A new multisymplectic unified formalism for second-order classical field theories

We present a new multisymplectic framework for second-order classical field theories which is based on an extension of the unified Lagrangian-Hamiltonian formalism to these kinds of systems. This model provides a straightforward and simple way to define the Poincar\'e-Cartan form and clarifies the construction of the Legendre map (univocally obtained as a consequence of the constraint algorithm). Likewise, it removes the undesirable arbitrariness in the solutions to the field equations, which are analyzed in-depth, and written in terms of holonomic sections and multivector fields. Our treatment therefore completes previous attempt to achieve this aim. The formulation is applied to describing some physical examples; in particular, to giving another alternative multisymplectic description of the Korteweg-de Vries equation.


Introduction
Higher-order field theories are relevant in physics and applied mathematics because they appear in many of important situations; for instance, the standard gravitational theories, in particular Hilbert's Lagrangian for gravitation, are of this kind; as well as string theories, Podolsky's generalization of electromagnetism, the different forms of the Korteweg-de Vries equation in fluid theory, and other interesting models in physics. As a consequence, many works are devoted to the development of the formalism of these kinds of theories and their application to many models in mechanics and field theory (a long but non-exhaustive list of references can be found in [8,39,40]).
In higher-order mechanical systems and field theories, the formalism shows explicit dependence on accelerations or higher-order derivatives of the generalized coordinates of position, or in the higher order derivatives of the fields. Thus, for Lagrangian systems, if the Lagrangian function depends on derivatives of order k, the corresponding Euler-Lagrange equations are of order 2k. These kinds of systems are therefore modeled geometrically using higher-order tangent and jet bundles as the main tool (see, for instance, [10,14,18,19,30,31,36,38,45,46]).
In particular, as regards higher-order field theories, great efforts have been made to extend the classical multisymplectic framework developed for describing first-order field theories to this realm. The usual way to do this consists in generalizing the construction of the Poincaré-Cartan form for a higher-order Lagrangian density and in stating the Lagrangian formalism [2,3,25,27,28,32,33,44]. Nevertheless, this procedure involves from some ambiguity, since the definition of the Poincaré-Cartan form in a higher-order jet bundle is not unique, and while for the second-order case it is proved that all these forms are equivalent [45,46], this is not true for the general higher-order cases. These and other kinds of problems involving the non-uniqueness of the geometrical constructions also appear in the definition of the Legendre transformation associated with a higher-order Lagrangian and as well as a suitable choice of the multimomentum phase space for the Hamiltonian formalism of the theory [4,26,34,35].
A way to overcome these difficulties and simplify the formalism was recently achieved in [9] using the so-called Skinner-Rusk or Lagrangian-Hamiltonian unified formalism for field theories. The origin of this formalism is the seminal paper [47], where R. Skinner and R. Rusk present a new framework for first-order autonomous mechanical systems that compresses the Lagrangian and Hamiltonian formalisms into a single one. This was subsequently generalized to first-order non-autonomous dynamical systems [7,13], control systems [6], higher-order autonomous and non-autonomous mechanical systems [10,12,18,30,37,39,40,41], and first-order classical field theories [17,21,42,43]. Then, in [9] the authors present an extension of this formulation to higher-order field theories in order to develop an unambiguous framework for higher-order classical field theories. While this model allows us to simplify previous formulations, some arbitrary parameters appearing in the higher-order field equations and in the definition of the Legendre transformation must be fixed "ad-hoc". Another interesting approach to the higherorder unified formalism for field theory, but using infinite-order jet bundles, is given in [48].
In this paper, we present a modification of the model given in [9] by using finite higherorder bundles to avoid the latest ambiguities of the model, clarifying the construction of the Legendre map and the choice of the jet and the multimomentum bundles for the Lagrangian and the Hamiltonian formalisms, as well as the field equations in both formalisms. Our model is therefore a completion of the approaches given in [9,48]. Our treatment works for second-order field theories because we want it to be applied here and in future papers to describe the well known theories previously cited: gravitation, Korteweg-de Vries equation and other models in physics, all of which are of second-order. Another advantage of working at this order is that we can use the diffeomorphism among several geometric structures in order to avoid part of the ambiguity inherent to the theory. In any case, further work to generalize our results to higher-order cases is in progress.
The organization of the paper is as follows. First, in Section 2, we review the geometric structures of higher-order jet bundles, introduce the concepts of holonomic sections and multivector fields in order to state the field equations on these bundles, and define the space of symmetric multimomenta suitable for the Hamiltonian formalism. Section 3 is devoted to developing our proposal of the Lagrangian-Hamiltonian unified formalism for second-order field theories. After introducing the unified jet-multimomentum bundles and their relevant submanifolds where the formalism takes place, we state the field equations in the unified formalism using sections and multivector fields. Thanks to this unified framework, we establish the Lagrangian and Hamiltonian formalisms for second-order field theories (in Sections 4 and 5) for both the regular and singular (almost-regular) theories. Finally, in Section 6 we apply our model to describe two physical systems: the bending or deflection of a plate with clamped edges and the classical Korteweg-de Vries equation. A comparison of our results with those of previous papers is given in the last Section 7, where we also summarize our results and outlook.
All the manifolds are real, second countable and C ∞ . The maps and the structures are assumed to be C ∞ . Sum over repeated indices is understood. The usual multi-index notation introduced in [45] is used: a multi-index I is an element of Z m such that every component is positive, the ith position of the multi-index is denoted I(i), and |I| = m i=1 I(i) is the length of the multi-index, while I! = m i=1 I(i)!. Finally, an expression of the type |I| = k means that the expression (or the sum) is taken for every multi-index of length k. The same applies for inequalities. (See [45], §6.1 for details).
2 Geometric structures of higher-order jet bundles 2.1 Higher-order jet bundles. Coordinate total derivatives (See [45] for details).
Let M be an orientable m-dimensional smooth manifold, and let η ∈ Ω m (M ) be a volume form for M . Let E π −→ M be a bundle with dim E = m + n. If k ∈ N, the kth-order jet bundle of the projection π, J k π, is the manifold of the k-jets of local sections φ ∈ Γ(π); that is, equivalence classes of local sections of π by the relation of equality on every partial derivative up to order k. A point in J k π is denoted by j k x φ, where x ∈ M and φ ∈ Γ(π) is a representative of the equivalence class. We have the following natural projections: if r k, Observe that π s r • π k s = π k r , π k 0 = π k (where J 0 π is canonically identified with E), π k k = Id J k π , andπ k = π • π k .
Local coordinates in J k π are introduced as follows: let (x i ), (1 i m) be local coordinates in M , and (x i , u α ), (1 α n), local coordinates in E adapted to the bundle structure. Let φ ∈ Γ(π) be a section with coordinate expression φ(x i ) = (x i , φ α (x i )). Then, local coordinates in J k π are (x i , u α , u α I ), where Using these coordinates, the local expressions of the natural projections are π k r (x i , u α , u α I ) = (x i , u α , u α J ) ; π k (x i , u α , u α I ) = (x i , u α ) ;π k (x i , u α , u α I ) = (x i ) .
If ψ ∈ Γ(π) is a section, we denote the kth prolongation of φ to J k π by j k φ ∈ Γ(π k ). In natural coordinates of J k π, if φ(x i ) = (x i , φ α (x i )), the k-jet lifting of φ is given by Definition 1 Let E π −→ M be a bundle, x ∈ M , φ ∈ Γ(π) a section in x, and v ∈ T x M . The kth holonomic lift of v by φ is defined as In local coordinates, if v ∈ T x M is given by The vector space (π k+1 k ) * (TJ k π) j k+1 x π has a canonical splitting as a direct sum of two subspaces: (π k+1 k ) * (TJ k π) j k+1 , where (j k φ) * T x M denotes the set of kth holonomic lifts of tangent vectors in T x M by φ. As a consequence, the vector bundle (π k+1 k ) * τ J k π : (π k+1 k ) * TJ k π → J k π has a canonical splitting as a direct sum of two subbundles (π k+1 k ) * TJ k π = (π k+1 k ) * V (π k ) ⊕ H(π k+1 k ) (π k+1 k ) * τ J k π G G J k π , where H(π k+1 k ) is the union of the fibres (j k φ) * (T x M ), for x ∈ M . Now, if X(π k+1 k ) denotes the module of vector fields along the projection π k+1 k , the submodule corresponding to sections of (π k+1 is denoted by X v (π k+1 k ), and the submodule corresponding to sections of (π k+1 k ) * τ J k π H(π k+1 k ) is denoted by X h (π k+1 k ). The splitting for the bundles given above induces the following canonical splitting for the module X(π k+1 k ): An element of the submodule X h (π k+1 k ) is called a total derivative. Definition 2 Given a vector field X ∈ X(M ), a section φ ∈ Γ(π) and a point x ∈ M , the kth holonomic lift of X by φ, j k X ∈ X h (π k+1 k ), is defined as In local coordinates, if X ∈ X(M ) is given by X = X i ∂ ∂x i , then, bearing in mind the local expression (1) of the kth holonomic lift for tangent vectors, the kth holonomic lift of X is Finally, the coordinate total derivatives are the holonomic lifts of the local vector fields ∂/∂x i ∈ X(M ), which are denoted by d/dx i ∈ X(π k+1 k ), and whose coordinate expressions are

