On the k-Symplectic, k-Cosymplectic and Multisymplectic Formalisms of Classical Field Theories

The objective of this work is twofold: First, we analyze the relation between the k-cosymplectic and the k-symplectic Hamiltonian and Lagrangian formalisms in classical field theories. In particular, we prove the equivalence between k-symplectic field theories and the so-called autonomous k-cosymplectic field theories, extending in this way the description of the symplectic formalism of autonomous systems as a particular case of the cosymplectic formalism in non-autonomous mechanics. Furthermore, we clarify some aspects of the geometric character of the solutions to the Hamilton-de Donder-Weyl and the Euler-Lagrange equations in these formalisms. Second, we study the equivalence between k-cosymplectic and a particular kind of multisymplectic Hamiltonian and Lagrangian field theories (those where the configuration bundle of the theory is trivial).


Introduction
The k-symplectic and k-cosymplectic formalisms are the simplest geometric frameworks for describing classical field theories. The k-symplectic formalism [13,25] (also called polysymplectic formalism) is the generalization to field theories of the standard symplectic formalism in autonomous mechanics, and is used to give a geometric description of certain kinds of field theories: in a local description, those whose Lagrangian and Hamiltonian functions do not depend on the coordinates in the basis (in many of them, the space-time coordinates). The foundations of the k-symplectic formalism are the k-symplectic manifolds intoduced in [2,3,4]. The k-cosymplectic formalism is the generalization to field theories of the standard cosymplectic formalism for non-autonomous mechanics, [21,22], and it describes field theories involving the coordinates in the basis on the Lagrangian and on the Hamiltonian. The foundations of the k-cosymplectic formalism are the k-cosymplectic manifolds introduced in [21,22]. One of the advantages of these formalisms is that only the tangent and cotangent bundle of a manifold are required for their development. (A brief review of k-symplectic and k-cosymplectic geometry is given in Section 2.2). Other different polysymplectic formalisms for describing field theories have been proposed in [10,11,15,23,26,27,30].
In these formalisms, the field equations (Hamilton-de Donder-Weyl and Euler-Lagrange equations) can be written in a geometrical way using integrable k-vector fields. However, although integral sections of integrable k-vector fields (i.e., integrable distributions) that are solutions to the geometrical field equations are proved to be solutions to the Hamilton-de Donder-Weyl or the Euler-Lagrange equations, the converse is not always true. This also occurs when other geometric descriptions of classical field theories in terms of multivector fields are considered (see [7,8,28] for details in the case of multisymplectic field theories). Here we prove that, in the k-cosymplectic formalism, every solution to the Hamilton-de Donder-Weyl equations is, in fact, an integral section of an integrable k-vector field that is a solution to the geometrical field equations in the Hamiltonian formalism. Nevertheless, in the k-symplectic Hamiltonian formalism, this is no longer true, unless some additional conditions on the solutions to the Hamilton-de Donder-Weyl are required. All these features are discussed in Sections 2.3, 2.4, 2.5, 3.2, and 3.3.
After reviewing the k-cosymplectic Hamiltonian formalism in Section 2.4, Section 2.5 contains other relevant results of this work. In particular, the relation between the k-cosymplectic and the k-symplectic Hamiltonian formalism is studied here, proving the equivalence between ksymplectic Hamiltonian systems and a class of k-cosymplectic Hamiltonian systems: the so-called autonomous k-cosymplectic Hamiltonian systems. This generalizes the situation in classical mechanics, where the symplectic formalism for describing autonomous Hamiltonian systems can be recovered as a particular case of the cosymplectic Hamiltonian formalism when systems described by time-independent Hamiltonian functions are considered.
A more general geometric framework for describing classical field theories is the multisymplectic formalism [5,12,24], first introduced in [16,17,18], which is based on the use of multisymplectic manifolds. In particular, jet bundles are the appropriate domain for stating the Lagrangian formalism [31], and different kinds of multimomentum bundles are used for developing the Hamiltonian description [9,14,19]. (A brief review of multisymplectic Hamiltonian and Lagrangian field theories is given in Sections 4.1, 4.2, and 5.1).
