A Poisson algebra for abelian Yang-Mills fields on Riemannian manifolds with boundary

We define a family of observables for abelian Yang-Mills fields associated to compact regions $U \subseteq M$ with smooth boundary in Riemannian manifolds. Each observable is parametrized by a first variation of solutions and arise as the integration of gauge invariant conserved current along admissible hypersurfaces contained in the region. The Poisson bracket uses the integration of a canonical multisymplectic current.


Introduction
In Classical Covariant Field Theory two desirable conditions are required for a family of observables: In one side we require this functions to separate solutions of the Euler-Lagrange equations. On the other hand we need the Jacobi identity in order to have a Lie (Poisson) bracket. It is a known problem to characterize those theories accomplishing these two requirements, as pointed out in [8,12] and others. There are two main difficulties. In one hand, under locality assumptions, Jacobi identity is well established but generically there are few observables associated conservation laws given by Noether's First Theorem, see for instance [4]. On the other hand, extending to non-locality of variations of solutions, we may provide enough observables, see for instance [15], nevertheless the Jacobi identity does not necessarily hold, see [17].
For linear theories there are no such difficulties so that we can obtain results such as Theorem 4.6. For instance in Lorentzian globally hyperbolic spacetimes, Maxwell equations [19] exhibit a family of observables, related to the Aharomov-Bohm effect, an a Poisson bracket constructed with Peierls method for local variables. We provide a similar set of observables for the abelian Yang-Mills (YM) fields on Riemannian manifolds. This could be mentioned as the novelty introduced in this work, although our aim is to prepare the scenario for non-abelian (non-linear) YM fields. We adopt the Lagrangian approach of the variational bicomplex formalism, see [23,22,18] rather than the Hamiltonian multysimplectic formalism approach to describe non abelian YM fields, see [10,9].
We consider regions U with smooth boundary ∂U both contained in a n−dimensional Riemannian manifold, usually n = 4. Here we avoid the complications of corners in ∂U which will be treated elsewhere. For a principal bundle we take solutions of the Yang-Mills (YM) equations for the abelian U (1) structure group. We are interested in defining a family of observables for YM solutions in U , η ∈ A U , of the integral form f Σ (η) = Σ jη * F defined for a 3−dimensional compact Riemannian admissible smooth hypersurface i Σ : Σ / / U with volume form ν Σ , where admissibility means ∂Σ ⊆ ∂U , see [7]. observable currents, are horizontal (n − 1)−forms, F ∈ Ω n−1,0 (JY | U ), in the jet bundle JY associated to sections of the affine bundle Y → M of connections. The local invariance condition is then assumed by imposing d h F | U = 0, when restricted to the locus of the YM equations E L . d h is the horizontal differential, see the notation of the variational bicomplex formalism recalled in Appendix A. Notice that we adopt the definition of helicity for hypersurfaces embedded properly in general compact regions U , rather than considering cylinder regions with space-like slices, Σ × [t 1 , t 2 ], this is related fo the General Boundary Formalism for field theories, see [5] and references therein.
The idea to implement this family of observables is to define the relative helicity from hydrodynamics properly adapted to YM fields as a local observable.
In order to motivate this definition we recall the notion of helicity from magnetohydrodynamics. For a divergence (non-autonomous) free vector field, ξ = ξ(t) ∈ X ∂Σ (Σ) in a three-dimensional Riemannian manifold Σ tangent to the boundary ∂Σ, helicity is defined as where one considers the vector field v = v(t), as a potential in Σ. Helicity of ξ measures globally the degree of self-linking of its flow. Helicity remains an invariant for every ν Σ −preserving diffeomorphism of Σ that carries the boundary ∂Σ into itself, where ν Σ is given by the volume form on Σ. The situation can be dually described in terms of 1−forms. If α = g(v, ·) where g is the Riemannian metric on Σ, then under the additional topological condition, H 2 dR (Σ) = 0, there exists a potential α ∈ Ω 1 (Σ) such that dα = ι ξ ν Σ . Here helicity reads as It does depend just on the vorticity dα although for its definition the potential 1− form α or the vector field v, respectively, may intervene. If we adopt v ∈ X ∂Σ (Σ) divergence-free or d ⋆ Σ (α) = 0, respectively, then the property of isovorticity holds for v(t) for the magnetic potential, as well as for any solution of the Euler equation of hydrodynamics. This means that ξ(t 2 ) can be constructed as the image of ξ(t 1 ) under a diffeomorphism and if we consider a space-time domain Σ × [t 1 , t 2 ], then helicity remains constant (does not depend on the parameter t of the non-autonomous flow). To review this concepts see for instance [2,11].
Under the assumption of simple connectednes of Σ, then the Lie algebra of divergence-free vector fields, have a bilinear form, relative helicity, defined as Notice that helicity is [α, α] Σ and also that [·, ·] is symmetric under the assumption of closedness for Σ.
Then the helicity for abelian YM fields could be defined as where ⋆ Σ is the Hodge operator associated to the induced Riemannian metric g on Σ. Hence we could define helicity as in (1) for the vector fields v, ξ defined Nevertheless, this notion of helicity would depend on the gauge fixing choice, therefore can not be generalized as a gauge invariant observable. Moreover we do not get a local d h −closedness condition for an observable current: where L is the Lagrangian density. We will rather try to define the relative helicity of YM fields. Take η ′ = η 0 + ϕ ′ ∈ A U any other solution. Take a first variation of solutions ϕ, let us define