Holonomic sections and multivector fields
(See appendix A for the terminology and notation on multivector fields in a manifold).
In natural coordinates, if ψ ∈ Γ(π k ) is given by ψ(x i ) = (x i , ψ α , ψ α I ) (1 |I| k), then the condition for ψ to be holonomic of type r gives the system of partial differential equations or, equivalently, Definition 4 A multivector field X ∈ X m (J k π) is holonomic of type r, 1 r k, if 1. X is integrable.
3. The integral sections ψ ∈ Γ(π k ) of X are holonomic of type r.
In particular, a multivector field X ∈ X m (J k π) is holonomic of type 1 (or simply holonomic) if it is integrable,π k -transverse and its integral sections ψ ∈ Γ(π k ) are the kth prolongations of sections φ ∈ Γ(π).
In natural coordinates, if X ∈ X m (J k π) is a locally decomposable andπ k -transverse multivector field locally given by with f i non-vanishing local functions. Then, the condition for X to be holonomic of type r gives the following equations: Hence, the local expression of a locally decomposable holonomic multivector field of type r is In the particular case r = 1, the local expression is Remark: It is important to point out that a locally decomposable andπ k -transverse multivector field X satisfying the local equations (4) may not be holonomic of type r, since these local equations are not a sufficient or necessary condition for the multivector field to be integrable. However, we can assure that if such a multivector field admits integral sections, then its integral sections are holonomic of type r. In first-order theories, these equations are equivalent to the so-called semi-holonomy (or SOPDE) condition [22].

The space of 2-symmetric multimomenta
For the sake of simplicity, in the following we restrict ourselves to the case k = 2, that is, the second-order case, which is our main goal in this paper. However, all the results that follow in this Section can be stated in the general situation.
In addition, the bundle Λ m 2 (J 1 π) is diffeomorphic to the union of the affine maps from J 1 uπ 1 to (Λ m M )π1 (u) , where u ∈ J 1 π is an arbitrary point; that is, Using this identification and the fact that J 2 π is embedded into J 1π1 , we can define a canonical pairing between the elements of J 2 π and the elements of Λ m 2 (J 1 π) as a fibered map over J 1 π, defined as follows C : there exists a pairing function associated to the canonical pairing C and the volume form η ∈ Ω m (M ), denoted by C : i m, 1 α n, be a local chart in E adapted to the bundle structure and such that η = dx 1 ∧ . . . ∧ dx m ≡ d m x. Then, the induced natural coordinates in Observe that dim Λ m 2 (J 1 π) = m + n + 2nm + nm 2 + 1. In these coordinates, the Liouville m and (m + 1)-forms have the following local expressions Finally, the pairing function C associated to C and η has the following coordinate expression According to the results in [46], let us consider the submanifold of Λ m 2 (J 1 π) defined by This submanifold is π J 1 π -transverse, and therefore fibers over J 1 π. Let π † J 1 π : J 2 π † → J 1 π and π † J 1 π =π 1 • π † J 1 π : J 2 π † → M be the canonical projections. Natural coordinates in J 2 π † adapted to the bundle structure are (x i , u α , u α i , p, p i α , p I α ), where |I| = 2. Then, the natural embedding j s : J 2 π † ֒→ Λ m 2 (J 1 π) is given in coordinates by The submanifold J 2 π † ֒→ Λ m 2 (J 1 π) is called the extended 2-symmetric multimomentum bundle.
All the geometric structures defined above for Λ m 2 (J 1 π) can be restricted to J 2 π † . In particular, let us denote Θ s the pull-back of the Liouville m and (m + 1)-forms to J 2 π † , which we call the symmetrized Liouville m and (m + 1)-forms. Bearing in mind the local expressions (5) of the Liouville m and (m + 1)-forms, and (7) of the canonical embedding j s : J 2 π † ֒→ Λ m 2 (J 1 π), the coordinate expressions of Θ s 1 and Ω s 1 are An important fact concerning the pull-back of the multisymplectic (m+1)-form Ω 1 to J 2 π † is that it is multisymplectic in J 2 π † . Since Ω s 1 = −dΘ s 1 is obviously closed, it suffices to show that it is 1-nondegenerate, that is, i(X)Ω s 1 = 0 if, and only if, X = 0. Expressed it in coordinates: let X ∈ X(J 2 π † ) be a generic vector field locally given by Then, taking into account the coordinate expression (8) of the (m + 1)-form Ω s 1 , the m-form i(X)Ω s 1 is locally given by where d m−2 x jk = i(∂/∂x k )d m−1 x j . From this coordinate expression it is clear that i(X)Ω s 1 = 0 if, and only if, X = 0. Hence Ω s 1 is multisymplectic.

Geometrical setting
Let E π −→ M be the configuration bundle describing a classical field theory, where M is a m-dimensional orientable smooth manifold with fixed volume form η ∈ Ω m (M ) and E is a (m + n)-dimensional smooth manifold. Let L ∈ Ω m (J 2 π) be a second-order Lagrangian density for this theory, that is, aπ 2 -semibasic m-form on J 2 π. L being aπ 2 -semibasic m-form, we can write L = L · (π 2 ) * η, where L ∈ C ∞ (J 2 π) is the second-order Lagrangian function associated to L and η.
According to [7,21,40], let us consider the fiber bundles The bundles W and W r are called the extended 2-symmetric jet-multimomentum bundle and the restricted 2-symmetric jet-multimomentum bundle, respectively.
These bundles are endowed with the canonical projections In addition, the natural quotient map µ : J 2 π † → J 2 π ‡ induces a natural projection (that is, a surjective submersion) µ W : W → W r . Thus, we have the following diagram Let (U ; x i , u α ) be a local chart of coordinates in E adapted to the bundle structure and such that η = dx 1 ∧ . . . ∧ dx m ≡ d m x. Then, we denote by ((π 3 ) −1 (U ); x i , u α , u α i , u α I , u α J ) and ((π 1 • π † J 1 π ) −1 (U ); x i , u α , u α i , p, p i α , p I α ) the induced local charts in J 3 π and J 2 π † , respectively, with |I| = 2 and |J| = 3. Thus, (x i , u α , u α i , p i α , p I α ) are the natural coordinates in J 2 π ‡ , and the coordinates in W and W r are (x i , u α , u α i , u α I , u α J , p, p i α , p I α ) and (x i , u α , u α i , u α I , u α J , p i α , p I α ), respectively.
The bundle W is endowed with some canonical structures.
and Ω s 1 ∈ Ω m+1 (J 2 π † ) be the symmetrized Liouville forms. Then we define the following forms in W which are called the second-order unified canonical forms.
Bearing in mind the local expressions (8) of the forms Θ s 1 and Ω s 1 , and taking into account that the projection ρ 2 is locally given by , we obtain the coordinate expression of the unified canonical forms, which are Observe that, although Ω s 1 is multisymplectic, the (m + 1)-form Ω is premultisymplectic, since it is closed and 1-degenerate. Indeed, for every X ∈ X V (ρ 2 ) (W) we have i(X)Ω = 0. This is easy to check in coordinates: the C ∞ (W)-module X V (ρ 2 ) (W) is locally given by with |I| = 2 and |J| = 3, and, bearing in mind the local expression (11) for Ω, we have Hence, (W, Ω) is a premultisymplectic manifold of degree m + 1, and ker Ω = X V (ρ 2 ) (W).
The second canonical structure in W is the following: The second-order coupling m-form in W is the ρ M -semibasic m-formĈ ∈ Ω m (W) defined as follows: for every (j 3 AsĈ is a ρ M -semibasic m-form, there exists a functionĈ ∈ C ∞ (W) such thatĈ =Ĉρ * M η. Bearing in mind the local expression (9) of C s , the coordinate expression of the second-order coupling form isĈ We denoteL = (π 3 2 • ρ 1 ) * L ∈ Ω m (W). Since the Lagrangian density is aπ 2 -semibasic form, we have thatL is a ρ M -semibasic m-form, and thus we can writeL =L · ρ * M η, wherê L = (π 3 2 • ρ 1 ) * L ∈ C ∞ (W) is the pull-back of the Lagrangian function associated with L and η. Then, we define a Hamiltonian submanifold Since bothL andĈ are ρ M -semibasic m-forms, the submanifold W o is defined by the constraint C −L = 0. In local coordinates, bearing in mind the local expression (14) ofĈ, the constraint function is .
(Proof ) W o is obviously 1-codimensional, since it is defined by a single constraint function.
To see that Φ = µ W • j o : W o → W is a diffeomorphism, we show that it is one-to-one. First, for every (j 3 Now, using the previous expression for ( Hence, by definition of W o , we have L(π 3 2 (j 3 . Locally, from the third equality we obtain . Then p(ω 1 ) = p(ω 2 ), and ω 1 = ω 2 . Furthermore, µ W • j o is surjective. In fact, given (j 3 x φ, [ω]) ∈ W r , we wish to find (j 3 . It suffices to take [ζ] such that, in local coordinates of W, This ζ exists as a consequence of the definition of W o . Now, since µ W •j o is a one-to-one submersion, then, by equality on the dimensions of W o and W r , it is a one-to-one local diffeomorphism, and thus a global diffeomorphism. Finally, in order to prove that W o is µ W -transversal, it is necessary to check if L(X)(ξ) ≡ X(ξ) = 0, for every X ∈ ker µ W * and every constraint function ξ defining W o . Since W o is defined by the constraintĈ −L = 0 and ker µ W * = ∂/∂p , we have As a consequence of Proposition 1, the submanifold W o induces a sectionĥ ∈ Γ(µ W ) defined asĥ = j o • Φ −1 : W r → W, which is called a Hamiltonian section of µ W or a Hamiltonian µ W -section. This section is specified by giving the local Hamiltonian function . Observe thatĥ satisfies ρ r 1 = ρ 1 •ĥ and ρ r 2 = µ • ρ 2 •ĥ. Hence, we have the following commutative diagram: with local expressions Finally, we generalize the definition of holonomic sections and multivector fields to the unified setting.
3. The integral sections ψ ∈ Γ(ρ r M ) of X are holonomic of type r in W r .

