Discrete dynamics in implicit form

A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie groupoid $G$ may be described in terms of Lagrangian implicit difference equations of the corresponding cotangent groupoid $T^*G$. Other situations include finite difference methods for time-dependent linear differential-algebraic equations and discrete nonholonomic Lagrangian systems, as particular examples.


Introduction
Lie algebroids and groupoids have deserved a lot of interest in recent years since these concepts generalize the traditional framework of tangent bundles and its discrete version, cartesian product of manifolds, to more general situations. In particular, it is well-known that the geometric description of the Euler-Lagrange equations of a mechanical system determined by a Lagrangian function L, relies on the intrinsic geometry of the tangent bundle T Q, the velocity phase space of a configuration manifold Q. In the case when the Lagrangian L is invariant under the action of a Lie group G, the description, in this case, relies on the geometry of the quotient space T Q/G and the equations describing the dynamics are called Lagrange-Poincaré equations [4]. In this sense, Weinstein [29] showed that the common geometric structure of the Lagrange-Poincaré equations is essentially the same as the one of the Euler-Lagrange equations, namely that of a Lie algebroid. In the case of Lagrangian systems on the usual tangent bundles, the tangent bundle carries a canonical Lie algebroid structure which is given by the usual Lie algebra of vector fields on Q. In the case of reduced lagrangian systems, we use the Atiyah algebroid T Q/G → Q/G to describe the evolution equations.
In [18], it is described geometrically discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids, in particular, the type of equations analyzed include the classical discrete Euler-Lagrange equations, the discrete Euler-Poincaré and discrete Lagrange-Poincaré equations. These results have applications for the construction of geometric integrators for continuous Lagrangian systems (reduced or not) [9,24]. On the other hand, in [26,27], W.M. Tulczyjew proved that it is possible to interpret the ordinary Lagrangian and Hamiltonian dynamics as Lagrangian submanifolds of suitable special symplectic manifolds. These results were successfully extended to general Lie algebroids in [12]. More generally, in [14,15,16] implicit differential equations are considered. More precisely, an implicit differential equation on a manifold Q is a submanifold D ⊂ T Q. Given such a datum, the problem of integrability is discussed, looking for a subset S ⊆ D where, for any v ∈ S there exists a curve γ : I → Q such thatγ(0) = v andγ(t) ∈ D for any t ∈ I. An algorithm for extracting the integrable part of an implicit differential equation is formulated. More precisely, if D ⊂ T Q is an implicit differential equation, the following sequence {(C k , D k )} is considered where τ Q : T Q → Q is the canonical projection. Clearly, D k ⊆ D k−1 and C k−1 ⊆ C k for any k. Under some smoothness assumptions, it is proved in [15] that if the algorithm stabilizes, that is, there exists k such that Dk = Dk −1 , then Dk ⊆ D is the (possibly empty) integrable part of D. The relation with Lagrangian mechanics is the following one. Given a Lagrangian L : T P → R, one can consider the implicit differential equation S L ⊆ T (T * P ) using the image of the differential of L, dL(T P ) ⊂ T * (T P ) and the canonical isomorphism A M : T * (T P ) → T (T * P ). If we apply the integrability algorithm, the first step will give us the solution of the Euler-Lagrange equations. Indeed, if (q i , v i ) are local coordinates on T P then Taking as starting point the results of discrete mechanics on Lie groupoids [10,18] and a remark on implicit differential equations on Lie algebroids in [11], in this paper, submanifolds of a Lie groupoid G are interpreted as systems of implicit difference equations on G. We study the problem of integrability of these implicit difference equations, proposing an algorithm which extracts the integrable part (backward and/or forward) of this type of systems and analyze the geometric properties of them. In our opinion, this approach offers greater conceptual clarity. Discrete Lagrangian systems (unconstrained and nonholonomic) are studied within the framework of implicit difference equations. Moreover, for the unconstrained case, the resulting implicit difference equation is a Lagrangian submanifold, thus obtaining a discrete version of the results in [12]. Other examples include finite difference methods for time-dependent linear differential algebraic equations. The results about implicit difference equations are useful for systems more general than the studied on this paper; in particular, discrete optimal control theories, discrete systems with external constraints ... See for instance [10,20].