Then for gauge translations
Thus for every couple η, ϕ where η ∈ A U and ϕ is a first variations of solutions, we consider the antisymmetric component of the relative helicity or simply ϕ−helicity, In Section 4 we formalize this construction in the language of the variational bicomplex, see Appendix A.

Variational bicomplex formalism for abelian YM fields
Let P → M be a principal bundle on a Riemannian manifold (M, g) with structure group G = U (1) and U ⊆ M a region with smooth boundary. Let π : Y → M with Y = J 1 P/G be the affine bundle whose sections Γ(Y ) are the G−covariant connections on P.
For abelian YM, the Lagrangian density L = L ν ∈ Ω n,0 (JY ) is defined by the Lagrangian the Euler-Lagrange equations, i.e. the YM equations whose locus is the prolongation In a local coordinate chart, The space of solutions over U is Thus solutions η satisfy jη * E(L) = 0. The linearized equations for any local evolutionary vector field V ∈ Ev(JY ) are where I : Ω n,k (JY ) → Ω n,k (JY ) is the integration by parts operator. In local coordinates this linearized equations read as Let F U ⊆ Ev(JY ) de the Lie subalgebra of those evolutionary vector fields satisfying the linearized Euler-Lagrange equations. The Lie algebra F U will turn out to be our model for variations of YM solutions. For example the radial evolutionary vector field R = a u a ∂ ∂u a whose prolongation is is a local symmetry, R ∈ F U . The multisymplectic current has the property stated in the following general Lemma.
define the Lie subalgebra of locally Hamiltonian first variations asF LH U ⊆ F U .
2. We define the Lie algebraĜ U of gauge first variations as those X ∈ F U satisfying locally the presymplectic degeneracy condition, i.e.
For instance, the radial vector R ∈ F U defined in (6) is not locally hamiltonian, since it satisfies the Liouville condition ι jR d v Ω L = Ω L , rather than condition (7).
In the second part of Definition 2.2 we may also have adopted X ∈ Ev(JY ) instead of X ∈ F U and as is stated in the following assertion.
Notice that the locally Hamiltonian condition is stronger than the property exhibited in Proposition 2.3 for every variation of solutions. ThusĜ U ⊆F LH U .
which by hypothesis and by anticommutativity of To see that in fact G U ⊆F LH U , just apply vertical derivation to (8). Take V ∈F LH U , then [V, X] apply vertical derivation to equation (9) with V = X ′ and the condition of d h −exactness for ι jX d v ι jV Ω L implies the d h −exactness of ι j[X,V ] Ω L holds. Therefore [V, X] ∈Ĝ U . Form Proposition 2.3 it follows also the following assertion holds, then ̟ ij X j = d dx j ρ ij holds in E L for each i, j = 1, . . . , n, recall that dx j ∧ ν ji = ν i . Definition 2.7 (Gauge with boundary condition).

The Lie subalgebra
of locally Hamiltonian first variations with null boundary conditions, consists of the of those V ∈F LH U satisfying (7) with when evaluated in E L , F U . In particular L jV Ω L | ∂U = 0.

The Lie ideal of gauge variations with null boundary conditions
Which means that there is no gauge action in the boundary.
The following assertions are mentioned in the definition.

