Lagrangian reduction of nonholonomic discrete mechanical systems by stages

In this work we introduce a category $LDP_d$ of discrete-time dynamical systems, that we call discrete Lagrange--D'Alembert--Poincar\'e systems, and study some of its elementary properties. Examples of objects of $LDP_d$ are nonholonomic discrete mechanical systems as well as their lagrangian reductions and, also, discrete Lagrange-Poincar\'e systems. We also introduce a notion of symmetry group for objects of $LDP_d$ and a process of reduction when symmetries are present. This reduction process extends the reduction process of discrete Lagrange--Poincar\'e systems as well as the one defined for nonholonomic discrete mechanical systems. In addition, we prove that, under some conditions, the two-stage reduction process (first by a closed and normal subgroup of the symmetry group and, then, by the residual symmetry group) produces a system that is isomorphic in $LDP_d$ to the system obtained by a one-stage reduction by the full symmetry group.


Introduction
Mechanical systems are dynamical systems that are used to model a wide variety of aspects of the real world, from the falling apple to the movement of astronomical objects, including machinery and billiards (see, for instance, [17] and [1]). One of the flavors of Mechanics -Lagrangian or Variational Mechanics-describes the evolution of a mechanical system using a variational principle defined in terms of a function, the Lagrangian, L : T Q → R, where Q is the configuration manifold of the system. Nonholonomic mechanical systems, which describe systems containing rolling or sliding contact (such as wheels or skates), add constraints -in the form of a non-integrable subbundle D ⊂ T Q-to the variational principle (see, for instance, [3] and [6]).
Numerical integrators and discrete mechanical systems. As in many applications it is essential to predict the evolution of a mechanical system, the equations of motion that can be derived from the corresponding variational principle must be solved. Solving these ordinary differential equations can be quite difficult in practice, so numerical integrators are used to find approximate solutions to those equations. The standard methods for numerically approximating solutions of ODEs do not necessarily preserve the structural characteristics of the solutions of the equations of motion of mechanical systems (see [18]). Discrete mechanical systems were introduced as a way of modeling discrete-time analogues of mechanical systems; the evolution of a discrete mechanical system is also defined in terms of a variational principle for the discrete Lagrangian L d : Q × Q → R; this formalism is extended to deal with more general systems, including forced discrete systems as well as discrete nonholonomic ones (see [24] and [9]). The equations of motion of discrete mechanical systems are algebraic equations whose solutions are numerical integrators for the corresponding continuous system. In many cases, these integrators have very good 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 a mechanical 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 process is usually known as the reduction of the given system and the resulting system is known as the reduced system. In the case of Classical Mechanics, this idea seems to go back as far as the work of Lagrange. Over time, it has become a technique that has been applied in both the Lagrangian and Hamiltonian formalisms, for unconstrained 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 systems with and without constraints (see, for instance, [21], [27], [20] and [13]).
It is well known that, in most instances, the reduction of a mechanical system is not a mechanical system but, rather, a more general dynamical system: that is, while the dynamics of a mechanical system on Q is defined using a variational principle for the Lagrangian, defined on T Q in the continuous case or on Q × Q in the discrete case (and, maybe, other additional data), the dynamics of the reduced system is determined by a function that is usually not defined on a tangent bundle or a Cartesian product (of a manifold with itself). This can be problematic if one expects to analyze the reduced system with the same techniques as the original one. That issue has usually been solved by passing from the family of mechanical systems to a larger class of dynamical systems, where there is a reduction process that is closed within this larger class. Such is the case, for example, of mechanical systems on Lie algebroids and Lie groupoids (see [19] and [20]). In this paper we follow this guiding principle, but choose the larger class following ideas adapted from [7] and [14].
Reduction by stages. Sometimes, it may be convenient to eliminate part of the symmetric behavior of a mechanical system, while keeping some residual symmetry that could be analyzed at a later point, if so desired. In this case a second reduction step to eliminate the residual symmetry is possible. A natural question is, in that case, whether the result of this two-stage reduction is equivalent to the full reduction of all the symmetries at one time. The equivalence of the two-stage and one-stage reduction processes has been established in several cases. For instance, for Lagrangian systems without constraints by H. Cendra, J. Marsden and T. Ratiu in [7], for Lagrangian systems with nonholonomic constraints by H. Cendra and V. Díaz in [5], for Hamiltonian systems by J. Marsden et al. in [26] and, for unconstrained discrete mechanical systems by the authors in [14].
