A new canonical affine BRACKET formulation of Hamiltonian Classical Field theories of first order

It has been a long standing question how to extend the canonical Poisson bracket formulation from classical mechanics to classical field theories, in a completely general, intrinsic, and canonical way. In this paper, we provide an answer to this question by presenting a new completely canonical bracket formulation of Hamiltonian Classical Field Theories of first order on an arbitrary configuration bundle. It is obtained via the construction of the appropriate field-theoretic analogues of the Hamiltonian vector field and of the space of observables, via the introduction of a suitable canonical Lie algebra structure on the space of currents (the observables in field theories). This Lie algebra structure is shown to have a representation on the affine space of Hamiltonian sections, which yields an affine analogue to the Jacobi identity for our bracket. The construction is analogous to the canonical Poisson formulation of Hamiltonian systems although the nature of our formulation is linear-affine and not bilinear as the standard Poisson bracket. This is consistent with the fact that the space of currents and Hamiltonian sections are respectively, linear and affine. Our setting is illustrated with some examples including Continuum Mechanics and Yang-Mills theory.


Introduction
It is well-known that the Hamilton equations of classical mechanics are naturally formulated in terms of canonical geometric structures, namely, canonical symplectic forms and canonical Poisson brackets. Given the configuration manifold Q of the mechanical system, its momentum phase space T * Q carries a canonical symplectic form which induces a vector bundle isomorphism ♯ : T * (T * Q) → T (T * Q). Given the Hamiltonian H ∈ C ∞ (T * Q) of the system, the associated Hamilton equations of motion are determined by the Hamiltonian vector field X H intrinsically defined in terms of the canonical symplectic form as X H = ♯dH, i.e., we have the commutative diagram The canonical Poisson formulation (1.2) of Hamilton's equation has been at the origin of many developments in the understanding of the geometry and dynamics of Hamiltonian systems and their quantization, as well as of many generalizations. In particular, the Poisson formulation is very appropriate for developing the geometric description of the Poisson reduction of a Hamiltonian system which is invariant under the action of a symmetry Lie group (see, for instance, [49]).
When extending the geometric setting from classical mechanics to classical field theories, it is crucial to identify the geometric structures playing the role of these canonical structures. While the field-theoretic analogue of the canonical symplectic structure is well-known to be given by the canonical multisymplectic form on a space of finite dimension, the extended momentum phase space, it has been a long standing question how to extend the canonical Poisson bracket formulation from classical mechanics to classical field theories, in a completely general, intrinsic, and canonical way. Several contributions have been made in this direction as we will review below. In this paper will shall provide an answer to this question by constructing explicitly such a canonical bracket, giving its algebraic properties, and showing that it allows a canonical formulation of the Hamilton equations for field theories that naturally extends the formulation (1.2) of classical mechanics. Such canonical bracket structures could be used as a starting point of a covariant canonical quantization.
One key difference between the canonical bracket that we propose and the canonical Poisson bracket of classical mechanics is its linear-affine nature. This is compatible with the fact that the set of currents and Hamiltonian sections are, respectively, linear and affine spaces for field theories. This linear-affine nature already arises in the case of time-dependent Hamiltonian Mechanics.
While in the present paper we focus on the extension to field theory of the canonical geometric setting (1.2)-(1.4) governing the evolution equations, which is of finite dimensional nature, one may also focus on the geometric structures related to the space of solutions of these equations. For a large class of field theories, this infinite dimensional space admits a Poisson (or a presymplectic) structure. This is not a new theory and we will quote an old paper by Peierls [50] (for a large list of contributions, see the recent papers [13,14] and the references therein). In this setting, the boundary conditions of the theory play an important role [44]. In fact, in [44], Margalef-Bentabol and Villaseñor introduce the so-called "relative bicomplex framework" and develop a geometric formulation of the covariant phase space methods with boundaries, which is used to endow the space of solutions with (pre)symplectic structures. These ideas are used to discuss formulations of Palatini gravity, General Relativity and Holst theories in the presence of boundaries [4,5,6]. On the other hand, a classic research topic has been the relationship between finite and infinite dimensional approximation to Classical Field Theories (see Section 6).
Before explaining the main difficulties that emerge in the process of finding a canonical bracket formulation that extends (1.2)-(1.4) to classical field theories, we quickly review below the previous contributions to the geometric formulation of Hamiltonian Classical Field Theories.
1.1. Previous contributions on the geometric formulation of Hamiltonian classical field theories of first order. The geometric description starts with the choice of the configuration bundle π : E → M , whose sections are the fields of the theory. The case of time-dependent mechanics corresponds to the special situation E = Q × R → R. For classical field theories, there are two useful generalisations of the notion of momentum phase space: the restricted multimomentum bundle M 0 π and the extended multi-momentum bundle Mπ. Both are vector bundles on E with vector bundle projections denoted ν 0 : M 0 π → E and ν : Mπ → E, and there is a canonical line bundle projection µ : Mπ → M 0 π. The main property of the extended multimomentum bundle is that it admits a canonical multisymplectic structure ω Mπ of degree m + 1 (m being the dimension of M ). The restricted multi-momentum bundle M 0 π, however, does not admit a canonical multisymplectic structure.
An important difference with Hamiltonian Mechanics is that for Hamiltonian Classical Field Theories we don't have a Hamiltonian function, but a Hamiltonian section h : M 0 π → Mπ of the canonical projection µ : Mπ → M 0 π, see [11]. The corresponding evolution equations are the Hamilton-deDonder-Weyl equations. They form a system of partial differential equations of first order on M 0 π with space of parameters M and whose solutions are sections of the projection π • ν 0 : M 0 π → M . More precisely, if we denote by (1.5) These equations go back, at least, to work of Volterra [52,53]. In the literature, we can find the following geometric descriptions of the Hamilton-deDonder-Weyl equations (1.5) associated to h: • From h and the canonical multisymplectic structure one can produce a non-canonical multisymplectic structure ω h = h * (ω Mπ ) on M 0 π. Then, using ω h , a special type of Ehresmann connections can be introduced on the fibration π • ν 0 : M 0 π → M (which, in the present paper, will be called Hamiltonian connections for h) and the solutions of the evolution equations are the integral sections of these connections (see [19,21]; see also [25,26]). The multisymplectic structure ω h can also be used to directly characterize the sections of the projection π • ν 0 : M 0 π → M which are solutions of the Hamilton-deDonder-Weyl equations for h (see [11,19,21,25,26]). From ω h , one can also define the "multisymplectic pseudo-brackets" and "multisymplectic brackets" of (m−1)-Hamiltonian forms which may be considered the field version of