Bundle-theoretic methods for higher-order variational calculus

We present a geometric interpretation of the integration-by-parts formula on an arbitrary vector bundle. As an application we give a new geometric formulation of higher-order variational calculus.


Introduction
Our results. The main motivation of this paper is to clarify the geometry of higher-order variational calculus. In order to study this topic we had to introduce new geometric tools and prove some results which we quickly sketch below.
First, we observe that, given a vector bundle σ : E → M , it is possible to characterize two canonical morphisms Υ k,σ : T k,k σ → σ and υ k,σ : T k,k σ → T k σ, where the bundle of semi-holonomic vectors T k,k σ consists of all elements of T k T k E which project to the elements of T 2k M ⊂ T k T k M (i.e., to holonomic vectors). Morphisms Υ k,σ and υ k,σ provide an elegant geometric description of the k th order integration-by-parts formula on the bundle σ. A more precise formulation is provided by Theorem 3.3 below.
Then, we use this result to obtain a comprehensive and, to certain extent, simpler geometric interpretation of the standard procedure of deriving both the Euler-Lagrange equations (forces) and the boundary terms (generalized momenta) for a k th order variational problem. Our description is "natural" in the sense that it mimics the standard way of deriving the Euler-Lagrange equations. We start from a homotopy γ(t, s) in M and then transform the variation of the action t 1 s=0 t 1 t 0 L(t k t γ(t, s)) d t to extract the integral and the boundary terms. Here t k t γ stands for the k th jet of a curve γ = γ(t) at a point t ∈ R. In particular, t 1 t=t 0 γ(t) is the tangent vector to the curve γ at t 0 . On the computational level this procedure consists basically of two steps: • reversing the order of differentiation to get t k t t 1 s γ(t, s) from t 1 s t k t γ(t, s), • performing the k th order integration by parts to extract t 1 s=0 γ(t, s) from t k t t 1 s=0 γ(t, s).

Preliminaries
Higher tangent bundles. Throughout the paper we shall work with higher tangent bundles and use the standard notation T k M for the k th tangent bundle of the manifold M . Points in the total space of this bundle will be called k-velocities. An element represented by a curve γ : [t 0 , t 1 ] → M at t ∈ [t 0 , t 1 ] will be denoted by t k γ(t) or t k t γ(t). The k + 1 st tangent bundle is canonically included in the tangent space of the k th tangent bundle (see, e.g., [21]): The composition of this injection with the canonical projection τ T k M : TT k M → T k M defines the structure of the tower of higher tangent bundles: The canonical projections from higher to lower-order tangent bundles will be denoted by τ k s : T k M → T s M (for k ≥ s). Instead of τ k 0 : T k M → M we simply write τ k and, instead of τ 1 0 = τ 1 : TM → M , just use the standard symbol τ . The cotangent fibration is denoted by τ * : T * M → M .
Another important constructions are the iterated tangent bundles T (k) M := T . . . TM and the iterated higher tangent bundles T n 1 . . . T nr M . Elements of the latter will be called (n 1 , . . . , n r )velocities. For notational simplicity we shall also denote the iterated higher tangent functor T n 1 T n 2 . . . T nr by T n 1 ,...,nr .
Of our particular interest will be the bundles T k T l M = T k,l M . These bundles admit natural projections to lower-order tangent bundles which will be denoted by τ (k,l) (k ′ ,l ′ ) : T k T l M → T k ′ T l ′ M (for k ≥ k ′ and l ≥ l ′ ). (Iterated higher) tangent bundles are subject to a number of natural inclusions such as already mentioned ι 1,k : T k+1 M ⊂ TT k M , ι l,k : T k+l M ⊂ T l T k M , ι k : T k M ⊂ T (k) M , etc., which will be used extensively.
Occasionally, when dealing with manifolds other than M , or when it can lead to confusions, we will add a suffix to the maps ι ... ... , τ ... ... , etc., to emphasize which manifold we are working with, e.g., τ k l = τ k l,M , etc. Given a smooth function f on a manifold M one can construct functions f (α) on T k M , with 0 ≤ α ≤ k, the so called (α)-lifts of f (see [17]), defined by f (γ(t)).
The functions f (k) : T k M → R and f (1) : TM → R are called the complete lift and the tangent lift of f , respectively. By iterating this construction we obtain functions f (α,β) : on M gives rise to the so-called adapted coordinate systems (x a,(α) ) 0≤α≤k on T k M and (x a,(ǫ) ) ǫ on T n 1 . . . T nr M where the multi-index ǫ = (ǫ 1 , . . . ǫ r ) is as before, and x a,(α) , x a,(ǫ) are obtained from x a by the above lifting procedure. Within this notation we easily find that the canonical inclusion ι k : T k M → T (k) M is given by where ǫ ∈ {0, 1} k . Let us remark that in the definition of f (α) we follow the convention of [5,20], whereas the original convention of [17] is slightly different, since it contains a normalizing factor 1 α! in front of the derivative.