Linear theory
Recall that each fiber of π : Y → M is an affine bundle modeled over a linear bundle π L : Since the space of YM solutions A U is an affine space, take a fixed connection Here ⋆ denotes the Hodge star operator. In addition, there exists V ϕ ∈ F U , such that Eventhought equation (5) imposes a condition on-shell, i.e. on As a complementary definition to (10) we may define for every solution, η ∈ A U , and every first variation of solutions, V ∈ F U the section Here we use the isomorphism, depending on a fixed connection, The following properties hold The following assertion holds as an observation that will follow from Lemma 4.3.
The following assertion holds for linear theories.

Lemma 3.2. For every solution, η ∈ A U , and every first variation of solutions,
If we want to consider the gauge classes on A U we can consider the gauge representatives consisting of Lorentz gauge fixing conditions, i.e. for every η = η 0 + ϕ ∈ A U there exists a gauge related Recall the Hodge-Morrey-Friedrichs L 2 −ortogonal decomposition, see [20], for null normal components we have, where . Given a fixed point, η 0 ∈ A U , the linear space of Lorentz gauge fixing, d ⋆ ϕ = 0, defines a linear subspace of the space of solutions modulo gauge, The following results of this section recover the usual characterizations of gauge transformations as translations by exact forms.
Proof. If we calculate the square of the L 2 −norm, dη X 2 2 = U dη X ∧ ⋆dη X , of dη X where ⋆ stands for the Hodge star operator for the Riemannian metric g, then we get Therefore dη X = 0.

Proposition 3.4.
For every solution, η ∈ A U , and every gauge first variation with null boundary condition, X ∈ G U , the induced 1−form η X in the base region U , η X , defined as in (11), is exact. Therefore η X ∈ G U .
Proof. We solve the Poisson BVP for ψ : Notice that the necessary integral condition for the Poisson equation U d ⋆ η X dν = 0 follows from the boundary condition η X | ∂U = 0. Thusη X = η X − dψ is a solution of d ⋆ dη X = 0 with Lorentz gauge fixing condition d ⋆η X = 0 and Dirichlet boundary condition. Recall (13). Since η X ∈Ĝ U , according to Lemma 3.3, dη X = 0 and dη X = 0. There are two cases: Case 1. The normal component ∂ψ/∂x n | ∂U does not vanish. Here in local coordinates, ∂U = {x n = 0}. Thenη X is harmonic (dη X = 0 and d ⋆η In any caseη X is exact and so is η X .

Proposition 3.5. Take any solution η, and any variation
According to the argument given in Proposition 3.4 we just need to show that the pullback i * ∂U η X ∈ Ω 1 (∂M ) is null for the inclusion i ∂U : ∂U → U . Then η X − dψ would have null Dirichlet condition and would be exact for suitable ψ.
Notice that the following boundary conditions are in general different objects: Since X ∈ F LH U , then we are assuming a boundary condition on X, namely We claim that indeed i * ∂U η X = 0. Recall that, according to Lemma 2.6, for Therefore X j (jη)| ∂U = 0 for j = 1, . . . , n − 1, hence Dirichlet boundary conditions hold for η X . There exists a smooth function f : U → R such that i * ∂U df = i * ∂U η X = 0, and ∂f ∂x n | ∂U = X n (jη). If has both Neumann and Dirichlet conditions on ∂U . We just need to refine the choice of f , so that jX ′ | ∂U = 0. Hence X ′ ∈ G U . Theorem 3.6. There is an inclusion of the gauge quotients of Lie algebras, Proof. By the Second Isomorphism Theorem for Lie algebraŝ Notice that where Ψ is the Lie algebra morphism defined as the composition in the diagram below.
By the first isomorphism theorem, there exists an induced monomorphismΨ and a commutative diagram given by X → X − X ′ . Therefore we have the required inclusion Hence, the demand in the proof of Proposition 3.5 for i * ∂U η X to be null is equivalent to demanding η X to lie into Ω 1 D (U ). Thus, η X defines a relative cohomology class [η X ] ∈ H 1 dR (U, ∂U ). Further considerations actually explain that [η X ] = 0.
then η V may be gauge translated by an exact form dψ so thatη V = η V + dψ has no normal components along ∂U and satisfies d ⋆η V = 0 as well as the linearized YM equation, d ⋆ dη V = 0.
Notice that the induced linearized solution X dψ ∈ F U in fact belongs tô

Consider the boundary conditions linear map dr
where the codomain is the linear space of Dirichlet-Neumann boundary conditions modulo gauge Recall the isomorphisms : : Notice that dr ∂U [η] and dr ∂U [η] have the same image.
Remark that we have the commutative diagram where is the linear space of boundary conditions of solutions modulo gauge. Here we use axial gauge fixing in a tubular neighborhood of ∂U as well as the linear map r ∂U (ϕ) = ϕ D ⊕ ϕ N is defined in (16). The linear map p is induced by p η (V ) = η + ϕ where ϕ ∈ Ω 1 (U ) is a coclosed linearized solution, d ⋆ dϕ = 0 such that η V = ϕ, see notation (11). By composing the projection p η with the covering e η we get the map exp η : (17) suggest that Hamiltonian first variation modulo gauge, F LH U /G U is a Lie algebra isomorphic as linear space to the tangent space of the moduli space A U /G U at η.
The following assertion related to Proposition 3.7 explains how the relative cohomology codifies the description of A U /G U with respect to the boundary conditions.