Preliminaries: Lie algebroids and Lie groupoids
In this Section, we will recall the definition of a Lie groupoid and some generalities about them are explained (for more details, see [13]).
A groupoid over a set M is a set G together with the following structural maps: • A pair of surjective maps α : G → M , the source, and β : G → M , the target. These maps define the set of composable pairs • A multiplication m : G 2 → G, to be denoted simply by m(g, h) = gh, such that -α(gh) = α(g) and β(gh) = β(h).
A groupoid G over a set M will be denoted simply by the symbol G ⇒ M . The groupoid G ⇒ M is said to be a Lie groupoid if G and M are manifolds and all the structural maps are differentiable with α and β differentiable submersions. If G ⇒ M is a Lie groupoid then m is a submersion, ǫ is an injective immersion and i is a diffeomorphism. Moreover, if x ∈ M , α −1 (x) (resp., β −1 (x)) will be said the α-fiber (resp., the β-fiber) of x.
Using the above facts, we may introduce a Lie algebroid structure ([[·, ·]], ρ) on AG, which is defined by for X, Y ∈ Γ(τ ) and x ∈ M . We recall that a Lie algebroid A over a manifold M is a real vector bundle τ : A → M together with a Lie bracket [[·, ·]] on the space Γ(τ ) of the global cross sections of τ : A → M and a bundle map, called the anchor map, such that if we also denote by ρ : Γ(τ ) → X(M ) the homomorphism of C ∞ (M )-modules induced by the anchor map then is called a Lie algebroid over M (see [13]).
Next, we will present some examples of Lie groupoids and their associated Lie algebroids which will be useful for our purposes.
Any Lie group G is a Lie groupoid over {e}, the identity element of G, and the Lie algebroid associated with G is just the Lie algebra g of G. On the other hand, given a manifold M , the product manifold M × M is a Lie groupoid over M , called the pair or banal groupoid, in the following way: α is the projection onto the first factor and β is the projection onto the second factor, m((x, y), (y, z)) = (x, z), The cotangent groupoid. (See [5], for details). Let G ⇒ M be a Lie groupoid. If A * G is the dual bundle to AG then the cotangent bundle T * G is a Lie groupoid over A * G. The projectionsβ andα, the partial multiplication ⊕ T * G , the identity sectionǫ and the inversionĩ are defined as follows, β(µ g )(X) = µ g ((T ǫ(β(g)) l g )(X)), for µ g ∈ T * g G and X ∈ A β(g) G, Note thatǫ(A * G) is just the conormal bundle of M ∼ = ǫ(M ) as a submanifold of G. In addition, A * G is endowed with a linear Poisson structure. For this Poisson structure on A * G,α : T * G → A * G is a Poisson map andβ : T * G → A * G is an anti-Poisson (for more details, see [5,13]).

Discrete dynamics in implicit form
Motivated by the studies of implicit differential equations in [14,15,16] and its generalization to Lie algebroids in [11], we introduce the notion of implicit difference equations on Lie groupoids and the corresponding notion of integrability.
An admissible sequence on the Lie groupoid G ⇒ M is a mapping γ G : I ∩ Z −→ G such that β(γ G (i)) = α(γ G (i + 1)) for all i, i + 1 ∈ I ∩ Z. Here I is an interval on R.
A solution of an implicit difference equation E ⊂ G is an admissible sequence Let E be a submanifold of the Lie groupoid G ⇒ M . Then E is said to be If these conditions hold for all g, we say that E is forward integrable, backward integrable or integrable, respectively. 1) E is forward integrable if and only if for each g ∈ E exists at least an h ∈ E such that (g, h) ∈ G 2 or, equivalently, E ⊆ β −1 (α(E)).