Tangent lifts of vector bundles and canonical pairings.
Let us pass to another important tool for our analysis, namely the (iterated) higher tangent bundles of vector bundles. Let σ : E → M be a vector bundle. It is clear that σ may be lifted to vector bundles T k σ : Throughout the paper we denote by (x a ) the coordinates on the base M , by (y i ) the linear coordinates on the fibers of σ, and by (ξ i ) the linear coordinates on the fibers of the dual bundle σ * : E * → M . Natural weighted coordinates on (iterated higher) tangent lifts of σ and σ * are constructed from x a , y i , ξ j by the lifting procedure mentioned above. They are denoted by adding the proper degree to a coordinate symbol. Degrees will be denoted by bracketed small Greek letters: (ǫ), (α), (β), etc.
The natural pairing ·, · σ between E and E * induces a non-degenerated pairing between TE and TE * over TM : i.e., ·, · Tσ is the tangent lift of ·, · σ . In a similar way, ·, · σ can be lifted to non-degenerate pairings on higher tangent prolongations of E and E * .
(c) ·, · T k σ = ·, · (k) σ , i.e., the pairing ·, · T k σ is the complete lift of ·, · σ (up to the isomorphism Proof. We get (2.3) by iterated differentiation of the function ·, · σ : (x a , ξ i ), (x a , y i ) → i ξ i y i . Taking into account (2.2) we find (2.4). Since the tangent lift of a non-degenerate pairing is nondegenerate, so it is for ·, · T (k) σ . The non-degeneracy of the pairings ·, · T (k) σ and ·, · T k σ can also be easily seen from the above local formulas. Finally, (c) follows immediately from the definition of the complete lift and the local expression (2.4) of ·, · T k σ .
We stress that the above lifting procedure can be expressed in the framework of Weil functors and Frobenius algebras [26,13].
Canonical flip κ k and its dual ε k . It is well-known that the iterated tangent bundle TTM admits an involutive double vector bundle isomorphism (called the canonical flip) κ : TTM −→ TTM, which intertwines the projections τ TM : TTM → TM and Tτ : TTM → TM (see, e.g., [16]). Such a notion of canonical flip can be generalized to a family of isomorphisms which map the projection T k τ : Morphisms κ k can be also defined inductively as follows: κ 1 := κ and κ k+1 := Tκ k • κ T k M T k+1 TM , i.e., as the unique morphism making the diagram commutative. The local description of κ k is very simple. If x a,(α,ǫ) are natural coordinates on T k TM and x a,(ǫ,α) are natural adapted coordinates on TT k M (with 0 ≤ α ≤ k and ǫ = 0, 1), then x a,(α,ǫ) corresponds to x a,(ǫ,α) via κ k .
Introduce now the dual ε k : T * T k M → T k T * M of the canonical flip κ k defined via the equality, where V ∈ T k TM and Ψ ∈ T * T k M is a vector such that both pairings make sense (cf. [2]). Formula (2.6) shows that κ k and ε k are "adjoint" to each other with respect to canonical pairings, as schematically shown by the commutative diagram ; ; In the coordinates (x a,(α) , p a,(α) = ∂ x a,(α) ) on T * T k M and (x a,(α) , p (α) a ) on T k T * M (adapted from standard coordinates (x a,(α) ) on T k M , and (x a , p a ) on T * M , respectively), we find from (2.4) that

The main result
In this section the construction of the vector bundle morphisms Υ k,σ and υ k,σ , associated with an integer k and a vector bundle σ : E → M , will be described. These morphisms are closely related with the geometric integration-by-parts procedure and will play a crucial role in the geometric construction of the Euler-Lagrange equations in the next Section 4.
In agreement with our previous notation we shall denote by T (k) E = T 1,...,1 E the subbundle of semi-holonomic velocities in T (k) E = T 1,...,1 E.
In future considerations the bundles T k,k E and T (k) E will be of our special interest. Let us remark that although, in general, there is no canonical projection T (k) E → T k E, there exists a natural projection P k : T (k) E → T k E. It is defined as the left inverse to the canonical inclusion ι k E but considered as a map to T k E, as is explained in the following proposition.