Multisymplectic models allow us to describe a higher variety of field theories than the kcosymplectic or k-symplectic models, since for the latter the configuration bundle of the theory must be a trivial bundle; however, this restriction does not oocur for the former. Another goal of this paper is to show the equivalence between the multisymplectic and k-cosymplectic descriptions, when theories with trivial configuration bundles are considered, for both the Hamiltonian and Lagrangian formalisms. In this way we complete the results obtained in [20], where an initial analysis about the relation between multisymplectic, k-cosymplectic and k-symplectic structures was carried out. This study is explained in Sections 4.3, and 5.2.
All manifolds are real, paracompact, connected and C ∞ . All maps are C ∞ . Sum over crossed repeated indices is understood.
2 k-symplectic and k-cosymplectic Hamiltonian formalisms 2.1 k-vector fields and integral sections (See [21] and [29] for details). If M is a differentiable manifold, let T 1 k M = T M ⊕ k . . . ⊕T M be the Whitney sum of k copies of T M , and τ 1 M : T 1 k M −→ M its canonical projection. T 1 k M is usually called the k-tangent bundle or tangent bundle of k 1 -velocities of M .
Giving a k-vector field X is equivalent to giving a family of k vector fields X 1 , . . . , X k on M obtained by projecting X onto every factor; that is, For this reason we will denote a k-vector field by X = (X 1 , . . . , X k ).
Definition 2 An integral section of the k-vector field X = (X 1 , . . . , X k ) passing through a point A k-vector field is said to be integrable if there is an integral section passing through every point of M .
Remark: k-vector fields in a manifold M can also be defined more generally as sections of the bundle Λ k (TM) → M (i.e., the contravariant skew-symmetric tensors of order k in M). The k-vector fields defined in Definition 1 are a particular class: the so-called decomposable or homogeneous k-vector fields, which can be associated with distributions on M. We remark that a k-vector field X = (X 1 , . . . , X k ) is integrable if, and only if, {X 1 , . . . , X k } define an involutive distribution on M. (See [7] for a detailed exposition on these topics).
Then (M, ω A , V ) is called a k-symplectic manifold.
The canonical model for this geometrical structure is ((T 1 k ) * Q, ω A , V ), where Q is a ndimensional differentiable manifold and (T 1 k ) * Q = T * Q⊕ k . . . ⊕T * Q is the Whitney sum of k copies of the cotangent bundle T * Q, which is usually called the k-cotangent bundle or bundle of k 1 -covelocities of Q. We use the following notation for the canonical projections: (here π A is the canonical projection onto the A th -copy T * Q of (T 1 k ) * Q). So, if q ∈ Q and (α 1 q , . . . , α k q ) ∈ (T 1 k ) * Q, we have If (q i ), 1 ≤ i ≤ n, are local coordinates on U ⊆ Q, the induced local coordinates (q i , p A i ) on (π 1 Q ) −1 (U ) = (T 1 k ) * U are given by The canonical k-symplectic structure in (T 1 k ) * Q is constructed as follows: we define the differential forms where θ is the Liouville 1-form on T * Q and ω = −dθ is the canonical symplectic form on T * Q.
Obviously ω A = −dθ A . In local coordinates we have The canonical k-symplectic manifold is (( 3. The forms η A and Ω A are closed, and V is integrable. Then, (M, η A , Ω A , V) is said to be a k-cosymplectic manifold.
For every k-cosymplectic structure (η A , Ω A , V) on M , there exists a family of k vector fields {R A } 1≤A≤k , which are called Reeb vector fields, characterized by the following conditions The canonical model for these geometrical structures is (R k × (T 1 k ) * Q, η A , Ω A , V). If (t A ) are coordinates in R k , and (q i ) are local coordinates on U ⊂ Q, then the induced local coordinates Considering the canonical projections (submersions), we have the commutative diagram: In particular, if t = (t 1 , . . . , t k ) ∈ R k , q ∈ Q and (t, The canonical k-cosymplectic structure in R k × (T 1 k ) * Q is constructed as follows: we define the differential forms Obviously Ω A = −dΘ A . In local coordinates we have The canonical k-cosymplectic manifold is ( . Moreover, the Reeb vector fields are which are defined intrinsically in R k × (T 1 k ) * Q and span locally the vertical distribution with respect to the projectionπ 2 ; i.e., the distribution generated by ker (π 2 ) * .
Finally, taking into account (1), (4), and the commutativity of the diagram (3), we have that Furthermore, the vector fields spanning the distributions V on R k × (T 1 k ) * Q, and V on (T 1 k ) * Q are alsoπ 2 -related.

