Variational Constrained Mechanics on Lie Affgebroids

In this paper we discuss variational constrained mechanics (vakonomic mechanics) on Lie affgebroids. We obtain the dynamical equations and the aff-Poisson bracket associated with a vakonomic system on a Lie affgebroid ${\mathcal A}$. We devote special attention to the particular case when the nonholonomic constraints are given by an affine subbundle of ${\mathcal A}$ and we discuss the variational character of the theory. Finally, we apply the results obtained to several examples.


Introduction
Lie algebroids have deserved a lot of interest in recent years (from a theoretical and applied point of view). In the context of Mechanics, an ambitious program was proposed by Weinstein [34] in order to develop geometric formulations of the dynamical behavior of Lagrangian and Hamiltonian systems on Lie algebroids. In the last years, this program has been actively developed by many authors. In fact, a Klein's formalism for unconstrained Lagrangian systems on Lie algebroids has been discussed and a symplectic formulation of Hamiltonian mechanics on these objects has been developed (see [20,23,24,30]). The main notion is that of prolongation of a Lie algebroid over a map introduced by Higgins and Mackenzie [14]. An alternative approach, using the linear Poisson structure on the dual bundle of a Lie algebroid, was discussed in [12].
An interesting kind of mechanical systems are those subject to external linear constraints. For these systems, one may derive the dynamical equations using the Lagrange-D'Alembert principle (nonholonomic mechanics) or using a constrained variational principle (vakonomic mechanics). The resultant equations are, in general, different. Constrained Lagrangian systems (variational or not) have application in many different areas: engineering, optimal control theory, mathematical economics, sub-Riemannian geometry, motion of microorganisms, etc. For a geometrical treatment of standard mechanical systems subject to external linear constraints we remit to the monographs [2,5] and references therein.
More recently, several authors discuss the more general class of nonholonomic Lagrangian (Hamiltonian) systems subject to linear constraints on Lie algebroids (see [7,8,27,28]). In the same Lie algebroid setting, other authors [16] consider variational constrained mechanical systems. In another direction, a unified approach of nonholonomic and vakonomic mechanics, using general algebroids instead of just Lie algebroids, was developed in [9].
As a consequence of all these investigations, one deduces that there are several reasons for discussing unconstrained (constrained) Mechanics on Lie algebroids: i) The inclusive nature of the Lie algebroid framework. In fact, under the same umbrella, one can consider standard unconstrained (constrained) mechanical systems, (nonholonomic and vakonomic) Lagrangian systems on Lie algebras, unconstrained (constrained) systems evolving on semidirect products or (nonholonomic and vakonomic) Lagrangian systems with symmetries.
ii) The reduction of a (nonholonomic or vakonomic) mechanical system on a Lie algebroid is a (nonholonomic or vakonomic) mechanical system on a Lie algebroid. However, the reduction of an standard unconstrained (constrained) system on the tangent (cotangent) bundle of the configuration manifold is not, in general, an standard unconstrained (constrained) system.
iii) The theory of Lie algebroids gives a natural interpretation of the use of quasicoordinates (velocities) in Mechanics (particularly, in nonholonomic and vakonomic mechanics).
On the other hand, in [10,26] an affine version of the notion of a Lie algebroid structure was introduced. The resultant geometric object is called a Lie affgebroid structure. A Lie affgebroid structure on an affine bundle A is equivalent to a Lie algebroid structure on the bidual bundle to A such that the section of the affine dual to A induced by the constant map on A equal to 1 is a 1-cocycle.
Lie affgebroid structures may be used to develop a time-dependent version of unconstrained Lagrangian and Hamiltonian equations on Lie algebroids (see [11,17,25,26,31]). In addition, in [15] the authors present a geometric description of Lagrangian and Hamiltonian systems on Lie affgebroids subject to affine nonholonomic constraints. If we apply this general theory to the particular case when the Lie affgebroid is the 1-jet bundle of local sections of a fibration τ : Q → R then one recovers some results obtained in [4,18,19] for standard time-dependent nonholonomic Lagrangian systems subject to affine constraints. The same reasons for discussing unconstrained (constrained) mechanics on Lie algebroids are also valid for discussing unconstrained (constrained) mechanics on Lie affgebroids.
On the other hand, in [33] the authors discuss standard time-dependent vakonomic dynamics and its relation with presymplectic geometry. More recently, in [1] a geometric approach to time-dependent optimal control problems is proposed. This formulation is based on the Skinner and Rusk formalism for Lagrangian and Hamiltonian systems. Some applications are also presented. The aim of this paper is to extend these formulations to the Lie affgebroid setting or, in other words, to discuss vakonomic mechanics on Lie affgebroids.