Aims. The main purpose of the present work is to establish the same equivalence described in the previous paragraph for nonholonomically constrained discrete-time mechanical systems. In this respect, the paper is an extension of [14] to the constrained setting that parallels [5] in the discrete-time context. We stress that, for the 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 are completely independent of the discretization process chosen to produce the discrete dynamical system in question, if one was used at all. As an aside, we also mention that, at the moment, the very interesting subject of geometric discretization of nonholonomic mechanical systems should be regarded as "work in progress", without conclusive hard results on the quality of the numerical integrators obtained.
Constructions and results. Except for a few special cases, the reduced system obtained from a nonholonomic discrete mechanical system via the general reduction process defined in [13] is not a discrete mechanical system, constrained or not. So, as we mentioned above, the first step is to construct a family of dynamical systems that contains all the systems of interest -nonholonomic discrete mechanical systems as well as their reductions. The discrete Lagrange-D'Alembert-Poincaré 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 nonholonomic infinitesimal variation chaining map P (see Definition 3.3) as well as a regular submanifold D d ⊂ E × M , the kinematic constraints, and a subbundle D ⊂ p * 1 T E (where p 1 : E × M → E is the projection), the variational constraints. All such systems are discrete-time dynamical systems whose trajectories are determined by a variational principle. Examples of DLDPSs include the nonholonomic discrete mechanical systems (Example 3.9) and, when they are symmetric, their reductions by the procedure defined in [13] (Section 3.2); also, the discrete Lagrange-Poincaré systems considered in [14] are DLDPSs. A convenient notion of morphism between DLDPSs is introduced and a category LDP d is so defined. The category LP d of discrete Lagrange-Poincaré 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 on the 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 additional piece of data: an affine discrete connection on a certain principal G-bundle; interestingly, we prove that the reduced systems obtained using different affine discrete connections are always isomorphic in LDP d (Proposition 5.14). Also, the reduction mapping M → M/G is a morphism in LDP d ; Corollary 5.16 and Theorem 5.17 prove that the reduction mapping determines a bijective correspondence between the trajectories of M and those of M/G. It is important to notice that both the notion of symmetry group and the reduction process extend the ones already in use for nonholonomic discrete mechanical systems as well as for discrete Lagrange-Poincaré systems.
When G is a symmetry group of the DLDPS M and H ⊂ G is a closed and normal subgroup, H is a symmetry group of M, so we can consider the reduced system M H := M/H using a discrete affine connection A H d . Then, under a condition on A H d , we prove that G/H is a symmetry group of M H , so that we can consider a new reduced system M G/H := M H /(G/H). One of the main results of the 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 as well as some basic results on principal bundles. Section 3 introduces the DLDPSs and their dynamics. Section 3.2 shows that both nonholonomic discrete mechanical systems 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 4 introduces the category LDP d whose objects are DLDPSs. Symmetries and a reduction 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 on a Lie group G. Finally, Section 6 establishes the equivalence between the two-stage and the single-stage reduction process, under appropriate conditions.
Future work. It would be very interesting to connect the analysis of this paper with a discretization process for continuous mechanical systems. This would allow, for instance, the estimation of the error made when using a DLDPS as an approximation of a (continuous) mechanical system. Indeed, a first step would be to tackle this same problem with no constraints, that is, for discrete Lagrange-Poincaré systems ([14]). It should be noted that this error analysis is only known for unconstrained systems (see [29]) and forced mechanical systems (see [10] and [12]). Another avenue for exploration would be the study of possible Poisson structures in DLDPSs: even though DLDPSs do not have a canonical Poisson structure, some of them do (those coming from discrete mechanical systems, for instance) and it would be interesting to see how those structures behave under the reduction process.
Notation. Throughout the paper many spaces are Cartesian products. In general we denote the corresponding projections by p k : N j=1 X j → X k and the obvious adaptations. Also, l X and r X will denote left and right smooth actions of a Lie group on the manifold X. If G acts on the left on X we denote the corresponding quotient map by π X,G : X → X/G.

Revision of some discrete tools
In this section we review some basic notions and results about affine discrete connections and smooth fiber bundles.
