Higher-order Variational Calculus on Lie algebroids

The equations for the critical points of the action functional defined by a Lagrangian depending on higher-order derivatives of admissible curves on a Lie algebroid are found. The relation with Euler-Poincar\'e and Lagrange Poincar\'e type equations is studied. Reduction and reconstruction results for such systems are established.


Introduction
In this paper we study optimization problems defined by a cost functional which depends on higher-order derivatives of admissible curves on a Lie algebroid. Examples of this type of problems are the optimal control of dynamical systems where the system to be controlled is a mechanical system, and hence depends on accelerations [2,1,11,8], trajectory planning problems in control theory [26], key-framed animations in computer graphics [37], and in general, problems of interpolation and approximation of curves on Riemannian manifolds [3,27]. In many of these examples the presence of symmetries is used to reduce the difficulty of the problem.
The advantage of the Lie algebroid approach is its inclusive nature, under the same formalism we can describe many systems which are apparently different [41,29,12,21] and hence it allows a unified description even in the case of a reduced system when symmetries are present in the problem. An alternative approach consists in a case by case study using Euler-Poincaré and Lagrange-Poincaré reduction techniques as in [18,17].
Previous work on the variational description of first-order Lagrangian systems defined on Lie algebroids provides a convenient departure point for the generalization presented here. In [36] it was shown that the Euler-Lagrange equations for a Lagrangian system defined on a Lie algebroid are the critical points for the action functional defined on an adequate Banach manifold of admissible curves satisfying boundary conditions. Such a manifold structure is a foliated one, the leaves being the E-homotopy classes of admissible curves.
It is frequently argued that the variational principle for reduced systems, and in general for systems on Lie algebroids, is a constrained variational principle, in the sense that some additional constraints are imposed to the admissible variations. The point of view of [36] and the present paper is different. The set of admissible curves where the action is defined is endowed with a reasonable Banach manifold structure. The tangent space to such manifold of admissible curves is precisely the whole set of infinitesimal admissible variations. Therefore no additional constraint to the infinitesimal variations is imposed, or in other words, they are as constrained as in the standard case when formulated directly in the tangent bundle instead of on the base manifold. In such standard case the relevant topology on the set C 1 (I, T M ) is the one induced by the manifold structure of C 2 (R, M ), which is a foliated structure where each leaf (connected component) is a the set of curves of the formγ with γ in a given homotopy class. A similar construction can be performed in the case of admissible curves on a Lie algebroid E by using the notion of E-homotopy [13,36].
Whether the reader agrees with the above argument or not, it should be clear that the variational principle stated here is a bona fide standard variational principle, as it is needed in optimization problems: the solutions of the higher-order Euler-Lagrange equations are the critical points of the action functional which is a smooth function on a Banach manifold. In this sense one should notice that some generalized variational principles that appeared in the literature [20,24] do not satisfy this property.
Description of the results and organization of the paper. For a Lie algebroid E we consider the set E k of (k − 1)-jets of admissible curves on E. Given a function L∈C ∞ (E k ), which will be called the Lagrangian, we want to find the maxima/minima/critical points of a cost/action functional S defined by S(a) = t1 t0 L(a k (t)) dt among the set of admissible curves which are E-homotopic to a given admissible curve a 0 and satisfy some boundary conditions. Finite variations a s (t) = a(s, t) are given by E-homotopies and infinitesimal variations are just the s-derivative of a s . E-homotopies are special morphisms of Lie algebroids φ = α(s, t)dt + β(s, t)ds which satisfy β(s, t 0 ) = β(s, t 1 ) = 0. This condition corresponds to the fixed endpoint condition in the standard case. The corresponding infinitesimal variation depends only on the values of σ(t) = β(0, t) and are denoted Ξ k a0 σ. Since each E-homotopy class is a Banach manifold [13,36], we can properly talk about critical points of the function S.
The differential equations satisfied by the critical points are called the higherorder Euler-Lagrange equations, and generalize the first-order Euler-Lagrange equations defined by Weinstein [41] (see also [29,21]). It will be shown that, in particular, this equations are the higher-order Euler-Poincaré equations when the Lie algebroid is a Lie algebra [18], the higher-order Lagrange-Poincaré equations when the Lie algebroid is the Atiyah algebroid associated to a principal bundle [17], or the higher-order Euler-Poincaré equations with advected parameters when the Lie algebroid is an action algebroid [18,17], in addition to the standard higher-order Euler-Lagrange equations when the Lie algebroid is the tangent bundle to a manifold.
One of the advantages of the formalism of Lie algebroids is that morphisms of Lie algebroids can serve to relate Lie algebroids of the different types mentioned above. When two Lagrangians are related by a morphism of Lie algebroids, the corresponding variational problems are also related and this allows to easily deduce correspondences between the critical points of the associated action functionals, which amounts to reduction theorems showing, among other results, that the standard higher-order Euler-Lagrange equations for a higher-order left-invariant Lagrangian (with/without parameters) on a Lie group reduce to the higher-order Euler-Poincaré equations of the reduced Lagrangian (with/without advected parameters) on the Lie algebra, or that the standard higher-order Euler-Lagrange equations for a higher-order invariant Lagrangian on a principal bundle reduce to the higher-order Lagrange-Poincaré equations of the reduced Lagrangian on the Atiyah algebroid.
The paper is organized as follows. In Section 2 we will introduce the necessary preliminary results about higher-order tangent bundles and Lie algebroids, and we will fix some notation. In Section 3 we will define the space of jets of admissible curves and we will study its basic properties. In Section 4 we will find the properties and the local expression of the variational vector fields defined by E-homotopies, and its relation to the complete lift of a section of E. In order to find an intrinsic expression of the Euler-Lagrange equations, we will define in Section 5 two differential operators, the variational operator and the Cartan operator. This will be done by introducing the vertical endomorphism. In Section 6 we will find the critical points of the action functional in terms of the variational operator and we will deduce its coordinate expression, as well as those of the Cartan form and the Legendre transformation. Also a version of Noether's theorem follows easily from the variational character of the equations. The relevant typical examples are given in Section 7. Finally, in Section 8 we will study the transformation properties of critical points under morphisms of Lie algebroids which readily amount to reduction results among the different kind of equations for different Lie algebroids.