Field equations for sections
The Lagrangian-Hamiltonian problem for sections associated with the system (W r , Ω r ) consists in finding holonomic sections ψ ∈ Γ(ρ r M ) satisfying the following condition In natural coordinates, let ψ ∈ Γ(ρ r M ) be locally given by ψ( . Then, bearing in mind the coordinate expression (16) of Ω r , we obtain the following system of partial differential equations for the component functions of ψ Observe that equations (21) give partially the holonomy condition for the section ψ, but since we required this condition from the beginning, these equations are automatically satisfied.
Notice also that equations (20) do not involve any partial derivative of the component functions of ψ: they are pointwise algebraic conditions that must be fullfilled for every section ψ ∈ Γ(ρ r M ) solution to the field equation (17). These equations arise from the ρ r 2 -vertical part of the vector fields X ∈ X(W r ), as shown in the following result.
(Proof ) This result is easy to prove in coordinates. In the natural coordinates of W r , the C ∞ (W r )-module of ρ r 2 -vertical vector fields is given by Then, bearing in mind the local expression (16) of Ω r , we have Thus, in both cases we obtain a ρ r M -semibasic m-form.
As a consequence of this result, we can define the submanifold where every section ψ ∈ Γ(ρ r M ) solution to the equation (17) must take values. This submanifold is called the first constraint submanifold of the premultisymplectic system (W r , Ω r ), and has codimension nm(m + 1)/2.
As we have seen in the proof of Lemma 1, the submanifold W c ֒→ W r is locally defined by the constraints (20). In combination with equations (19), we have the following.
Proposition 2 A solution ψ ∈ Γ(ρ r M ) to equation (17) takes values in a nm-codimensional submanifold W L ֒→ W c which is identified with the graph of a bundle map FL : J 3 π → J 2 π ‡ over J 1 π defined locally by (Proof ) Since W c is defined locally by the constraints (20), it suffices to prove that these contraints, together with the remaining local equations for the section ψ ∈ Γ(ρ r M ) to be a solution to the equation (17), give rise to the local functions defining the bundle map given above, and thus to the submanifold W L .
Replacing p I α by ∂L/∂u α I in equations (19), we obtain Therefore, these constraints define a submanifold W L ֒→ W c , which can be identified with the graph of a map FL : J 3 π → J 2 π ‡ given by The bundle map FL : J 3 π → J 2 π ‡ is called the restricted Legendre map associated with the Lagrangian density L. Observe that dim W L = dim J 3 π = m + n + mn + nm(m + 1)/2 + nm(m + 1)(m + 2)/6.

Remark:
The terminology "Legendre map" is justified, since FL is a fiber bundle morphism from the Lagrangian phase space to the Hamiltonian phase space that identifies the multimomenta coordinates with functions on partial derivatives of the Lagrangian function, and thus generalizes the Legendre map in first-order field theories (see [24,20]), and first-order and higher-order mechanics (see [1] for first-order mechanics and [18] for the higher-order setting).
According to [46], we can give the following definition.
Otherwise, the Lagrangian density is said to be singular.
Hence, a second-order Lagrangian density L ∈ Ω m (J 2 π) is regular if, and only if, the restricted Legendre map FL : J 3 π → J 2 π ‡ associated to L is a submersion onto J 2 π ‡ . This implies that there exist local sections of FL, that is, maps σ : U → J 3 π, with U ⊂ J 2 π ‡ an open set, such that FL • σ = Id U . If FL admits a global section Υ : J 2 π ‡ → J 3 π, then the Lagrangian density is said to be hyperregular.
and the equality holds if, and only if, m = 1. Therefore, unlike in higher-order mechanics or first-order field theories, the Legendre map cannot be a local diffeomorphism due to dimension restrictions.
Computing the local expression of the tangent map to FL in a natural chart of J 3 π, the regularity condition for the Lagrangian density L is equivalent to where |I| = |K| = 2. That is, the Hessian of the Lagrangian function associated to L and η with respect to the highest order velocities is a regular matrix at every point, which is the usual definition for a regular Lagrangian density.
Note that since W r is diffeomorphic to the submanifold W o ֒→ W (Proposition 1), and W o is defined locally by the constraint p + p i α u α i + p I α u α I −L = 0, the restricted Legendre map FL : J 3 π → J 2 π ‡ can be extended in a canonical way to a map FL : J 3 π → J 2 π † , defining FL * p as the pull-back of the local Hamiltonian function −Ĥ. This enables us to state the following result, which is a straightforward consequence of Proposition 2 and satisfying FL = µ • FL.
The bundle map FL : J 3 π → J 2 π † is the extended Legendre map associated with the Lagrangian density L. An important result concerning both Legendre maps is the following.
Following the same patterns as in [15] for first-order mechanical systems, the proof of this result consists in computing in a natural chart of coordinates the local expressions of the Jacobian matrices of both maps FL and FL. Then, observe that the ranks of both maps depend on the rank of the Hessian matrix of the Lagrangian function with respect to the highest order velocities, and that the additional row in the Jacobian matrix of FL is a combination of the others. Since it is just a long calculation in coordinates, we omit the proof of this result.
Notice that the component functions u α J with |J| = 3 of the section ψ ∈ Γ(ρ r M ) are not yet determined, since the coordinate expression of the field equation (17) does not give any condition on these functions. In fact, these functions are determined by the equations (18) and (19). Indeed, since the section ψ ∈ Γ(ρ r M ) must take values in the submanifold W L given by Proposition 2, then by replacing the local expression of the restricted Legendre map in equations (18) and (19) we obtain the second-order Euler-Lagrange equations for field theories: Finally, observe that since the section ψ ∈ Γ(ρ r M ) must take values in the submanifold W L ֒→ W r , it is natural to consider the restriction of equation (17) to the submanifold W L ; that is, to restrict the set of vector fields to those tangent to W L . Nevertheless, the new equation may not be equivalent to the former. The following result gives a sufficient condition for these two equations to be equivalent. (17) is equivalent to (Proof ) We prove this result in coordinates. First of all, let us compute the coordinate expression of a vector field X ∈ X(W r ) tangent to W L . Let X be a generic vector field locally given by Then, since W L is the submanifold of W r defined locally by the nm + nm(m + 1)/2 constraint functions ξ i α , ξ I α with coordinate expression gives the following relation on the component functions of X Hence, the tangency condition enables us to write the component functions , then the equation (17) gives in coordinates where the terms (· · · ) contain a long expression with several partial derivatives of the component functions and the Lagrangian function, which is not relevant in this proof. On the other hand, if we take a vector field Y tangent to W L , then we must replace the component functions G i α and G I α by G i α and G I α in the previous equation, thus obtaining Finally, if ψ is holonomic, then equations (21) are satisfied, and the last two terms of both i(X)Ωr and i(Y )Ω r vanish, thus obtaining Hence, we have i(X)Ωr = 0 if, and only if, i(Y )Ω r = 0. Remarks: • Observe that, contrary to first-order field theories [21], the holonomy condition is not recovered from the coordinate expression of the field equations. Moreover, in this case, unlike in higher-order time-depending mechanical systems [40], not even a condition for the holonomy of type 2 can be obtained. This is due to the constraints p ij α − p ji α = 0 introduced in Section 2.3 to define both the extended and restricted 2-symmetric multimomentum bundles. Hence, the full holonomy condition is necessarily required in this formalism.
It is important to point out that, although the holonomy condition cannot be obtained from the field equation, a holonomic section ψ ∈ Γ(ρ r M ) satisfies equations (21). Hence, a holonomic section can be a solution to the equation (17).
• The regularity of the Lagrangian density seems to play a secondary role in this formulation, because the holonomy of the section solution to the equation (17) is necessarily required, regardless of the regularity of the Lagrangian density given. Nevertheless, recall that the Euler-Lagrange equations (25) may not be compatible if the Lagrangian density is singular, and thus the regularity of L still determines if the section ψ ∈ Γ(ρ r M ) solution to the equation (17) lies in W L or in a submanifold of W L . If L is singular, in the most favourable cases, there exists a submanifold W f ֒→ W L where the section ψ takes values.