The paper is organized as follows. In Section 2, we recall some well-known facts about the geometry of Lie affgebroids and about the unconstrained Hamiltonian formalism on Lie affgebroids. In Section 3, we obtain the vakonomic equations and the vakonomic bracket for a constrained mechanical system on a Lie affgebroid A. We devote special attention to the particular case when the constraints are given by an affine subbundle of A. We also discuss, in this section, the variational character of the theory. In section 4, we apply the results obtained in the paper to several examples. In fact, we develop a Skinner-Rusk formalism on Lie affgebroids. We also consider vakonomic Mechanics on a Lie affgebroid A, for the particular case when A is the 1-jet bundle of local sections of a fibration over R. As a consequence, we recover some previous results in the literature. We also discuss optimal control systems as vakonomic systems on Lie affgebroids. The paper ends with our conclusions and a description of future research directions.

Hamiltonian formalism on Lie affgebroids
2.1. Lie affgebroids. Let τ A : A → Q be an affine bundle with associated vector bundle τ V : V → Q. Denote by τ A + : A + = Aff (A, R) → Q the dual bundle whose fibre over x ∈ Q consists of affine functions on the fibre A x . Note that this bundle has a distinguished section 1 A ∈ Γ(τ A + ) corresponding to the constant function 1 on A. We also consider the bidual bundle τ e A : A → Q whose fibre at x ∈ Q is the vector space A x = (A + x ) * . Then, A may be identified with an affine subbundle of A via the inclusion i A : A → A given by i A (a)(ϕ) = ϕ(a), which is an injective affine map whose associated linear map is denoted by i V : V → A. Thus, V may be identified with a vector subbundle of A.
A Lie affgebroid structure on A consists of a Lie algebra structure and an affine map ρ A : A → T Q, the anchor map, satisfying the following conditions: [10,26]).
If ([[·, ·]] V , D, ρ A ) is a Lie affgebroid structure on an affine bundle A then (V, [[·, ·]] V , ρ V ) is a Lie algebroid, where ρ V : V → T Q is the vector bundle map associated with the affine morphism ρ A : A → T Q (for the definition and properties of Lie algebroids we remit to [22] Let τ A : A → Q be a Lie affgebroid modelled on the Lie algebroid τ V : V → Q. Suppose that (x i ) are local coordinates on an open subset U of Q and that {e 0 , e α } is a local basis of Γ(τ e A ) in U which is adapted to the 1-cocycle 1 A , i.e., such that 1 A (e 0 ) = 1 and 1 A (e α ) = 0, for all α. Note that if {e 0 , e α } is the dual basis of {e 0 , e α } then e 0 = 1 A . Moreover, since 1 A is a 1-cocycle, we have that Denote by (x i , y 0 , y α ) the corresponding local coordinates on A. Then, the local equation defining the affine subbundle A (respectively, the vector subbundle V ) of A is y 0 = 1 (respectively, y 0 = 0). Thus, (x i , y α ) may be considered as local coordinates on A and V .
The standard example of a Lie affgebroid is the 1-jet bundle τ 1,0 : J 1 τ → Q of local sections of a fibration τ : Q → R. It is well known that τ 1,0 is an affine bundle modelled on the vector bundle π = (π Q ) |V τ : V τ → Q, where V τ is the vertical bundle of τ . Moreover, if t is the usual coordinate on R and η is the closed 1-form on Q given by η = τ * (dt) then we have the identification J 1 τ ∼ = {v ∈ T Q | η(v) = 1} (see, for instance, [32]). Note that V τ = {v ∈ T Q | η(v) = 0}. Thus, the bidual bundle J 1 τ to τ 1,0 : J 1 τ → Q may be identified with the tangent bundle T Q to Q and, under this identification, the Lie algebroid structure on π Q : T Q → Q is the standard Lie algebroid structure and the 1-cocycle 1 J 1 τ on π Q : T Q → Q is just η.
2.2. The Hamiltonian formalism. Suppose that (τ A : A → Q, τ V : V → Q) is a Lie affgebroid. Then, we consider the prolongation T e A V * of the bidual Lie algebroid ) the Lie algebroid structure on T e A V * (for the definition of the Lie algebroid structure on the prolongation of a Lie algebroid over a fibration, we remit to [14,20]).