2.1.
Higher-order tangent bundles. Let M be a manifold. For a curve γ : R → M , defined in some open interval containing the origin in R, we denote by [γ] k = j k 0 γ the k-jet of γ at 0, which is said to be the kth-order velocity of γ or simply the k-velocity of γ. The set of k-velocities of curves in M is a manifold T k M , known as the kth-order tangent manifold to M . For k = 1 we have T 1 M = T M , the tangent bundle to M . The projections τ M k,l : See [16,14] for more information.
A vector tangent to T k M can be described by a 1-parameter family of curves γ : R 2 → M defined locally in a neighborhood of the origin in R 2 . Concretely, the family γ s (t) = γ(s, t) defines a curve in T k M by fixing s and taking k-jets [γ s ] k .
There is a canonical injective immersion i M k,1 : 1 . Such an immersion allows to identify (k + 1)velocities with the vectors tangent to T k M which are in the image of i M k,1 . The canonical flip map on T k M will be denoted by χ k : T k T M → T T k M . It can be easily defined in terms of 1-parameter families of curves by Fixing local coordinates x i in M , we have an induced system of local coordinates In what concerns to Variational Calculus and Mechanics, it is convenient to think of a Lie algebroid as a generalization of the tangent bundle of M . One regards an element a of E as a generalized velocity, and the actual velocity v is obtained when applying the anchor to a, i.e., v = ρ(a). A curve a : [t 0 , t 1 ] → E is said to be admissible, or an E-path, ifγ(t) = ρ(a(t)), where γ(t) = τ (a(t)) is the base curve.
The Lie algebroid structure is equivalent to the existence of a degree 1 derivation, d : Sec(∧ k E * ) → Sec(∧ k+1 E * ), which is a cohomology operator d 2 = 0, and is known as the exterior differential on E. A morphism of Lie algebroids is a vector bundle map Φ : We may also define the Lie derivative with respect to a section σ of E as the operator d σ : Sec(∧ k E * ) → Sec(∧ k E * ) given by d σ = i σ • d + d • i σ . Along this paper, the symbol d stands for the exterior differential on a Lie algebroid while the non-slanted symbol d stands for the standard exterior differential on a manifold. A local coordinate system (x i ), i = 1, . . . , n = dim(M ), in the base manifold M and a local basis {e α }, α = 1, . . . , m = rank(E), of sections of E determine a local coordinate system (x i , y α ) on E. The anchor and the bracket are locally determined by the structure functions ρ i α and C α βγ on M given by The exterior differential d on the Lie algebroid is locally determined by where {e α } is the dual basis of {e α }.
The E-tangent to a fibration. [29,30,15]. Let E be a Lie algebroid over a manifold M and π : P → M be a fibration. The E-tangent to P is the Lie algebroid τ E P : T E P → P whose fibre at p is the vector space , the anchor is ρ(b, v) = v and the bracket is determined by the bracket of projectable sections. We will use the redundant notation (p, b, v) to denote the element (b, v) ∈ T E p P . The projection onto the second factor (p, b, v) → b is a morphism of Lie algebroids and will be denoted by T π : T E P → E.
Given local coordinates (x i , u A ) on P and a local basis {e α } of sections of E, we can define a local basis {X α , V A } of sections of T E P by Locally, the Lie brackets of the elements of the basis are and, therefore, the exterior differential is determined by First-order variational vector fields. Within the context of Variational Calculus on Lie algebroids, a variation of an admissible curve a : [t 0 , t 1 ] → E is associated to a morphism of Lie algebroids α(s, t)dt + β(s, t)ds : T R 2 → E such that a(t) = α(0, t) and β(s, t 0 ) = 0, β(s, t 1 ) = 0. The variational vector field ∂α ∂s (0, t) is determined by σ(t) = β(0, t) and it is denoted Ξ a σ(t). In local coordinates, it has the expression .
where a and σ have local expression a(t) = (γ i (t), a α (t)) and σ(t) = σ α (t)e α (γ(t)). See [5,36]. The variational vector field can also be defined in terms of the canonical involution of the bundle T E E. See [15,36] for the details.