Proposition 3.2. Consider a semi-holonomic vector
where Ψ ∈ T k E * ⊂ T (k) E * lies over v k , defines a canonical projection P k : T (k) E → T k E, i.e., P k •ι k E = id T k E . Moreover, P k is a vector bundle morphism and it can be expressed in local coordinates as P k x a,(ǫ) , y i,(ǫ) = x a,(α) , y i,(α) , is the arithmetic average of all the coordinates of total degree α.
Proof. It follows immediately from the properties of the non-degenerate pairing ·, · T k σ (see Proposition 2.1).
The map Υ k,σ . The result below describes the construction of a certain canonical and universal vector bundle morphism Υ k,σ from T k,k σ to σ. The precise sense of the word universal will be given later in Lemma 3.4. Informally speaking, any other morphism T k,k σ → σ can be derived in an easy way from T k,k . In the next Section 4 we show that this morphism is directly connected with integration by parts in the procedure of deriving the Euler-Lagrange equations.
To fix some notation denote an element Φ ∈ T k,k E ⊂ T k T k E by Φ (k,k) and its projections to lower-order velocities by where m, n ≤ k. Observe that since Φ lies over some 2k-velocity, say v 2k ∈ T 2k M , then all the elements Φ (m,n) project under T m T n σ to a fixed m + n-velocity v m+n ∈ T m+n M ⊂ T m T n M independently on the numbers in the sum m + n. In particular, different elements Φ (m,n) with m + n fixed, belonging a priori to different bundles T m T n E, can be added together in the vector bundle T (m+n) σ : T (m+n) E → T (m+n) M , which contains all of them. Denote by Φ the following element of T (k) E: Now we are ready to state the main result of this section.