Let µ : A + → V * be the canonical projection given by x is the linear map associated with the affine map ϕ and h : V * → A + be a Hamiltonian section of µ, that is, µ • h = Id. Now, we consider the prolongation T e A A + of the Lie algebroid A over τ A + : A + → Q with vector bundle projection τ It is easy to prove that the pair (Th, h) is a Lie algebroid morphism between the Lie algebroids τ We denote by λ h and Ω h the sections of the vector bundles (T e A V * ) * → V * and and Ω e A being the Liouville section and the canonical symplectic section, respectively, associated with the Lie algebroid A (see [20]). Note that On the other hand, let η : T e A V * → R be the section of (T e A V * ) * → V * given by We remark that if pr 1 : T e A V * → A is the canonical projection on the first factor then (pr 1 , τ * V ) is a morphism between the Lie algebroids τ Let (x i ) be local coordinates on an open subset U of Q and {e 0 , e α } be a local basis of Γ(τ e A ) on U adapted to 1 A . Denote by (x i , y 0 , y α ) the induced local coordinates on A and by (x i , y 0 , y α ) the dual coordinates on A + . Then, (x i , y α ) are local coordinates on Then η = Y 0 and, from (2.1) and the definition of the map Th, it follows that Thus, it is easy to prove that the pair (Ω h , η) is a cosymplectic structure on the is the Reeb section of (Ω h , η) (that is, i R h Ω h = 0 and i R h η = 1), then its integral curves (i.e., the integral curves of ρ τ * V e A (R h )) are just the solutions of the Hamilton equations for h, for i ∈ {1, . . . , m} and α ∈ {1, . . . , n}.
Next, we will present an alternative approach in order to obtain the Hamilton equations. For this purpose, we will use the notion of an aff-Poisson structure on an AV-bundle which was introduced in [10] (see also [11]).
Let τ Z : Z → Q be an affine bundle of rank 1 modelled on the trivial vector bundle τ Q×R : Q × R → Q, that is, τ Z : Z → Q is an AV-bundle in the terminology of [11]. Then, we have an action of R on the fibres of Z. This action induces a vector field X Z on Z which is vertical with respect to the projection τ Z : Z → Q.
On the other hand, there exists a one-to-one correspondence between the space of sections of τ Z : Z → Q, Γ(τ Z ), and the set and (x i , s) are local fibred coordinates on Z such that X Z = ∂ ∂s and h is locally defined by h(x i ) = (x i , −H(x i )), then the function F h on Z is locally given by F h (x i , s) = −H(x i ) − s, (for more details, see [11]). Now, an aff-Poisson structure on the AV-bundle τ Z : Z → Q is a bi-affine map, , which satisfies the following properties: where {·, ·} V is the affine-linear part of the bi-affine bracket. iii) If h ∈ Γ(τ Z ) then the map {h, ·} : , is an affine derivation.
Condition iii) implies that, for each h ∈ Γ(τ Z ) the linear part {h, ·} V : C ∞ (Q) → C ∞ (Q) of the affine map {h, ·} : Γ(τ Z ) → C ∞ (Q) defines a vector field on Q, which is called the Hamiltonian vector field of h (see [11]). In [11], the authors proved that there is a one-to-one correspondence between aff-Poisson brackets {·, ·} on τ Z : Z → Q and Poisson brackets {·, ·} Π on Z which are X Z -invariant, i.e., which are associated with Poisson 2-vectors Π on Z such that L XZ Π = 0. This correspondence is determined by Using this correspondence one may prove the following result.
the dual vector bundle to A (resp., to V ) and by µ : A + → V * the canonical projection. Then:

a Hamiltonian section then the Hamiltonian vector field
of h with respect to the aff-Poisson structure is a vector field on V * whose integral curves are just the solutions of the Hamilton equations for h.

Vakonomic mechanics on Lie affgebroids
3.1. Vakonomic equations and vakonomic bracket. Let τ A : A → Q be a Lie affgebroid of rank n over a manifold Q of dimension m. We consider an embedded Now, suppose that e is a point of M, with τ M (e) = x, that (x i ) are local coordinates on an open subset U of Q, x ∈ U , and that {e 0 , e α } is a local basis of Γ(τ e A ) on U adapted to the 1-cocycle 1 A . Denote by (x i , y 0 , y α ) (respectively, (x i , y α )) the corresponding local coordinates for A (respectively, The rank of the (m × (n + m))-matrix ∂Φ A ∂x i , ∂Φ A ∂y α is maximun, that is,m. Then, using that τ M : M → Q is a submersion, we can suppose that the (m ×m)-matrix is regular. Then, we will use the following notation (y α ) = (y A , y a ), for 1 ≤ α ≤ n, Now, using the implicit function theorem, we obtain that there exist an open , an open subset W ⊆ R m+n−m and smooth real functions Consequently, (x i , y a ) are local coordinates on M.
