D d , D nh )thus, as noted in Remark 5.4, G is also a symmetry group of M LL in LDP d .In order to reduce M LL by G we need a discrete affine connection on the trivialprincipal bundle G → G/G = { [ e ] } . It is easy to check that A d ( g , g ) := g g − issuch a connection and, for completeness sake, we recall that, by Proposition 5.14,all the reduced systems M LL /G obtained using different discrete connections areisomorphic in LDP d . Thus, we have the reduced space M r := M LL / ( G, A d ).The total space of the fiber bundle underlying M r is e G G = ( G × G ) /G where the G -action is l G × Gg ( g , g ) := ( gg , gg g − ). The corresponding reduction morphismΥ A d : G × G → e G G × { [ e ] } is given by Υ A d ( g , g ) = ( π G × G,G ( g , g g − ) , [ e ]).It is easier to work with an isomorphic model of M r . Let e η : G × G → G bedefined by e η ( g , g ) := g − g g ; as e η is G -invariant for the action l G × Gg defined aboveit induces a smooth map η : ( G × G ) /G → G that turns out to be a diffeomorphism.If we view η as an isomorphism of the fiber bundle ( G × G ) /G → { [ e ] } onto G → { e } we can use Lemma 4.7 to obtain M η ∈ ob LDP d that is isomorphic to M r in LDP d . Explicitly, M η = ( E η , L ηd , D ηd , D η , P η ), where E η → M η is the fiber bundle G → { e } , L ηd = ℓ d —where ℓ d ( W ) = L d ( e, W )—, D ηd = S d × { e } —where W ∈ S d if and only if ( e, W ) ∈ D d —, D ηW = dR W ( e )( d ) and P η (( W , e ) , ( W , e ))( δW ,
0) = dL W ( δh )where ( δW ,
0) = ( − dR W ( e )( δh ) , ∈ e p ( D η ). With this information, the evolu-tion in M η can be determined using the section ν ηd defined in (3.3). In this case it is easy to describe all possible affine discrete connections: their domain can beextended to G × G and have discrete connection form A hd ( g , g ) = g h − g − for a fixed h ∈ G . Proposition 5.24. W k ∈ G for k = 0 , . . . , N − is a trajectory of M η if and onlyif ( W k ∈ S d for k = 0 , . . . , N − ,R ∗ W k +1 ( T W k +1 ℓ d ) − L ∗ W k ( T W k − ℓ d ) ∈ d ◦ for k = 0 , . . . , N − . Proof.
Indeed, by Proposition 3.8, W k is a trajectory of M η if and only if ( W k ∈ S d for k = 0 , . . . , N − ,ν ηd (( W k , e ) , ( W k +1 , e ))( D η ( W k +1 ,e ) ) = 0 for k = 0 , . . . , N − . As, in this case, ν ηd (( W k , e ) , ( W k +1 , e ))( D η ( W k +1 ,e ) ) = ν ηd (( W k , e ) , ( W k +1 , e ))( − dR W k +1 ( e )( d ))=( dℓ d ( W k +1 ) ◦ dR e g k ( e ) − dℓ d ( W k ) ◦ dL W k ( e ))( d )=( R ∗ W k +1 ( dℓ d ( W k +1 )) − L ∗ W k ( dℓ d ( W k )))( d ) , the statement follows. (cid:3) Remark 5.25.
When the discrete path W · is the reduction of the discrete path g · , that is, when W k = p ◦ η ◦ Υ A d ( g k , g k +1 ) = g − k g k +1 , by Corollary 5.16, W · isa trajectory of M η if and only if g · is a trajectory of M LL and, by Example 3.9, ifand only if g · is a trajectory of the discrete nonholonomic system ( G, L d , D d , D nh ). Remark 5.26.
The result of Proposition 5.24 is part (iv) of Theorem 3.2 in [11].The complete result can be read off Corollary 5.16 applied to M LL and M η . Example 5.27.