Poisson-Lie algebra of Hamiltonian Observables
Definition 4.1 (Hamiltonian observable currents). We say that an observable current F ∈ Ω n−1,0 (JY | U ) is a Hamiltonian observable current if there exist V ∈ F U and a residual form σ F such that the following relation holds when restricted to E L and evaluated on W ∈ F U , We denote the space of Hamiltonian observable currents over U as HOC U .The evolutionary vector field V , is actually a locally Hamiltonian first variation, i.e. V ∈F LH U . If in addition in (18) we have the boundary condition then we call F a Hamiltonian observable current with boundary condition. Here V ∈ F LH U . We denote the space these kind of observable currents as HOC U . Definition 4.2 (Helicity current). Suppose that ϕ ∈ Ω 1 (U ) is a solution of the linearized YM equation, d ⋆ dϕ = 0. Define the ϕ−helicity current as where R ∈ F U was defined in (6). More expilicitly Form the very definition and the multysimplectic formula it can be seen that Remark that we could have defined observable currents, F ϕ , for any divergencefree ϕin U , d⋆ϕ = 0, with evolutionary Hamiltonian vector field, V ϕ ∈ Ev(JY | U ), rather than in restricting ourselves to Hamiltonians first variations in F U , just as the observables considered in [19]. Nevertheless, if we had adopted this definition, then we would have to restrict the domain of F ϕ and evaluate only ob solutions η ′ = η 0 + ϕ ′ ∈ A U with Lorentz gauge fixing (12), ϕ ′ ∈ L U in order to have local invariance d h F ϕ | EL = 0.
From the following assertion it follows that helicity currents are Hamiltonian observable currents restricted to U , that is F ϕ ∈ HOC U .