Field equations for multivector fields
The Lagrangian-Hamiltonian problem for multivector fields associated with the premultisymplectic manifold (W r , Ω r ) consists in finding a class of locally decomposable holonomic multivector fields {X } ⊂ X m (W r ) satisfying the following field equation According to [16], we have: (27) exists only on the points of the submanifold W c ֒→ W r defined by The submanifold W c ֒→ W r is the so-called compatibility submanifold for the premultisymplectic system (W r , Ω r ). Observe that we denoted this submanifold by W c , which is the notation used for the first contraint submanifold defined in (22). Indeed, both submanifolds are equal. In order to prove this, recall that the first constraint submanifold is defined locally by the constraints p I α − ∂L/u α I = 0. Hence, it suffices to prove that the compatibility submanifold given in Proposition 5 is defined locally by the same contraints.
In fact, in natural coordinates, the coordinate expression for the local Hamiltonian function H is given by (15), and thus we have Now, bearing in mind that ker Ω is the (nm(m + 1)/2 + nm(m + 1)(m + 2)/6)-dimensional C ∞ (W)-module locally given by (12), the functions i(Z)dĤ for Z ∈ ker Ω have the following coordinate expressions Therefore, the submanifold W c ֒→ W r is locally defined by the nm(m + 1)/2 constraints p I α − ∂L/∂u α I = 0. In particular, it is equal to the submanifold defined in (22), and we have dim W c = dim W r − nm(m + 1)/2 = m + n + 2mn + nm(m + 1)/2 + nm(m + 1)(m + 2)/6 . Now we compute the coordinate expression of the equation (27) in a local chart of W r . From the results in [22], a representative X of a class of locally decomposable, integrable and ρ r M -transverse m-vector fields {X } ⊂ X m (W r ) can be written in coordinates where f is a non-vanishing local function. Taking f = 1 as a representative of the equivalence class, the equation (27) gives the following system of equations The m additional equations alongside the dx i are a straightforward consequence of the others and the tangency condition that follows, and thus we omit them. Therefore, the multivector field X is locally given by where the functions F α i,j , G i α,j and G I α,j must satisfy the equations (29), (30) and (31). Note that most of the component functions remain undetermined, and that there can be several different functions satisfying the referred equations. However, recall that the statement of the problem requires the class of multivector fields to be holonomic. In coordinates, this implies that equations (4) are satisfied with k = 3 and r = 1, and thus the multivector field X has the following coordinate expression with G i α,j and G I α,j satisfying (30) and (31). Observe that the equations (32) are a compatibility condition for the multivector field X , which state that the multivector field solution to the field equation (27) exists only at support on the submanifold W c . Hence, we recover in coordinates the result stated in Proposition 5.
Let us analyze the tangency of the multivector field X along the submanifold W c ֒→ W r . From [22] we know that the necessary and sufficient condition for X = X 1 ∧ . . . ∧ X m ∈ X m (W r ) to be tangent to W c is that X j is tangent to W c for every j = 1, . . . , m.
Therefore, since the submanifold W c ֒→ W r is locally defined by the constraint functions ξ K α = p K α − ∂L/∂u α K , we must check if the condition L(X j )(ξ K α ) ≡ X j (ξ K α ) = 0 holds on W c for every 1 j m, 1 α n, |K| = 2. Computing, we obtain Hence, the tangency condition enables us to determinate all the functions G K α,j , since we obtain nm 2 (m + 1)/2 equations, one for each function. Now, taking into account equations (31) and the coefficients G K α,j that we have determined, we obtain Hence, the tangency condition for the multivector field X along W c gives rise to mn new constraints defining a submanifold of W c that coincides with the submanifold W L introduced in Proposition 2. Now we must study the tangency of X along the new submanifold W L . After a long but straightforward calculation, we obtain Therefore, the tangency condition along the submanifold W L enables us to determinate all the functions G i α,k . Now, taking into account equations (30), we have These n equations are the second-order Euler-Lagrange equations for a locally decomposable holonomic multivector field. Observe that ifL is a regular Lagrangian density, then the Hessian ofL with respect to the second-order velocities is regular, and we can assure the existence of a local multivector field X solution to the equation (27), defined at support on W L ֒→ W r and tangent to W L . A global solution is then obtained using partitions of the unity.
If the Lagrangian density is not regular, then the above equations may or may not be compatible, and may give rise to new constraints. In the most favourable cases there exists a submanifold W f ֒→ W L (where we admit W f = W L ) where we have a well-defined holonomic multivector field at support on W f , and tangent to W f , solution to the equation Therefore, we can state the following result.

If ψ is given locally by
, then the component functions of ψ satisfy equations (18) and (19), that is, the following system of n + nm partial differential equations 3. ψ is a solution to the equation where Λ m ψ ′ : M → Λ m TW r is the canonical lifting of ψ.
4. ψ is an integral section of a multivector field contained in a class of locally decomposable holonomic multivector fields {X } ⊂ X m (W r ), tangent to W L , and satisfying the equation (27), that is, i(X )Ω r = 0 .
(1 ⇔ 2) From the results in Section 3.2, the field equation (17) gives in coordinates the equations (18), (19), (20) and (21). As stated in the aforementioned Section, the equations (20) are the local constraints defining the first constraint submanifold W c ֒→ W r . In addition, since we assume that the section ψ ∈ Γ(ρ r M ) is holonomic, the equations (21) are satisfied. Therefore, the equation (17) is locally equivalent to equations (18) and (19), that is, to equations (34).
then its canonical lifting to Λ m TW r is locally given by where d/dx j is the jth coordinate total derivative, and the 1 is at the jth position. Then, the inner product i(Λ m ψ ′ )(Ω r • ψ) gives, in coordinates, where the terms (· · · ) along the forms dx i involve of partial derivatives of the Lagrangian function and of the rest of component functions. Now, requiring this last expression to vanish, we obtain equations (18), (19), (20) and (21), along with m additional equations which are a combination of those. Same comments as in the proof of the previous item apply. In particular, equations (20) are the local constraints defining the first constraint submanifold W c ֒→ W r , and equations (21) are automatically satisfied because of the holonomy assumption. Therefore, the equation (35) is locally equivalent to equations (18) and (19), that is, to equations (34).
(2 ⇔ 4) From the results in this Section, if X ∈ X m (W r ) is a generic locally decomposable multivector field locally given by (28), then, taking f = 1 as a representative of the equivalence class, the field equation (27) is locally equivalent to the equations (29), (30), (31) and (32). As already stated, equations (32) give, in coordinates, the compatibility submanifold W c obtained using the constraint algorithm in [16]. On the other hand, since the multivector field X is assumed to be holonomic, then equations (29) are satisfied. Hence, the field equation (27) is locally equivalent to equations (30) and (31).
Let γ ∈ Γ(ρ r M ) be an integral section of X locally given by γ( . Then, the condition of integral section is locally equivalent to the following system of equations Replacing these equations in (30) and (31), we obtain the following system of partial differential equations for the component functions of γ Since the multivector field X is holonomic and tangent to W L , the first equations are identically satisfied. Thus, the condition of γ to be an integral section of a locally decomposable holonomic multivector field X ∈ X m (W r ), tangent to W L , and satisfying the equation (27) is locally equivalent to equations (34).