It is immediate that the discrete Suslov system described in Ex-ample 5.23 is an LL -system, so that M LL ∈ ob LDP d can be reduced by G = SO (3) and the connection A d ( g , g ) := g g − , defining a reduced system M r := M LL / ( G, A d ). Just as it was described above, the diffeomorphism η : ( G × G ) /G → G defined by η ( π G × G,G ( g , g )) := g − g g can be used to define M η ∈ ob LDP d with the property that η turns out to be an isomorphism between M r and M η .Explicitly, the fiber bundle E η → M η is G → { } , L ηd ( W ) = ℓ d ( W ) = − Tr( J W ), D ηd = S d × { } = Cay( d ) × { } , D ηW, = dR W (1)( d ) = { δhW ∈ R × : δh ∈ d } and,for any δW ∈ D ηW , , P η (( W , , ( W , δW ,
0) = − W δW W − . This discrete system M η provides the usual (reduced) description of the discreteSuslov system as a dynamical system on SO (3). Specializing Proposition 5.24 tothe current setting, we have that a discrete path W · in G is a discrete trajectory of M η if and only if ( W k ∈ S d = Cay( d ) for all k = 0 , . . . − ( W k +1 J − J W tk +1 ) + ( J W k − W tk J ) ∈ d ⊥ for all k = 0 , . . . , where we identify so (3) ∗ with so (3) using the inner product h A , A i := Tr( A A t );in particular, d ◦ corresponds to d ⊥ .Just as in the continuous case, it is possible to give an alternative model for M η as a dynamical system in (a submanifold of) g ∗ . Recall that the ( − ) dis-crete Legendre transform of L d is the map F − L d : G × G → T ∗ G defined by ONHOLONOMIC DISCRETE LAGRANGIAN REDUCTION BY STAGES 27 F − L d ( g , g ) := − D L d ( g , g ). Using the trivialization λ : T ∗ G → G × g ∗ definedby λ ( α g ) := ( g, L ∗ g ( α g )) we see that λ ( F − L d )( g , g ) = λ ( − D L d ( g , g )) = ( g , L ∗ g ( − D L d ( g , g ))) . If we define p : G × G → g ∗ by p ( g , g ) := L ∗ g ( − D L d ( g , g )), for any ξ ∈ g wehave p ( ξ ) = L ∗ g ( − D L d ( g , g ))( ξ ) = − D L d ( g , g )( dL g ( e )( ξ ))= − dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 L d ( g exp( sξ ) , g ) = − dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ℓ d (exp( − sξ ) g − g )= dℓ d ( W )( T e R W ( ξ )) = R ∗ W ( dℓ d ( W ))( ξ )for W := g − g . Thus, we define the reduced Legendre transform L : G → g ∗ by(5.9) L ( W ) := p = R ∗ W ( dℓ d ( W )) . In what follows we assume that L is a diffeomorphism . Then, applying Lemma 4.7to L and M η , we see that there is M S ∈ ob LDP d such that L is an isomorphismfrom M η into M S . Explicitly, E S → M S is g ∗ → { } , L Sd = ℓ d ◦ L − , D Sd = L ( S d ), D Sp = d ( L , L − ( p ) , e )( D η ( L − ( p ) ,e ) ) = R ∗L − ( p ) ( d L ( L − ( p )))( d ) and P S (( p k − , p k , δp k ,
0) = R ∗L − ( p k − ) ( d L ( L − ( p k − )))( δh )if δp k = − R ∗L − ( p k ) ( d L ( L − ( p k )))( δh ).A direct application of Proposition 3.8 to M S ∈ ob LDP d leads to the followingresult. Proposition 5.28.
A discrete path p · in g ∗ is a discrete trajectory of M S if andonly if (5.10) ( p k ∈ L ( S d ) , for k = 0 , . . . ( p k +1 − Ad ∗L − ( p k ) ( p k )) ∈ d ◦ for k = 0 , . . . , where Ad ∗ g := L ∗ g ◦ R ∗ g − . The system (5.10) is known as the discrete Euler–Poincar´e–Suslov equations (seeTheorem 3.3 in [11]).