2.1. Affine discrete connections. Let l Q : G × Q → Q be a smooth left action of the Lie group G on the manifold Q. We consider several other actions of G; for example, we have the G actions l Q×Q and l Q×Q2 on Q×Q defined by l Q×Q g (q 0 , q 1 ) := (l Q g (q 0 ), l Q g (q 1 )) and l Q×Q2 g (q 0 , q 1 ) := (q 0 , l Q g (q 1 )). We also consider the left G-action on itself given by l G g (g ′ ) := gg ′ g −1 . Definition 2.1. Let γ : Q → G be a smooth G-equivariant map with respect to l Q and l G , Γ := {(q, l Q γ(q) (q)) : q ∈ Q} and Hor ⊂ Q × Q be an l Q×Q -invariant submanifold 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 the level of A d . As in this paper the only type of discrete connection that we consider is the affine, we will simply call them discrete connections.
Given a discrete connection A d on π Q,G : Q → Q/G, the space Proof. See point 1 of Proposition 2.4 in [15]. Proposition 2.3. Let A d be a discrete connection with level γ and domain U on the principal G-bundle π Q,G : Proof. See Proposition 2.5 in [15].
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 where g ∈ G is the element that appears in Proposition 2.3.
In what follows we consider the open set Definition 2.5. Let A d be a discrete connection on the principal G-bundle π Q,G : That is h q0 d (r 1 ) = h d (q 0 , r 1 ) := (q 0 , q 1 ) ⇔ (q 0 , q 1 ) ∈ Hor A d and π Q,G (q 1 ) = r 1 . In addition we define h q0 d := p 2 • h q0 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 are smooth 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 0 , r 1 ) := (l Q g (q 0 ), r 1 ), 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 0 , g 1 ∈ G, Proof. All of the following references are from [15] and must be adapted to affine discrete connections. Point 1 is Lemma 3.2 (smoothness of A d ) and Point 2 in 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 0 , q 1 ) ∈ Q × Q : A(q 0 , q 1 ) = e} defines an affine discrete connection with level set γ(q) := A(q, q) −1 and whose discrete connection 1-form is A.

Principal bundles.
Here we review a few basic notions and results on principal bundles. We refer to Section 9 of [14] and its references for additional details.
Remark 2.9. When a Lie group G acts on the fiber bundle (E, M, φ, F ) and on the 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 −1 hg is known as the conjugate bundle and is denoted by 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 : Proof. This is Proposition 2.6 in [14] adapted to affine discrete connections.
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 Ψ A d and Φ A d to appropriate open sets, where the results of the Proposition 2.10 hold. We will ignore this point and keep working as if A d were globally defined in order to avoid a more involved notation.
We have the commutative diagram 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, Proof. This is Lemma 2.8 in [14] adapted to affine discrete connections.
All together, we have the following commutative diagram Proposition 2.13. Let ρ : X → Y be a principal G-bundle, Z ⊂ X a G-invariant regular 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 regular submanifold.
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 −1 (U ) → U × G that is G-equivariant (for l U×G g (u, g ′ ) := (u, gg ′ )) and that p 1 • Φ U = ρ| ρ −1 (U) .
As Z ⊂ X is a regular submanifold and ρ −1 (U ) is an open subset of X, Z ∩ ρ −1 (U ) is a regular submanifold of ρ −1 (U ). Then, as Φ U is a diffeomorphism, Let i : U → U × G be given by i(u) := (u, e), where e is the identity of G; it is easy to check that S ∩ U = i −1 ( Z). Then i is smooth and, furthermore, for each is a regular submanifold of U (see Theorem 6.30 in [22]).

Discrete Lagrange-D'Alembert-Poincaré systems
In this section we introduce a type of discrete-time dynamical system that contains, among other examples, all nonholonomic discrete mechanical systems as well as their reductions, as defined in [13].
Given a DLDPS M = (E, L d , D d , D, P) we have the vector bundle ( p 34 * (D)) * → C ′′ (E). Let ν d be the smooth section of this bundle defined by The next result characterizes the trajectories of a DLDPS in terms of its equations of motion.