2) E backward integrable if and only if for each g ∈ E exists at least an
Conversely, if g ∈ E there exists a g 1 ∈ E such that (g, g 1 ) ∈ G 2 . Since g 1 ∈ E and using again the hypothesis, there is g 2 ∈ E satisfying (g 1 , g 2 ) ∈ G 2 . Therefore, we can construct a sequence With a similar argument we deduce 2), and 3) is a direct consequence of 1) and 2).
This example can be generalized to bisections on Lie groupoids. We recall that a bisection on a Lie groupoid is a map σ : M → G which is a section of α and such that β • σ is a diffeomorphism. Given a bisection, is a solution with γ(0) = g.
A similar result holds forẼ f andẼ b . Thus,Ẽ f andẼ b are forward and backward integrable, respectively. Now, it is easy to prove the following From Proposition 3.2, we have the following obvious relations Propositions 3.2 and 3.5 suggest a method for extracting the integrable part of an implicit difference equation under some regularity hypothesis.
Let E ⊂ G be an implicit difference equation of a Lie groupoid G ⇒ M .
For forward integrable part. Consider the sequence of sets of M )), . . . and the sequence of subsets of G For integrable part. Consider the sequence of sets of M It may happen that after a finite number of steps, the sets in two consecutive steps in the different sequences are equal: Then It is straightforward to see that if this happens then the subsequent steps in the different sequences are all equal.
. Therefore, using Proposition 3.2, we have that E is forward integrable at g.
Conversely, suppose that E is forward integrable at g. Then, there exists an admissible sequence γ G : The corresponding results for backward integrability and integrability are proved in a similar way.
Remark 3.7. We have to stress that the concept of implicit difference equation can be to generalized to the setting of a pair of manifolds G and M endowed with two surjective submersions α, β : G → M . The corresponding notions of admissible sequences, solutions, integrable equations and the integrability algorithm can be straightforwardly generalized. This more general approach will be useful in Section 4.4. ⋄ Remark 3.8. For some concrete applications (see, for instance, the next example) instead of a unique subset E of the Lie groupoid G ⇒ M , we have a sequence of subsets {E k } 0≤k≤∞ . A generalization of this type is also useful for discretization of time-dependent systems. It is not hard to extend our setting for this situation and also to extract the integrable part of a sequence of this type. For example, the forward integrable part is extracted using the following procedure. Consider the k-sequence of sets of M defined for k ≥ 0: , and the subsets If the algorithm stabilizes in a sequence of subsets {[E k ] f } we will call this set the backward integrable part of the sequence {E k }. ⋄ Example 3.9. Finite difference methods for time-dependent linear differential-algebraic equations The proposed algorithm is also useful for numerical methods for differential-algebraic equations (DAEs). These systems of equations appear in a great number of applications: mechanical systems with constraints, electrical networks, chemical reacting systems... See, for instance, [3,22]. These types of systems take the form F (t, x,ẋ) = 0 including as a particular example the explicit differential equations. The system that we will study are the so-called time-dependent linear differential-algebraic equations: Consider an explicit Euler approximation to this system where h > 0 is the step size and A k = A(t 0 +kh), B k = B(t 0 +kh) and b k = b(t 0 +kh) and x 0 , t 0 initial conditions. Define the subsets of the pair groupoid R n × R n We consider V k the image of the linear map associated with A k . Then, V k = R n (in this case, ) or there exists a matrix Q k ∈ L(R n , R n ) such that the kernel of its linear map is just V k , i.e., Q k A k = 0. Then, one may easily prove that It is possible to continue the algorithm, considering α([E k ] 1 f ) = [C k ] 1 f , but in many cases of interest (as, for instance, discretizations of index one DAEs) we have that the matrix A k + Q k+1 B k+1 is regular, then f and the algorithm stops in this step (see [23] for more details).
4. Implicit systems defined by discrete Lagrangian functions 4.1. Implicit systems defined by discrete Lagrangian functions. Lagrangian submanifolds. Let L : G → R be a discrete Lagrangian on a Lie groupoid G ⇒ M . Then, S L = dL(G) ⊆ T * G is clearly a Lagrangian submanifold of the symplectic manifold T * G and S L is also an implicit difference equation on the cotangent groupoid T * G ⇒ A * G. If we apply the forward integrability algorithm to S L , then there is an interesting relation with the discrete Euler-Lagrange equations for L (see [18,25] and the Appendix A for more details on discrete Lagrangian Mechanics on Lie groupoids), which we describe in this section. Proposition 4.1. Let G ⇒ M be a Lie groupoid and L : G → R be a discrete Lagrangian on G. The first step, (S L ) 1 f , to obtain the forward integrable part of S L = dL(G) are the points dL(g) ∈ S L such that there exists an element h ∈ G, satisfying (g, h) ∈ G 2 and, in addition, (g, h) is a solution of the discrete Euler-Lagrange equations, that is, As a consequence, ifβ (resp.α) is the target (resp. source) map of the cotangent groupoid T * G ⇒ A * G defined in (2.4), then Proof. Using the definition of the source mapα : T * G → A * G (see (2.4)) and the fact that we have that Therefore, (S L ) 1 f = S L ∩ (β) −1 (C 0 f ) is the set of points dL(g) ∈ S L such that there exists h ∈ G, with (g, h) ∈ G 2 and d [L • l g + L • r h • i] (ǫ(x)) |AxG = 0.
Let L : G → R be a hyperregular discrete Lagrangian function (see Appendix A). This implies that the discrete Lagrangian evolution operator Υ L : G → G is given by is a solution of the implicit difference equation S L and Υ T * G (0) = dL(g).
Thus, we have proved the following result. On the other hand, ifΥ L = F + L • (F − L) −1 : A * G → A * G is the discrete Hamiltonian evolution operator then one observes that graphΥ L = Φ(G) = (α,β)(S L ).
Using that S L is a Lagrangian submanifold of T * G and that (α,β) : T * G → A * G × A * G is a Poisson map, A * G denoting A * G endowed with the linear Poisson structure changed of sign, then (α,β)(S L ) is a coisotropic submanifold of A * G×A * G (this is a particular case of [28, Corollary 2.2.5]).
From the previous discussion, we can conclude the following result (see also [29]).
Remark 4.5. T * (Q × Q) ⇒ T * Q is a Lie groupoid which is symplectomorphic to the pair groupoid T * Q × T * Q ⇒ T * Q when we consider the symplectic form The isomorphism is given by Note that (Ψ • dL)(q 0 , q 1 ) = (−D 1 L(q 0 , q 1 ), D 2 L(q 0 , q 1 )). ⋄ Now, suppose that L : Q × Q → R is hyperregular. Then is a discrete Lagrangian evolution operator. In addition, we have that the image of the map On the other hand, we can consider the discrete Hamiltonian evolution operator We remark that, sinceΥ L : T * Q → T * Q is a symplectomorphism, we deduce (Id,Υ L )(T * Q) is a Lagrangian submanifold of (T * Q × T * Q, (−Ω Q , Ω Q )).