Theorem 3.3 (Bundle-theoretic integration by part). Let
(c) υ k,σ satisfies the recurrence formulas where in the last term we consider Φ (k−1,k) as a semi-holonomic vector in T k−1,k−1 TE ⊃ T k−1,k E; (d) Υ k,σ and υ k−1,σ are related by the "bundle-theoretic integration by parts formula" (e) morphism Υ k,· commutes with the tangent functor T up to the canonical isomorphism κ k,k,E : is commutative; (f) morphism υ k,· commutes with the tangent functor T up to the canonical isomorphisms κ k,k,E : Lemma 3.4 is a natural continuation of Theorem3.3, but we decided to keep it separated since its proof is rather technical (see Appendix).
Proof of Theorem 3.3. We shall prove first that Υ k,σ is a well-defined mapping and, simultaneously part (b). To this end, we proceed by induction with respect to k for arbitrary σ : E → M .
Consider k = 1. Take Φ = Φ (1,1) ∈ TTE and let v 2 := TTσ(Φ) ∈ T 2 M . TTσ has the structure of a double vector bundle with the projections Tτ E and τ TE onto TE. Vectors Φ (0,1) = τ TE (Φ) and It follows that their difference with respect to the vector bundle structure Tσ : TE → TM belongs to TE M ∼ = E × M TM . Hence, from the properties of the pairing ·, · Tσ , In other words Υ 1,σ is well-defined and satisfies (3.5). Assume now that the assertion holds for every l ≤ k − 1 and every vector bundle σ. Observe that, for every l, is a functorial vector bundle morphism over τ 2l l : T 2l M −→ T l M ⊂ T (l) M being the combination, with constant coefficients, of functorial vector bundle morphisms Φ → Φ (i,l−i) . In other words, given a vector bundle σ ′ : E ′ → M ′ and a vector bundle morphism α : σ → σ ′ , it holds .
This fact, combined with functoriality of ·, · σ and the inductive assumption, guarantees that for every l ≤ k − 1. In particular, we may take as α in (3.13), the following vector bundle morphisms Thanks to our inductive assumption the right-hand side later equals Now we can use equalities (3.14) and (3.15) to bring the above expression to the form This sums up to formula (3.6) and assures that Υ k,σ is indeed correctly defined. The proof of (a) is simple. Locally, Φ and t k ξ (in canonical induced graded coordinates on T k,k E and T k E * ) are given by where α = 0, 1, . . . , 2k and β, β ′ = 0, 1, . . . , k.
We have shown that Φ, t k ξ T (k) σ , which has a coordinate form of a polynomial in ξ (β) i and y j,(β ′ ,β ′′ ) , actually does not depend on ξ we know that y i is the coordinate of total degree k in Φ. We conclude that the formula (3.4) for Υ k,σ is true.
To prove (c) we use the standard decomposition of binomial coefficients k+1 i+1 = k i + k i+1 to obtain In the first expression on the right-hand side of this equality we easily recognize formula (3.3), whereas, in the second, formula (3.2) k−1 .
The proof of (d) is very similar. We decompose Now on the right-hand side of this equality we easily recognize formulas (3.3) and (3.2) k−1 . Finally, to prove (e) and (f), observe that the tangent functor T commutes (up to canonical isomorphisms) with the projections τ k,k i,k−i and P k . Thus it commutes (up to canonical isomorphisms) with Υ k,· and υ k,· which are linear combinations of the compositions of the mentioned projections.
Remark 3.5. Note that in formula (3.4) only the coordinates of the highest degree (k) matter but the coordinates of lower degrees may be non-trivial. For example, if k = 2, we have Φ = Φ (0,2) − 2Φ (1,1) + Φ (2,0) ∈ TTE and Φ (0,2) ∈ T 2 E, interpreted as an element of TTE by means of ι 1,1 : T 2 E → TTE, equals to x a,(0) , x a,(1) , x a,(1) , x a, (2) , y i,(0,0) , y i,(0,1) , y i,(0,1) , y i,(0,2) . Similarly for Φ (2,0) . Therefore, in coordinates, Φ looks like Φ ∼ x a,(0) , x a,(1) , x a, (2) , y i,(0,0) , y i,(1,0) , y i,(0,1) , y i, (1,1) , Remark 3.6. Observe that, for any k and l, the bundle T l,l T k,k E is the pullback of T l,l T k,k E with respect to the canonical inclusion T l,l ι k,k • ι l,l T 2k M : T 2l,2k M ⊂ T l,l T k,k M , i.e., Consider now the inclusion T k+l,k+l E ⊂ T l,l T k,k E. Any semi-holonomic vector X ∈ T k+l,k+l E lying over v 2k+2l ∈ T 2k+2l M ⊂ T k+l,k+l M is mapped via this inclusion to an element lying over an element in T 2l T 2k M ⊂ T l,l T k,k M . Thus we have the canonical inclusion T k+l,k+l E ⊂ T l,l T k,k E. Therefore the inductive formula (3.6) can be described as follows: Expressing Υ k,σ as a composition of morphisms Υ 1,· according to (3.6) we can obtain the more general formula Υ k+l,σ = Υ k,σ • Υ l, T k,k σ T k+l,k+l E , i.e., (3.16) The proof is left to the reader. In light of (3.9), we can interpret formulas (3.16) and (3.17) as follows: integration by parts k + l times can be obtained as the composition of k times and l times integration by parts.

Applications to variational calculus
In this section we use Theorem 3.3 to give a geometric construction of the force and momentum in higher-order variational calculus (Theorem 4.2). As a consequence, in Theorem 4.3, we obtain necessary and sufficient conditions (Euler-Lagrange equations and transversality conditions) for a curve to be an extremal of the higher-order variational Problem 4.1.
Formulation of the problem. Consider a Lagrangian function L : T k M → R and the associated action where γ : [t 0 , t 1 ] → M is a path and t k γ(t) ∈ T k γ(t) M its k th prolongation. The set of admissible paths t k γ will be denoted by ADM([t 0 , t 1 ], T k M ).
By an admissible variation of an admissible path t k γ(t) we will understand a curve δt k γ(t) ∈ T t k γ(t) T k M . Observe that every admissible variation δt k γ(t) can be obtained from a homotopy χ(·, ·) : [t 0 , t 1 ] × (−ǫ, ǫ) → M such that χ(t, 0) = γ(t) and δt k γ(t) = t 1 s=0 t k χ(t, s). We see that δt k γ is generated by δγ(t) = t 1 s=0 χ(t, s) ∈ T γ(t) M , i.e., a vector field along γ(t), in the sense that So, δγ can be called a generator of the variation δt k γ. Given an admissible variation δt k γ we may define the differential of the action S L in the direction of this variation: Define now a natural projection which sends an admissible path t k γ to the pair consisting of its initial and final (k − 1)-velocity. Its tangent map TP sends a variation δt k γ ∈ T t k γ T k M to the pair (δt k−1 γ(t 0 ), δt k−1 γ(t 1 )) ∈ TT k−1 M × TT k−1 M . Now we are ready to formulate the following variational problem. Let us comment the above formulation. In our approach to variational problems we study the behavior of the differential of the action functional in the directions of admissible variations (differential approach), rather to compare the values of the action on nearby trajectories (integral approach). Hence, solutions of Problem 4.1 are only the critical, not extremal, points of the action functional (4.1). The philosophy of understanding a variational problem as the study of the differential of the action restricted to the sets of admissible trajectories and admissible variations allows one to treat the unconstrained and constrained cases in a unified way (see, e.g., [7,9,11]).
Higher-order variational calculus. Let S L be the action functional (4.1) and δt k γ the variation (4.2). Define the force F L,γ (t) ∈ T * M and the momentum M L,γ (t) ∈ T k−1 T * M along γ(t) by Theorem 4.2. The differential of the action S L in the direction of the variation δt k γ equals Theorem 4.3 below is an immediate consequence of formula (4.5).