the Poisson bracket for functions in Classical Mechanics (see [28]). • Using an auxiliary Ehresmann connection ∇ on the configuration bundle π : E → M and the Hamiltonian section h, one may produce a Hamiltonian energy H ∇ : M 0 π → R associated to h and ∇ and a non-canonical multisymplectic structure on M 0 π which allow us to describe the solutions of the evolution equations in a geometric form (see [11]; see also [25,26]). In addition, it is possible to consider a suitable space of currents (a vector subspace of m − 1-forms on M 0 π which are horizontal with respect to the projection π • ν 0 : M 0 π → M ) and one may introduce a "Poisson bracket" of a current and an Hamiltonian energy associated to ∇. This bracket allows the description of the Hamilton-deDonder-Weyl equations as in Hamiltonian Mechanics (see [12]). • The Hamiltonian section h induces a canonical extended Hamiltonian density F h , which is a smooth π * (Λ m T * M )-valued function defined on Mπ see [10,32]; see also [24] for the particular case when a volume form on M is fixed. Then, using F h and the canonical multisymplectic structure ω Mπ one write intrinsically a system of partial differential equations on Mπ whose solutions are sections of the fibration π • ν : Mπ → M . The projection, via µ, of these sections are the solutions of the Hamilton-deDonder-Weyl equations for h (see [24]). • From the configuration bundle π : E → M one can construct the phase bundle P(π), an affine bundle over M 0 π, and the differential dh : M 0 π → P(π) of the Hamiltonian section h, as a section of P(π). In addition, an affine bundle epimorphism A : J 1 (π • ν 0 ) → P(π) from the 1-jet bundle of the fibration π •ν 0 : M 0 π → M onto P(π) may be also introduced. Then, the solutions of the Hamilton-deDonder-Weyl equations are the sections s 0 : M → M 0 π whose first prolongation j 1 s 0 : M → J 1 (π • ν 0 ) is contained in the submanifold A −1 (dh(M 0 π)) (see [32,33]; see also [35,47] for the particular case of time-dependent Hamiltonian Mechanics).
1.2. The problem. The previous comments lead naturally to the following question: Does there exist a completely canonical geometric formulation of the Hamilton-deDonder-Weyl equations which is analogous to the standard Poisson bracket formulation of time-independent Hamiltonian Mechanics?
A possible answer to this question could be the geometric formulation developed in [12] (see Section 1.1). However, this formulation is not canonical since most of the constructions in [12] depend on the chosen auxiliary connection in the configuration bundle. In fact, in a previous paper [46] Marsden and Shkoller justify the use of this connection in the geometric formulation of the theory and one may find, in that paper (see [46], page 554), the following cite: It is interesting that the structure of connection is not necessary to intrinsically define the Lagrangian formalism (as shown in the preceding references), while for the intrinsic definition of a covariant Hamiltonian the introduction of such a structure is essential. Of course, one can avoid a connection if one is willing to confine ones attention to local coordinates.
However, in our paper, we will construct a bracket that does not use any auxiliary objects such as a connection in the configuration bundle and which is completely canonical, thereby giving an affirmative answer to the question above.
1.3. Answer to the problem and contributions of the paper. In order to give an affirmative answer to the question in Section 1.2, we will use the following previous contributions and results: • The construction of the phase space P(π) associated with the configuration bundle π : E → M and the differential of a Hamiltonian section h : M 0 π → Mπ as a section of P(π) (see [32]). • The affine bundle epimorphism A : J 1 (π • ν 0 ) → P(π) which was also introduced in [32].
• The notion of a Hamiltonian connection associated with a Hamiltonian section h. This type of objects were already considered in [19,21] in order to characterize the solutions of the Hamilton-deDonder-Weyl equations for h (although the authors of these papers did not use the terminology of a Hamiltonian connection).
We will combine the previous constructions as follows.
As a first step, we consider the affine bundle isomorphism where Ker A is the kernel of the affine bundle epimorphism A : J 1 (π • ν 0 ) → P(π) and ♯ aff =Â −1 , withÂ : J 1 (π•ν 0 )/ Ker A → P(π) the affine bundle isomorphism induced by A. Then, we introduce the section Γ h : M 0 π → J 1 (π • ν 0 )/ Ker A The next step is to introduce a suitable space of currents O (a vector subspace of (m − 1)-forms on M 0 π which are horizontal with respect to the fibration π • ν 0 : M 0 π → M ), in such a way that the restriction of the standard exterior differential to O takes values in the space of sections of the vector bundle ( The dual vector bundle (J 1 (π • ν 0 )/ Ker A) + is chosen so that these differentials can be canonically paired with the Hamiltonian connections Γ h , thereby extending to the field-theoretic context the pairing dF, X H between the differential dF of an observable and the Hamiltonian vector field X H , see (1.3). This is our motivation for introducing the space of currents O and although it is different to the motivation in [12], O just coincides with the space of currents in [12] (see Remark 3.14). Now, if Γ(µ) is the space of Hamiltonian sections, we can define the linear-affine canonical bracket Then, one may prove that the evolution of any current α 0 ∈ O along a solution s 0 : M → M 0 π of the Hamilton-deDonder-Weyl equations for h is given by Conversely, if s 0 : M → M 0 π is such that (1.7) holds for all α 0 ∈ O, then s 0 is a solution of Hamilton-deDonder-Weyl equations. The canonical bracket formulation (1.7) is the field-theoretic analogue to the canonical Poisson bracket formulation (1.2) of classical mechanics.
The previous tasks are performed in Section 3.4 (see Theorem 3.15). Here again, the analogy with the canonical Poisson formulation of classical mechanics (see (1.2) and (1.3)) is evident.
It is important to note the affine character of the canonical bracket {·, ·} in (1.6): the space Γ(µ) of Hamiltonian sections is an affine space modelled over the vector space Γ((π • ν 0 ) * (Λ m T * M )). Recalling that the canonical Poisson bracket on T * Q induces a Lie algebra structure on C ∞ (T * Q), a new question arises: What are the algebraic properties of the bracket {·, ·} : Related to this question, we will prove that O admits a canonical Lie algebra structure {·, ·} O (see Theorem 4.2) and that the linear map The previous results will be applied to the following examples: time-dependent Hamiltonian systems, Continuum Mechanics (including fluid dynamics and nonlinear elasticity) and Yang-Mills theories. Some of the constructions developed in the paper are illustrated in the Diagram in Appendix §C.
1.4. Structure of the paper. The paper is structured as follows. In Section 2, we review the geometric formulation of the Hamilton-deDonder-Weyl equations using the multisymplectic structure on the phase space induced by the Hamiltonian section. In Section 3, we introduce the canonical linear-affine bracket {·, ·} : O × Γ(µ) → Γ((π • ν 0 ) * (Λ m T * M )) and we formulate the Hamilton-deDonder-Weyl equations using this bracket. In particular, we describe the evolution of a current along a solution of the Hamilton-deDonder-Weyl equations. In Section 4, we introduce a Lie algebra structure on O and we prove that {·, ·} induces a representation of the Lie algebra O on the affine space Γ(µ) of Hamiltonian sections. In Section 5, we apply the previous results to several examples. The paper closes with three appendices. In the first one, we review the definition of the 1-jet bundle associated with a fibration, in the second one, we discuss the vertical lift of a section of a vector bundle as a vertical vector field on the total space and, in the third one, we present a Diagram which illustrates most of the relevant constructions in the paper.