Next, consider the Whitney sum of A + and A, that is, A + ⊕ Q A and the canonical projections pr 1 : and the restrictions π 1 = pr 1|W 0 and π 2 = pr 2|W 0 . Also denote by ν : W 0 → Q the canonical projection. Now, we take the prolongation τ Moreover, we can prolong π 1 : W 0 → A + to a morphism of Lie algebroids Tπ 1 : If (x i , y 0 , y α ) are the local coordinates on A + induced by the dual basis {e 0 , e α } of the local basis {e 0 , e α } of Γ(τ e A ), then (x i , y 0 , y α , y a ) are local coordinates for W 0 and we may consider the local basis for (ϕ, a) ∈ W 0 and ν(ϕ, a) = x, where ρ i 0 and ρ i α are the components of the anchor map ρ e A with respect to the local basis {e 0 , e α }. Now, one may consider on the Lie algebroid τ ν where Ω e A is the canonical symplectic section on T e A A + . The local expression of Ω is On the other hand, if pr 1 : T e A W 0 → A is the canonical projection on the first factor, then we can introduce the section η ∈ Γ((τ ν e A ) * ) defined by η = (pr 1 , ν) * 1 A .
Now, let L : A → R be a Lagrangian function on A and denote byL the restriction of L to the constraint submanifold M.

The Pontryagin Hamiltonian
Thus, one can consider the presymplectic 2-section Ω W0 on T e A W 0 defined by In local coordinates, using (3.1), (3.2) and (3.3), we deduce that First, we will obtain the local expression of the vakonomic problem. In general, a section X satisfying the equations (3.5) cannot be found in all points of W 0 . Thus, we consider the points where (3.5) have sense. We define In local coordinates, we deduce that W 1 is characterized by the equations Moreover, a direct computation, using (3.2) and (3.4), proves that the local expression of any section X satisfying the equations (3.5) is of the form with Υ 0 and Υ a arbitrary functions. Consequently, the vakonomic equations are Remark 3.2. The motion equations for the vakonomic mechanics may be also expressed as follows Note that in contrast to equations (3.6), equations (3.7) are expressed in terms of the global Lagrangian L : A → R. Thus, the equations (3.6) stress how the information given by the Lagrangian L outside M is irrelevant to obtain the vakonomic equations. This is in contrast with what happens in nonholonomic mechanics (see [15]). ⋄ Then we have a system of (n −m) equations with (n −m) unknowns (the functions Υ a ). Thus, if we denote by R ab and µ b the functions it is clear that the above system has a solution Υ a if the matrices R = (R ab ) and R µ = (R ab ; µ b ) have the same rank. Note that even if the above system has a unique solution (i.e., if the matrix R = (R ab ) is regular), the solution X (Υ0,Υ a ) |W1 is not, in general, unique (since the function (Υ 0 ) |W1 is still arbitrary).
To solve the above problem, we consider a suitable submanifold W ′ 1 of W 1 whose intrinsic definition is In local coordinates, the submanifold W ′ 1 is given by the equation is a fibration and, therefore, we can consider the prolongation T e A W ′ 1 of the Lie algebroid A over ν ′ 1 . Moreover, we have the following result.
On the other hand, , for all b, and, consequently, λ a = 0, for all a. Thus, Z = λ 0 P 0 (w ′ 1 ) and The converse is proved in a similar way.
Remark 3.4. We remark that the condition det (R ab ) = 0 implies that the matrix ∂ϕ a ∂y b a,b=m+1,...,n is regular. Thus, using the implicit theorem function, we deduce that there exist open subsetsW 0 ⊆ W 0 ,W ⊆ R m+n and smooth real functions µ a :W → R, a =m+1, . . . , n, such that W 1 ∩W 0 is locally defined by the equations y a = µ a (x i , y α ), a =m + 1, . . . , n.
Therefore, we may consider (x i , y 0 , y α ) as local coordinates on W 1 and, consequently, from (3.8), we obtain that (x i , y α ) are local coordinates on W ′ 1 . Thus, a local basis of sections of T

⋄
Proceeding as in the proof of Proposition 3.3, we deduce the following result.
) solution of the vakonomic problem (L, M). In fact, ζ 1 is the Reeb section of (Ω W ′ 1 , η W ′ 1 ), that is, ζ 1 is characterized by the conditions i ζ1 Ω W ′ 1 = 0 and i ζ1 η W ′ 1 = 1. The above results suggest us to introduce the following definition. In what follows, we will suppose that (L, M) is a regular vakonomic system on the Lie affgebroid A. Then, from Theorem 3.5, we have that the vakonomic problem has a unique solution which is the Reeb section ζ 1 of the cosymplectic structure (Ω W ′ 1 , η W ′ 1 ). First, we will give the local expression of the solution section ζ 1 . Suppose that (x i , y α ) are local coordinates on W ′ 1 as in Remark 3.4 and that we have that (see (3.4)) Thus, we obtain that Now, we will introduce an aff-Poisson bracket on the AV-bundle determined by the constraint submanifolds W 1 and W ′ 1 . For this propose, we define the application If (x i , y 0 , y α ) (respectively, (x i , y α )) are local coordinates on W 1 (respectively, W ′ 1 ) as in Remark 3.4, we deduce that the local expression of µ 1 : Moreover, we have the following result. Proof. It is easy to prove that µ 1 : W 1 → W ′ 1 is an AV-bundle (see Section 2.2). In fact, if w = (ϕ, a) ∈ (W 1 ) x , with x ∈ Q, and t ∈ R then To define an aff-Poisson bracket on µ 1 we will introduce a Poisson bracket on W 1 which is invariant with respect to X W1 . Here, X W1 is the infinitesimal generator of the principle action of R on W 1 .