Lemma 4.3.
The ϕ−helicity current, F ϕ ∈ Ω n−1,0 (JY | U , defines a locally Hamiltonian observable current with Hamiltonian V ϕ ∈ F LH U whenever d⋆dϕ = 0. Proof. Recall the notation in (10). Notice that the relation d v F ϕ + ι jVϕ Ω L = 0 is valid off-shell. Therefore we have On the other hand a general formula (9) states that Define the family of ϕ−helicity observables as We see that f ϕ Σ is related to the anti-symmetric component of the helicity as bilinear form, see Section 1, in the sense of (3). Notice also that [·, ·] Σ is not necessarily symmetric, unless ∂Σ = 0. Hence f ϕ Σ not necessarily equals 0.
We say that f ϕ Σ is a Hamiltonian observable with Hamiltonian v ϕ in the sense that the following formal identity holds: Let us explain this formal notation. Any first variation of solutions, W ∈ F U , encodes a variation of any fixed solution η ∈ A U , which we denote as w = δη, for a one-parameter family of smooth solutions φ ε ∈ A U . This means that = jW (jη). In the r.h.s. we have an evaluation of a symplectic form, While in the l.h.s. we have With this notation we suggest that we are modeling a Lie derivative L w (·) in the tangent space of the moduli space A U /G U , while w = δη corresponds to local vector fields near [η] ∈ A U /G U . If X ∈ G U corresponds to a first variation of a one-parametric family of gauge equivalent solutions, φ ε , then L x f ϕ Σ = 0, which follows from jX| ∂U = 0. Thus f ϕ Σ is well defined for the gauge class [V ϕ ] ∈ F LH U /G U . Lemma 4.5. Consider the linear space Proof. Let ϕ, ϕ ′ be 1−forms as in the hypothesis. As in the proof of Lemma 4.4, recall that There are gauge translations X = X ψ , X ′ = X ψ ′ ∈ G U , ψ, ψ ′ : U → R such that the gauge translations V, V ′ are divergence-free, see for instance the Appendix [5]. Recall that V, V ′ are defined by ϕ + dψ, ϕ ′ + dψ ′ , respectively. Hence Denoteφ ∈ Ω 1 (U ) as the a 1−form such that [V, V ′ ] = Vφ. In local coordinates: Recall that divergence-free vector fields form a Lie algebra, that is d ⋆φ = 0. Therefore for every variation of solutions w associated to every W ∈ F U . Hence f We claim that f ΣU yields a family of local observables sufficiently rich to separate solutions. Suppose that we consider a non-gauge variation v = δη of a solution η ∈ A U . More precisely, take a one-parametric family of solutions η ε = η + εϕ encoded by the symmetry V ∈ F U , that is d dε ε=0 j(η ε ) = jV (jη). Without loss of generality we can also suppose that V = V ϕ with d ⋆ dϕ = 0.  Hence Remark that for every YM solution η ∈ A U and for every variation V ∈ F U , if ϕ = jη * V , then and jV ϕ | jη = jV | jη . Thus we could index the family {f ϕ Σ } as {f V Σ } where we take the index V in F LH U . We summarize the results exposed in this section in the following result, which allows us to regard the family of observables {f V Σ } as a "Darboux local coordinate system" for our gauge field theory.
such that the following assertions hold: is an observable that satisfies the Hamilton's equation (see notation (22) and (20)): Notice that the following commutative diagram of Lie algebra morphisms and vertical exact sequences summarizes our results where C U denote subset of the the constant observable currentŝ is generated by the Lie algebra The proof of the following assertion follows from the fact that the space of boundary conditions of solutions, LŨ ⊆ L ∂U , is a Lagrangian subspace with respect to the symplectic form ω ∂U,L , see [3].  (U, ∂U )) and for its complement, Σ ′ = U − Σ ⊆ ∂U, the corresponding observables uniquely define an observable associated to the oriented and closed (n − 2)−dimensional boundary ∂Σ ⊆ ∂U .
The Lie algebra f ∂U := {f ϕ ∂Σ : Σ ⊆ ∂U }/R will suffice to separate boundary conditions, while the Lie algebras f ΣU corresponding to 0 = [Σ] ∈ H n−1 (U, ∂U ) will be necessary if we want to separate solutions in the fibers of r ∂U : L U → LŨ ⊆ L ∂U . This happens when H 1 dR (U, ∂U ) = 0 according to Proposition 3.8. This also allows us to consider the fibers of r ∂U : L U → LŨ as the symplectic leafs the coisotropic linear space L U . This image has been described in detail for the moduli space A U /G U of non-abelian YM solutions in the two dimensional case, see for instance [21].

Gluing observable currents
Suppose that a region U is obtained by gluing U 1 , U 2 along the closed hypersurfaces Σ 1 ⊆ U 1 , Σ 2 ⊆ U 2 , ∂Σ 1 = ∅ = ∂Σ 2 . This includes an isometry of Σ 1 with Σ 2 together with the compatibility of normal derivatives of the metric. We also suppose that the principal bundle P o ver U is induced bu the corresponding principal bundle P 1 , P 2 over U 1 , U 2 . From the projection map p : U 1 × U 2 → U we fix base points η ∈ A U obtained by gluing p * η i ∈ A Ui , i = 1, 2.
Suppose that V i ∈ F Ui , i = 1, 2 satisfy the continuity gluing condition along and denote those couples (V 1 , V 2 ) satisfying (24) as Notice that the continuity gluing condition (24) is trivially satisfied for the gauge Lie algebras so that we could define LetĜ Σ U1 ⊆Ĝ U1 denote those gauge variations whose jet vanish along the boundary components of ∂U 1 except for Σ 1 . Similarly defineĜ Σ U2 . If we define There is a commutative diagram of linear maps as follows. Recall the gluing procedure for abelian YM see [5]. The doted arrow is a Lie algebra morphism.
From the Lagrangian embedding of LŨ with respecto to the symplectic structure, ω ∂U,L , it follows that the Dirichlet conditions along Σ 1 and Σ 2 completely determine the Neumann conditions in U 1 and U 2 , respectively. Here we consider an axial gauge fixing for solutions in ∂U satisfying also the Lorentz gauge fixing condition in ∂U , see Appendix in [5]. This means that the continuous gluing condition (24) will suffice to reconstruct the first variation ϕ = V ϕ1 #V ϕ2 for V ϕi ∈ F Ui , i = 1, 2 disregarding the normal derivatives along Σ. This proves the following assertion Theorem 5.1. Gluing of first variations There is an isomorpmhism of Lie algebras

Outlook: Further problems
We just remark further directions of research we will be concerned about the results we have obtained. In the first place wit is highly desirable to see whether or not f V V ′ observables can be defined for non abelian (non-linear) YM equations and if will suffice to separate solutions just as in Theorem 4.6. The existence of a Jacobi bracket needs also to be verified in this case. Non-local first variations may be required to get enough observables to separate solutions. Gluing properties for observables need also to be developed and explained in detail. In a further article we will perform the continuous gluing of currents in relation to HOC U , as well as the gluing (f Σ1U1 )# Σ (f Σ2U2 ). Finally, considerations of Riemannian manifolds with corners may introduce further difficulties in the results we hay established for the smooth boundary case.