k-symplectic Hamiltonian systems
Consider the k-symplectic manifold ((T 1 k ) * Q, ω A , V ), and let H ∈ C ∞ ((T 1 k ) * Q) be a Hamiltonian function. ((T 1 k ) * Q, H) is called a k-symplectic Hamiltonian system. The Hamilton-de Donder-Weyl equations (HDW-equations for short) for this system are the set of partial diferential equations: where ψ : , is a solution. We denote by X k H ((T 1 k ) * Q) the set of k-vector fields X = (X 1 , . . . , X k ) on (T 1 k ) * Q which are solutions to the equations In a local system of canonical coordinates, each X A is locally given by then, using (2), we obtain that the equation (8) is equivalent to the equations The existence of k-vector fields that are solutions to (8) is assured, and in a local system of coordinates they depend on n(k 2 − 1) arbitrary functions. Nevertheless, they are not necessarily integrable, and hence the integrability conditions imply that the number of arbitrary functions will in general be less than n(k 2 − 1).
is a solution to the HDW-equations (7) if, and only if, and therefore (10) are the HDW-equations (7).
Remark: It is important to point out that the equations (7) and (8) are not equivalent, because there is no way to prove that every solution to the HDW-equations (7) is an integral section of some integrable k-vector field of X k H ((T 1 k ) * Q), unless some additional conditions are required. In particular, we could assume the following condition (which holds for a large class of mathematical applications and physical field theories): (7), is said to be an admissible solution to the HDW-equations for a k-symplectic Hamiltonian system is an admissible k-symplectic Hamiltonian system if all the solutions to its HDW-equations are admissible.

Proposition 2 Every admissible solution to the HDW-equations (7) is an integral section of an
(Proof ): Let ψ : R k → (T 1 k ) * Q be an admissible solution to the HDW-equations (7). By hypothesis, Im ψ is a k-dimensional closed submanifold of (T 1 k ) * Q. As ψ is an embedding, we can define a k-vector field X| Im ψ (at support on Im ψ), and tangent to Im ψ, by which is a solution to (8) on the points of Im ψ, since (10) holds on these points as a consequence of (7) and (11). Furthermore, by hypothesis, Im ψ is a closed submanifold of (T 1 k ) * Q; therefore we can extend this k-vector field X| Im ψ to an integrable k-vector field X ∈ X k H ((T 1 k ) * Q) in such a way that this extension is a solution to the equations (8) (remember that these equations have solutions everywhere on (T 1 k ) * Q), and which obviously has ψ as an integral section. This extension is made at least locally, and then the global k-vector field is constructed using partitions of unity.
In this way, for admissible k-symplectic Hamiltonian systems, the field equations (8) are a geometric version of the HDW-equations (7).

k-cosymplectic Hamiltonian systems
is called a k-cosymplectic Hamiltonian system. The HDW-equations for this system are the set of partial diferential equations: where the solutionsψ(t) Since R A = ∂/∂t A and η A = dt A , then we can write locally the above equations as follows In a local system of coordinates,X A are locally given bȳ and, using (2), we obtain that the equations (13) are equivalent to the equations The existence of k-vector fields that are solutions to (14) is assured, and in a local system of coordinates they depend on n(k 2 − 1) arbitrary functions, but for integrable solutions the number of arbitrary functions is, in general, less than n(k 2 − 1).
is an integral section ofX, we have that and therefore we obtain that (16) are the HDW-equations (7).
Furthermore we have: be a section of the projectionπ k that is a solution to the HDW-equations (12). We have thatψ is an injective immersion and Imψ is a closed Then the construction of the integrable k-vector field in R k × (T 1 k ) * Q, which hasψ as integral section and is a solution to (13), follows the same pattern as in proposition 2.
So the equations (13) are a geometric version of the HDW-equations(12).