On the other hand, note that the restriction H W1 to W 1 of the Pontryagin Hamiltonian H W0 verifies that X W1 (−H W1 ) = −1. Therefore, there exists h 1 ∈ Γ(µ 1 ) such that F h1 = −H W1 . In fact, h 1 is the inclusion of W ′ 1 into W 1 . Moreover, we have the following result. Proof. We know that the Hamiltonian vector field {h 1 , ·} al vak of h 1 with respect to the vakonomic bracket is given by Then, from (3.9), (3.10), (3.11) and Remark 3.4, we deduce that this vector field is just ρ ν ′ 1 e A (ζ 1 ) (see (3.9)).
Next, let h ′ 1 , h ′′ 1 be two sections of µ 1 : W 1 → W ′ 1 and suppose that Then, using (3.10), we have that Since A is a Lie affgebroid, it follows that A + is the total space of an AV-bundle over V * with projection µ : A + → V * and, moreover, the linear Poisson structure Π A + on A + (induced by the Lie algebroid structure of A) defines an aff-Poisson bracket {·, ·} : Γ(µ) × Γ(µ) → C ∞ (V * ) on the AV-bundle µ : A + → V * (see Theorem 2.1).
On the other hand, we may consider the applications (π 1 ) |W1 : • µ 1 . In fact, using (3.10) and Remark 3.4, we can prove the following result. Corollary 3.9. If (L, M) is a regular vakonomic system on A, then the pair ((π 1 ) |W1 , µ • (π 1 ) |W ′ 1 ) is a local aff-Poisson isomorphism of AV-bundles, that is: ii) The restriction of (π 1 ) |W1 to each fibre of µ 1 : W 1 → W ′ 1 is an affine isomorphism over the corresponding fibre of µ : A + → V * and iii) If h ′ 1 , h ′′ 1 ∈ Γ(µ 1 ) and h ′ , h ′′ ∈ Γ(µ) satisfy that This implies that the Hamiltonian vector fields of h 1 and h are ((π 1 ) |W1 , µ • (π 1 ) |W ′ 1 )-related. Therefore, if γ ′ 1 : where G is a bundle metric on A and V a function on Q. We also denote by G the bundle metric induced on A + and we suppose that the 1-cocycle 1 A has constant norm equal to 1. Moreover, using the metric, we can identify A and V . In fact, we have the affine bundle morphism I : A → V given by Then, if G is the restriction to V of G, the Lagrangian L may be written as follows and the Euler-Lagrange equations (that is, the vakonomic equations for the system (L, A)) reduce to Next, suppose that the constraint submanifold M of the vakonomic system is an affine subbundle B of A, that is, we have an affine bundle B over Q with associated vector bundle τ U B : U B → Q and the corresponding inclusions i B : B → A and i U B : U B → V . Furthermore, assume that 1 G A ∈ Γ(τ B ). Then, we can choose an special coordinate system adapted to the structure of the problem as follows. In fact, we consider local coordinates (x i ) on an open subset U of Q and an orthonormal local basis of Γ(τ V ), {e A , e a }, adapted to the decomposition V = U ⊥,Ḡ being the orthogonal subbundle to U B with respect to the bundle metricḠ. Thus, we deduce that {1 G A = e 0 , e A , e a } is an orthonormal local basis of Γ(τ e A ) adapted to the affine subbundle B. Denote by (x i , y 0 , y A , y a ) the corresponding local coordinates on A and by (x i , y 0 , y A , y a ) the dual local coordinates on A + . Note that the equations which define to B as an affine subbundle of A are y A = 0. Therefore, the vakonomic system (L, B) is regular and the local expression of the vakonomic equations is y 0 = 1, y A = 0, y a = y a , y 0 = 1 2 (y a ) 2 + V(x i ).

The variational point of view. Let τ A :
A → Q be a Lie affgebroid modelled on the Lie algebroid τ V : V → Q and L : A → R be a Lagrangian function on A.