Hamiltonian Classical Field Theories of first order
In this section, we review some basic constructions and results on Hamiltonian Classical Field Theories of first order (for more details, see [11]).
2.1. The restricted and extended multimomentum bundle associated with a fibration. The configuration bundle of a classical field theory is a fibration π : E → M , that is, a surjective submersion from E to M . We assume dim M = m and dim E = m + n.
The extended multimomentum bundle Mπ associated with the configuration bundle π : E → M is the vector bundle over E whose fiber at the point y ∈ E is y π is the 1-jet bundle of the fibration π : E → M (see Appendix A). It is well-known that Mπ may be identified with the vector bundle Λ m 2 (T * E) over E, whose If (x i , u α ) are local coordinates on E which are adapted with the fibration π, then γ ∈ Λ m 2 (T * y E) reads locally for γ ∈ Λ m 2 (T * E) and Y 1 , . . . , Y m ∈ T γ Λ m 2 (T * E), with ν : Mπ → E the vector bundle projection. From (2.2), λ Mπ has the local expression The canonical multisymplectic structure ω Mπ on Mπ is the (m + 1)-form given by 3) It is clear that ω Mπ is closed and non-degenerate, that is, the vector bundle morphism The restricted multimomentum bundle M 0 π is the vector bundle Local coordinates on M 0 π are (x i , u α , p i α ).
There is a canonical projection µ : Mπ → M 0 π given by Remark 2.1. Note that this last statement implies that M 0 π ≃ Mπ/ ∼. This situation recalls a particular case in the Poisson realm, the quotient of a symplectic manifold by a proper and free action of a symmetry Lie group inherits a Poisson structure. In this formalism, the quotient of the extended multimomentum bundle also has a new version of a multi-Poisson structure (see for example [8]) which is defined via a Lie algebroid structure on a subbundle of Λ m T * M 0 π, when the base manifold M is orientable. Note that, in such a case, if we fix a volume form on M , we have an action of the real line R on M 0 π, which preserves the multisymplectic structure, and Mπ is the space of orbits of this action. For the definition and details on the construction of the multi-Poisson structure, we refer to [9]. ⋄

2.2.
Hamilton-deDonder-Weyl equations. Given a configuration bundle π : E → M , a Hamiltonian section h : M 0 π → Mπ is a smooth section of the canonical projection The local expression of h is Using the Hamiltonian section, we can define the (m + 1)-form ω h on M 0 π given by (2.5) The local expression of ω h is Note that if m = 1, ω h is degenerate and the rank of its kernel is 1. On the other hand, if m ≥ 2, ω h is non-degenerate which implies that it is multisymplectic.
Proof. Using the local expression s 0 (