Autonomous k-cosymplectic Hamiltonian systems
Following a terminology analogous to that in mechanics, we define: Observe that the condition in definition 6 means that H does not depend on the variables t A , and thus H =π * 2 H for some H ∈ C ∞ ((T 1 k ) * Q). For an autonomous k-cosymplectic Hamiltonian system, the equations (13) become Therefore: We have the following result for solutions to the Hamilton-de Donder-Weyl equations: be an autonomous k-cosymplectic Hamiltonian system and let ((T 1 k ) * Q, H) be its associated k-symplectic Hamiltonian system . Then, every sectionψ : Letψ : R k → R k ×(T 1 k ) * Q be a section of the projectionπ k , which in coordinates is expressed . Then, ifψ is a solution to the HDW-equations (12), from (19) we obtain that ψ is a solution to the HDW-equations (7).
Conversely, consider a map ψ : Hence, if ψ is a solution to the HDWequations (7), from (19) we obtain thatψ is a solution to the HDW-equations (12).
For k-vector fields that are solutions to the geometric field equations (8) and (18) we have: be an autonomous k-cosymplectic Hamiltonian system and let ((T 1 k ) * Q, H) be its associated k-symplectic Hamiltonian system. Then every k-vector , which is defined as follows (see [1], p. 374, for this construction in mechanics): for every p ∈ ( ). Therefore,X A is the vector field tangent tō γ Ā p at (t 0 , p). In natural coordinates, if X A is locally given by (9), thenX A is locally given bȳ Observe thatX A areπ 2 -projectable vector fields, and (π 2 ) * XA = X A . In this way we have defined a k-vector fieldX = (X 1 , . . . ,X k ) in R k × (T 1 k ) * Q. Therefore, taking (6) into account, Now, ifψ is an integral section ofX, the equations (17) hold forψ(t) = (t,ψ i (t),ψ A i (t)) and, as ( i , this is equivalent to saying that the equations (11) hold for ψ(t) = (ψ i (t), ψ A i (t)); that is, ψ is an integral section of X.
Remark: The converse statement is not true. In fact, the k-vector fields that are solution to the geometric field equations (18) are not completely determined, as the equations (16) show, and then there are k-vector fields in X k H (R k × (T 1 k ) * Q) that are notπ 2 -projectable (in fact, it suffices to take their undetermined component functions to be notπ 2 -projectable). However, we have the following particular result: Proposition 7 Let ((T 1 k ) * Q, H) be an admissible k-symplectic Hamiltonian system, and (R k × (T 1 k ) * Q, H) its associated autonomous k-cosymplectic Hamiltonian system. Then, every inte- is an integrable k-vector field, denote byS the set of its integral sections (i.e., solutions to the the HDW-equations (12)). Let S be the set of maps ψ : R k → (T 1 k ) * Q associated with these sections by Theorem 3, which are admissible solutions to the HDW-equations (7), by the hypothesis that ((T 1 k ) * Q, ω A , H) is an admissible k-symplectic Hamiltonian system. Then, by proposition 2 we can construct an integrable k-vector field X ∈ X k H ((T 1 k ) * Q) for which S is its set of integral sections (which are admissible solutions to the HDW-equations (7)).
3 k-symplectic and k-cosymplectic Lagrangian formalisms (See [25,29] for details on the construction of this formalism).

Canonical structures in the bundles
For a vector Z q ∈ T q Q, and for A = 1, . . . , k, we define its vertical A-lift, (Z q ) V A , at the point (v 1q , . . . , v k q ) ∈ T 1 k Q, as the vector tangent to the fiber (τ 1 Q ) −1 (q) ⊂ T 1 k Q, which is given by . Then, the canonical k-tangent structure on T 1 k Q is the set (S 1 , . . . , S k ) of tensor fields of type (1, 1) defined by In local coordinates we have The Liouville vector field ∆ ∈ X(T 1 k Q) is the infinitesimal generator of the following flow Now, consider the manifold J 1 π R k of 1-jets of sections of the trivial bundle π R k : R k ×Q → R k , which is diffeomorphic to R k × T 1 k Q, via the diffeomorphism given by where We consider the extension of S A to R k × T 1 k Q, which we denote byS A , and they have the same local expressions (20). Finally, we introduce the Liouville vector field∆ ∈ X(R k × T 1 k Q), which is the infinitesimal generator of the following flow and in local coordinates it has the form∆