Next, we will show how to obtain the Euler-Lagrange equations on the Lie affgebroid A from a variational point of view. We define the set of A-paths as follows that is, as the set of admissible curves in A. Then, for two fixed points x, y ∈ Q, denote by Adm([t 0 , t 1 ], A) y x the set of A-paths with fixed base endpoints equal to x and y. Now, if i V : V → A is the canonical inclusion, we consider as infinitesimal variations the complete lifts of sections of τ V : V → Q which vanish at the points x and y, that is, Note that if {e 0 , e α } is a local basis of Γ(τ e A ) andX ∈ Γ(τ V ) is locally given bȳ X =X α e α , then (i V •X ) c |A is the vector field on A given by x the set of A-paths on M with fixed base endpoints equal to x and y, respectively, that is, . In this case, we are going to consider infinitesimal variations (that is, complete lifts (i V •X ) c |A , withX ∈ Γ(τ V )) tangent to the constraint submanifold M and we assume that there exist enough infinitesimal variations of this kind (that is, we are studying the so-called normal solutions of the vakonomic problem). If M is locally given by the equations y A − Ψ A (x i , y a ) = 0, for A = 1, . . . ,m, we deduce that the infinitesimal variations must satisfy Now, let y A be the solution of the differential equationṡ (3.14) Then, from (3.13), we have that Aa )X A . Using this equality, we deduce that Finally, using (3.14) and the fact that Since the variationsX a are free, we conclude that the equations are for all 1 ≤ i ≤ m, 1 ≤ A ≤m andm + 1 ≤ a ≤ n, with y a = ∂L ∂y a − y A ∂Ψ A ∂y a , that is, the vakonomic equations for the system (L, M) on τ A : A → Q (see (3.6)).

4.1.
Skinner-Rusk formalism on Lie affgebroids. Consider on a Lie affgebroid τ A : A → Q a vakonomic system (L, M) with M = A, that is, a free system. In this case, W 0 = A + ⊕ Q A and the Pontryagin Hamiltonian H W0 : A + ⊕ Q A → R is defined by H W0 (ϕ, a) = ϕ(a) − L(a). Moreover, the precosymplectic structure (Ω W0 , η) on T e A W 0 is given by where pr 1 : A + ⊕ Q A → A + is the canonical projection on the first factor, Tpr 1 : In local coordinates, we have that Then, the submanifold W ′ 1 ⊆ A + ⊕ Q A is locally characterized by Thus, if (pr 2 ) |W1 : W 1 → A is the restriction to W 1 of the canonical projection on the second factor and γ 1 : I → W 1 is a solution of the vakonomic equations, then (pr 2 ) |W1 • γ 1 is a solution of the Euler-Lagrange equations for L.
Note that in the standard case, that is, if A = J 1 τ , this procedure is the Skinner-Rusk formulation for time-dependent mechanics (see [1,6]).

4.2.
The 1-jet bundle of local sections of a fibration. Let τ : Q → R be a fibration and τ 1,0 : J 1 τ → Q be the associated Lie affgebroid modelled on the vector bundle π = (π Q ) |V τ : V τ → Q (see Section 2.1). If (t, q i ) are local fibred coordinates on Q then { ∂ ∂t , ∂ ∂q i } is a local basis of sections of π Q : T Q → Q. Denote by (t, q i ,ṫ,q i ) the corresponding local coordinates on T Q. Then, the (local) structure functions of T Q with respect to this local trivialization are given by C k ij = 0 and ρ i j = δ ij , for i, j, k ∈ {0, 1, . . . , n}. Moreover, (t, q i ,q i ) are the corresponding local coordinates on J 1 τ . Now, let M ⊆ J 1 τ be a constraint submanifold such that τ 1,0|M : M → Q is a surjective submersion and L : J 1 τ → R be a Lagrangian function. Suppose that the constraint submanifold M is locally defined by the equationsq A = Ψ A (t, q i ,q a ), where we use the following notation (t, q i ,q i ) = (t, q i ,q A ,q a ).
Thus, if we apply the results of the Section 3.1 to this particular case, we recover some results obtained in [1]. In particular, using (3.6) and (4.1), it follows that the vakonomic equations reduce to where (t, q i , p, p i ) are the local coordinates on T * Q induced by the local basis {dt, dq i }. If these equations are written using the Lagrange multipliers (see (3.7)) then they coincide with the equations obtained in [33].
On the other hand, if (L, M) is a regular vakonomic system on τ 1,0 : J 1 τ → Q, then the AV-bundle µ 1 :

Optimal control systems as vakonomic systems on Lie affgebroids.
Let τ A : A → Q be a Lie affgebroid and C a fibred manifold over the state manifold π : C → Q. We also consider a section σ : C → A along π and an index function l : C → R. The triple (l, π, σ) is an optimal control system on the Lie affgebroid A.