Proof. As M S is constructed out of M η using the diffeomorphism L and Lemma 4.7,we know that ν Sd (( L ( W k ) , , ( L ( W k +1 ) , d L ( W k +1 )( δW k +1 ) , ν ηd (( W k , e ) , ( W k +1 , e ))( δW k +1 k, . In fact, under the usual regularity conditions on ℓ d , L is a local diffeomorphism and care mustbe taken, restricting the constructions to appropriate open subsets, where L is diffeomorphism. Using the computation of ν ηd developed in the proof of Proposition 5.24, ν Sd (( p k , , ( p k +1 , D S ( p k +1 , )= ν Sd (( L ( W k ) , , ( L ( W k +1 ) , d L ( W k +1 , e )( D η ( W k +1 ,e ) ))= ν ηd (( W k , e ) , ( W k +1 , e ))( D η ( W k +1 ,e ) )=( R ∗ W k +1 ( dℓ d ( W k +1 )) − L ∗ W k ( dℓ d ( W k )))( d )=( p k +1 − Ad ∗ W k ( p k ))( d ) = ( p k +1 − Ad ∗L − ( p k ) ( p k ))( d )As M S ∈ ob LDP d , the result follows from Proposition 3.8. (cid:3) Example 5.29.
The discrete Legendre transform L : G → so (3) ∗ associated tothe (reduced) discrete Suslov system M η described in Example 5.27 can be easilycomputed using (5.9): for W ∈ G and ξ ∈ so (3), L ( W )( ξ ) = dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ℓ d (exp( sξ ) W ) = dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 − Tr( J exp( sξ ) W ) = − Tr( J ξW )= −
12 (Tr( J ξW ) + Tr(( J ξW ) t )) = −
12 (Tr( W J ξ ) − Tr( J W t ξ ))= 12 Tr(( W J − J W t ) ξ t ) = h W J − J W t , ξ i , and we conclude that L ( W ) = W J − J W t . It is easy to check that L is a local diffeo-morphism, but it is not globally injective (nor onto). Still, the previous argumentscan be applied locally to a pair of domains where L restricts to a diffeomorphism.Hence, L can be used together with Lemma 4.7 to construct M S ∈ ob LDP d thatis isomorphic to M η . Proposition 5.28 provides the equations of motion for M S .Let p · be a discrete path in so (3) ∗ ≃ so (3), and define W k := L − ( p k ) ∈ G for all k . Then, by Proposition 5.28, p · is a trajectory of M S if and only if ( p k ∈ L ( S d ) for k = 0 , . . . , (equivalently) W k ∈ S d ,p k +1 − Ad W k ( p k ) ∈ d ⊥ for k = 0 , . . . where, as before, d ◦ is identified with d ⊥ . As, for any ξ ∈ so (3),Ad ∗ W k ( p k )( ξ ) = p k (Ad W k ( ξ )) = p k ( W k ξW − k ))= 12 Tr( p k ( W k ξW − k ) t ) = 12 Tr( W tk p k W k ξ t )=( W tk p k W k )( ξ )we see that p k +1 − Ad ∗ W k ( p k ) = p k +1 − W tk p k W k . Then, the discrete Euler–Poincar´e–Suslov equations (5.10) for M S are ( p k ∈ L ( S d ) for k = 0 , . . . ,p k +1 − W tk p k W k ∈ d ⊥ for k = 0 , . . . . This expression matches the ones given in (6.11) of [11] and (29) of [16].6.
Reduction by two stages
Having introduced a category of DLDPSs, a notion of symmetry group for aDLDPS and a process of reduction for such symmetric objects that is closed in thecategory, we study the problem of reduction by stages in this section. In other
ONHOLONOMIC DISCRETE LAGRANGIAN REDUCTION BY STAGES 29 words, when G is a symmetry group of M and H is a subgroup of G we wantto compare the result of the reduction M /G with that of the iterated reduction( M /H ) / ( G/H ) whenever possible.6.1.