A new canonical bracket formulation of Hamiltonian Classical Field Theories of first order
As we reviewed in the Introduction, the phase space of momenta for classical mechanics is the cotangent bundle T * Q of the configuration space Q, a smooth manifold of dimension n. The cotangent bundle T * Q carries a canonical symplectic structure which induces a vector bundle isomorphism ♭ : T (T * Q) → T * (T * Q) over the identity with inverse denoted ♯ : T * (T * Q) → T (T * Q). The Hamiltonian is a real C ∞ -function on T * Q and the Hamiltonian vector field is given in terms of the differential dH and the vector bundle isomorphism ♯ as X H = ♯dH, see Diagram (1.1). For field theories, we don't have a Hamiltonian function, but a Hamiltonian section h : M 0 π → Mπ of the canonical projection µ : Mπ → M 0 π. So, the following questions arise when extending the previous construction to field theories: Question 1: What is the differential of h? Question 2: Where does the differential of h take values?
We will answer these questions in §3.1 by showing that the differential dh of h is a section of the phase bundle P(π) associated with the fibration µ : Mπ → M 0 π. The bundle P(π) was introduced in [32] and was used there to discuss a Tulczyjew triple for Classical Field Theories of first order. This will allow us to define the field-theoretic analogue to the vector bundle isomorphism ♯ in §3.2 and the field-theoretic analogue Γ h to the Hamiltonian vector field X H in §3. 3. In particular, we will show that Γ h can be identified with the equivalence class of Hamiltonian Ehresmann connections associated to h.
Going back to Classical Hamiltonian Mechanics, we recall that the set of observables is the space C ∞ (T * Q) and that the Hamilton equations can be equivalently formulated in the Poisson bracket form (1.2) with respect to the canonical Poisson bracket giving by the formulas (1.3). In view of this formulation, we need to find a suitable space of currents for field theories (the observables in field theories) such that their differentials take values in a bundle dual to the target bundle of Γ h , this is the goal of §3.4. From this a canonical bracket can be obtained between currents and Hamiltonian sections. This construction is carried out in §3.5.
3.1.1. The differential of F h and the extended phase bundle. Note that F h may be considered, in a natural way, as a m-form on Mπ. Thus, we can take its exterior differential and we obtain a (m + 1)-form on Mπ which is a section of the vector bundle Λ m+1 2 (T * (Mπ)) → Mπ. Now, it is easy to prove that the vector bundles Λ m+1 2 (T * (Mπ)) and V * (π • ν) ⊗ (π • ν) * (Λ m T * M ) are isomorphic. In fact, an isomorphism forθ ∈ Λ m+1 2 (T * γ (Mπ)) and γ ∈ Mπ. Note that iŨθ ∈ Λ m 1 (T * γ (Mπ)), therefore, it induces an element of Λ m (T * π(ν(γ)) M ). We denote by d v F h the section of the vector bundle V * (π •ν)⊗(π •ν) * (Λ m T * M ) → Mπ induced by the differential of F h . In local coordinates, if and Note that if µ : Mπ → M 0 π is the canonical projection, Φ is a m-form on M and Φ v ∈ Γ(V µ) is the vertical lift to Mπ (see Appendix B) then, using (3.2) and (B.1), we deduce that This property of d v F h motivates the definition of the following affine subbundle of Definition 3.1. The extended phase bundle of the configuration bundle π : Note that P(π) is modelled over the vector bundle V ( P(π)) whose fiber at the point γ ∈ Mπ is We remark that an elementÃ of P(π) has the following local form A = (dp +Ã α du α +Ã α i dp i α ) ⊗ d m x and a generic elementν of V ( P(π)) has the local form ν = (Ã α du α +Ã α i dp i α ) ⊗ d m x. Therefore, the local coordinates on P(π) and V ( P(π)) are (x i , u α , p, p i α ;Ã α ,Ã α i ). In addition, (3.5) 3.1.2. The differential of a Hamiltonian section and the phase bundle. Now, given a point x ∈ M , we can consider an action of the abelian group Λ m T * x M on the fiber (π • ν) −1 (x) defined as follows.
The tangent and cotangent lift of the previous action induces a fibred action of the vector bundle (π • ν) * (Λ m T * M ) on the vector bundles V (π • ν) and Note that, using (3.6), (3.7) and (B.1), it follows that for In a similar way, ifθ ∈ V * γ (π • ν) ⊗ Λ m T * π(ν(γ)) M then the cotangent lift is forŨ ′ ∈ V Φ·γ (π • ν). From (3.4) and (3.8), we deduce that this action restricts to the extended phase bundle P(π), and to the vector bundle V ( P(π)). In local coordinates, we have . Taking the quotient with respect to the action, we can introduce the following definition.
We note that P(π) is an affine bundle over M 0 π = Mπ/π * (Λ m T * M ) modelled over the vector bundle V (P(π)) = V ( P(π)) (π • ν) * (Λ m T * M ) . This bundle is isomorphic to the vector bundle , an isomorphism being given by with γ ∈ Mπ and µ(γ) = γ 0 . Local coordinates on P(π) and V (P(π)) are It is clear that there exists a one-to-one correspondence between the space of sections of the affine bundle P(π) → M 0 π and the set of sections of the extended phase bundle P(π) associated with π, which are (π • ν) * (Λ m T * M )-equivariant. So, if h : M 0 π → Mπ is a Hamiltonian section then, using (3.5), it is easy to see that the vertical differential d v F h is (π •ν) * (Λ m T * M )-equivariant and, therefore, it induces a section dh : M 0 π → P(π) of the phase bundle P(π). We can thus write the following definition. defined by the following commutative diagram where µ : Mπ → M 0 π andμ : P(π) → P(π) are the canonical projections.
The local expression of dh is (3.12) So, we have given an answer to Questions 1 and 2 stated above.
Note that Γ(µ) and Γ(P(π)) are affine spaces modelled over the vector spaces Γ((π •ν 0 ) * (Λ m T * M )) and Γ(V (P(π))), respectively, and d is an affine map. Later in the paper, we shall use the corresponding linear map d l : -valued function on M 0 π then it may be considered as a section of the vector bundle Λ m 1 (T * (M 0 π)) → M 0 π. So, we can take the standard differential dF 0 and we obtain a section of the vector bundle

This vector bundle is isomorphic to
) and, therefore, it induces an element of Λ m (T * π(ν 0 (γ 0 )) M ). We denote by d l F 0 the section of the vector bundle Comments on the next steps. The differential dh : M 0 π → P(π) of a Hamiltonian section is the field theoretic analogue to the differential dH : T * Q → T * (T * Q) of a Hamiltonian function in classical mechanics. In the next section, we will introduce a quotient affine bundle J 1 (π • ν 0 )/ KerA → M 0 π which is the field theoretic analogue to the tangent bundle T (T * Q) of the phase space in classical mechanics. Recall that using the canonical symplectic structure of T * Q, one can define a canonical vector bundle isomorphism and the Hamiltonian vector field X H on T * Q associated with a Hamiltonian function H ∈ C ∞ (T * Q) is given by X H = ♯ • dH. So, a natural question arises: Question 3: Does there exist an affine bundle isomorphism ♯ aff : P(π) → J 1 (π • ν 0 )/ Ker A which, in the presence of a Hamiltonian section h : M 0 π → Mπ, allows us to introduce a distin- In the next Section 3.2, we will give an affirmative answer to Question 3 and we will discuss the relation between Γ h and the solutions of the Hamilton-deDonder-Weyl equations for h. The section Γ h will play the role of X H in Hamiltonian Mechanics.