Theorem 4.3. A curve γ is a solution of Problem 4.1 if and only if it satisfies the following Euler-Lagrange (EL) equation
F L,γ (t) = 0 (4.6) and the transversality conditions Proof of Theorems 4.2 and 4.3. Let us calculate the variation of the action S L in the direction δt k γ: Now we can use formula (3.9) with Φ (k,k) = t k Λ L (t k γ(t)), Φ (0,k) = Λ L (t k γ(t)) and Φ (k,k−1) = t k λ L (t k γ(t)), since the element Φ (k,k) := t k Λ L (t k γ(t)) ∈ T k T k T * M is a semi-holonomic vector as it projects to t k t k γ = t 2k γ ∈ T 2k M . We get In the first summand we recognize the force F L,γ (t) defined by (4.3). To the second we can apply the equality We conclude that where the momentum M L,γ (t) is defined by (4.4). Thus the variation (4.5) reads Integrating the above expression over [t 0 , t 1 ] we get (4.5), concluding the proof of Theorem 4.2. Theorem 4.3 follows easily, as δt k−1 γ(t) = κ k−1 (t k−1 δγ(t)) and ε k−1 is dual to κ k−1 , in light of equation (2.6).
Remark 4.4. The process of constructing the EL equations (4.6) starting from the Lagrangian function L can be followed on the diagram Similarly, the geometric construction of the momenta (4.4) corresponds to the diagram (4.10) Above we used dotted arrows to denote the objects associated with the Lagrangian function, whereas dashed arrows are maps defined only along the images of γ(t).
Note also that the map Υ k,τ * allows us to define higher-order EL equations with external forces. Namely, given an external force, i.e., a map F : [t 0 , t 1 ] → T * M , we can consider equation When F (t) = 0, the equation above reduces to the EL equation (4.6).

Local form of the forces and momenta.
We shall now derive the local form of the force (4.3) and momentum (4.4).
Let us first calculate the force. Consider a trajectory γ(t) ∼ (x a (t)). The differential d L(t k γ(t)) ∈ T * T k M is given by p a,(α) = ∂L ∂x a,(α) (t k γ(t)), hence using (2.7) we calculate the coordinates of Λ L (t k γ(t)) ∈ T k T * M , namely, ∂x a,(k−β) , and using (3.4) we find that our formula (4.3) of the force takes the following well-known form Concerning momentum, a direct (i.e., by means of formulas (3.2) and (4.10)) derivation of its local form is a quite complicated computational task, so that we will obtain it by using formula (4.8), instead.