One important case happens when the section σ : C → A along π is an embedding. In such a case, we have that the image M = σ(C) is a submanifold of A. Moreover, since σ : C → M is a diffeomorphism, we can define a Lagrangian function L : M → R by L = l • σ −1 . Therefore, it is equivalent to analyze the optimal control problem defined by (l, π, σ) (applying the Pontryagin maximum principle) that to study the vakonomic problem on the Lie affgebroid τ A : A → Q defined by (L, M).
In the general case (when σ : C → A is not, in general, an embedding), we consider the subset J A C of the product manifold A × T C defined by Next, we will show that J A C admits a Lie affgebroid structure. Let τ π e A : T e A C → C be the prolongation of the bidual Lie algebroid A of A over the fibration π : C → Q.
On the other hand, let φ : Note that if pr 1 : T e A C → A is the canonical projection on the first factor then (pr 1 , π) is a morphism between the Lie algebroids τ π e A : T e A C → C and τ e A : A → Q and, moreover, we have that (pr 1 , π) * (1 A ) = φ. Since 1 A is a 1-cocycle of τ e A : A → Q, we deduce that φ is a 1-cocycle of the Lie algebroid τ π e A : T e A C → C and, using the fact that (1 A ) | e Ax = 0, for all x ∈ Q, we have that φ |T e A p C = 0, for all p ∈ C.
In addition, it follows that On the other hand, let τ π V : T V C → C be the prolongation of the Lie algebroid (V, [[·, ·]] V , ρ V ) over the fibration π : C → Q. Then, it is easy to prove that φ −1 {0} = T V C. Therefore, we conclude that J A C is an affine bundle over C with projection τ π A : J A C → C defined by τ π A (a, v) = π C (v), where π C : T C → C is the canonical projection. Moreover, the affine bundle τ π A : J A C → C admits a Lie affgebroid structure such that its bidual Lie algebroid is just (T If A = J 1 τ is the 1-jet bundle of local sections of a fibration τ : Q → R, it is easy to prove that the prolongation of J 1 τ ∼ = T Q over π : C → Q is just T C. Thus, J A C = {X ∈ T C | dt(X) = 1} ∼ = J 1 (τ • π), t being the usual coordinate on R. Under these identifications, the constraint submanifold is Therefore, we recover the construction developed in [1].
Example 4.1. We consider the following mechanical problem (see [3,4,15,21,29]). A (homogeneous) sphere of radius r > 0, mass m and inertia mk 2 about any axis rolls without sliding on a horizontal table which rotates with time-dependent angular velocity about a vertical axis through one of its points. Apart from the constant gravitational force, no other external forces are assumed to act on the sphere. The configuration space of the sphere is Q = R 3 ×SO(3) and the Lagrangian of the system corresponds with the kinetic energy K(t, x, y;ṫ,ẋ,ẏ, ω x , ω y , ω z ) = 1 2 (mẋ 2 + mẏ 2 + mk 2 (ω 2 x + ω 2 y + ω 2 z )), where (ω x , ω y , ω z ) are the components of the angular velocity of the sphere.
Since the ball is rolling without sliding on a rotating table then the system is subjected to the affine constraints: where Ω(t) is the angular velocity of the table. Moreover, it is clear that Q = R 3 × SO(3) is the total space of a trivial principal SO(3)-bundle over R 3 and the bundle projection π : Q → R 3 is just the canonical projection on the first factor. Therefore, we may consider the corresponding Atiyah Lie algebroid T Q/SO(3) over R 3 (see [20,22]).
Since the Atiyah Lie algebroid T Q/SO(3) is isomorphic to the product manifold (X, u), where X is a vector field on R 3 and u : R 3 → R 3 is a smooth map. Therefore, a global basis of sections of T R 3 × R 3 → R 3 is Moreover, φ : T R 3 × R 3 → R given by φ(t, x, y;ṫ,ẋ,ẏ, ω x , ω y , ω z ) =ṫ is a 1cocycle in the corresponding Lie algebroid cohomology and, then, it induces a Lie affgebroid structure over and its bidual Lie algebroid A is just the Atiyah Lie algebroid T R 3 × R 3 . Note that the Lie affgebroid structure on A = R × T R 2 × R 3 is a special type of Lie affgebroid structure called Atiyah Lie affgebroid structure (see Section 9.3.1 in [17] for a general construction). Thus, (t, x, y;ẋ,ẏ, ω x , ω y , ω z ) may be considered as local coordinates on A and V .
It is clear that the Lagrangian function and the nonholonomic constraints are defined on the Atiyah Lie affgebroid A ≡ R × T R 2 × R 3 (since the system is SO(3)invariant). In fact, we have a nonholonomic system on the Atiyah Lie affgebroid A ≡ R × T R 2 × R 3 (see [15] for more details).