Lagrangian formalism
Now we recover the Lagrangian field equations and geometric structures from the unified formalism. The results remain the same for both regular and singular Lagrangian densities. Thus, no distinction will be made in this matter.

General setting
In order to establish the field equations in the Lagrangian formalism, we must define the Poincaré-Cartan m and (m + 1)-forms in J 3 π. Since a unique Legendre map is recovered in the unified framework, we can give the following definition: and Ω s 1 ∈ Ω m+1 (J 2 π † ) be the symmetrized Liouville forms in J 2 π † . The Poincaré-Cartan forms in J 3 π are defined as Θ L = FL * Θ s 1 ∈ Ω m (J 3 π) and The Poincaré-Cartan forms can also be recovered directly from the unified formalism. In fact: Lemma 2 Let Θ = ρ * 2 Θ s 1 and Θ r =ĥ * Θ be the canonical m-forms defined in W and W r , respectively. Then, the Poincaré-Cartan m-form satisfies Θ = ρ * 1 Θ L and Θ r = (ρ r 1 ) * Θ L .
(Proof ) A straightforward computation leads to this result. For the first statement we have and from this the second statement follows: Observe that, as the pull-back of a form by a function and the exterior derivative commute, this result also holds for the Poincaré-Cartan (m + 1)-form Ω L .
Using the natural coordinates (x i , u α , u α i , u α I , u α J ) in J 3 π, and bearing in mind the local expression (8) of Θ s 1 , and (24) of the extended Legendre map, the local expression of the Poincaré-Cartan m-form is An important fact regarding the Poincaré-Cartan (m + 1)-form Ω L is that it is 1-degenerate when m > 1, regardless of the regularity of the Lagrangian density. Indeed, since the Legendre map FL : J 3 π → J 2 π ‡ is a submersion with dim J 3 π > dim J 2 π ‡ , and rank(FL) = rank( FL), there exists a non-zero vector field X ∈ X(J 3 π) which is FL-related to 0 ∈ X(J 2 π † ), that is, (Proof ) Since ρ L 1 is a surjective submersion, the equality dim J 3 π = dim W L implies that it is also an injective immersion, and therefore a diffeomorphism.

Field equations for sections
Proposition 7 Let ψ ∈ Γ(ρ r M ) be a holonomic section solution to the equation (17). Then the section ψ L = ρ r 1 • ψ ∈ Γ(π 3 ) is holonomic, and is a solution to the equation (Proof ) By definition, a section ψ ∈ Γ(ρ r M ) is holonomic if the section ρ r 1 • ψ ∈ Γ(π 3 ) is holonomic. Hence, ψ L = ρ r 1 • ψ is clearly a holonomic section. Now, since ρ r 1 : W r → J 3 π is a submersion, for every vector field X ∈ X(J 3 π) there exist some vector fields Y ∈ X(W r ) such that X and Y are ρ r 1 -related. Observe that this vector field Y is not unique because the vector field Y + Y o , with Y o ∈ ker Tρ r 1 is also ρ r 1 -related with X. Thus, using this particular choice of ρ r 1 -related vector fields, we have Since the equality ψ * i(Y )Ω r = 0 holds for every Y ∈ X(W r ), it holds, in particular, for every Y ∈ X(W r ) which is ρ r 1 -related with X ∈ X(J 3 π). Hence we obtain The following diagram illustrates the situation of the above Proposition: Observe that Proposition 7 states that every section solution to the field equations in the unified formalism projects to a section solution to the field equations in the Lagrangian formalism, but it does not establish an equivalence between the solutions. This equivalence does exist, due to the fact that the map ρ L 1 : W L → J 3 π is a diffeomorphism. In order to establish this equivalence, we first need the following technical result.

Lemma 3
The Poincaré-Cartan forms satisfy (ρ L 1 ) * Θ L = j * L Θ r and (ρ L 1 ) * Ω L = j * L Ω r (Proof ) Since the exterior derivative and the pull-back commute, it suffices to prove the statement for the m-forms. We have Now we can state the remaining part of the equivalence between the solutions of the Lagrangian and unified formalisms.
is an embedding, for every vector field X ∈ X(W r ) tangent to W L , there exists a unique vector field Y ∈ X(W L ) which is j L -related with X. Hence, let us assume that X ∈ X(W r ) is tangent to W L . Then we have Applying Lemma 3 we obtain where Z ∈ X(J 3 π) is the unique vector field related with Y by the diffeomorphism ρ L 1 . Hence, as ψ * L i(Z)ΩL = 0, for every Z ∈ X(J 3 π) by hypothesis, we have proved that the section However, from Proposition 4 we know that if ψ ∈ Γ(ρ r M ) is a holonomic section, then the last equation is equivalent to the equation (17), that is, ψ * i(X)Ωr = 0 , for every X ∈ X(W r ) .
Let us compute the local equation for the section ψ L = ρ r 1 • ψ ∈ Γ(π 3 ). Assume that the section ψ ∈ Γ(ρ r M ) is given locally by ψ(x i ) = (x i , u α , u α i , u α I , u α J , p i α , p I α ). Since ψ is a holonomic section solution to equation (17), it must satisfy the local equations (18), (19) and (21). The equations (21) are automatically satisfied as a consequence of the assumption of ψ being holonomic. Now, taking into account that ψ takes values in the submanifold W L ∼ = graph(FL), the equations (18) and (19) can be ρ r 1 -projected to J 3 π, thus giving the following system of n partial differential equations for the component functions of the section ψ L = ρ r where the section ψ L is locally given by ψ L (x i ) = (x i , u α , u α i , u α I , u α J ). Finally, since ψ L is holonomic in J 3 π, there exists a section φ ∈ Γ(π) with local expression φ(x i ) = (x i , u α (x i )) satisfying j 3 φ = ψ L . Then, the above equations can be rewritten as follows Therefore, we obtain the Euler-Lagrange equations for a second-order field theory.

Field equations for multivector fields
Lemma 4 Let X ∈ X m (W r ) be a multivector field tangent to W L ֒→ W r . Then there exists a unique multivector field X L ∈ X m (J 3 π) such that X L • ρ r Conversely, if X L ∈ X m (J 3 π), then there exists a unique multivector field X ∈ X(W r ) tangent to W L such that (Proof ) Since the multivector field X is tangent to W L , there exists a unique multivector field X o ∈ X m (W L ) which is j L -related to X , that is, Λ m Tj L • X o = X • j L . Furthermore, since ρ L 1 : W L → J 3 π is a diffeomorphism, there is a unique multivector field X L ∈ X m (J 3 π) which is ρ L 1 -related to X o ; that is, The converse is proved reversing this reasoning.
The above result states that there is a 1-to-1 correspondence between the set of multivector fields X ∈ X m (W r ) tangent to W L and the set of multivector fields X L ∈ X m (J 3 π), which makes the following diagram commutative As a consequence, we obtain the following result: Theorem 2 Let X ∈ X m (W r ) be a locally decomposable holonomic multivector field solution to the equation (27) (at least on the points of a submanifold W f ֒→ W L ) and tangent to W L (resp. tangent to W f ). Then there exists a unique locally decomposable holonomic multivector field X L ∈ X m (J 3 π) solution to the equation (at least on the points of S f = ρ L 1 (W f ), and tangent to S f ).
Conversely, if X L ∈ X m (J 3 π) is a locally decomposable holonomic multivector field solution to the equation (38) (at least on the points of a submanifold S f ֒→ J 3 π, and tangent to S f ), then there exists a unique locally decomposable holonomic multivector field X ∈ X m (W r ) which is a solution to the equation (27) (at least on the points of (ρ L 1 ) −1 (S f ) ֒→ W L ), and tangent to W L (resp. tangent to W f ).
(Proof ) Applying Lemmas 2 and 4, we have Hence, X L is a solution to the equation i(XL)ΩL = 0 if, and only if, X is a solution to the equation i(X )Ω r = 0. Now we must prove that X L is holonomic if, and only if, X is holonomic. Observe that, following the same reasoning as above, we have Hence, X L isπ 3 -transverse if, and only if, X is ρ r M -transverse. Now, let us assume that X ∈ X m (W r ) is holonomic, and let ψ ∈ Γ(ρ r M ) be an integral section of X . Then, the section ψ L = ρ r 1 • ψ ∈ Γ(π 3 ) is holonomic by definition, and we have where ψ ′ : M → Λ m TW r is the canonical lifting of ψ to Λ m TW r . That is, ψ L is an integral section of X L . Hence, if X is holonomic, then X L is holonomic.
For the converse, let us assume that X L ∈ X m (J 3 π) is holonomic, and let ψ L ∈ Γ(π 3 ) be an integral section of X L . Then, the section Therefore, the section ψ = j L • (ρ L 1 ) −1 • ψ L is holonomic. Finally, since the multivector field X is tangent to W L , there exists a unique multivector field X o ∈ X m (W L ) satisfying Λ m Tj L • X o = X • j L . In addition, since the map ρ L 1 is a diffeomorphism, X L and X o are (ρ L 1 ) −1 -related; that is, Hence, ψ is an integral section of X . Therefore, X is holonomic if, and only if, X L is holonomic.
Let X L ∈ X m (J 3 π) be a locally decomposable multivector field. From the results in [22] we know that X L admits the following local expression (39) Taking f = 1 as a representative of the equivalence class, since X L is required to be holonomic, it must satisfy the equations (4) with k = 3 and r = 1, that is, In addition, X L is a solution to the equation (38). Bearing in mind the local equations for the multivector field X , we obtain that the local equations for the component functions of Theorem 3 The following assertions on a section φ ∈ Γ(π) are equivalent: 1. j 3 φ is a solution to equation (36), that is,