Residual symmetry.
Let G be a symmetry group of M = ( E, L d , D d , D , P ) ∈ ob LDP d and H ⊂ G a closed normal subgroup. The following result proves that H is a symmetry group of M . Proposition 6.1.
Let G be a symmetry group of M ∈ ob LDP d . If H ⊂ G is aclosed Lie subgroup, then H is a symmetry group of M .Proof. By Lemma 5.7 of [14] we have that if G is a Lie group that acts on thefiber bundle ( E, M, φ, F ) and H ⊂ G is a closed Lie subgroup, then H acts onthe fiber bundle ( E, M, φ, F ) so that condition 1 in Definition 5.3 is valid. Theremaining conditions follow from the fact that G satisfies them and that H acts bythe restriction of the corresponding actions of G . (cid:3) In what follows we consider the action of the group
G/H on the system obtainedafter having reduced the symmetry of H . As a first step we recall the statementof Lemma 7.1 in [14], that establishes that G/H acts on the fiber bundle obtainedafter the first reduction stage.
Lemma 6.2.
Let G be a Lie group that acts on the fiber bundle ( E, M, φ, F ) and H ⊂ G be a closed normal subgroup. Define the maps l e H E π G,H ( g ) ( π E × H,H ( ǫ, w )) := π E × H,H ( l Eg ( ǫ ) , l Gg ( w )) ,l M/Hπ
G,H ( g ) ( π M,H ( m )) := π M,H ( l Mg ( m )) . Then l e H E , l M/H and the trivial right action on F × H define an action of G/H onthe fiber bundle ( e H E , M/H, p M/H , F × H ) . As in Section 5, these actions induce ”diagonal” actions on C ′ ( e H E ), C ′′ ( e H E )through definitions (5.1) and (5.2) and ”lifted” actions on the spaces T e H E and f p ∗ ( T C ′ ( e H E )) through definitions (5.3) and (5.5).The reduction of M by H requires the choice of a discrete connection A Hd on theprincipal H -bundle π M,H : M → M/H . It turns out that, under some conditionson A Hd that we explore next, G/H is a symmetry group of ˇ M := M / ( H, A Hd ). Lemma 6.3.
Let G be a Lie group that acts on M by the action l M in such a waythat π M,G : M → M/G is a principal G -bundle. Assume that H ⊂ G is a closednormal subgroup and that A Hd is a discrete connection on the principal H -bundle π M,H : M → M/H , whose domain U is G -invariant by the diagonal action l M × M .Then, the following statements are equivalent.(1) For each g ∈ G and ( m , m ) ∈ U , A Hd ( l Mg ( m ) , l Mg ( m )) = g A Hd ( m , m ) g − . (2) The submanifold Hor A Hd ⊂ M × M is G -invariant by the action l M × M .Proof. The proof is analogous to the proof of Lemma 7.2 in [14]. (cid:3)
Proposition 6.4.