Jets of admissible curves
Consider a Lie algebroid 1 (τ : E → M, ρ, [ , ]). A curve a : I⊂R → E is said to be an admissible curve in E, or an E-path, if it satisfies ρ•a =γ, where γ = τ •a is the base curve.
Definition 3.1. For k∈N, we denote by E k the set of (k − 1)-jets of admissible curves on E: These spaces were introduced by Colombo and Martín de Diego in [9]. See also [24] for a simple exposition.
Remark. Notice the grading E 1 = E, E 2 ⊂T E, and in general E k ⊂T k−1 E. The notation is suggested by the intended use in higher-order mechanics, where the elements in E are already considered as quasi-velocities (i.e. 1-quasi-velocities are in E 1 ≡ E). For instance, in the standard case E = T M , we have E 1 = T M , E 2 = T 2 M , etc. In our notation, plain indices will indicate the space where the object is defined, while indices between parenthesis will indicate jet prolongation or the number of derivatives.
To gain some intuition, it is convenient to describe the situation locally. Taking local coordinates (x i , y α ) on E, an admissible curve a(t) = (γ i (t), a α (t)) is determined by the function a α (t) and the initial value γ i (0), since the function γ i (t) can be determined as the solution of the initial value problemẋ i = ρ i α (x)a α (t) with initial condition x(0) = γ(0). Therefore, the (k − 1)-jet of a(t) corresponds just to the (k − 1)-jet of the function a α (t) together with the initial value γ i (0), the 0-jet of γ i (t). The natural coordinates where Ψ i r are smooth functions depending also smoothly on ρ i α and its partial derivatives up to order r−1, obtained by taking total derivatives of the admissibility conditionẋ i = ρ i α a α . Conversely, given a point (x i 0 , y α 1 , . . . , y α k )∈R n × R k·m the admissible curve given by a α (t) = k−1 j=0 1 j! y α j+1 t j and the solution γ i (t) of the initial value problemẋ i = ρ i α (x)a α (t), x i (0) = x i 0 , is an admissible curve whose (k − 1)-jet has coordinates as given in (3.1) with d r−1 a α dt r−1 (0) = y α r for r = 1, . . . , k − 1. It follows that E k is a smooth submanifold of T k−1 E with dimension dim E k = n + km, and that we can take a local coordinate system (x i , y α r ) of the form given above, x i = x i (0) , y α r = y α (r−1) . We will denote by τ k,l : E k → E l the restriction of the natural jet bundle projection τ E k−1,l−1 : T k−1 E → T l−1 E. Then it is straightforward to prove the following result.
Proposition 3.2. The set E k is a submanifold of T k E. The dimensión of E k is dim(E k ) = dim(M ) + k· rank(E). For k > l > 0 we have that τ k,l : E k → E l is a smooth fibre bundle. For k = l + 1 it is an affine bundle. See Section 7 bellow for the construction of E k for some concrete examples of Lie algebroids.
For an admissible curve a : R → E we will denote by a k : R → E k the natural jet-prolongation of a to E k , given by Notice that with the above notations a k (t) = a (k−1) (t).
Remark. The manifold E k+1 can also be defined as a subset of the E-tangent to E k by the following inductive procedure. Starting with E 1 = E, and once we have constructed E k , we define E k+1 as follows. Consider the submersion τ k,0 : E k → M and the E-tangent T E E k to E k with respect to such projection. Then we define . For instance, E 2 = {(a, a, V ) ∈ T E E}, is the set of admissible elements, denoted by Adm(E) in [29]. This construction allows to consider E k+1 as a submanifold E k+1 ⊂T E E k . However, it is preferable to consider E k as a separate manifold, and to define an embedding into T E E k as described in the next paragraph.
Canonical inclusion into the E-tangent. There exists a canonical injective immersion In terms of the natural jet-prolongation of admissible curves, the canonical immersion i k,1 is given by The canonical immersion can be considered as a section of T E E k along τ k+1,k . We will use the symbol T = i k,1 to denote such a section, that is For k = 1 the section T was already introduced in [29]. The associated derivation d T maps functions defined on E k to functions defined on E k+1 , and will be called the total time derivative operator. The reason for that name is that for every function F ∈C ∞ (E k ) and every admissible curve a we have More generally To write more compact expressions we will use the notation V 0 α = X α . In local coordinates we have the local expressions When E is a Lie algebroid, the operator d T acts also on p-forms on E k producing a p-form on E k+1 . For the dual basis Jet prolongation of admissible maps. Consider a second Lie algebroid τ ′ : An admissible map induces a map Φ k : E k → E ′k by jet-prolongation as follows: if a is an admissible curve then Φ•a is admissible and When we look at E k as a submanifold of

Variational vector fields and complete lifts
As it was explained above, a vector tangent to the k-tangent space to a manifold is determined by a 1-parameter family of curves. Accordingly, a vector tangent to E k is determined by a 1-parameter family α(s, t) of admissible curves in E, by the same procedure: In the calculus of variations on Lie algebroids the families α(s, t) of admissible curves are given by morphisms of Lie algebroids φ : T R 2 → E of the form α(s, t)dt+ β(s, t)ds. The variational vector field d ds α s k (t) s=0 can be determined in terms of σ(t) = β(0, t) and its derivatives up to order k. It is a vector field along a k (t), where a(t) = α(0, t), and will be denoted Ξ k a σ(t). Indeed, let us find the local expression of the variational vector field associated to the morphism αdt + βds. In local coordinates, the family α is (γ i (s, t), α µ (s, t)) and the family β is (γ i (s, t), β µ (s, t)). The fact that αdt + βds is a morphism amounts to where the local structure functions ρ i µ and C µ νγ are evaluated at the point γ(s, t). The curve a s k (t) in E k defined by the family α is given by Taking into account the equations (4.1) we have It conclusion, we have It follows that Ξ k a σ is a differential operator in σ of order k (depends on [σ] (k) ) and a differential operator in a of order Remark. In the classical notation of the calculus of variations we have . . , k, as it should be expected.
The variational vector field Ξ k a σ can alternatively be defined directly in terms of σ and a, without reference to the morphism φ, by jet-prolongation as follows.
Proposition 4.1. Let a be an admissible curve and let σ be a section of E along τ •a. Consider the associated first order variational vector field Ξ a σ : Proof. It is enough to prove the proposition for t = 0; for t = t 0 =0 we can just consider the curvesā(t) = a(t 0 + t) andσ(t) = σ(t 0 + t).
We take α(s, t) a family of admissible curves such that α(0, t) = a(t) and find a complementary β(s, t) such that αdt + βds : T R 2 → E is a morphism of Lie algebroids with β(0, t) = σ(t). On one hand, by construction, and the result follows.
As an immediate consequence of the above proposition we have It follows that Ξ k a σ projects to the vector field ρ(σ), i.e. T τ k,0 •Ξ k a σ = ρ(σ), and hence it make sense the following definition.
Definition 4.2. Let a be an admissible curve and σ be a section along γ = τ •a. The section σ k a of T E E k along a k given by σ k a (t) = a k (t), σ(t), Ξ k a σ(t) is said to be the kth-order complete lift of σ with respect to a.
It is easy to see that every element in T E E k is of the form σ k a (0) for some section σ.
Complete lift of a section of E. The concept of variational vector field is related to the concept of complete lift of a section of E.
Let η be a section of E and (Φ s , ϕ s ) its flow [13,36]. The map Φ s k : E k → E k is a flow in E k which defines a vector field X k η ∈T E k . This vector field is τ k,l -projectable over X l η for k > l > 0, and τ k,0 -projectable over ρ(η). Therefore it allows to define a section η k ∈ Sec(T E E k ) by The section η k will be called the kth-order complete lift of the section η. For k = 1 the section η 1 coincides with the complete lift η c of the section η as defined in [29].
Proof. It is enough to prove the proposition over basic forms θ∈ Sec(E * ), and over forms of the type d r (dF ) for F ∈C ∞ (E) and r = 0, . . . , k − 1 (we have omitted the pullbacks for simplicity), since they generate the set of sections of (T E E k ) * . For θ∈ Sec(E * ), in local coordinates if θ = θ α e α and η = η α e α , we have We will prove that for any function F ∈C ∞ (E) and r = 0, . . . , k we have Then the left hand side evaluated at A is equal to and the right hand side evaluated at A is and both expressions are equal.
Moreover, since Z projects to η it follows that this last relation holds also for r = 0. Therefore Z − η k+1 = 0.
Notice that as a consequence of the above property we have that Relation of complete lifts with variational vector fields. The relation between complete lifts of sections and variational vector fields is as follows.
Let a be an admissible curve over γ. Consider a section η of E and define the section σ along γ by σ(t) = η(γ(t)). Consider on one hand the complete lift σ k a of σ with respect to a, and on the other the complete lift η k of the section η. Then (4.6) σ k a = η k •a k . Indeed, it is shown in [36] that the section η and the admissible curve a define a morphism φ = α(s, t)dt + β(s, t)ds = Φ s (a(t))dt + η(ϕ s (γ(t)))ds. At s = 0 we have α(0, t) = a(t), β(0, t) = η(γ(t)) = σ(t) and the vector field d ds a k s s=0 follows. Conversely, given σ(t) along γ(t) we can find a time-dependent section η such that η(t, γ(t)) = σ(t). Using the construction for the time-dependent section as in [36] we also have σ k (t) = η k (t, a k (t)).
It follows that if the coordinate expression of η is η = η α e α then the coordinate expression of the complete lift is Simmilarly where σ = σ α (t)e α (γ(t)).