Proof. Let (δǫ · , δm · ) be a nonholonomic infinitesimal variation over (ǫ · , m · ) with fixed endpoints. A straightforward but lengthy computation using Definition 3.5 shows that As the δǫ k ∈ D (ǫ k ,m k+1 ) are arbitrary, the result then follows by Definition 3.7.
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 nonholonomic mechanical system is a collection (Q, In an analogous way to what happens with discrete mechanical systems and the discrete Lagrange-Poincaré systems in [14] (Example 3.12), a discrete nonholonomic mechanical system can be seen as a discrete Lagrange-D'Alembert-Poincaré system with φ = id Q (so that C ′ (E) = Q × Q), P = 0, the same D d as kinematic constraints and D : In this case, a discrete path in C ′ (E) can be identified with path q · = (q 0 , . . . , 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 − 1 and . . , N − 1 for k = 1, . . . N − 1, which are the same equations of motion of the discrete nonholonomic 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 trajectories of a DLDPS is guaranteed in a neighborhood of a given trajectory.

3.2.
Nonholonomic discrete mechanical systems with symmetry. Symmetries of a nonholonomic discrete mechanical system (Example 3.9) were considered in [13]. Even more, a reduction process was developed there so that a new discretetime dynamical system -called the reduced system-was constructed starting from a symmetric nonholonomic discrete mechanical system and whose dynamics captured the essential features of that of the original system. Unfortunately, that reduced system is not usually a nonholonomic discrete mechanical system. The goal of this section is to recall those constructions and results from [13] and prove that, indeed, the reduced system can be interpreted as a DLDPS whose trajectories in the sense of Definition 3.7 are the same as those of the reduced system (in the sense of [13]).
Let l Q be a left G-action on Q such that π Q,G : Q → Q/G is a principal G-bundle and fix a discrete connection A d on this bundle. In this case, the commutative diagram (2.2) turns into where G := (Q × G)/G with G acting on Q by l Q and on G by conjugation and A Lie group G is a symmetry group of the discrete nonholonomic mechanical system (Q, L d , D d , D nh ) if π Q,G : Q → Q/G is a principal bundle, L d and D d are invariant 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 mapĽ d : The next result of [13] relates the variational principle that describes the dynamics of (Q, L d , D d , D nh ) with a variational principle for its reduced system defined on G × (Q/G).
Theorem 3.11. Let G be a symmetry group of the discrete nonholonomic me- and v k := π Q×G,G (q k , w k ) be the corresponding discrete paths in Q/G, G and 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 nh q k for all k.
(2) (v k , r k+1 ) ∈Ď d for all k and dŠ d (v · , r · )(δv · , δr · ) = 0 for all (δv · , δr · ) such that for k = 0, . . . , N − 1 and where δq · is a fixed-endpoint variation on q · such that δq k ∈ D nh q k for all k. Remark 3.12. Theorem 3.11 is part of Theorem 5.11 in [13]; this last result requires 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 have omitted 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 the discrete-time dynamical system on G × (Q/G) whose trajectories are discrete paths that satisfy the variational principle of point 2 in Theorem 3.11. Next we construct a DLDPS that will, eventually, be equivalent to this reduced system. We define the fiber bundle φ : ) defines a subbundle of T (C ′ ( G)) (this is a special case of Lemma 5.8 in Section 5.2).
In order to define the NIVCMP ∈ hom( p 34 * (Ď), ker(dp Q/G )) we consider the . ThatP is well defined follows from Lemma 3.13.
In this way, we associate a DLDPS M := (E,Ľ d ,Ď d ,Ď,P) to the reduced system and 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 statements are true.
Proof. See point 2 in Lemma 5.1 for point 1 and Lemma 5.10 for points 2 and 3.
The following result of [14] proves that all discrete paths in The following result compares the nonholonomic infinitesimal variations with fixed endpoints on the discrete path (v · , r · ) in C ′ (E) = C ′ ( G) for the system M, with the variations defined in point 2 of Theorem 3.11 on the same discrete path.
. Then, the following statements are true.