After some computations the equations of motion for this nonholonomic system may be written as followṡ where c is a constant, together witḧ Now, we pass to an optimization problem. Assume full control over the motion of the center of the sphere and consider the cost function L(t, x, y;ẋ,ẏ, ω x , ω y , ω z ) = 1 2 (ẋ) 2 + (ẏ) 2 and the following optimal control problem: Given two points q 0 , q 1 ∈ Q, find an optimal control curve (t, x(t), y(t)) on the reduced space that steer the system from q 0 and q 1 and minimizes 1 0 1 2 (ẋ) 2 + (ẏ) 2 dt subject to the constraints defined by equations (4.2).
Note that this problem is equivalent to the optimal control problem defined by the section σ : and the index function l(t, x, y; u 1 , u 2 ) = 1 2 ((u 1 ) 2 + (u 2 ) 2 ). Since σ is obviously an embedding, we deduce the equivalence between both problems.
A necessary condition for optimality of the problem is given by the corresponding vakonomic equations. In this case, we will denote by y 1 =ẋ, y 2 =ẏ, y 3 = ω x , y 4 = ω y , y 5 = ω z .
After some computations, we obtain that the vakonomic equations are Moreover, it is easy to prove that the vakonomic system is regular. Therefore, there exists a unique solution of the vakonomic equations on the submanifold W ′ 1 which is determined by the following conditions Thus, it follows that (t, x, y; y 1 , y 2 , y 3 , y 4 , y 5 ) (respectively, (t, x, y; y 0 , y 1 , y 2 , y 3 , y 4 , y 5 )) are local coordinates on W ′ 1 (respectively, on W 1 ). Then, the local expression of the Hamiltonian H W1 is H W1 (t, x, y; y 0 , y 1 , y 2 , y 3 , y 4 , y 5 ) = −y 0 −  Optimal control of affine control systems. Let τ A : A → Q be a Lie affgebroid. Suppose that the constraint submanifold M of the vakonomic system is an affine subbundle B of A, that is, we have an affine bundle B over Q with associated vector bundle τ U B : U B → Q and the corresponding inclusions i B : B → A and i U B : U B → V . Choose now a coordinate system adapted to this affine subbundle B. That is, take local coordinates (x i ) on an open subset U of Q and an local basis of Γ(τ V ), {e A , e a }, adapted to the decomposition V =Ũ B ⊕ U B , whereŨ B is an arbitrary complementary subspace. Thus, {e 0 , e A , e a } is an local basis of Γ(τ e A ) adapted to the affine subbundle B, where 1 A (e 0 ) = 1. Denote by (x i , y 0 , y A , y a ) the corresponding local coordinates on A and by (x i , y 0 , y A , y a ) the dual local coordinates on A + . Note that the equations which define B as an affine subbundle of A are y A = 0.
The affine control problem given by the drift section e 0 and the input sections e a is defined by the following equation on Q,ẋ i = ρ i 0 + y a ρ i a , where the coordinates y a are playing the role of the set of admissible controls. Now, consider a functionL : B −→ R as a performance index. The equations of motion of the optimal control problem defined by (L, B) are precisely the vakonomic equations. In the selected coordinate system are: for all 1 ≤ i ≤ m, 1 ≤ γ ≤ n, 1 ≤ A ≤m andm + 1 ≤ a ≤ n, with y a = ∂L ∂y a . Example 4.2. Consider a particle of unit mass in a planar inverse-square law gravitational field which has thrusters in the "x, y" directions (see [13]). Then, the equations of motion are: q 1 = v 1 ,q 2 = v 2 ,v 1 = −q 1 (q 2 1 + q 2 2 ) −3/2 + u 1 ,v 2 = −q 2 (q 2 1 + q 2 2 ) −3/2 + u 2 defined on M = (R 2 − {(0, 0)}) × R 2 . The objective will be to drive the particle to a given circular orbit with minimum energy. Therefore, let us take L = 1 2 (u 2 1 + u 2 2 ). Now, choose a global basis of sections of T (R × M ) −→ R × M adapted to this affine control system: where {e 0 ; e 3 , e 4 } defines a affine subbundle of R × T M −→ R × M determining the initial affine control system.

Conclusions and future work
Variational constrained Mechanics is discussed in the Lie affgebroid setting. We obtain the vakonomic equations and the vakonomic bracket associated with a constrained mechanical system on a Lie affgebroid. The variational character of the theory is analyzed. Vakonomic systems subjected to affine constraints are of special interest. Other examples are also discussed.
In this paper we only consider normal solutions of the vakonomic problems. It would be interesting to extend the results of the paper for abnormal solutions.