The vertical endomorphism and the variational operator
Working with forms and their time derivatives, it is useful to have an operator which formally resembles integration, that is, it is a kind of inverse of d T . This is the role of the vertical endomorphism, which is an endomorphism of the vector bundle T E E k . We introduce the vertical endomorphism as an auxiliary tool to define two differential operators, the variational operator and the Cartan operator, which will be fundamental in the development of the calculus of variations. 5.1. Vertical endomorphism. We will define the vertical endomorphism S by its action on sections of the dual bundle. We will make no notational distinction between S and its dual S * , both will be written S.
The set of sections of (T E E k ) * is generated (with coefficients in C ∞ (E k )) by sections of the form (T τ k,r+1 ) ⋆ d r T θ for sections θ∈ Sec(E * ) and r = 0, . . . , k. In particular, if {e α } is a local basis of Sec(E * ), then a local basis of Sec( T e α } for r = 0, . . . , k. Therefore, it is sufficient to give the action of S over sections of such a form. To simplify the notation, in what follows we will omit the pullbacks We define the vertical endomorphism as the (1, 1) tensor field S : We have to check the consistency of the above formula when taking linear combinations of sections. Obviously there is no problem when taking sums, so that we have to check consistency when multiplying by functions on the base. If we take f θ, with f ∈C ∞ (M ), then , consistently with the given definition.
A section λ of (T E E k ) * is said to be τ k,r -semibasic if it vanishes over elements in the kernel of T τ k,r . In other words, λ is τ k,r -semibasic if there exists a map λ : T E E k → (T E E r ) * such that λ(Z) = λ , T τ k,r (Z) for every Z∈T E E r . In particular, λ is τ k,0 -semibasic if there exists a mapλ : T E E k → E * such that η(Z) = λ , T τ k,0 (Z) for every Z∈T E E r . It is clear that the space of τ k,l -semibasic forms is generated (over C ∞ (E k )) by the family of sections of the form d r T θ for r = 0, . . . , l.
The image of S l is the set of τ k,k−l -semibasic forms. In particular, we have that S k+1 = 0.
Proof. For r = 0, . . . , k, a simple calculation shows that It follows that S p (λ) = 0 if and only λ can be written as a linear combination (with coefficients in C ∞ (E k )) of forms of the type d r T θ with r = 0, . . . , p − 1. Therefore S p (λ) = 0 if and only if λ is τ k,p−1 -semibasic. Moreover, from this expression we have that the image of S p are τ k,k−p -semibasic forms. A simple argument shows that every τ k,k−p -semibasic form is in the image of S p .

Proposition 5.2. For any section λ∈
Proof. It is sufficient to prove the proposition for a section λ of the form λ = d r T θ for r = 0, . . . , k − 1. Using the definition of S we have Coordinate expression. Even though it is not needed in this paper, we will find the local expression of the vertical endomorphism. We take a local basis {e α } of sections of E and the dual basis {e α } of sections of E * . Then we have a local basis {X α , V r α } of sections of T E E k and the dual basis {X α , V α r } of sections of (T E E k ) * . We have to find the image by S of such sections. From the definition it follows that S(X α ) = 0. To find the expression of S(V α 1 ) we use equation (3.5), Finally, in order to find S(V α r ) for r≥2 we recall equation (3.5), V α r = d T V α r−1 , and using Proposition 5.2 we have for r≥2. It easily follows by induction that Therefore we found Remark. A more explicit local expression for S can be obtained as follows. The value of S(V α r ) can be written in the form Therefore we can also write In more compact way