(1) Given a fixed-endpoint variation δq · in D nh on q · , the infinitesimal variation (δv · , δr · ) defined in point 2 of Theorem 3.11 by (3.7) is a nonholonomic infinitesimal variation on (v · , r · ) with fixed endpoints (in the sense of Definition 3.5) for M. (2) Given a nonholonomic infinitesimal variation (δv · , δr · ) on (v · , r · ) with fixed endpoints (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 nh q k and let (δv · , δr · ) be the variation defined by (3.7) in terms of δq · . Let We want to see that (δv · , δr · ) is a nonholonomic infinitesimal variation on (v · , r · ) with fixed endpoints. Recall that and given that δq · is a fixed-endpoint variation, we notice that Thus, (δv · , δr · ) satisfies conditions (3.2). By construcion of the discrete path (v · , r · ), r k = p Q/G (v k ) for all k and since 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.
Proof. The proof that F satisfies the points 1 to 4 and 7 in Definition 4.1 and the last assertion of the statement is the same as in the proof of Lemma 4.5 in [14]. We want to prove that F satisfies points 5 and 6 of Definition 4.1. Since Υ ′ ∈ mor LDP d (M, M ′ ) and Υ ′′ ∈ mor LDP d (M, M ′′ ) and the previous diagram is commutative, we have that Proof. By hypothesis, (ǫ · , m · ) is a discrete path in C ′ (E). It follows from its definition, the fact that Υ is a morphism and (4.1) that (ǫ ′ · , m ′ · ) is a discrete path in C ′ (E).
In order to prove point 2, assume that (ǫ ′ · , m ′ · ) is a trajectory of M ′ . Then, An argument similar to the one used in the proof of point 1 shows that (ǫ · , m · ) satisfies the criticality condition in M, so that it is a trajectory of M, thus proving point 2.
The following result, whose proof is immediate, is useful when working with concrete DLDPSs.

Reduction of discrete Lagrange-D'Alembert-Poincaré systems
In this section we introduce the notion of symmetry group of a DLDPS and a reduction 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 and compare the dynamics of the reduced system to that of the original one.

Discrete Lagrange-D'Alembert-Poincaré 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 Also, we consider the G-actions on ker(dφ) ⊂ T E and on p 34 * T (C ′ (E)) ⊂ T C ′′ (E) given by  m1) is an isomorphism of vector spaces for every (ǫ 0 , m 1 ) ∈ C ′ (E).
is a principal G-bundle with structure group G.
Proof. This result is almost identical to Lemma 5.1 in [14], the only difference being that, 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 discrete connections.
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. Given a discrete path (v · , r · ) = ((v 0 , r 1 ), . . . , Proof. This is Proposition 5.2 in [14] except for using affine discrete connections instead of discrete connections, which doesn't alter the proof. To prove point 3 we start by noting that Proof. Assume that G is a symmetry group of M. Then, by definition, G acts on the fiber bundle φ : E → M . We have to prove that l Proving that l We defineĽ d : Lemma 5.7. The spaceĎ d : is the projection onto the first factor. In addition, rank(Ď) = rank(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-Poincaré system constructed in Example 3.9. Let G be a symmetry group of (Q, L d , D d , D nh ). As noted in Remark 5.4, G is a symmetry group of M. Let A d be a discrete connection on the principal G-bundle π Q,G : This DLDPS coincides with the one obtained 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 reduction construction of discrete nonholonomic mechanical systems introduced in [13]. The following result proves that given a discrete Lagrange-D'Alembert-Poincaré system with symmetry, the reduced systems obtained using different discrete connections are all isomorphic in LDP d . Proof. The proof is analogous to the proof of Proposition 5.14 in [14], using Lemmas 2.12 and 4.5 and Proposition 5.13.
Then, the following statements are equivalent.
Proof. The equivalence 1 ⇔ 2 was demonstrated in Proposition 3.8. The equivalence 3 ⇔ 4 follows from Proposition 3.8 applied to the system M/(G, A d ). The equivalence 1 ⇔ 3 follows from Theorem 5.15.
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 ( ǫ 0 , m 1 ) ∈ D d such that Υ A d ( ǫ 0 , m 1 ) = (v 0 , r 1 ). Then, there exists a unique trajectory (ǫ · , m · ) of M such that (ǫ 0 , m 1 ) = ( ǫ 0 , m 1 ) and Υ A d (ǫ k , m k+1 ) = (v k , r k+1 ) for all k.