Acknowledgments
The author was partially supported by CONACYT-México. He thanks J. A. Zapata since most of the results of this article arise as a particular application of the results sketched in a joint work [6]. He also thanks the referee of [6] who provided a lot of clarifications for the difficulties arising in non-linear field theories.

A Variational bicomplex fromalism
For convenience of the reader, in this section we fix notation by recalling basic definitions of the variational formalism for variational PDEs taken from [1,16,4,13,14,23]. Let M be an n−dimensional manifolds, and let π : Y → M be a fiber bundle with m−dimensional fiber F . Denote its sections or histories as Γ(Y | U ) where U ⊆ M is a compact domain with piecewise smooth boundary.
The k−jet bundle π k,0 : where i = 1, . . . , n; a = 1, . . . , m; and I = (i 1 , . . . , i n ) denotes a multiindex of degree |I| := i 1 + · · · + i n = 0, 1, . . . , k, i j ≥ 0, i j ∈ Z. For I = ∅, we define u a ∅ = u a . We denote the projection of the (k + 1)−jet onto the k−jet as π k+1,k : J k+1 Y → J k Y . For a section φ : M → Y , we denote its k−jet as Denote the space of p−forms on J k Y as Ω p (J k Y ). For the decomposition p = r + s, denote the space of r−horizontal and s−vertical forms on J k Y as Ω r,s (J k Y ), have as basis the (r + s)−forms ϑ a1 I1 ∧ · · · ∧ ϑ as Is ∧ dx j1 ∧ · · · ∧ dx jr , where The Cartan distribution on J k Y is generated by the basis of contact 1−forms (25) The vertical differential d v for F ∈ Ω 0 (J k Y ) defined as then we are forced to consider the horizontal differential with range in the The injective limit Ω r,s (JY ) := lim − →π * k+1,k Ω r,s (J k Y ), models the p forms in the We have the identities Hence, the following diagram commutes . . . . . . . . .
Derivations in the algebra of smooth functions on Ω 0 (JY ), are in correspondence with sections V ∈ Γ(π * ∞,0 (π v )), where π * ∞,0 (π v ) is the pullback under π ∞,0 : JY → Y of the vertical (vector) bundle, π v : Y v → Y , whose fiber at each (x, u) ∈ Y is consists of the vertical fibers Y v (x,u) = T (x,u) π −1 (x). In fact its prolongations where b a I := D (k) I v a + n j=1 u a jI a j , |I| ≤ k − 1, act as infinitesimal symmetries of the Cartan distribution in JY in the sense that Here the horizontal derivative operator D We will assume that V ∈ Γ(π * ∞,0 (π v )) has no horizontal component. Hence We call this space the space of evolutionary vector fields, where the functions v a are local in the sense that depend on a finite number of derivatives of u. For a first-order Lagrangian variational problem in a region U ⊆ M , the space of first variations of histories, v = δφ, for a fixed φ ∈ Γ(Y | U ), can be modeled as V ∈ V ∈ Γ π * 1,0 (π)| U : V ∈ Ev (U ⊆ JY ) where U ⊆ JY, is a neighborhood of the graph jφ(U ).
Consider the action functional on U ⊂ M , if we take the vertical derivative On the other hand, if we define the form Ω L = −d v Θ L ∈ Ω n−1,2 J 1 Y , or For a first variation δφ modeled by V ∈ Ev(JY | U ), let us consider the Cartan formula for vertical derivation Therefore In particular d h ι j 1 V Θ L = −ι j 2 V (d h Θ L ). Hence the variation for the action is Notice that in the Euler-Lagrange equations E a (L) = 0 arising from j 2 φ * E(L) = 0, the total horizontal derivations d/dx i are involved. Meanwhile the Euler-Lagrange equations mentioned in Proposition A.1 deal with partial horizontal derivations, ∂/∂x i , see [7,8].