Variational and Cartan operators.
Having in mind the procedure of integration by parts that will be needed in the calculus of variations, we can define two differential operators (see [4] for the standard case): the variational operator E, mapping sections of (T E E k ) * into sections of (T E E 2k ) * , given by and the Cartan operator S which maps sections of (T E E k ) * into sections of (T E E 2k−1 ) * given by . From the definition of E and S it is clear that Moreover, using the Proposition 5.2, a long but straightforward calculation shows that S(E(λ)) = 0, so that E(λ) is τ 2k,0 -semibasic, and also S k (S(λ)) = 0, so that S(λ) is τ 2k−1,k−1 -semibasic. Moreover S(d T λ) = λ.
Remark. In a more systematic way, one can proceed as in [4], where a family of operators D r was introduced to study complete lifts of vector fields and other related properties. For s = 0, . . . , k we define the operator D r mapping sections of (T E E k ) * into sections of (T E E 2k−r ) * be means of for λ ∈ Sec((T E E k ) * ). Following the arguments in [4], it is easy to see that the image of D r are τ 2k−r,k−r -semibasic sections, and that we have the relations From the second relation (5.10) it follows that only D 0 and D 1 are relevant, while the others can be determined inductively by D r+1 = 1 r S • D r for r≥1. The variational operator is E = D 0 and the Cartan operator is S = D 1 . Also from (5.10) for r = 0 we have S • D 0 = 0 so that E(λ) is τ 2k,0 -semibasic. Applying S k−1 to (5.10) for r = 1 we get so that S(λ) is τ 2k−1,k−1 -semibasic.

Variational calculus
Let J = [t 0 , t 1 ]⊂R and fix two points A 0 ∈E k−1 and A 1 ∈E k−1 . Given a Lagrangian function L ∈ C ∞ (E k ) we consider the action functional 2 L(a k (t)) dt restricted to admissible curves a in E such that a k−1 (t 0 ) = A 0 and a k−1 (t 1 ) = A 1 . We look for critical points of such a functional.
Of course, to speak about critical points, local maxima or local minima, has no meaning if we do not give explicitly a differential manifold structure to the set of such curves. In the case k = 1 it was shown in [36] that the relevant Banach manifold structure is the foliated one P(J, E), where each connected component (a leaf) is an E-homotopy class of admissible curves. Therefore, in the higher-order case we will use the same structure and we impose the additional conditions coming from the boundary conditions. In an alternative but equivalent way, we can restrict S to an E-homotopy leaf (which is a Banach submanifold) and we use differential calculus to find the critical points. It follows that finite variations are those defined by E-homotopies with the given boundary conditions.
We denote by m 0 , m 1 ∈M the base points m 0 = τ k−1,0 (A 0 ) and m 1 = τ k−1,0 (A 1 ). From the results in [36], the following result is easy to prove. Theorem 6.1. The set (6.2) P(J, E) A1 A0 = a∈P(J, E) a is C k and a k−1 (t 0 ) = A 0 , a k−1 (t 1 ) = A 1 is a Banach submanifold of P(J, E) m1 m0 . The tangent space to P(J, E) A1 A0 at a point a∈P(J, E) A1 A0 is Remark. If a∈P(J, E) m1 m0 is an admissible curve, the connected component of a in P(J, E) m1 m0 is the equivalence class H a of admissible curves in E which are E-homotopic to a. Similarly, if a∈P(J, E) A1 A0 , the connected component of a in P(J, E) A1 A0 is the equivalence class H ′ a ⊂H a of admissible curves in E which are E-homotopic to a and have higher-order contact with a at the endpoints.
2 In what follows the symbol S stands for the action functional. No confusion with the vertical endomorphism is possible since it will not be used explicitly. We will only need the operators E and S.
Given an admissible curve a we take a curve α s on P(J, E) A1 A0 with α 0 = a, and the corresponding E-homotopy α(s, t)dt + β(s, t)ds where α s (t) = α(s, t). Taking the derivative at s = 0, dt.
In the above expressions d stands for the standard exterior differential on a manifold. Using the exterior differential d of the algebroid T E E k and taking into account that σ k a (t) = (a k (t), σ(t), Ξ k a σ(t)) we have Let us consider the sections E(dL) and θ L = S(dL), which are related by Taking into account that E(dL) is τ k,0 -semibasic we will consider the associated map δL : E 2k → E * . Similarly, taking into account that θ L = S(dL) is τ 2k−1,k−1semibasic we will consider the associated map F L : Inserting this into the variation of the action we get After imposing the boundary conditions we arrive to Since σ is arbitrary, from the fundamental lemma of the Calculus of Variations, we find that a curve a is a critical point of the action if and only if it satisfies δL(a 2k (t)) = 0. We have proved the following result. In order to find the local expression of the Euler-Lagrange equations we just calculate the variations of the Lagrangian. To simplify the notation we will use the symbol δy α r for the expression δy α A straightforward calculation shows that, for r≥2, In the first term, we take into account that δy α r =σ α + C α βγ y β 1 σ γ and we perform a further integration by parts. Denoting by π α the coefficient of δy α 1 , Relabeling the sums we finally get Therefore, the Euler-Lagrange equations take the form of the system of ordinary differential equations (6.9) where the functions π α are given by (6.7).
The coordinate expressions of δL, F L and θ L can be obtained from the definition using the local expressions for the vertical endomorphism. However, from the integration by parts formula we know that the integrand is δL(a 2 k) , σ and that the boundary terms are F L (a 2k−1 ) , σ (k−1) , and hence from equation (6.8) we get that the local expression of δL is Taking the induced coordinates (x i , y α 1 , . . . , y α k−1 , µ 0 α , . . . , µ k−1 α ) on (T E E k ) * , the local expression of the Legendre transformation F L : , where the µ r α are given by the following functions Notice that the momenta p r α satisfy the relation which can serve to define them by recurrence starting from p k−1 α = ∂L ∂y α k . Also notice that p 0 α = π α . Finally, the coordinate expression of the Cartan form readily follows from that of F L , with p r α given by (6.11). Noether's theorem. An immediate consequence of our variational formalism is the following version of Noether's theorem.
Proof. Indeed, we have Therefore, for any solution a of the Euler-Lagrange equations we have so that G is constant along any solution of the Euler-Lagrange equations.
In particular, if η is an infinitesimal symmetry of the Lagrangian, i.e. d η k L = 0, then the function θ L , η 2k−1 ∈C ∞ (E 2k−1 ), the momentum in the direction of η, is a first integral.