Examples and perspectives
Tulczyjew's approach to higher-order geometric mechanics. The problem of geometric formulation of higher-order variational calculus on a manifold M has a few solutions. The first approach is due to Tulczyjew [20,21], who gave a geometric construction of a 0-order derivation E : Sec(T * T ∞ M ) → Sec(T * T ∞ M ) such that E(d L) = 0 are the higher-order Euler-Lagrange equations for any Lagrangian function L : T k M → R. Tulczyjew expressed his construction using the language of derivations and infinite jets. The latter allowed him to cover all orders k by a unique operator. Later Tulczyjew's theory was interestingly extended by Crampin, Sarlet and Cantrijn [4].
Another approach was inspired by Tulczyjew's papers [22,23] on the first order mechanics. The idea was to generate the EL equations from a Lagrangian submanifold. Two similar solutions where given by Crampin [3] and de Leon and Lacomba [14]. They constructed the equations from a Lagrangian submanifold in TT k−1 T * M or TT * T k−1 M generated by the Lagrangian L .
All these solutions have, however, some drawbacks. First of all, they describe only a part of the Lagrangian formalism, namely the EL equations, whereas the full structure of variational calculus should contain also momenta (boundary terms). Secondly, the correctness of these constructions is checked in coordinates. Note, however, that it is not the coordinate expression that defines the EL equations, but the opposite: we deduce the right local expression from the proper variational principle. Therefore a fully satisfactory geometric construction should somehow explain the steps performed while deriving the known form of the equations (as it is in our approach described in the previous Section 4), not give a black-box answer.
In later years Tulczyjew [24] extended his work to give a full description of higher-order Lagrangian formalism (i.e., including momenta) in the language of derivations. Another approach was communicated to us by Grabowska [6], who derived the k th order formalism as the first order formalism on T k−1 M . Her ideas are, to some extend, similar to these of [3,14], where the canonical inclusion T k M ⊂ TT k−1 M was also used. Another quite general approach to the topic was presented by A.M. Vinogradov and his collaborators in the framework of secondary calculus (see Section 3 of [25] and the references therein).
Below we relate Tulczyjew's approach to our results from the previous Section 4.
Comparison with Tulczyjew's formulas. In [21,24] Tulczyjew introduced graded derivations of degree 0 defined by means of two basic derivations, namely:  Recall that a derivation a of the algebra of differential forms Ω • (M ) is called a d * -derivation (resp. i * -derivation) if a commutes with de Rham differential (resp. if a vanishes on Ω 0 (M ) = C ∞ (M )) ( [21], see also [13], chapter 8). As the algebra Ω • (M ) is generated by Ω 0 (M ) and Ω 1 (M ), hence a d * -derivation (resp. i * -derivation) is fully determined by its values on Ω 0 (M ) (resp. Ω 1 (M )). The values of d T on Ω 0 (T k M ) are given in the above characterization of d T , while for ι Fn one defines ι Fn µ, v := µ, F n v for a 1-form µ and a tangent vector v on T k M . Note that F n = F n 1 and F 1 is the canonical higher almost tangent structure on T k M [15].
It turns out that the form E(d L) ∈ Ω 1 (T 2k M ) is vertical with respect to the projection τ 2k : T 2k M → M and that the form P(d L) ∈ Ω 1 (T 2k−1 M ) is vertical with respect to τ 2k−1 k−1 : T 2k−1 M → T k−1 M . Therefore taking the appropriate vertical parts of these forms we can define operators Tulczyjew's constructions to ours.