In natural coordinates, if φ is given by
is a solution to the second-order Euler-Lagrange equations given by (37), that is, 3. ψ L = j 3 φ is a solution to the equation is the canonical lifting of ψ.
4. j 3 φ is an integral section of a multivector field belonging to a class of locally decomposable holonomic multivector fields {X L } ⊂ X m (J 3 π) satisfying equation (38), that is, i(XL)ΩL = 0 .

General setting
In order to the describe the Hamiltonian formalism for second-order field theories using the results obtained in Section 3, we must distinguish between the regular and non-regular cases.
Let FL : J 3 π → J 2 π † be the extended Legendre map obtained in (24) and FL : J 3 π → J 2 π ‡ the restricted Legendre map obtained in (23). We denote P = Im( FL) = FL(J 3 π) ֒→ J 2 π † and P = Im(FL) = FL(J 3 π)  ֒→ J 2 π ‡ the image of the extended and restricted Legendre maps, respectively, which we assume to be submanifolds. We denoteπ P : P → M the natural projection, and FL o the map defined by FL =  • FL o .
Remark: In the hyperregular case, we have P = J 2 π ‡ and FL o = FL.
With the previous notations, we can give the following definition: Definition 11 A Lagrangian density L ∈ Ω m (J 2 π) is said to be almost-regular if 1. P is a closed submanifold of J 2 π ‡ .
2. FL is a submersion onto its image.
If the Lagrangian density is almost-regular, the Legendre map is a submersion onto its image, and therefore it admits local sections defined on the submanifold P ֒→ J 2 π ‡ . We denote by Γ P (FL) the set of local sections of FL defined on the submanifold P. Observe that if L is regular, then Γ P (FL) is exactly the set of local sections of FL.
As a consequence of Proposition 3, we have that P is diffeomorphic to P. This diffeomorphism is µ = µ • : P → P. This enables us to state (Proof ) It is clear that, given [ω] ∈ J 2 π ‡ , the sectionĥ maps every point (j 3 Thus, the crucial point is the ρ 2 -projectability of the local functionĤ. However, since a local base for ker Tρ 2 is given by with |I| = 2 and |J| = 3, then we have thatĤ is ρ 2 -projectable if, and only if, This condition is fulfilled when [ω] ∈ P = Im(FL), which implies that ρ 2 [ĥ((ρ r 2 ) −1 ([ω]))] ∈ P.
As in the unified setting, this Hamiltonian µ-section is specified by a local Hamiltonian function H ∈ C ∞ (P), that is, .
Remark: The Hamiltonian µ-section can be defined in some equivalent ways without passing through the unified formalism. First, we can define it as h = • µ −1 . From this, bearing in mind the definition of P and P as the image sets of the extended and restricted Legendre maps, respectively, we can also define the Hamiltonian µ-section as h = FL • σ, where σ ∈ Γ P (FL).

Hyperregular and regular Lagrangian densities
For the sake of simplicity, we assume throughout this Section that the Lagrangian density L ∈ Ω m (J 2 π) is hyperregular, and that Υ : J 2 π ‡ → J 3 π is a global section of FL. All the results stated also hold for regular Lagrangians, restricting to the corresponding open sets where the Legendre map admits local sections.
First, observe that if the Lagrangian density is hyperregular, then the local Hamiltonian function associated to the Hamiltonian µ-section h has the following coordinate expression where f α I (x i , u α , u α i , p i α , p I α ) = Υ * u α I . Therefore, the Hamilton-Cartan m and (m + 1)-forms have the following coordinate expression In addition, since Im(FL) = J 2 π ‡ , then the Hamiltonian sections h andĥ satisfy h • ρ r 2 = ρ 2 •ĥ, that is, the following diagram commutes Proposition 9 If the Lagrangian density is hyperregular, then the Hamilton-Cartan (m + 1)- (Proof ) A direct computation in coordinates leads to this result. Let Υ ∈ Γ(FL) be a global section of the restricted Legendre map, and assume that the local Hamiltonian function H is given locally by (40). Then we have the following local expression for dH dH = u α i dp i α + p i α du α i + f α I dp I α + p I α df α I − Observe that since H takes values in J 2 π ‡ = Im(FL), we have p I α − ∂L/∂u α I = 0. Thus, the expression of dH reads and therefore the Hamilton-Cartan (m + 1)-form is locally given by Now, since the C ∞ (J 2 π ‡ )-module of vector fields X(J 2 π ‡ ) is locally given by From this it is clear that i(X)Ωh = 0 if, and only if, X = 0, that is, Ω h is multisymplectic. Now we recover the field equations from the unified setting using the natural projection ρ r 2 : W r → J 2 π ‡ . First, the sections solution in the Hamiltonian formalism are recovered using the following result: Proposition 10 Let ψ ∈ Γ(ρ r M ) be a holonomic section solution to the equation (17). Then the section ψ h = ρ r 2 • ψ ∈ Γ(π ‡ J 1 π ) is a solution to the equation (Proof ) Since ρ r 2 : W r → J 3 π is a submersion, for every vector field X ∈ X(J 2 π ‡ ) there exist some vector fields Y ∈ X(W r ) such that X and Y are ρ r 2 -related. Observe that this vector field Y is not unique, the vector field Y + Y o , with Y o ∈ ker Tρ r 2 is also ρ r 2 -related with X. Thus, using this particular choice of ρ r 2 -related vector fields, we have Since the equality ψ * i(Y )Ω r = 0 holds for every Y ∈ X(W r ), in particular it holds for every Y ∈ X(W r ) which is ρ r 2 -related with X ∈ X(J 2 π ‡ ). Hence we obtain The diagram illustrating the situation of the above Proposition is the following: Let us compute the local equations for the section ψ h = ρ r 2 • ψ ∈ Γ(π ‡ J 1 π ). If the section ψ ∈ Γ(ρ r M ) is locally given by ψ( . Now, bearing in mind that the section ψ solution to the equation (17) must satisfy the local equations (18), (19) and (21), and that the section ψ takes values in the submanifold W L ∼ = graph(FL) and the local expression (40) of the Hamiltonian function H in the hyperregular case, we obtain the following system of partial differential equations for the section ψ h In order to recover the field equations for multivector fields, we first need the following technical result, which is similar to Lemma 4.

Lemma 6
Let X ∈ X m (W r ) be a multivector field tangent to W L ֒→ W r . Then there exists a unique multivector field X h ∈ X m (J 2 π ‡ ) such that Conversely, if X h ∈ X m (J 2 π ‡ ), then there exist multivector fields X ∈ X(W r ) tangent to W L such that X h • ρ r 2 • j L = Λ m Tρ r 2 • X • j L .
(Proof ) The proof of this result is analogous to the proof of Lemma 4, bearing in mind that As in the Lagrangian formalism, the previous result gives a correspondence between the set of multivector fields X ∈ X m (W r ) tangent to W L and the set of multivector fields X h ∈ X m (J 2 π ‡ ) such that the following diagram is commutative Nevertheless, observe that in the Hamiltonian formalism, the map ρ L 2 = ρ r 2 • j L : W L → J 2 π ‡ is a submersion (instead of a diffeomorphism, as in the Lagrangian setting), and thus the correspondence is not 1-to-1.
Theorem 4 Let X ∈ X m (W r ) be a locally decomposable, integrable and ρ r M -transverse multivector field solution to the equation (27) and tangent to W L . Then there exists a locally decomposable, integrable and (π ‡ J 1 π )-transverse multivector field X h ∈ X m (J 2 π ‡ ) solution to the equation Conversely, if X h ∈ X m (J 2 π ‡ ) is a locally decomposable, integrable and (π ‡ J 1 π )-transverse multivector field solution to the equation (43), then there exist locally decomposable, integrable and ρ r M -transverse multivector fields X ∈ X m (W r ) tangent to W L solution to the equation (27).
(Proof ) The proof of this result is analogous to the proof of Theorem 2.
Let X h ∈ X m (J 2 π ‡ ) be a locally decomposable multivector field given in the natural coordinates of J 2 π ‡ by Taking f = 1 as a representative of the equivalence class, since X h is a solution to the equation (43), we obtain that the local equations for the component functions of X h are The following assertions on a section ψ h ∈ Γ(π ‡ J 1 π ) are equivalent: 1. ψ h is a solution to equation (41), that is, 2. In natural coordinates, if ψ h is given by ψ h (x i ) = (x i , u α , u α i , p i α , p I α ), then its component functions are a solution to the equations (42), that is, is the canonical lifting of ψ.