Let G be a symmetry group of M = ( E, L d , D d , D , P ) ∈ ob LDP d and H ⊂ G be a closed normal subgroup. Choose a discrete connection A Hd on theprincipal H -bundle π M,H : M → M/H so that one of the conditions of Lemma 6.3holds. Then,
G/H is a symmetry group of ˇ M := M / ( H, A Hd ) = ( e H E , ˇ L d , ˇ D d , ˇ D , ˇ P ) .Proof. By Lemma 6.2,
G/H acts on the fiber bundle ( e H E , M/H, p M/H , F × H ).Thus, we have the G/H -action l C ′ ( e H E ) π G,H ( g ) ( v , r ) := ( l e H E π G,H ( g ) ( v ) , l M/Hπ
G,H ( g ) ( r )). Un-raveling the definitions, we have that, for g ∈ G ,(6.1) Υ A Hd ◦ l C ′ ( E ) g = l C ′ ( e H E ) π G,H ( g ) ◦ Υ A Hd . In addition, just as in the proof of Proposition 7.3 in [14], ˇ L d : e H E × ( M/H ) → R is l C ′ ( e H E ) -invariant.As ˇ D d := Υ A Hd ( D d ), for any g ∈ G , using (6.1) and the G -invariance of D d , wehave that l C ′ ( e H E ) π G,H ( g ) ( ˇ D d ) = l C ′ ( e H E ) π G,H ( g ) (Υ A Hd ( D d )) = Υ A Hd ( l C ′ ( E ) g ( D d )) = Υ A Hd ( D d ) = ˇ D d , so that ˇ D d is G/H -invariant.Also, as ˇ D := d Υ A Hd ( D ), for any g ∈ G , using (6.1) and the G -invariance of D ,we have l T C ′ ( e H E ) π G,H ( g ) ( ˇ D ) = dl C ′ ( e H E ) π G,H ( g ) ( d Υ A Hd ( D )) = d ( l C ′ ( e H E ) π G,H ( g ) ◦ Υ A Hd )( D )= d (Υ A Hd ◦ l C ′ ( E ) g )( D ) = d Υ A Hd ( dl C ′ ( E ) g ( D ))= d Υ A Hd ( l T C ′ ( E ) g ( D )) = d Υ A Hd ( D ) = ˇ D , so that ˇ D is G/H -invariant.The proof of the
G/H -invariance of ˇ P mimics the one given in Proposition 7.3of [14], adapted to the current context. (cid:3) Comparison with reduction by the full symmetry group.
Let G bea symmetry group of M = ( E, L d , D d , D , P ) ∈ ob LDP d . Then, if we choose adiscrete connection A Gd on the principal G -bundle π M,G : M → M/G we havethe reduced system M G := M / ( G, A Gd ) ∈ ob LDP d . If H ⊂ G is a closed normalLie subgroup, then H is a symmetry group of M (Proposition 6.1) and choosinga discrete connection A Hd on the principal H bundle π M,H → M/H we have thereduced system M H := M / ( H, A Hd ) ∈ ob LDP d . Last, if A Hd satisfies either one ofthe conditions that appear in Lemma 6.3, then G/H is a symmetry group of M H (Proposition 6.4). Then, choosing a discrete connection A G/Hd on the principal
G/H -bundle π M/H,G/G : M/H → ( M/H ) / ( G/H ), we have the discrete system M G/H := M H / ( G/H, A G/Hd ) ∈ ob LDP d .The goal of this section is to prove that M G and M G/H are isomorphic in
LDP d . The following diagram depicts the relation between the different discrete ONHOLONOMIC DISCRETE LAGRANGIAN REDUCTION BY STAGES 31
Lagrange–D’Alembert–Poincar´e systems and morphisms. M Υ A Gd (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ Υ A Hd " " ❊❊❊❊❊❊❊❊ M H Υ A G/Hd $ $ ❍❍❍❍❍❍❍❍❍ M G M G/H
At the ”geometric level” the corresponding manifolds and smooth maps are C ′ ( E ) Υ A Gd ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ Υ A Hd $ $ ■■■■■■■■■ C ′ ( e H E ) Υ A G/Hd & & ▲▲▲▲▲▲▲▲▲▲ C ′ ( e G E ) C ′ (cid:0) ] G/H e H E (cid:1) We can enlarge the previous diagram by adding the various diffeomorphismsassociated to a discrete connection and by taking into account the diagram (2.2).(6.2) C ′ ( E ) Υ A Gd (cid:127) (cid:127) π C ′ ( E ) ,G (cid:5) (cid:5) ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ π C ′ ( E ) ,H (cid:15) (cid:15) Υ A Hd (cid:30) (cid:30) C ′ ( E ) Hπ C ′ ( E ) H ,G/H (cid:15) (cid:15) F } } ③③③③③③③③③ Φ A Hd ∼ / / C ′ ( e H E ) π C ′ ( f HE ) ,G/H (cid:15) (cid:15) Υ A G/Hd % % ❑❑❑❑❑❑❑❑❑❑❑ C ′ ( e G E ) C ′ ( E ) G Φ A Gd ∼ o o C ′ ( E ) H G/HF ∼ o o Φ A Hd V ∼ / / C ′ ( e H E ) G/H Φ A G/Hd ∼ / / C ′ (cid:0) ] G/H e H E (cid:1) The following result introduces the new functions that appear in diagram (6.2)and explores their basic properties.