3.2.
The field-theoretic analogue to the canonical isomorphism ♯ : T * (T * Q) → T (T * Q). We will show that it is given by an affine bundle isomorphism ♯ aff : Let J 1 (π • ν 0 ) be the 1-jet bundle associated with the fibration π • ν 0 : M 0 π → M (see Appendix A). To define the quotient affine bundle, we shall use a construction in [32]. In this paper, the author introduced an affine bundle epimorphism over the identity of M 0 π. This epimorphism is constructed in several steps.
First, we consider the vector bundle monomorphism (3.14) This defines a vector bundle morphism If γ ∈ Mπ, the following commutative diagram illustrates the relation between♭ and♭.
We shall now use the vector bundle morphism♭ to construct A. (3.18) forŨ ∈ V γ (π • ν). Then, ifμ : P(π) → P(π) is the canonical projection, we set Note that A is well-defined and its local expression is This proves that A is an affine bundle epimorphism over the identity of M 0 π (for more details, see [32]).
Recall that the affine bundle . From the local expression (3.20), it follows that the kernel of A is a vector subbundle of V (J 1 (π • ν 0 )) which is locally characterized by We can thus consider the quotient affine bundle , we have that a local basis of sections for this vector bundle is . . , m} and α ∈ {1, . . . , n}. Note that in the quotient vector bundle Local coordinates associated to this basis of sections on the quotient vector bundle (π and, from (3.20), we deduce that the local expression ofÂ iŝ By definition, the affine bundle isomorphism If we consider the local coordinates (x i , u α , p i α ; A α , A α i ) on the phase bundle P(π) then, using (3.22), it follows that 3.3. The field-theoretic analogue to the Hamiltonian vector field X H : Let h : M 0 π → Mπ be a Hamiltonian section. We have seen that the differential of h is a section of the phase bundle P(π). So, we can define the section Using (3.12), (3.23) and the local basis of sections {ν i α , ν α } introduced above, we obtain that the local expression of Γ h is Now, we will show that Γ h plays the same role, in Hamiltonian Classical Field Theories of first order, that the Hamiltonian vector field associated with a Hamiltonian function in Classical Mechanics. This will give an affirmative answer to Question 3 in Section 3.1.
For this purpose, we will discuss the relation between Γ h and the solutions of the Hamilton-deDonder-Weyl equations for h. This uses the notion of a Hamiltonian connection (see [19,21]).
Let τ : N → B be an arbitrary fibration and H an Ehresmann connection on τ : the previous map induces a vector bundle isomorphism between N × B Λ r (T B) and Λ r H which we also denote by , the image χ H of χ by the previous map is called the horizontal lift of χ. If (b i , n α ) are local coordinates on N which are adapted to the fibration τ , then the horizontal lift reads locally Now, suppose that π : E → M is the configuration bundle of a Hamiltonian Classical Field Theory of first order, with m = dim M , and consider the fibration π • ν 0 : M 0 π → M . For a Hamiltonian section h : M 0 π → Mπ, we denote by ω h the (m + 1)-form on M 0 π given by (2.5).
Then, we may prove the following result.
Proof. Using that s 0 is horizontal with respect to H we deduce that This proves the result.
The previous result suggests the introduction of the following definition.
As a direct consequence of the definition, if locally the Hamiltonian section h is then from (2.6), we have the equivalence So, our definition of a Hamiltonian connection is equivalent to that introduced in [19,21]. Note that a Hamiltonian connection H for h may be identified with a section s H : (3.25) and (3.28), we obtain the following result.
From (3.21) and (3.28), we also get the following result.   The following commutative diagram illustrates the results obtained in Sections 3.1, 3.2, 3.3 It is the field-theoretic analogue to Diagram 1.1 for Hamiltonian Mechanics. The last step is to introduce a suitable space of currents for Hamiltonian Classical Field Theories of first order and a suitable canonical bracket formulation for the evolution of such currents along the solution of the Hamilton-deDonder-Weyl equations. This is the aim of the next two subsections.

3.4.
A suitable space of currents for Hamiltonian Classical Field Theories. We shall define a space of currents for Hamiltonian Classical Field Theories of first order, which plays the same role that the space of observables in Hamiltonian Mechanics.
Recall that in Hamiltonian Mechanics, the Hamiltonian vector field X H is a section of the vector bundle T (T * Q) → T * Q and the space of observables is the set C ∞ (T * Q) of real C ∞ -functions on T * Q. Given an observable F ∈ C ∞ (T * Q), we can consider a section dF (the differential of F ) of the dual bundle T * (T * Q) → T * Q to T (T * Q) → T * Q and the evolution of the observable F along a solution s : I ⊆ R → T * Q of Hamilton's equation is given as When written for all observables F , the previous equations are equivalent to the Hamilton equations.
Our goal is to carry out these construction for Hamiltonian Classical Field theories. As we have seen, given a Hamiltonian section h : M 0 π → Mπ, the object corresponding to the Hamiltonian vector field X H is the section Γ h of the quotient affine bundle J 1 (π • ν 0 )/ Ker A → M 0 π. So, we need to overcome the following two steps: Second step: Introduce a space O of currents and a differential operator on this space, such that the evolution of a current α 0 ∈ O along a solution s 0 : We will show that s 0 satisfies these equations for any α ∈ O if and only if s 0 is a solution of the Hamilton-deDonder-Weyl equations.
The following commutative diagram illustrates the situation.
We now consider the composition the two vector bundles isomorphisms A + and I defined above and show that A + • I can be expressed in a simple way, which allows to describe its image L explicitly.
Consider the vector bundle morphism♭ : in Section 3.2 which is characterized by Eq. (3.15) and has the local expression (3.16). Using (3.9), we deduce that♭ induces the vector bundle morphism ) over the identity of M 0 π given by is the canonical isomorphism between the fibers by γ and µ(γ) of the vector bundles is commutative Proposition 3.10. We have the equality This implies that the vector bundle In particular, a local basis of sections of the vector subbundle L is Therefore, using (3.30) and (3.31), it follows that So, from (2.1), (3.18) and (3.32), we conclude that This proves (3.34).
Second step: The previous result together with (3.29) suggests the introduction of the following definition.
Definition 3.11. The space of currents of a Hamiltonian Field Theory with configuration bundle which implies that dα ∈ Γ(L), see (3.36).
ii) Let Y be a section of the vector bundle V π → E, that is, Y is a vector field on E and (T y π)(Y (y)) = 0, ∀y ∈ E.
Conversely, suppose that α 0 is a current. The local expression of α 0 is α Thus, using (3.36), we deduce that This implies that Consequently, we have proved that there exists a local π-vertical vector field Y = Y α (x, u) ∂ ∂u α and a local π-semibasic (m − 1) Note that Y and α are unique. Then, this last fact also proves the global result.
Remark 3.14. (i) Note that O is a C ∞ (E)-module.
(ii) In [12], the authors consider as a space of currents the set of horizontal Poisson (m − 1)-forms on M 0 π. Moreover, they prove that a (m − 1)-form F of this type may be described as where X is a vertical vector field on E, α ′ is a π-semibasic (m − 1)-form on E and β ′ is a closed (π•ν 0 )-semibasic (m−1)-form on M 0 π. Now, it is easy to prove that, under the previous conditions, there exists a unique closed (m − 1)-form β on M such that β ′ = (π • ν 0 ) * (β) = (ν 0 ) * (π * (β)). So, if we take α = α ′ + π * β, we conclude that The previous discussion shows that O is just the space of currents which was considered in [12]. ⋄ 3.5. A suitable linear-affine bracket and the Hamilton-deDonder-Weyl equations. We consider the linear-affine bracket Assume that m ≥ 2, that the local expression of the Hamiltonian section h ∈ Γ(µ) is Then, using (3.25) and (3.39), we obtain the local expression of the linear-affine bracket (3.39) as Note that if we write the current as α 0 ( As we know, Γ(µ) is an affine space which is modelled over the vector space Γ((π•ν 0 ) * (Λ m T * M )). Therefore, using (3.39), it follows that the bilinear bracket associated with the linear-affine bracket {·, ·} is given by Here, d l F 0 is the vertical differential of F 0 (see (3.13)), Ker A is the vector bundle isomorphism associated with the affine bundle isomorphism ♯ aff : P(π) → J 1 (π • ν 0 )/ Ker A, and is the canonical projection. In local coordinates, we have Again, if we write the current as α 0 ( u)), the bilinear bracket takes the form On the other hand, if m = 1 then the space of currents is and, using (3.25) and (3.39), we deduce that the linear-affine bracket and the bilinear bracket are locally given by and for f 0 , g 0 ∈ C ∞ (M 0 π) and h ∈ Γ(µ).
The following result extends to the field-theoretic context the canonical Poisson bracket formulation of Hamilton's equations. (3.44) Proof. Suppose that m ≥ 2 and that The boundary of the configuration space E is just in such a way that π |∂E : ∂E → ∂M is again a fibration.
In a similar way, the restricted multimomentum bundle M 0 π is a manifold with boundary, A boundary condition for the Hamiltonian Classical Field theory is given by specifying a subbundle In such a case, we will consider only sections s 0 : A standard assumption in the literature for the subbundle B 0 is is the canonical inclusion and h : M 0 π → Mπ is the Hamiltonian section (see, for instance, [2,22,39,40]; see also [7] for boundary conditions in the Lagrangian formalism).
From (3.45), we deduce that among all the Hamiltonian connections we should only consider those whose restriction to ∂(M 0 π) × ∂M T (∂M ) takes values in the tangent bundle T B 0 , that is, H should induce a monomorphism of vector bundles This remark is sufficient for the purposes in this paper.
A more detailed discussion of boundary conditions for a Hamiltonian Classical Field theory of first order and its relation with the section Γ h of the quotient affine bundle J 1 (π • ν 0 )/ Ker A and with the theory of covariant Peierls brackets [50] in the space of the solutions will be postponed to a future publication (see the next Section 6).