Singular (almost-regular) Lagrangian densities
For singular (almost-regular) Lagrangian densities, only in the most favourable cases does there exists a submanifold W f ֒→ W L where the field equations can be solved. In this situation, the solutions in the Hamiltonian formalism cannot be obtained directly from the projection of the solutions in the unified setting, but rather by passing through the Lagrangian formalism and using the Legendre map. Recall that, in this case, the phase space of the system is P = Im(FL) ֒→ J 2 π ‡ .
Proposition 11 Let L ∈ Ω m (J 2 π) be an almost-regular Lagrangian density. Let ψ ∈ Γ(ρ r M ) be a solution to the equation (17). Then, the section is a solution to the equation ψ * h i(X)Ωh = 0 , for every X ∈ X(P) .
(Proof ) Since the Lagrangian density L is assumed to be almost-regular, then the map FL o is a submersion onto its image, P. Thus, for every vector field X ∈ X(P) there exist some vector fields Y ∈ X(J 3 π) such that X and Y are FL o -related. Using this particular choice of FL o -related vector fields, we have Then, using Proposition 7, we have proved since the last equality holds for every Y ∈ X(J 3 π) and, in particular, for every vector field FL o -related to a vector field in P.
The diagram for this situation is the following Now, assume that there exists a submanifold W f ֒→ W L and a multivector field X ∈ X m (W r ), defined at support on W f and tangent to W f , which is a solution to the equation (33). Now consider the submanifolds S f = ρ L 1 (W f ) ֒→ J 3 π and P f = FL(S f ) ֒→ P ֒→ J 2 π ‡ . Using Theorem 2, from the holonomic multivector field X ∈ X m (W r ) we obtain the corresponding holonomic multivector fields X L ∈ X m (J 3 π) solution to the equation (38) at support on S f . From this, one can prove that there are multivector fields in S f (perhaps only on the points of another submanifold), which are FL-projectable to P f . So we have the diagram Moreover, we can state the following result, which is the analogous theorem to Theorem 4 in the case of almost-regular Lagrangian densities.
Theorem 6 Let X ∈ X m (W r ) be a locally decomposable, integrable and ρ r M -transverse multivector field, defined at support on W f and tangent to W f , which is a solution to the equation (33). Then there exists a locally decomposable, integrable and (π ‡ J 1 π )-transverse multivector field X h ∈ X m (P), defined at support on P f and tangent to P f , which is a solution to the equation Conversely, if X h ∈ X m (P) is a locally decomposable, integrable and (π ‡ J 1 π )-transverse multivector field defined at support on P f and tangent to P f which is a solution to the equation (46), then there exist locally decomposable, integrable and ρ r M -transverse multivector fields X ∈ X m (W r ), defined at support on W f and tangent to W f , which are solutions to the equation (33).

Loaded and clamped plate
Let us consider a plate with clamped edges. We wish to determine the bending (or deflection) perpendicular to the plane of the plate under the action of an external force given by a uniform load. This system has been studied using a previous version of the unified formalism in [9], and can be modeled as a second-order field theory, taking M = R 2 as the base manifold (the plate) and the "vertical" bending as a fiber bundle E = R 2 × R π −→ R 2 (that is, the fibers are 1-dimensional).
We consider in M = R 2 the canonical coordinates (x, y) of the Euclidean plane, and in E = R 3 we take the global coordinates (x, y, u) adapted to the bundle structure. Recall that R 2 admits a canonical volume form η = dx ∧ dy ∈ Ω 2 (R 2 ).
Lagrangian-Hamiltonian formalism. Following Section 3.1, consider the fiber bundles with the natural coordinates introduced in the aforementioned Section. Observe that, in this example, we have dim J 3 π = 12 and dim J 2 π ‡ = 10, and therefore dim W = 18 and dim W r = 17.
Let ψ ∈ Γ(ρ r M ) be a section. Then the field equation (17) gives in coordinates the following system of equations Combining the second and third group of equations, we obtain the constraints defining the submanifold W L , and hence the Legendre map associated to this Lagrangian density, which is the fiber bundle map FL : J 3 π → J 2 π ‡ given locally by Observe that the tangent map of FL at every point j 3 φ ∈ J 3 π is given in coordinates by the 10 × 12 real matrix From this it is clear that rank(FL(j 3 φ)) = 10 = dim J 2 π ‡ . Hence, the restricted Legendre map is a submersion onto J 2 π ‡ , and thus the Lagrangian density L ∈ Ω 2 (J 2 π) is regular.
Finally, combining the first three groups of equations, we obtain the second-order Euler-Lagrange equation This is the classical equation for the bending of a clamped plate under a uniform load. Now, let X ∈ X 2 (W r ) be a locally decomposable bivector field given locally by (28). Then the equation (27) gives in coordinates the following system of equations Moreover, if we assume that X is holonomic, then we have the following additional equations F 1,2 = u (1,1) ; F 2,1 = u (1,1) ; F (2,0),1 = u (3,0) ; F (2,0),2 = u (2,1) F (1,1),1 = u (2,1) ; F (1,1),2 = u (1,2) ; F (0,2),1 = u (1,2) ; F (0,2),2 = u (0, 3) From the field equations, we deduce that the first constraint submanifold W c ֒→ W r is given in coordinates by the local constraints The tangency condition for the multivector field X along W c enables us to determine all the coefficients G I i , with i = 1, 2 and |I| = 2, in the following way Then, using the previous field equations, we obtain the following additional constraints which define a new submanifold W L ֒→ W r . Analyzing the tangency of the multivector field X along this new submanifold W L , we obtain the following equations Using again the field equations, we obtain the second-order Euler-Lagrange equation for a multivector field, which is F (3,0),1 + F (1,2),1 + F (2,1),2 + F (0,3),2 = q .
Observe that if ψ ∈ Γ(ρ r M ) is an integral section of X , then its component functions must satisfy the second-order Euler-Lagrange equation previously obtained for sections.

Korteweg-de Vries equation
In the following we derive the Korteweg-de Vries equation, usually denoted as the KdV equation for short, using the geometric formalism introduced in this paper. The KdV equation is a mathematical model of waves on shallow water surfaces, and has become the prototypical example of a non-linear partial differential equation whose solutions can be specified exactly. Many papers are devoted to analyzing this model and, in particular, some previous multisymplectic descriptions of it are available for instance [5,29,49]. A further analysis using a different version of the unified formalism is given in [48].
The usual form of the KdV equation is that is, a non-linear, dispersive partial differential equation for a real function y depending on two real variables, the space x and the time t. It is known that the KdV equation can be derived from a least action principle as the Euler-Lagrange equation of the Lagrangian density where y = ∂u/∂x. It is therefore clear that we can use our formulation to derive the Korteweg-de Vries equation as the field equations of a second-order field theory with a 2-dimensional base manifold and a 1-dimensional fiber over this base.
Hence, let us consider M = R 2 with global coordinates (x, t), and E = R 2 × R with natural coordinates adapted to the bundle structure, (x, t, u). In these coordinates, the canonical volume form in R 2 is given by η = dx ∧ dt ∈ Ω 2 (R 2 ).
The Hamiltonian µ W -sectionĥ ∈ Γ(µ W ) is specified by the local Hamiltonian function and the Hamilton-Cartan forms have the same expressions as in the previous example, replacing the local Hamiltonian function.
Let ψ ∈ Γ(ρ r M ) be a section. Then the field equation (17) gives in coordinates the following system of equations From these local equations, we obtain the coordinate expression of the Legendre map FL : J 3 π → J 2 π ‡ , which is The tangent map of FL at every point j 3 φ ∈ J 3 π is given in coordinates by From this it is clear that rank(FL(j 3 φ)) = 7 < 10 = dim J 2 π ‡ . Hence, the Lagrangian density L ∈ Ω 2 (J 2 π) is singular.
Finally, by combining the first three groups of equations, we obtain the second-order Euler-Lagrange equation for this field theory which, taking y = ∂u/∂x, is the usual Korteweg-de Vries equation.
The natural coordinates (x, t, u, u 1 , u 2 , p 1 , p 2 , p (2,0) , p (1,1) , p (0,2) ) in J 2 π ‡ induce coordinates (x, t, u, u 1 , u 2 , p 1 , p (2,0) ) in P, with the natural embedding  : P ֒→ J 2 π ‡ given locally by In these coordinates, the local Hamiltonian function that specifies the Hamiltonian section h is given by Therefore, the Hamilton-Cartan 2-form Θ h = h * Θ s 1 ∈ Ω 2 (P) is given locally by Now we recover the Hamiltonian field equations. If ψ ∈ Γ(ρ r M ) is a (holonomic) section solution to the field equation (17), then the section ψ h = FL • ρ r 1 • ψ ∈ Γ(π P ) is a solution to the equation (45). In coordinates, the component functions of ψ h must satisfy the following system of partial differential equations ∂u ∂x = u 1 ; 1 2 ∂u ∂t = p 1 + 3u 2 1 ; Finally, if X ∈ X 2 (W r ) is a locally decomposable 2-vector field solution to the equation (27), then there exists a locally decomposable 2-vector field X h ∈ X 2 (P) solution to the equation (46). If X h is locally given by