Lemma 6.5.
Under the previous conditions,(1) Φ A Hd : C ′ ( E ) H → C ′ ( e H E ) (see Proposition 2.10) is a G/H -equivariant dif-feomorphism. Then, it induces a smooth diffeomorphism Φ A Hd V : C ′ ( E ) H G/H → C ′ ( e H E ) G/H .(2) π C ′ ( E ) ,G : C ′ ( E ) → C ′ ( E ) G is a smooth H -invariant map. Then, it inducesa smooth map F : C ′ ( E ) H → C ′ ( E ) G .(3) F : C ′ ( E ) H → C ′ ( E ) G is a smooth G/H -invariant map. Then, it induces asmooth map F : C ′ ( E ) H G/H → C ′ ( E ) G . Also, F is a diffeomorphism.(4) Diagram (6.2) is commutative. Proof.
The proof is the same as that of Lemma 7.5 in [14]. (cid:3)
Theorem 6.6.
Consider the description given at the beginning of this section. Let F : C ′ ( ] G/H e H E ) → C ′ ( e G E ) be definided by F := Φ A Gd ◦ F ◦ (Φ A Hd V ) − ◦ (Φ A G/Hd ) − (see diagram (6.2) ). Then, F is an isomorphism in LDP d .Proof. From Proposition 2.10 both Φ A Gd and Φ A G/Hd are diffeomorphisms and, byLemma 6.5, both Φ A Hd V and F are diffeomorphisms. Thus, F is a diffeomorphism.As Υ A Hd and Υ A G/Hd are morphisms in
LDP d , the same is true for Υ A G/Hd ◦ Υ A Hd .Then, by Lemma 4.5 applied to Υ A Gd , Υ A G/Hd ◦ Υ A Hd and F , we conclude that F isan isomorphism in LDP d (cid:3) Theorem 6.7.
Consider the description given at the beginning of this section.(1) Let ( ǫ · , m · ) = (( ǫ , m ) , . . . , ( ǫ N − , m N )) be a discrete path in C ′ ( E ) . For k = 0 , . . . , N − we define the discrete paths ( v Hk , r Hk +1 ) := Υ A Hd ( ǫ k , m k +1 ) , ( v Gk , r Gk +1 ) := Υ A Gd ( ǫ k , m k +1 ) and ( v G/Hk , r
G/Hk +1 ) := Υ A G/Hd ( v Hk , r Hk +1 ) in C ′ ( e H E ) , C ′ ( e G E ) and C ′ ( ] G/H e H E ) respectively. Then, the following con-ditions are equivalent.(a) ( ǫ · , m · ) is a trajectory of M .(b) ( v G · , r G · ) is a trajectory of M G .(c) ( v H · , r H · ) is a trajectory of M H .(d) ( v G/H · , r G/H · ) is a trajectory of M G/H .(2) Let F : C ′ ( ] G/H e H E ) → C ′ ( e G E ) be the diffeomorphism defined in Theorem6.6. Then, F ( v G/Hk , r
G/Hk +1 ) = ( v Gk , r Gk +1 ) for all k .(3) The systems M G and M G/H are isomorphic in
LDP d .Proof. Point 1 is verified by Theorem 5.15, while point 3 follows from Theorem 6.6.The following computation proves point 2.( v Gk , r Gk +1 ) =Υ A Gd ( ǫ k , m k +1 ) = ( F ◦ Υ A G/Hd ◦ Υ A Hd )( ǫ k , m k +1 )=( F ◦ Υ A G/Hd )( v Hk , r Hk +1 ) = F ( v G/Hk , r
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