The affine representation of the Lie algebra of currents on the affine space of Hamiltonian sections
In this section, we prove that the space of currents O of a Hamiltonian Field theory of first order admits a Lie algebra structure and we show that the linear affine bracket {·, ·} introduced in Section 3.5 (see (3.39)) induces an affine representation of O on the affine space of Hamiltonian sections.
We first review the notion of an affine representation of a Lie algebra on an affine space (for more details, see [38]).
Let A be an affine space modelled over the vector space V . The vector space of affine maps of A on V , Aff(A, V ), is a Lie algebra and the Lie bracket on Aff(A, V ) is given by for ϕ, ψ ∈ Aff(A, V ), where ϕ l , ψ l : V → V are the linear maps associated with ϕ, ψ, respectively.
An affine representation of a real Lie algebra g on A is a Lie algebra morphism We first note that if π : E → M is a configuration bundle with dimM = 1 then it is easy to prove the following facts (see Section 5.1 for the particular case when M is the real line R and E = R × Q): • J 1 π is an affine subbundle of corank 1 of the tangent bundle T E → E which is modelled over the vertical bundle V π → E to the fibration π : E → M . • The restricted multimomentum bundle is just the dual bundle V * π → E to V π → E. • The extended multimomentum bundle is the cotangent bundle T * E → E of E.
• Γ(µ) is an affine space which is modelled over the vector space O = C ∞ (V * π) of the currents.
So, in this case, we have a Lie algebra structure on O = C ∞ (V * π). In fact, the Lie bracket on O is just the Poisson bracket {·, ·} l on V * π given by (3.43). Moreover, using (3.42) and (3.43), we deduce that the linear-affine bracket induces an affine representation of the Lie algebra (C ∞ (V * π), {·, ·} l ) on the affine space Γ(µ). More explicitly, we have Therefore, in the rest of this section, we will assume the following hypothesis: Assumption: In what follows, we will suppose that dimM ≥ 2.

Note that a section of the vector bundle
Mπ which is equivariant with respect to the fibred actions of π * (Λ m T * M ) on Mπ and on V (π • ν).
By applying this observation to ♯(dα 0 ), we denote bỹ the equivariant vector field on Mπ associated with the section ♯(dα 0 ). SinceH α 0 is equivariant, it follows that it is µ-projectable to a vertical vector field We now present a description ofH α 0 in terms of α 0 . Let♭ : V (π • ν) → M(π • ν) = Λ m 2 (T * (Mπ)) be the vector bundle monomorphism given by (3.14). Denote byL the image of V (π • ν) by♭, so that♭ : is a vector bundle isomorphism over the identity of Mπ. From (2.3), it follows that a local basis of Γ(L) is be the vector bundle morphism over the canonical projection µ : Mπ → M 0 π which is characterized by Eq. (3.15). If γ ∈ Mπ and J γ : is the canonical isomorphism between the fibers by γ and µ(γ) of the vector bundles V (π • ν) → Mπ and V (π • ν)/(π • ν) * (Λ m T * M ) → M 0 π then, using (3.17) and (3.33), it follows that the following diagram is commutative. This implies that or, in other words,H α 0 satisfies the following condition Note that, since ω Mπ is non-degenerate, (4.3) may be considered as a definition of the equivariant vector fieldH α 0 .
We now present the local expressions of the vector fieldsH α 0 and H α 0 . As we have seen (see Section 3.4), the local expression of an element α 0 ∈ O is Thus, using (2.3) and (4.3), we deduce that (4.5) Therefore, it follows that Following the terminology in [11] (see also [31]), Eq. (4.3) implies that µ * α 0 is a Hamiltonian (m − 1)-form and that the vector fieldH α 0 is a Hamiltonian vector field on the multisymplectic manifold (Mπ, ω Mπ ).

This equation suggests the introduction of the the (m
Using the local expressions Moreover, we can prove the following result. Proof. Using Theorem 3.13, we deduce that there exists and isomorphism between the C ∞ (E)modules O and Γ(V π) × Γ(Λ m−1 1 T * E). In addition, from (4.8), it follows that under the previous isomorphism, the bracket {·, ·} O is given by Note that if we write the observables locally as α 0 ( which is reminiscent of the local expression of the canonical Poisson bracket. As a final remark on the definition of the morphism ♭ and the bracket {·, ·} O on the currents, we can derive a version of the classical result that any Poisson structure on a manifold induces a Lie algebroid structure on the cotangent bundle of the manifold. In this case, we will obtain a Lie algebroid structure on the vector bundle L over M 0 π. Indeed, it is clear that the vector bundle V (π • ν) → Mπ admits a Lie algebroid structure. The Lie bracket in the space Γ(V (π • ν)) of sections is just the restriction of the standard Lie bracket to (π • ν)-vertical vector fields and the anchor map is the inclusion Γ(V (π • ν)) ֒→ X(Mπ). So, using the vector bundle isomorphism♭ : V (π • ν) →L ⊆ M(π • ν), we can induce a Lie algebroid structure on the vector bundleL. In fact, a direct computation proves that the Lie bracket in the space of sections ofL, Γ(L), is given by whereρ =♭ −1 : Γ(L) → Γ(V (π • ν)) is the anchor map.