k-symplectic Lagrangian formalism
Let L ∈ C ∞ (T 1 k Q) be a Lagrangian function. A family of forms θ A L ∈ Ω 1 (T 1 k Q), 1 ≤ A ≤ k, is introduced by using the k-tangent structure of T 1 k Q, as follows θ A L = dL • S A 1 ≤ A ≤ k , and hence we define ω A L = −dθ A L . In coordinates We can also define the Energy Lagrangian function associated to L, E L ∈ C ∞ (T 1 k Q), as E L = ∆(L) − L. Its local expression is Finally, the Legendre map F L : T 1 k Q −→ (T 1 k ) * Q was introduced by Günther [13], and we rewrite it as follows: for each A = 1, . . . , k. We have that F L is locally given by Furthermore, from (2) and (23) we obtain that The Lagrangian L is said to be regular if ( ) is a non-singular matrix at every point of T 1 k Q. Then, from (23) and (24) we get: The following conditions are equivalent: A Lagrangian function L is said to be hyperregular if the corresponding Legendre map F L is a global diffeomorphism. If L is regular, (T 1 k Q, L) is said to be a k-symplectic Lagrangian system. If L is not regular (T 1 k Q, L) is a k-presymplectic Lagrangian system. The Euler-Lagrange equations for L are: whose solutions are maps ϕ : R k → T 1 k Q that, as a consequence of the last group of equations (25), are first prolongations to .
Let X k L (T 1 k Q) be the set of k-vector fields Γ = (Γ 1 , . . . , Γ k ) in T 1 k Q, wich are solutions to locally, then Γ is a solution to (26) if, and only if, (Γ A ) i and If the Lagrangian is regular, the above equations are equivalent to The last group of these equations is the local expression of the condition that Γ is a sopde (see [25]), and hence, if it is integrable, its integral sections are first prolongations φ (1) : R k → T 1 k Q of maps φ : R k → Q, and using the first group of equations, we deduce that φ (1) are solutions to the Euler-Lagrange equations (25). If L is not regular then, in general, the equations (25) or (26) have no solutions anywhere in T 1 k Q, but they do in a submanifold S of T 1 k Q (in the most favourable situations). Moreover, solutions to (26) are not sopde necessarily.
We define admissible solutions to the Euler-Lagrange equations and admissible k-symplectic Lagrangian systems in the same way as in the Hamiltonian case (definition 5). Then the statement of Proposition 2 can be proved analogously for these admissible solutions. This proof holds for regular k-symplectic Lagrangian systems, and for the non-regular case the proof is still valid considering the submanifold S of (T 1 k ) * Q where the Lagrangian field equations have solutions.