The equation (5.8) is known as the discrete nonholonomic momentum evolution equation and has been considered, for instance, in [8] and [13]. 5.4. Discrete LL systems on Lie groups. In this section we review the notions of discrete and continuous LL system on a Lie group G and then show how, in the discrete case, LL systems are examples of DLDPSs. We also show that their reduced and "momentum description" on Lie(G) * are also examples of DLDPSs in a natural way and find their equations of motion, which agree with the ones that appear 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 a system can be described alternatively as a reduced system on g := Lie(G) with reduced lagrangian ℓ and constraint subspace d ⊂ g. Yet another description, using a (reduced) Legendre transform, is as a dynamical system on g * satisfying the Euler-Poincaré-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 the components 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,ġ) := 1 2 IdL g −1 (g)(ġ), dL g −1 (g)(ġ) , where L g is the left multiplication by g map in G, so that dL g −1 (g)(ġ) ∈ T e SO(3) = so(3) ≃ R 3 (with the Lie algebra operation given by the vector product ×), I is the inertia tensor of the body and ·, · is the canonical inner product of R 3 . The nonholonomic constraint D is determined by the subspace d := {ω ∈ R 3 : ω 3 = 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 the Euler-Poincaré-Suslov equations in so (3)  A discrete analogue of the LL systems has been considered by Yu. Fedorov and D. Zenkov in [11] and, also, by R. McLachlan and M. Perlmutter in [27]; the purpose of 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 on the 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 Example 5.23. The discrete version of the Suslov system has been extensively studied in [11] and [16] as a reduced discrete mechanical system on SO(3) and as a discrete dynamical system obeying the discrete Euler-Lagrange-Suslov equations. Here we follow the notation of [16] 1 . This nonholonomic discrete mechanical system is defined on the space G := SO(3), with discrete Lagrangian   is the mass tensor associated to the rigid body's inertia tensor I (which can be assumed to have component I 12 = 0). The infinitesimal variations are determined 1 We consider here only the system whose discrete Lagrangian originates in ℓ 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 − 1, ν η d ((W k , e), (W k+1 , e))(D η (W k+1 ,e) ) = 0 for k = 0, . . . , N − 2. As, in this case, , the statement follows.
Remark 5.25. When the discrete path W · is the reduction of the discrete path g · , that is, when W k = p 1 • η • Υ A d (g k , g k+1 ) = g −1 k g k+1 , by Corollary 5.16, W · is a trajectory of M η if and only if g · is a trajectory of M LL and, by Example 3.9, if and 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 Example 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 0 , g 1 ) := g 1 g −1 0 , 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 0 , g 1 )) := g −1 0 g 1 g 0 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 : δh ∈ d} and, for any δW 1 ∈ D η W1,1 , P η ((W 0 , 1), (W 0 , 1))(δW 1 , 0) = −W 0 δW 1 W −1 1 . This discrete system M η provides the usual (reduced) description of the discrete Suslov system as a dynamical system on SO(3). Specializing Proposition 5.24 to the current setting, we have that a discrete path W · in G is a discrete trajectory of M η if and only if . . , where we identify so(3) * with so(3) using the inner product A 1 , A 2 := 1 2 Tr(A 1 A t 2 ); 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 (−) discrete Legendre transform of L d is the map F − L d : G × G → T * G defined by F − L d (g 0 , g 1 ) := −D 1 L d (g 0 , g 1 ). Using the trivialization λ : T * G → G × g * defined by λ(α g ) := (g, L * g (α g )) we see that ).

Reduction by two stages
Having introduced a category of DLDPSs, a notion of symmetry group for a DLDPS and a process of reduction for such symmetric objects that is closed in the category, we study the problem of reduction by stages in this section. In other words, when G is a symmetry group of M and H is a subgroup of G we want to 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 a closed 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 the fiber bundle (E, M, φ, F ) and H ⊂ G is a closed Lie subgroup, then H acts on the fiber bundle (E, M, φ, F ) so that condition 1 in Definition 5.3 is valid. The remaining conditions follow from the fact that G satisfies them and that H acts by the restriction of the corresponding actions of G.
In what follows we consider the action of the group G/H on the system obtained after having reduced the symmetry of H. As a first step we recall the statement of Lemma 7.1 in [14], that establishes that G/H acts on the fiber bundle obtained after 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 HE π G,H (g) (π E×H,H (ǫ, w)) := π E×H,H (l E g (ǫ), l G g (w)),