(i) If h ∈ Γ(µ) is a Hamiltonian section then
is the extended Hamiltonian density associated with h and d v F h is the vertical differential of F h . So, we have where d l F 0 is the linear part of the differential of F 0 .
If α 0 , β 0 are currents and h ∈ Γ(µ) is a Hamiltonian section then, using (4.12) and (4.14), we obtain Now, using (4.14) and the fact the vector fieldsH α 0 andH β 0 are µ-projectable on the vector fields H α 0 and H β 0 , respectively, we deduce So, from (4.16), we obtain that Finally, using (4.12) and the fact that which proves the result.

Proof. (of Lemma 4.4) It is clear that the mapH :
Thus, using (4.3) and the fact that ω Mπ is closed, we deduce that But, since the vector fieldH α 0 on Mπ is µ-projectable over the vector field H α 0 on M 0 π, it follows that Therefore, using (4.7), we obtain that So, from (4.3), it follows that and, since ω Mπ is non-degenerate, this implies the result.

Time-dependent Hamiltonian Mechanics.
In this section, we will use the following terminology. Let τ : V → P be a vector bundle. Then, we can consider the vector bundle The sections of this vector bundle are just the time-dependent sections of τ : V → P . For this reason, the vector bundle id R × τ : R × V → R × P will be called the time-dependent vector bundle associated with τ : V → P . For time-dependent Mechanics, the base space of the configuration bundle is the real line R, that is, we have a fibration π : E → R. This fibration is trivializable but not canonically trivializable. In fact, if one choses a reference frame one may trivialize the fibration. This means that E may be identified with a global product R × Q and, under this identification, π : E ≃ R × Q → R is the canonical projection on the first factor. For simplicity, in what follows, we will assume that this is our starting point although all the constructions in this section may be extended, in a natural way, if we don't chose a reference frame (for an affine formulation of frame-independent Mechanics, we remit to [34,35,36,37,41,47].) So, if the configuration bundle is trivial, the 1-jet bundle J 1 π = J 1 (pr 1 ) → E = R × Q may be identified with the affine subbundle of Thus, J 1 (pr 1 ) is isomorphic to the vector bundle R × T Q → R × Q.
An Ehresmann connection H : (R×Q)× R T R → H ⊆ T (R×Q) on the fibration pr 1 : R×Q → R is completely determined by a vector field Γ on R × Q satisfying dt, Γ = 1.
In fact, the horizontal subbundle associated with the connection is of rank 1 and generated by the vector field Γ.
The extended (resp. restricted) multimomentum bundle may be identified with the cotangent bundle T * (R × Q) = R × R × T * Q (resp. the vector bundle R × T * Q) and, under this identification, the multisymplectic structure on R × R × T * Q is just the canonical symplectic structure ω (R×Q) on R × R × T * Q.
We have a principal R-action on T * (R × Q) = R × R × T * Q given by for (t, p, α q ) ∈ R×T * q Q and p ′ ∈ R. The principal bundle projection is just the canonical projection µ : Note that R × T * Q admits a Poisson structure of corank 1 which is induced by the canonical symplectic structure on T * Q. In fact, the R-action on the extended multimomentum bundle preserves the symplectic form and the canonical projection µ is a Poisson map.
Remark 5.1. We recall that a cosymplectic structure on a manifold P of odd dimension 2p + 1 is a couple (ω, θ), where ω is a closed 2-form, θ is a closed 1-form and θ ∧ ω p is a volume form on P . The Reeb vector field Γ associated with the structure (ω, θ) is the vector field on P which is completely characterized by the conditions ⋄ Note that the 2-form ω H on R × T * Q is given by where ω Q is the canonical symplectic structure on T * Q. So, the Reeb vector field Γ H of the cosymplectic structure (ω H , η) on R × T * Q is with X H(t,·) the Hamiltonian vector field on T * Q associated with the function Thus, if (t, q i , p i ) are canonical coordinates on R × T * Q then and the integral curves t → (t, q i (t), p i (t)) of Γ H are just of the solution of the Hamilton equations for H, that is, Therefore, Γ H is the evolution vector field associated with the time-dependent Hamiltonian function H : R × T * Q → R (for more details see, for instance, [15,20]).
In addition, the vector bundle V * (π • ν) may be identified with the time-dependent cotangent bundle to R × T * Q, Under the previous identification, the principal R-action on Moreover, the extended phase bundle P(π) is P(π) = {(t, dp |p + γ) | (t, p) ∈ R and γ ∈ T * (T * Q)}.
In other words, P(π) may be identified with the time-dependent vector bundle associated with the vector bundle id R × π T * Q : and the principal R-action on P(π) ≃ R × (R × T * T * Q) is given by p ′ · (t, (p,γ)) = (t, p + p ′γ ) for p ′ , t, p ∈ R andγ ∈ T * (T * Q).
Thus, the phase bundle P(π) may be identified with the time-dependent cotangent bundle to T * Q id R × π T * Q : P(π) ≃ R × T * T * Q → R × T * Q. Then, the differential dh of h is just the vertical differential d v H (with respect to the projection pr 1 : R × T * Q → R) of the Hamiltonian function H, that is, On the other hand, the vector bundle L, which was introduced in Proposition 3.10, is isomorphic to the cotangent bundle to R × T * Q.
Moreover, as we know, the 1-jet bundle to the projection π • ν 0 : R × T * Q → R may be identified with the affine subbundle of T (R × T * Q) given by or, equivalently, with the time-dependent tangent bundle to T * Q Under all the previous identifications, the affine bundle isomorphism A : J 1 (π • ν 0 ) ≃ R × T T * Q → P(π) ≃ R × T * T * Q is given by with ω Q the canonical symplectic structure of T * Q. Thus, in this case, the vector subbundle Ker A is trivial and this implies that there exists a unique Hamiltonian connection for the hamiltonian section h. In fact, the horizontal subbundle of such a connection is generated by the evolution vector field Γ H .
In addition, if ♯ aff : P(π) → J 1 (π • ν 0 ) is the inverse morphism of A : J 1 (π • ν 0 ) → P(π) it is clear that, under the identification of J 1 (π • ν 0 ) with the affine subbundle of T (R × T * Q) given by (5.1), the image of the section dh of P(π) is just the evolution vector field Γ H .
On the other hand, the space O of currents is the set of smooth real functions on R × T * Q (the space of observables in Classical Mechanics) and it is clear that the space Γ(µ) of sections of the projection may be also identified with C ∞ (R × T * Q). Then, the linear-affine bracket is given by This bracket was considered in [34,35,47]. In addition, the bracket {·, ·} O on the space of observables O ≃ C ∞ (R×T * Q) is just the standard Poisson bracket {·, ·} R×T * Q induced by the canonical symplectic structure of T * Q. In other words, Finally, the affine representation R of the Lie algebra (C ∞ (R × T * Q), {·, ·} R×T * Q ) on the affine space Γ(µ) ≃ C ∞ (R × T * Q) is given by