k-cosymplectic Lagrangian formalism and autonomous k-cosymplectic Lagrangian systems
Let We can also define the Energy Lagrangian function associated to L, E L ∈ C ∞ (R k × T 1 k Q) as E L =∆(L) − L, whose local expression is Finally, the Legendre map F L : R k × T 1 k Q −→ R k × (T 1 k ) * Q, is defined as follows: for each A = 1, . . . , k; and it is locally given by It is obvious that (6), (24) and (29)).
is not singular at every point of R k × T 1 k Q. Then, from (5), (28) and (29) we deduce the following proposition (See [22]): The following conditions are equivalent: A Lagrangian function L is said to be hyperregular if the corresponding Legendre map F L is a global diffeomorphism. If L is regular, (R k × T 1 k Q, L) is said to be a k-cosymplectic Lagrangian system. If L is not regular, (R k × T 1 k Q, L) is a k-precosymplectic Lagrangian system.
The Euler-Lagrange equations are (25), but now the Lagrangian is L = L(t B , q j , v j B ), and their solutions are sectionsφ : In a local system of natural coordinates, if thenΓ is a solution to (30) if, and only if, When L is regular, we obtain that (Γ A ) i = v i A , and the last equation can be written as follows thenΓ is a sopde (see [22]), and hence, if it is integrable, its integral sections are holonomic and they are solutions to the Euler-Lagrange equations for L. If L is not regular, the existence of solutions to the equations (25) for L or to (30) is not assured, in general, except in a submanifold of T 1 k Q (in the most favourable situations). Moreover, solutions to (30) are not sopde necessarily.
Definition 7 A k-cosymplectic (or k-precosymplectic) Lagrangian system is said to be au- Now, all the results obtained in Section 2.5 can be stated and proved in the same way, considering the systems (R k ×T 1 k Q, L) and (T 1 k Q, L) instead of (R k ×(T 1 k ) * Q, H) and ((T 1 k ) * Q, H). Finally, the k-symplectic and k-cosymplectic Lagrangian and Hamiltonian systems are related by means of the Legendre maps F L and F L.
, is a multisymplectic manifold if Ω is closed and 1-nondegenerate; that is, for every p ∈ M, and X p ∈ T p M, we have that i(Xp)Ωp = 0 if, and only if, X p = 0.
A very important example of multisymplectic manifold is the multicotangent bundle Λ k T * Q of a manifold Q, which is the bundle of k-forms in Q, and is endowed with a canonical multisymplectic (k + 1)-form. Other examples of multisymplectic manifolds which are relevant in field theory are the so-called multimomentum bundles: let π : E → M be a fiber bundle, (dim M = k, dim E = n + k), where M is an oriented manifold with volume form ω ∈ Ω k (M ), and denote by (t A , q i ) (1 ≤ A ≤ k, 1 ≤ n) the natural coordinates in E adapted to the bundle, such that which is the bundle of k-forms on E vanishing by the action of two π-vertical vector fields. This is called the extended multimomentum bundle, and its canonical submersions are denoted by We can introduce natural coordinates in Mπ adapted to the bundle π : E → M , which are denoted by (t A , q i , p A i , p), and such that ω = d k t. Then, denoting d k−1 t A = i ∂ ∂t A d k t, the elements of Mπ can be written as Mπ is a subbundle of Λ k T * E, and hence Mπ is also endowed with canonical forms. First we have the "tautological form" Θ ∈ Ω k (Mπ), which is defined as follows: let (x, α) ∈ Λ k 2 T * E, with x ∈ E and α ∈ Λ k 2 T * x E; then, for every X 1 , . . . , X m ∈ T (x,α) (Mπ), we have Θ((x, α))(X 1 , . . . , X m ) := α(x)(T (x,α) κ(X 1 ), . . . , T (x,α) κ(X m )) (34) Thus we define the multisymplectic form and the local expressions of the above forms are Consider π * Λ k T * M , which is another bundle over E, whose sections are the π-semibasic k-forms on E, and denote by J 1 π * the quotient Λ k 2 T * E/π * Λ k T * M . J 1 π * is usually called the restricted multimomentum bundle associated with the bundle π : E → M . Natural coordinates in J 1 π * (adapted to the bundle π : E → M ) are denoted by (t A , q i , p A i ). We have the natural submersions specified in the following diagram

Multisymplectic Hamiltonian formalism
The Hamiltonian formalism in J 1 π * presented here is based on the construction made in [5] (see also [6] and [9]).
Definition 9 A section h : J 1 π * → Mπ of the projection µ is called a Hamiltonian section.
The differentiable forms Θ h := h * Θ and Ω h := −dΘ h = h * Ω are called the Hamilton-Cartan k and (k + 1) forms of J 1 π * associated with the Hamiltonian section h. (J 1 π * , h) is said to be a Hamiltonian system in J 1 π * .
In natural coordinates we have that , and H ∈ C ∞ (U ), U ⊂ J 1 π * , is a local Hamiltonian function. Then we have The field equations for these multisymplectic Hamiltonian systems can be stated as whereψ : M → J 1 π * are sections of the projectionσ that are solutions to these equations. In natural coordinates, writingψ(t) = (t,ψ i (t),ψ A i (t)), we have that this equation is equivalent to the Hamilton-de Donder-Weyl equations for the multisymplectic Hamiltonian system (J 1 π * , h) We denote by X k h (J 1 π * ) the set of k-vector fieldsX = (X 1 , . . . ,X k ) in J 1 π * which are solution to the equations i(X)Ωh = i(X1) . . . i(Xk)Ωh = 0 , i(X)ω = i(X1) . . . i(Xk)ω = 1 , (we denote by ω = d k t the volume form in M and its pull-backs to all the manifolds. The contraction of k-vector fields and forms is the usual one between tensorial objects).
In a system of natural coordinates, the components ofX are given by (15), then i(X)ω = 1 leads to (X A ) B = 1, for every A, B = 1, . . . , k, and hence the other equation (39) gives The existence of k-vector fields that are solutions to (39) is assured, and in a local system of coordinates they depend on n(k 2 − 1) arbitrary functions, but the number of arbitrary functions for integrable solutions is, in general, less than n(k 2 − 1).