Applications to mechanics on algebroids.
In [10] we showed that with every almost-Lie algebroid structure on the bundle σ : E → M , one can canonically associate an infinite tower of graded bundles equipped with a family of graded-bundle relations The relation κ k is of special kind -it is dual of a vector bundle morphism ε k : T * E k → T k E * . A natural example of such a structure is provided by the higher tangent bundles E k = T k M together with the canonical flips κ k : T k TM → TT k M . Another example is E k := T k e G, the higher tangent space at identity e ∈ G to a Lie group G. Both examples should be considered as the extreme cases of what should we call a higher algebroid. Except for k = 1 there is no Lie bracket on sections of E k . It is the relation κ k which is responsible for the algebraic structure on E k . More general examples can be obtained by reducing higher tangent bundles of Lie groupoids.
Given a smooth function L : E k → R one can naturally define a variational problem on E k . Such problems cover, on one hand, the standard variational problems like Problem 4.1 (in which case E k = T k M ), and, on the other hand, the reduction of invariant higher-order variational problems on a Lie groupoid. We showed in [10] that for such problems an analog of Theorem 4.2 holds, as well. Thus, we can characterize the variation of an action by means of the force Υ k,σ * t k ε k d L(t k γ(t)) and momentum υ k−1, is the tower projection (5.1), σ * : E * → M is the dual of σ, while Υ k,σ * and υ k,σ * are the same maps introduced in Section 3.
Example: Riemannian cubic polynomials. Let us consider one of the simplest, but interesting, second-order variational problem: given an integer n ≥ 2 and points a = x 1 < x 2 < . . . < x n = b on the real line R, and values y 1 , y 2 , . . . , This problem has a unique solution, called a complete cubic spline [1], which is a piece-wise cubic polynomial P determined uniquely by the following properties: it is a polynomial of degree ≤ 3 on each interval [x i , x i+1 ], 1 ≤ i ≤ n − 1, P ′ (a) = v a , P ′ (b) = v b , and it has continuous second derivatives at each "slope" x 2 , . . . , x n−1 . Note that the EL equations (4.11) read as f (4) = 0, i.e., f is locally a polynomial of degree ≤ 3.
Above example generalizes to a variational problem on any Riemannian manifold (M, g). Indeed, let ∇ denote the Levi-Civita connection for the metric g and, for a smooth curve γ : R → M denote by D t the covariant derivative ∇γ (t) along γ. Following [18] we define Locally, for γ(t) ∼ (x a (t)), where Γ c ab (x) are the Christoffel symbols of the metric g andẋ a (resp.ẍ a ) are (resp. second-order) derivatives of x a (t) at t = 0. Therefore (5.2) is indeed a function on T 2 M .
We shall now compute the second-order EL equations associated with the Lagrangian L given by (5.2). To this end, recall two fundamental properties of the Levi-Civita connection: which hold for any vector fields X, Y, Z on the manifold M ..
Note that although the Lie bracket [X, Y ] is defined for vector fields, to calculate its value at a point p ∈ M it is enough to know vectors X(Y )(p) := TY •X(p) ∈ T Y (p) TM and Y (X)(p) ∈ T X(p) TM (see, e.g., [13]). According, we introduce the following notion: for a vector X ∈ T p M and a vector A ∈ T X TM lying over Y := Tτ M (A) by an A-extension of X around p we will understand any (local) vector field X on M such that X(p) = X and Y ( X)(p) = A. This means that A is tangent to the graph of X at X.

A Appendix
Proof of Lemma 3.4. Let us explain that the functoriality of a vector bundle morphism T k,k E → E means that actually we have a family F = {F E } of vector bundle morphisms as in (3.12), parameterized by vector bundles σ : E → M , such that for any morphism f : E 1 → E 2 between vector bundles σ i : We shall derive the local coordinate form of F E . Observe first that the base map of F E , denoted by F E : T 2k M → M , has to be functorial as well. In other words F E is a natural transformation between Weil functors T 2k and Id. It follows from [12] that such a transformation corresponds to a unique homomorphism between the corresponding Weil algebras, namely h : D 2k = R[ν]/ ν 2k+1 → R given by 1 → 1 and ν → 0. We conclude that F E = τ 2k is the canonical bundle projection.
Every vector bundle can be locally described as E = M × V , where M is the base and V is the model fiber. In this case T k,k E = T 2k M × T k T k V and hence F E must be of the form is a smooth function. It follows from (A.1) that L v 2k does not depend on v 2k and hence, locally, F E is of the form Now we shall find the coordinate form of L. Since every vector space V is a vector bundle over a one-point base, it follows that F induces a functorial morphism Any functorial linear map D k,k ⊗ V → V is of the form f ⊗ id V for some fixed linear map f : D k,k → R. If we denote c αβ := f (ν α ν ′ β ), then we conclude that general local form of where (x a,(r) , y i,(α,β) ) are the adapted coordinates on T k,k E (as defined in Preliminaries) induced from the standard coordinates (x a , y i ) on E and we have underlined the coordinates in the co-domain.
for some coefficients a 0 , . . . , a k ∈ R. Therefore, F E must have the desired form being a linear combination of morphisms Υ s,σ given locally by (3.4).
A slight modification of the proof above shows that Υ k,σ has no canonical extension to T k T k E.
Corollary A.1. Any functorial vector bundle morphism : T k T k E → E covering τ (k,k) : T k T k M → M . In particular, Υ k,σ has, in general, no canonical extension to a functorial vector bundle morphism on T k T k E.
Proof. Just go over the same reasoning of the proof of Lemma 3.4. By considering a trivial vector bundle E = M × V and endomorphisms of the form f = f 0 × id V : E → E, where f 0 ∈ C ∞ (M ), we find that a general local form of F E : T k T k E → E has to be F E (x a,(α ′ ,β ′ ) , y i,(α,β) ) = (x a = x a,(0,0) , y i = 0≤α,β≤k c αβ y i,(α,β) ).