4.3.
Example: a discrete singular Lagrangian. Consider the discretization of the degenerate Lagrangian L : T R 2 −→ R given by L(x, y,ẋ,ẏ) = 1 2ẋ 2 + 1 2 x 2 y (see [2]). Take, for instance, a consistent discretization of this lagrangian: The algorithm produces the sequence: Composing the source and the target mapsα andβ of the cotangent groupoid T * G ⇒ A * G as defined in Equation (2.4) with i * Dc : AG * → D * c , the dual map of the canonical inclusion i Dc : D c → AG, we have the maps   , (g, h) is a solution of the discrete non-holonomic Euler-Lagrange equations, that is, In other words, ) is a solution of the discrete nonholonomic Euler-Lagrange eqns.} = {(g, h) ∈ G 2 ∩ (M c × M c ) |β Dc (dL(g)) =α Dc (dL(h))} Proof. The proof is analogous to that of Proposition 4.1, taking into account the version of the discrete nonholonomic Euler-Lagrange equations given by Equation (B.4).

4.5.
Example: The discrete Chaplygin sleigh [7,8]. To illustrate the results contained in the previous section, we will describe a discretization of the Chaplygin sleigh system (previously considered in [10]). The Chaplygin sleigh system describes the motion of a rigid body sliding on a horizontal plane. The body is supported at three points, two of which slide freely without friction while the third is a knife edge, a constraint that allows no motion orthogonal to this edge (see [21]).
4.6. An approach to implicit discrete Hamiltonian systems. Along the paper, we have focused our attention to the case of discrete lagrangian mechanics including this theory in the setting of discrete implicit systems. Of course, we can also adopt a dual point of view, that is, the hamiltonian formalism. In the continuous setting, the hamiltonian formalism is specified given a hamiltonian function H on A * G equipped with its canonical linear Poisson bracket. The dynamics is given by the associated hamiltonian vector field X H . In the sequel, we will show how to obtain a lagrangian submanifold of the symplectic groupoid T * G ⇒ A * G directly from the flow of X H .
Assume that G ⇒ M is a Lie groupoid with corresponding algebroid τ : AG → M . Then, we have already explained (see (2.4)) that the cotangent bundle T * G is a groupoid over A * G. Moreover, there is a Poisson structure on A * G (induced from the Lie algebroid structure on AG) such that the source mapα : T * G → A * G (resp. the target mapβ : T * G → A * G) is a Poisson (resp. anti-Poisson) map.
Let H : A * G −→ R be a Hamiltonian function. Then, since A * G is a Poisson manifold, we have the hamiltonian vector field X H . On the other hand, consider the pull-back of the 1-form dH ∈ Ω 1 (A * G) by the source mapα. Using the canonical symplectic structure on T * G, we have the corresponding Hamiltonian vector field Xα * H . It is easy to see that both vector fields areα-related. Indeed, if Π A * G (resp. discrete Lagrangian L h d : Q × Q → R given by where σ 01 (t) is the unique solution of the Euler-Lagrange equations for L which satisfies the boundary conditions σ 01 (0) = q 0 and σ 01 (h) = q 1 .
The Legendre transformations, F ± L h d and FL, for L h d and L respectively, are related as follows (see [17]) In [17] it is proved that for a small time-step h, the following diagrams relate the flow ψ h H : T * Q → T * Q at time h of the Hamiltonian vector field X H ∈ X(T * Q) and the discrete Lagrangian evolution operator Υ L h d : Here Ψ : T * (Q × Q) → T * (Q × Q) is the isomorphism described in (4.1). Thus, It would be interesting to extend this result for a general regular Lagrangian L : AG → R on the associated Lie algebroid AG of a Lie groupoid G ⇒ M . Previously, its is necessary to describe the exact discrete Lagrangian L h d : G → R induced by L (see [19] for more details).
The results of this section motivate that we may introduce, in the general context of a Lie groupoid G, the notion of an implicit discrete Hamiltonian system as a Lagrangian submanifold of T * G.