Relation with the k-cosymplectic Hamiltonian formalism
In order to compare the multisymplectic and the k-cosymplectic formalisms of field theory, from now on we consider the case when π : E → M is the trivial bundle R k × Q → R k . Then we can establish relations among the canonical multisymplectic form on Mπ ≡ Λ k 2 T * (R k × Q), the canonical k-symplectic structure on (T 1 k ) * Q, and the canonical k-cosymplectic structure on R k × (T 1 k ) * Q (see also [20]). First recall that in M = R k we have the canonical volume form ω = dt 1 ∧ . . . ∧ dt k ≡ d k t. Then: (Proof ): 1. Consider the canonical embedding ı t : Q ֒→ R k ×Q given by i t (q) = (t, q), and the canonical submersion ρ 2 : R k × Q → Q. We can define the map , . . . , ∂ ∂t k (t,q) , X ∈ X(Q) (note that t A and p are now global coordinates in the corresponding fibres). The inverse ofΨ is given by . Thus,Ψ is a diffeomorphism. LocallyΨ is written as the identity.
2. It is a straighforward consequence of the above item because Next, using a procedure analogous to that in the above proof, we can give the

Relationship between the canonical geometric structures in
Starting from the canonical forms Θ and Ω in Mπ ≃ R k × R × (T 1 k ) * Q we can define the forms θ A on (T 1 k ) * Q, 1 ≤ A ≤ k, by Then for X, Y ∈ X((T 1 k ) * Q), we get the 2-forms ω A on (T 1 k ) * Q given as From (36) we obtain the local expressions Furthermore, we have the involutive distribution V = ker (π 1 Q ) * , and hence (ω A , V ; 1 ≤ A ≤ k) is the canonical k-symplectic structure in (T 1 k ) * Q. Conversely, starting from this k-symplectic structure in (T 1 k ) * Q we can obtain the canonical Relationship between the canonical geometric structures in R k × R × (T 1 k ) * Q and in R k × (T 1 k ) * Q.
In an analogous way, we can also relate the canonical geometric structures in R k ×R×(T 1 k ) * Q and in R k × (T 1 k ) * Q. In fact, denoting by i : then from the canonical forms Θ and Ω in Mπ ≃ R k × R × (T 1 k ) * Q we can define the forms Θ A on R k × (T 1 k ) * Q as follows: forX ∈ X(R k × (T 1 k ) * Q), and 1 ≤ A ≤ k, (These forms have the same coordinate expressions as θ A and ω A ). Furthermore, although the 1forms η A are canonically defined on R k × (T 1 k ) * Q, we can recover them from the multisymplectic form Ω as follows: forX ∈ X(R k × (T 1 k ) * Q), whose coordinate expressions are η A = dt A . These forms can also be defined by introducing the canonical embedding α 1 x , . . . , α k x ) → (t, 1, 0 x , . . . , 0 x ) and then making Furthermore, we have the involutive distribution V = ker (π 2 ) * = ∂ ∂t A , and hence (η A , Relationship between the canonical geometric structures in J 1 π * ≃ R k × (T 1 k ) * Q.
It is important to point out that, as the bundle µ : k ) * Q is trivial, then Hamiltonian sections can be taken to be global sections of the projection µ by giving a global Hamiltonian function H ∈ C ∞ (R k × (T 1 k ) * Q). Then we can also relate the non-canonical multisymplectic form with the k-cosymplectic structure in R k × (T 1 k ) * Q as follows: starting from the forms Θ h and Ω h in R k × (T 1 k ) * Q, we can define the forms Θ A and Ω A on R k × (T 1 k ) * Q as follows: forX,Ȳ ∈ X(R k × (T 1 k ) * Q), and 1 ≤ A ≤ k, and the 1-forms η A = dt A are canonically defined.
Conversely, starting from the canonical k-cosymplectic structure on R k × (T 1 k ) * Q, and from H, we can construct So we have: The multisymplectic form and the 2-forms of the canonical k-cosymplectic structure on J 1 π * ≃ R k × (T 1 k ) * Q are related by (48) and (49).
Finally, the following result about the solutions to the Hamiltonian equations establishes the complete equivalence between both formalisms:

is a solution to the equations (13) if, and only if, it is also a solution to the equations (39); that is, X
(Proof ): The proof is immediate, bearing in mind that in natural coordinates the solutions to the equations (13) and (39) are partially determined by the equations (16) and (40) respectively, and these are equivalent.