Examples
We will consider in this section some typical examples of Lie algebroids and we will show the form of the Euler-Lagrange equations.
7.1. The standard case, parameters and quasi-velocities. In the standard case E = T M taking a coordinate basis e i = ∂ ∂x i we recover the standard higherorder Euler-Lagrange equations [16,14,40,39]. The same equations hold when we consider a system defined by a Lagrangian L λ ([γ] k ) ≡ L(λ, [γ] k ), depending on additional parameters λ∈Λ. In this case the Lie algebroid is E = Λ × T M → Λ × M with ρ(λ, v) = (0 λ , v) and the bracket is the bracket of vector fields on M depending on the variables λ as parameters.
Also, the formalism developed here allows naturally to use a different local basis {e i = ρ j i ∂ ∂x j } of vector fields in M . In such case the associated coordinates y i 1 are called quasi-velocities and the local expressions for the Euler-Lagrange equations that we have got is the Euler-Lagrange equations written in quasi-velocities, which are sometimes called the higher-order Hammel equations. In that expressions, the structure functions C i jk are the so called Hammel's transpositional symbols, defined by the equation For more information about quasi-velocities and their use in Mechanics see [29,6] and for their use in dynamic optimal control see [2]. 7.2. Systems with holonomic constraints. Let E⊂T M be an integrable subbundle (i.e. an integrable regular distribution on M ). A curve a : R → E is admissible if and only if the base curve γ = τ •a : R → M is contained into an integral leaf of E and a =γ. If we denote by F the foliation defined by E, so that E = T F , then it follows that E k = T k F , that is, it is the set of k-jets of curves contained in the leaves of F . Given a Lagrangian L∈C ∞ (E k ) an admissible curve a =γ is a critical point of the action if and only if γ is a critical point of the restriction of the Lagrangian to the k-tangent T k (F γ(t0) ) to the leaf F γ(t0) which contains γ. This follows from standard optimization results. Therefore the Euler-Lagrange equations can be obtained as the Euler-Lagrange equations for the restriction of the Lagrangian to every leaf, usually called the holonomic Euler-Lagrange equations.
In local coordinates (x i ) = (q a , w A ) adapted to the foliation, so that the leaves are given by the equations w A = k A = constant, we have coordinates (q a (r) , w A ), r = 0, . . . , k, in E k and the Euler-Lagrange equations read that is, the standard Euler-Lagrange equations on the variables q depending on w A as parameters. Alternatively, one can take a local basis {e a , e A } of vector fields adapted to the distribution E, i.e. E = span({e a }) and then the Euler-Lagrange equations can be written as δL(a(t) , e a = 0, whereL is any extension of L∈C ∞ (E) to a function on T M . In this way we get an expression of the Euler-Lagrange equations with holonomic constraints written in terms of quasi-velocities. 7.3. Lagrangian systems on Lie algebras. A Lie algebra g can be considered as a Lie algebroid over a singleton g → {e}. The anchor vanishes from where it follows that every curve ξ : R → g is admissible and hence we have that g k is just the cartesian product of k copies of g, [ξ] k−1 ≡ ξ(0),ξ(0),ξ(0), . . . , d k−1 ξ dt k−1 (0) ∈g × · · · × g. A section of g → {e} is just an element of g and a local basis {e α } of g provides global coordinates (ξ 1 , ξ 2 , . . . , ξ k ) in g k . Variational vector fields are of the form The Euler-Lagrange equations for a Lagrangian L∈C ∞ (g k ) are (7.2)π + ad * ξ π = 0, where π is given by (6.7), which in the present case takes the global form In this expression δL δξr stands for the globally defined partial derivative of L with respect to ξ r given by These equations are called the higher-order Euler-Poincaré equations [18,10]. They can be interpreted as the equations for parallel transport of the momentum π with respect to the canonical g-connection on g defined by D ξ ζ = [ξ, ζ]. See [34] for the details in the first order case. The same kind of equations are obtained in the case of a bundle of Lie algebras τ : K → M where the bracket depends on the variables in M , that is, we can have different Lie algebra structures on the fibres K m for different m∈M . 7.4. Systems with advected parameters. We consider a Lie algebra g acting on a manifold M by means of a morphism of Lie algebras g → X(M ); ξ → ξ M . The trivial bundle τ = pr 1 : E = M × g → M is endowed with a Lie algebroid structure with anchor ρ(m, ξ) = ξ M (m) and where the bracket is induced naturally by the bracket on g (the bracket of constant sections is just the constant section corresponding to the bracket on g).
The Euler-Lagrange equations take the form of the so called Euler-Poincaré equations with advected parameters where π is given by a expression similar to (7.3) but depending also on m, and ρ m is the restriction ρ m : g → T m M of ρ to the fibre over m∈M , i.e. ρ m (ξ) = ξ M (m). To find some general enough expression for the Euler-Lagrange equations we can consider the case of a trivial bundle where m is the base curve of v. Therefore admissible curves are of the form (ṁ(t), ξ(t)) for m(t) a curve in M and ξ(t) a curve in g, and we have the identification E k ≡ T k M × g k , given by ∂L ∂ṁ π + ad * ξ π = 0, where π is given by a expression similar to (7.3) and π M is In other words, the equations look like the ordinary Euler-Lagrange equations on M together with the Euler-Poincaré equations on g. In the case of a non-trivial bundle one has to use a principal connection on Q to obtain global expressions for the Euler-Lagrange equations (see [17,8] for the details).