Conclusion
We have developed an algorithm which permits to extract the integrable part of an implicit difference equation. This algorithm produces a sequence of submanifolds (or generally subsets) which encodes where there exist well defined solutions of the discrete dynamics. As an application, our method allows us to easily analyze the case of discrete (nonholonomic) Lagrangian systems from an implicit point of view, in particular the situation of singular lagrangians, and we will expect that it will be an useful tool for discrete optimal control theory. We will discuss this topic in a future paper.
Finally, in our opinion, the developments about implicit systems defined by Hamiltonian functions open doors to future research in the theory of generating functions for Poisson morphisms (using standard symplectic techniques), the theory of discrete hamiltonian systems in a Poisson context [1], the Dirac theory of constraints in the discrete setting, a discrete Hamilton-Jacobi theory... among other research topics.
Appendix A. Discrete Lagrangian Mechanics on Lie groupoids A discrete Lagrangian system consists of a Lie groupoid G ⇒ M (the discrete space) and a discrete Lagrangian L : G → R.
For g ∈ G fixed, we consider the set of admissible sequences: We may identify the tangent space to C N g at γ G with Now, we define the discrete action sum associated to the discrete Lagrangian L : G → R by Hamilton's principle requires that this discrete action sum be stationary with respect to all the infinitesimal variations. This requirement gives the following alternative expressions for the discrete Euler-Lagrange equations (see [18]): for all sections X of τ : AG → M . Alternatively, we may rewrite the discrete Euler-Lagrange equations as Thus, we may define the discrete Euler-Lagrange operator D DEL L : G 2 → A * G from G 2 to A * G, the dual of AG. This operator is given by Let Υ : G → G be a smooth map such that: -graph(Υ) ⊆ G 2 , that is, (g, Υ(g)) ∈ G 2 , for all g ∈ G (Υ is a second order operator). -(g, Υ(g)) is a solution of the discrete Euler-Lagrange equations, for all g ∈ G, that is, (D DEL L)(g, Υ(g)) = 0, for all g ∈ G.

(A.3)
A discrete Lagrangian L : G → R is said to be regular if and only if the Legendre transformation F + L is a local diffeomorphism (equivalently, if and only if the Legendre transformation F − L is a local diffeomorphism). In this case, if (g 0 , h 0 ) ∈ G × G is a solution of the discrete Euler-Lagrange equations for L then, one may prove (see [18]) that there exist two open subsets U 0 and V 0 of G, with g 0 ∈ U 0 and h 0 ∈ V 0 , and there exists a (local) discrete Lagrangian evolution operator Υ L : U 0 → V 0 such that: (i) Υ L (g 0 ) = h 0 , (ii) Υ L is a diffeomorphism and (iii) Υ L is unique, that is, if U ′ 0 is an open subset of G, with g 0 ∈ U ′ 0 , and Υ ′ L : U ′ 0 → G is a (local) discrete Lagrangian evolution operator then Υ L|U0∩U ′ 0 = Υ ′ L|U0∩U ′ 0 . Moreover, if F + L and F − L are global diffeomorphisms (that is, L is hyperregular) then Υ L = (F − L) −1 • F + L.
If L : G → R is a hyperregular Lagrangian function, then pushing forward to A * G with the discrete Legendre transformations, we obtain the discrete Hamiltonian evolution operator,Υ L : A * G → A * G which is given bỹ (A.4)

Appendix B. Discrete nonholonomic Lagrangian systems
A discrete nonholonomic Lagrangian system on a Lie groupoid G ⇒ M is a Lagrangian discrete L : G → R, a vector subbundle D c (the constraint distribution) of the Lie algebroid AG of G and a discrete constraint embedded submanifold M c of G such that dim M c = dim D c .