Multisymplectic Lagrangian systems
(For details, see [7] and the references quoted therein). Consider the first-order jet bundle π E : J 1 π → E, which is also a bundle over M with projectionπ : J 1 π −→ M , and is endowed with natural coordinates (t A , q i , v i A ), adapted to the bundle structure. A Lagrangian density is aπ-semibasic k-form on J 1 π, and hence it can be expressed as L = Lω, where L ∈ C ∞ (J 1 π) is the Lagrangian function associated with L and ω. Using the canonical structures of J 1 π, we can define the Poincaré-Cartan k and (k + 1)-forms, which have the following local expressions: (J 1 π, L) is said to be a Lagrangian system. The Lagrangian system and the Lagrangian function are regular if Ω L is a multisymplectic (k + 1)-form. Elsewhere they are singular (or nonregular), and Ω L is a pre-multisymplectic form. The regularity condition is locally equivalent to The Lagrangian field equations can be stated as where φ : M → E are sections of the projection π, and φ 1 : M → J 1 π are their canonical liftings, which are solutions to these equations. In natural coordinates, writing φ(t) = (t, φ i (t)), we have that this equation is equivalent to the Euler-Lagrange equations (25) for the Lagrangian L. Furthermore, we denote by X k L (J 1 π) the set of k-vector fieldsΓ = (Γ 1 , . . . ,Γ k ) in J 1 π, that are solutions to the equations i(Γ)ΩL = 0 , i(Γ)ω = 1 (50) In a system of natural coordinates the components ofΓ are given by (31), thenΓ is a solution to (50) if, and only if, (Γ A ) B = 1, for every A, B = 1, . . . , k, and (Γ A ) i and (Γ A ) i B satisfy the equations (32). When L is regular, we obtain that (Γ A ) i = v i A , and the equations (33 hold; thenΓ is a sopde, and hence, if it is integrable, its integral sections are holonomic and they are solutions to the Euler-Lagrange equations for L. If L is not regular, the existence of solutions to the equations (25) for L or to (50) is not assured, in general, except in a submanifold of J 1 π (in the most favourable situations). Moreover, solutions to (50) are not sopde necessarily.
Finally, Θ L ∈ Ω 1 (J 1 π) being π E -semibasic, we have a natural map F L : J 1 π → Mπ, given by F L(ȳ) = Θ L (ȳ) ;ȳ ∈ J 1 π which is called the extended Legendre map associated to the Lagrangian L. The restricted Legendre map is F L = µ • F L : J 1 π → J 1 π * . Their local expressions are Moreover, we have F L * Θ = Θ L , and F L * Ω = Ω L . Observe that the Legendre transformations F L defined for the k-cosymplectic and the multisymplectic formalisms are the same, as their local expressions (28) and (51) show.

Relation between multisymplectic and k-cosymplectic Lagrangian systems
In the particular case E = R k × Q, we have J 1 π ≃ R k × T 1 k Q and we can define the Energy Lagrangian function E L as In this particular case, as in the Hamiltonian case, we can relate the non-canonical Lagrangian multisymplectic (or pre-multisymplectic) form Ω L with the non-canonical Lagrangian k-cosymplectic (or k-precosymplectic) structure in R k × T 1 k Q constructed in Section 3.3 as follows: starting from the forms Θ L and Ω L in J 1 π ≃ R k × T 1 k Q, we can define the forms Θ A L and Ω A L on R k × T 1 k Q, as follows: for X, Y ∈ X(R k × T 1 k Q), and 1 ≤ A ≤ k, and the 1-forms η A = dt A are canonically defined.
Conversely, starting from the Lagrangian k-cosymplectic (or k-precosymplectic) structure on R k × T 1 k Q, and from E L , we can construct on R k × T 1 k Q ≃ J 1 π So we have proved that: Theorem 8 The Lagrangian multisymplectic (or pre-multisymplectic) form and the Lagrangian 2-forms of the k-cosymplectic (or k-precosymplectic) structure on J 1 π ≡ R k × T 1 k Q are related by (52) and (53).
The discussion in the above section about the Lagrangian equations proves the following result, which establishes the complete equivalence between both formalisms: Theorem 9 A k-vector fieldΓ = (Γ 1 , . . . ,Γ k ) in J 1 π ≃ R k × T 1 k Q is a solution to the equations (50) if, and only if, it is also a solution to the equations (30); that is, we have that X k L (R k × T 1 k Q) = X k L (R k × T 1 k Q).