Reduction and reconstruction
It was proved in [36] that a morphism of Lie algebroids Φ : E → E ′ defines a map between the path spacesΦ : P(J, E) → P(J, E ′ ) by compositionΦ(a) = Φ•a. If Φ is fiberwise injective (surjective) thenΦ is an immersion (submersion). Moreover, variational vector fields are mapped in a simple manner T Φ Φ•Ξ a σ = Ξ Φ•a (Φ•σ). As a consequence, a morphism induces relations between critical points of functions defined on path spaces, in particular between solutions of Euler-Lagrange equations for a first-order Lagrangian L in E and a first-order Lagrangian L ′ = L•Φ in E ′ , producing easy reduction and reconstruction results.
This correspondence can be easily extended to the higher-order case as follows. Consider a Lagrangian L∈C ∞ (E k ) and a Lagrangian L ′ ∈C ∞ (E ′k ) which are related by the map Φ k : E k → E ′k , that is, L = L ′ • Φ k . Then the associated action functionals S on P(J, E) and S ′ on P(J, E ′ ) are related byΦ, that is,

For the boundary conditions, if
, so that the corresponding variational problems Reduction theorems easily follows by considering fiberwise surjective morphisms of Lie algebroids. For the sake of clarity, we will omit any reference to the boundary conditions. Theorem 8.1 (Reduction). Let Φ : E → E ′ be a fiberwise surjective morphism of Lie algebroids. Consider a Lagrangian L on E k and a Lagrangian L ′ on E ′k such that L = L ′ • Φ k . If a is a solution of the Euler-Lagrange equations for L then a ′ = Φ • a is a solution of Euler-Lagrange equations for L ′ .
A0 . If Φ is fiberwise surjective, thenΦ is a submersion, from where it follows thatΦ maps critical points of S into critical points of S ′ , i.e. solutions of the Euler-Lagrange equations for L into solutions of Euler-Lagrange equations for L ′ .
We can reduce partially a system and then reduce again the obtained system. The final result obviously coincides with the system obtained by the total reduction. Theorem 8.2 (Reduction by stages). Let Φ 1 : E → E ′ and Φ 2 : E ′ → E ′′ be fiberwise surjective morphisms of Lie algebroids. Let L, L ′ and L ′′ be Lagrangian functions on E k , E ′k and E ′′k , respectively, such that L ′ •Φ k 1 = L and L ′′ •Φ k 2 = L ′ .
Then the result of reducing first by Φ 1 and later by Φ 2 coincides with the reduction by Φ = Φ 2 • Φ 1 .
As an example of the above situation we can consider a system with a group of symmetry G. If N is closed normal subgroup of G we can reduce first by N and later by G/N . Alternatively we can reduce directly by the full group G. The result of both procedures is the same. See [7,12] for the first order case.
For the reconstruction of solutions we have the following result, which is immediate from the variational formalism.
The reconstruction procedure can be understood as follows. Consider a fiberwise surjective morphism Φ : E → E ′ and the associated reduction mapΦ : P(J, E) → P(J, E ′ ). Given an E ′ -path a ′ ∈ P(J, E ′ ) solution of the dynamics defined by the Lagrangian L ′ , we look for an E-path a ∈ P(J, E) solution of the dynamics for the Lagrangian L = L ′ • Φ, such that a ′ =Φ(a). For that, it is sufficient to find a map Υ : P(J, E ′ ) → P(J, E) such thatΦ • Υ = id P(J,E ′ ) . Indeed, given the E ′ -path a ′ solution for the reduced Lagrangian L ′ , the curve a = Υ(a ′ ) is an E-path and satisfy Φ • a = a ′ . From Reconstruction Theorem 8.3 we deduce that a is a solution of the Euler-lagrange equations for the original Lagrangian.
When specifying the boundary conditions is necessary, the map Υ must restrict to a map Υ A1 A0 : P(J, E ′ ) . Of course, one can define several maps Υ, and different maps will produce different E-paths a for the same E ′ -path a ′ . In most cases of interest, Φ is fiberwise bijective, so thatΦ is a local diffeomorphism, and the reconstruction process can be done with the help of some global diffeomorphismΦ : P(J, E) → P × P(J, E ′ ), for some manifold P , such that pr 2 •Φ =Φ. Then fixing p∈P we define Υ p (a ′ ) =Φ −1 (p, a ′ ). In this way the set of solutions to which a ′ can be lifted is parametrized by P . See bellow for some explicit examples of application of this procedure (and some extensions).
Next, we present some examples where the reduction process indicated above can be applied.
Assume that L is a left-invariant Lagrangian function on T k G, i.e.
for every element h∈G and every curve g(t) on G. Then we can define a Lagrangian L ′ on g k by means of where g e is the solution of the differential equationġ = T ℓ g ξ(t) with initial value g e (0) = e. In other words we have L([g] k ) = L ′ ([g −1ġ ] k−1 ) for every curve g(t) in G, or equivalently L = L ′ •Φ k . It follows that if a is a solution of the higherorder Euler-Lagrange for L in the Lie group G then a ′ = Φ•a is a solution of the higher-order Euler-Poincare for L ′ on g. Moreover, since Φ is fiberwise bijective every solution can be found in this way, so that the higher-order Euler-Lagrange equations on the group reduce to the higher-order Euler-Poincaré equations on the Lie algebra.
For the reconstruction process we note that the mapΦ : P(J, T G) → G × P(J, g) given byΦ(g,ġ) = (g(t 0 ), T ℓ g −1ġ) is a global diffeomorphism. Its inverse is the mapΦ −1 (g 0 , ξ) = (g,ġ) where g(t) is the integral curve of the left-invariant timedependent vector field X(t, g) = T ℓ g ξ(t), with initial value g(t 0 ) = g 0 . Thus the map Υ g0 (ξ) =Φ −1 (g 0 , ξ) provides a reconstruction map. Notice that the curve g(t) above is g(t) = g 0 g e (t) where g e (t) is the integral curve of X with g e (t 0 ) = e.

8.2.
Lie groupoid reduction. For integrable Lie algebroids, the variational principle developed here can be obtained via a reduction of an ordinary higher-order variational principle with holonomic constraints. In the first-order case this was the argument used by Weinstein [41] to obtain a variational principle on (integrable) Lie algebroids. Here we consider the higher-order case.
Consider a Lie groupoid G over M with source s and target t, and denote by E its Lie algebroid, E = A(G). Denote by T s G → G the kernel of T s with the structure of Lie algebroid as integrable subbundle of T G. The integral manifolds of T s G are the s-fibres of the groupoid, so that T s G is the distribution tangent to the foliation S = ∪ m∈M s −1 (m), and hence (T s G) k = (T S) k = T k S.
The map Φ : T s G → E given by left translation to the identities, Φ(v g ) = T ℓ g −1 (v g ) is a fiberwise bijective morphism of Lie algebroids. As a consequence, if L is a Lagrangian function on E k and L is the associated left invariant Lagrangian on (T s G) k , then the solutions of the higher-order Euler-Lagrange equations for L (which are the holonomic Euler-Lagrange equations) project by Φ to solutions of the higher-order Euler-Lagrange equations on the Lie algebroid E.