Conclusions and further research
We develop a new multisymplectic framework for describing higher-order field theories, and, in particular, second-order ones which are the most relevant in physics (to the best of our knowledge, the most interesting higher-order models and theories in physics are second-order). This model is based on the extension of the so-called Skinner-Rusk unified formalism from mechanical systems to higher-order field theory, and thereby complements previous papers such as [9,48], in which analogous but different formulations are given.
The key points of the formalism are as follows: • The Skinner-Rusk formalism is a special case of what (in the modern terminology) is called a Dirac structure. It unifies in a single frame the Lagrangian and Hamiltonian formalisms, and hence gives a unified version of the Euler-Lagrange and the Hamilton equations.
In our case, the 4th-order Euler-Lagrange equations and the Hamilton-De Donder-Weil equations for field theories described by 2nd-order Lagrangian densities are stated in a combined form using both sections and multivector fields in a suitable fiber bundle over the configuration bundle of the theory, E π −→ M . This bundle is the restricted 2-symmetric jetmultimomentum bundle W r = J 3 π× J 1 π J 2 π ‡ , which is a quotient bundle of the extended 2symmetric jet-multimomentum bundle W = J 3 π × J 1 π J 2 π † , where J 2 π † is the 2-symmetric multimomentum bundle introduced in [46], and J 2 π ‡ = J 2 π † /Λ m 1 (J 1 π). The use of this bundle is the crucial point for univocally defining a Legendre map, and therefore the Poincaré-Cartan forms.
As usual, the physical information of the theory is given by a Lagrangian density, although the geometry is provided by the canonical multisymplectic form Ω 1 with which the 2symmetric multimomentum bundle is endowed. This enables us to construct the form Ω r which induces the geometry of W r . Thus, in the unified formalism the geometry and the physical information are separated.
• As is characteristic in the unified formalism, independently of the regularity of the Lagrangian density, Ω r is a premultisymplectic form in W r . Hence, the compatibility condition for the field equations and the subsequent tangency or consistent condition for their solutions allows us to determine univocally the Legendre map, thanks to the symmetry relation introduced in the highest-order multimomenta coordinates. This relation equals the number of highest-order multimomenta with the number of highest-order "velocities" in the Lagrangian density, and therefore enables us to establish a 1-to-1 correspondence between these two sets of coordinates, giving rise to the highest-order equations defining the Legendre map. If the Lagrangian is regular (in the sense given in Definition 9), then the constraint algorithm stops at the first level; otherwise it continues in the usual way.
Furthermore, as stated above, from the form Ω r we also recover the Poincaré-Cartan form of the Lagrangian formalism in an unambiguous way. Hence, the Lagrangian formalism for second-order field theories is stated straightforwardly for the regular and singular (almostregular) cases. In the same way, we can obtain the associated Hamiltonian formalism in both cases using the unambiguously defined Legendre map, and eventually a Hamiltonian section associated to the Lagrangian function.
• Despite what occurs in higher-order mechanics, the condition for the solutions to the field equations to be holonomic is not guaranteed (even in the regular case), and neither can it be obtained from the constraint algorithm. In higher-order field theory, this condition constitutes an additional requirement of the theory.
• The unified formalism developed in [9] is different from ours, since it uses J 2 π× J 1 π Λ m 2 (J 1 π) as the extended jet-multimomentum bundle, and, as pointed out in the introduction, some parameters appearing in the higher-order field equations (which are written in terms of sections and Ehresmann connections), and in the definition of the Legendre map remain undetermined and must be fixed "ad-hoc". This does not occur in our formalism; in fact, the constraint algorithm plays a crucial role in the determination of all these arbitrary parameters.
Furthermore, our formalism is different from the unified formalism developed in [48], where infinite-order jet bundles are used.
• In addition to analyzing the example of the loaded and clamped plate, we use this unified framework to give a multisymplectic description of the KdV equation, which is also different from the standard ones existing in the literature.
As further research, we intend to study the variational principles of second-order field theories from this perspective.
In the main, we wish to apply this formalism to provide a multisymplectic description of the Hilbert-Einstein theory of gravitation and other classical theories in theoretical physics. We believe that this formalism will be useful for studying new reduction procedures of the corresponding field equations, for example, or for developing new numerical techniques of integration of these equations using multisymplectic integrators.
Let M be a n-dimensional differentiable manifold. Sections of Λ m (TM) are called mmultivector fields in M (they are the contravariant skew-symmetric tensors of order m in M). We denote the set of m-multivector fields in M by X m (M).
If Y ∈ X m (M), for every p ∈ M, there exists an open neighbourhood U p ⊂ M and Y 1 , . . . , Y r ∈ X(U p ) such that Y| Up = 1≤i 1 <...<im≤r and m ≤ r ≤ dim M. Then, Y ∈ X m (M) is said to be locally decomposable if, for every p ∈ M, there exists an open neighbourhood U p ⊂ M and Y 1 , . . . , Y m ∈ X(U p ) such that A non-vanishing m-multivector field Y ∈ X m (M) and a m-dimensional distribution D ⊂ TM are locally associated if there exists a connected open set U ⊆ M such that Y| U is a section of Λ m D| U . If Y, Y ′ ∈ X m (M) are non-vanishing multivector fields locally associated with the same distribution D, on the same connected open set U , then there exists a non-vanishing function f ∈ C ∞ (U ) such that Y ′ | U = f Y| U . This fact defines an equivalence relation in the set of nonvanishing m-multivector fields in M, whose equivalence classes will be denoted by {Y} U . Then there is a one-to-one correspondence between the set of m-dimensional orientable distributions D in TM and the set of the equivalence classes {Y} M of non-vanishing, locally decomposable m-multivector fields in M.
If Y ∈ X m (M) is non-vanishing and locally decomposable, and U ⊆ M is a connected open set, the distribution associated with the class {Y} U is denoted by D U (Y). If U = M we write D(Y).
A non-vanishing, locally decomposable multivector field Y ∈ X m (M) is said to be integrable (resp. involutive) if its associated distribution D U (Y) is integrable (resp. involutive). Of course, if Y ∈ X m (M) is integrable (resp. involutive), then so is every other in its equivalence class {Y}, and all of them have the same integral manifolds. Moreover, Frobenius theorem allows us to state that a non-vanishing and locally decomposable multivector field is integrable if, and only if, it is involutive. Nevertheless, in many applications we have locally decomposable multivector fields Y ∈ X m (M) which are not integrable in M, but integrable in a submanifold of M. A (local) algorithm for finding this submanifold has been developed [22].
The particular situation in which we are interested is the study of multivector fields in fiber bundles. If π : M → M is a fiber bundle, we will be interested in the case where the integral manifolds of integrable multivector fields in M are sections of π. Thus, Y ∈ X m (M) is said to be π-transverse if, at every point y ∈ M, (i(Y)(π * β)) y = 0, for every β ∈ Ω m (M ) with β(π(y)) = 0. Then, if Y ∈ X m (M) is integrable, it is π-transverse if, and only if, its integral manifolds are local sections of π : M → M . In this case, if φ : U ⊂ M → M is a local section with φ(x) = y and φ(U ) is the integral manifold of Y through y, then T y (Im φ) = D y (Y).