8.3.
Euler-Poincaré reduction with advected parameters. Let G be a Lie group acting from the right on a manifold M . We consider the Lie algebroid E = M × T G → M × G where M is a parameter manifold, that is, the anchor is ρ(m, v g ) = (0 m , v g ) and the bracket is determined by the standard bracket of vector fields on G, i.e. of sections of T G → G, depending on the coordinates in M as parameters. Consider also the transformation Lie algebroid E ′ = M × g → M , where ρ(m, ξ) = ξ M (m), (ξ M being the fundamental vector field associated to ξ ∈ g). The map Φ : M × T G → M × g given by Φ(m, v g ) = (mg, g −1 v g ) is a morphism of Lie algebroids over the action map ϕ(m, g) = mg, and it is fiberwise bijective.
Consider We consider the Lagrangian L ′ on E ′k given by L ′ (m, [ξ] k−1 ) = L(m, [g e ] k ), where g e (t) is the solution of the initial value problemġ = T ℓ g ξ(t), g(t 0 ) = e. Then L ′ • Φ k = L. The variables m which initially are parameters are now dynamic variables due to the group action and are called advected parameters. In this way, solutions of the higher-order Euler-Lagrange equations for L (standard Euler-Lagrange equations with parameters) are mapped by Φ to solutions of the higher-order Euler-Lagrange equations for L ′ , i.e. the higher-order Euler-Poincaré equations with advected parameters.
For the reconstruction process we consider the global diffeomorphismΦ : P(J, M × T G) → G × P(J, M × g) given byΦ m, (g,ġ) = g(t 0 ), (mg, T ℓ g −1ġ) . Its inverse isΦ −1 g 0 , (m, ξ) = mg −1 , (g,ġ) where g(t) is the integral curve of the leftinvariant time-dependent vector field X(t, g) = T ℓ g ξ(t) with g(t 0 ) = g 0 . Thus the map Υ g0 (m, ξ) =Φ −1 g 0 , (m, ξ) provides a reconstruction map. 8.4. Lagrange-Poincaré reduction. We consider a Lie group G acting free and properly on a manifold Q, so that the quotient map p : Q → M = Q/G has the structure of a principal bundle. We consider the standard Lie algebroid structure on E = T Q and the Atiyah algebroid E ′ = T Q/G → M . The quotient map Φ : T Q → T Q/G, Φ(v) = [v] G is a Lie algebroid morphism and it is fiberwise bijective. Every G-invariant Lagrangian on T k Q defines uniquely a Lagrangian L ′ on E ′k such that L ′ • Φ k = L, explicitly given by Therefore every solution of the G-invariant Lagrangian on T k Q projects to a solution of the reduced Lagrangian on (T Q/G) k ≡ T k Q/G, and every solution on the reduced space can be obtained in this way. Thus, the higher-order Euler-Lagrange equations on the principal bundle reduce to the higher-order Lagrange-Poincaré equations on the Atiyah algebroid [17].
For the reconstruction process we consider the manifold Q × M P(J, T Q/G) defined by Q × M P(J, T Q/G) = { (q 0 , a)∈Q × P(J, T Q/G) | p(t 0 ) = τ (a(t 0 )) }. The mapΦ : P(J, T Q) → Q × M P(J, T Q/G) given byΦ(q,q) = (q(t 0 ), [q] G ) is a global diffeomorphism. Its inverse is given byΦ −1 (q 0 , a) = (q,q) where q(t) is the curve determined as follows. The curve a(t) is admissible, so that it is of the form a(t) = [q(t),q(t)] G ; if g 0 ∈G is the unique element in the group such thatq(t 0 ) = g 0 q 0 , then we define q(t) = g −1 0q (t). Therefore the map Υ q0 (a) = Φ −1 (q 0 , a) provides a reconstruction map, which in this case is defined as a map Υ q0 : P(J, T Q/G) m0 → P(J, T Q) q0 where m 0 = p(q 0 ).

Conclusions and outlook
In this paper we have obtained the Euler-Lagrange equations for a higher-order Lagrangian. The emphasis has been on the variational description using the tools of variational calculus on Lie algebroids developed in [36].
However, there are other many aspects of the theory that we have left out and will be studied in future. The variational structure strongly suggests that a symplectic formalism is possible, similar to Klein's formalism [25] of standard Lagrangian Mechanics and its generalization to Lie algebroids [29], and the corresponding Hamiltonian version [30,15]. It is interesting to study such formalisms and the transformation properties of the symplectic form.
On the other hand, a geometric version of Pontryagin maximum principle which allows reduction in the setting of Lie algebroids was studied in [31,32] (a direct proof was given in [23], based on needle variations [38] and the results in [36]). It is interesting to study the relation between that results with those on this paper and the proposed Hamiltonian versions already mentioned, on one hand, and with other 'variational principles' that appeared recently in the literature (Hamilton-Pontryagin [42], Clebsh-Pontryagin [19], etc.), on the other.
Concerning the optimality properties of the solutions, in this paper we have obtained only first order conditions, that is conditions for critical points of the action. To characterize those which are local maxima/minima we will study further optimality conditions. Since our results are based on a bona fide variational principle it is expected that the Hessian of the action contains such information. Moreover, when two Lagrangians are related by a morphism the Hessians at corresponding critical points should be also related.
Problems with time dependent Lagrangians can be treated by embedding E into T R × E as it is mentioned in [36]. However the formalism introduced in [35,22] for first-order systems can be easily generalized to the present case. All this problems and the possible generalizations to field theory, following the ideas in [33], will be studied elsewhere.