5.2.
A particular case: the configuration bundle is trivial and the base space is orientable. In this section we will assume that E = M × Q, π : E → M is the canonical projection pr 1 : M × Q → M on the first factor. In this case, the affine bundle can be identified with the vector bundle Let us further assume that M is orientable, with m = dimM ≥ 2, and fix a volume form vol ∈ Ω m (M ) on M . We denote by χ vol the m-vector on M which is characterized by the condition i(χ vol )vol = 1.
Using the volume form vol on M , we have Λ m T * x M ≃ R, ∀x ∈ M, and the vector bundle Λ m T * M → M may be trivialized as the trivial line vector bundle M × R → M . Using vol again, the reduced multimomentun bundle M 0 π is isomorphic to the vector bundle.
We can also identify it with T M ⊗ T * Q = ∪ (x,y)∈M×Q Lin(T * x M, T * y Q). As in the general case, we will denote by ν 0 : Λ m−1 T * M ⊗ T * Q → M × Q the vector bundle projection. We will see that this space admits a multisymplectic structure.
Proposition 5.2. Let λ M 0 π be the m-form on M 0 π given by where pr 2 : M ×Q → Q is the projection on the second factor. Then, ω M 0 π = −dλ M 0 π is a multisymplectic structure on M 0 π.

Proof.
A direct computation proves that the local expression of λ M 0 π is which implies that ω M 0 π is a multisymplectic structure on M 0 π.
On the other hand, the extended multimomentum bundle Mπ may be identified with the Withney sum of the vector bundles Λ m T . So, using the volume form vol, we deduce that Under this identification, the canonical multisymplectic structure ω Mπ on Mπ is where p is the canonical coordinate on R. Here, we also denote by ω M 0 π and vol the pullbacks to Mπ of ω M 0 π and vol, respectively.
Moreover, a Hamiltonian section h : M 0 π → Mπ is just a global Hamiltonian function H : M 0 π → R on M 0 π and the (m + 1)-form ω h on M 0 π is On the other hand, under the identification between Mπ and R × M 0 π and using (B.1) (see Appendix B), it follows that where p is the standard coordinate on R.
So, the extended phase bundle P(pr 1 ) is isomorphic to the affine bundle over Mπ ≃ R × M 0 π where V (pr 1 • ν) is the vertical bundle of the fibration pr 1 • ν : Mπ → M . Now, using the previous identifications, we have that the fibred action of π * (Λ m T * M ) on M(pr 1 ) is just the standard action of R on R × M 0 π. Therefore, since for (p, γ 0 ) ∈ R × M 0 (pr 1 ), we deduce that the phase bundle P(pr 1 ) is isomorphic to the vector bundle V * (pr 1 • ν 0 ). An isomorphism V * (pr 1 • ν 0 ) → P(pr 1 ) = P(pr 1 ) R between these spaces is given by ∈ P(pr 1 ) γ 0 , for γ 0 ∈ M 0 π, with p an arbitrary real number.
Thus, the image of the vertical differential d v H of H under the previous isomorphism is just the equivalence class induced by the vertical differential d v F h of the extended Hamiltonian density F h = p + H. This implies that, under the identification between V * (pr 1 • ν 0 ) and P(pr 1 ), the differential dh of h (as a section of the affine bundle P(pr 1 ) → M 0 (pr 1 )) is just d v H (as a section of the vector bundle V * (pr 1 • ν 0 ) → M 0 (pr 1 )).
Moreover, using that (♭ 0 ) * = A, it follows that the vector subbundle L of M(pr 1 • ν 0 ) introduced in Proposition 3.10 is In addition, as we know (see first step in Section 3.4), we have that On the other hand, under the identification between M(pr 1 ) and R × M 0 (pr 1 ), the projection µ : M(pr 1 ) → M 0 (pr 1 ) is just the canonical projection on the second factor. Thus, the affine space Γ(µ) is isomorphic to the vector space C ∞ (M 0 (pr 1 )) and the linear-affine bracket given by (3.39) may be considered as a bracket

Continuum Mechanics.
In this section we develop the formulation of Continuum Mechanics as a Canonical Hamiltonian Field Theory. This covers the case of fluid mechanics and nonlinear elasticity. We shall assume that the reference configuration of the continuum is described by a manifold B of dimension N , N = 2, 3 possibly with boundary, and we suppose that the continuum evolves in a N dimensional manifold Q without boundary, the ambient manifold, typically R N . The elements x ∈ B denote the labels of the material points of the continuum, whereas the elements u ∈ Q denote the current positions of these material points. The evolution of the continuum is described by a map ϕ : [0, T ] × B → Q, where [0, T ] is the interval of time. Hence, u = ϕ(t, x) describes the position of the material point x at time t. We shall assume that for each t fixed, the map x ∈ B → ϕ(t, x) ∈ Q is a smooth embedding. Boundary conditions will be described in §5.3.4

Lagrangian and Hamiltonian formulations in continuum mechanics.
Continuum mechanics is usually written either as a Lagrangian field theory or as an infinite dimensional classical Lagrangian or Hamiltonian system. While the infinite dimensional description is more classical, the field-theoretic description is especially useful for the derivation of multisymplectic integrators for fluid and elasticity ( [16,17,18,42,45]). In the field theoretic Lagrangian description, the map ϕ is interpreted as a section φ ∈ Γ(π) of the trivial fiber bundle π : , ϕ(t, x)). The equations of motion are given by the Euler-Lagrange equations for a given Lagrangian density L : J 1 π → Λ m T * M , m = N + 1. Since the bundle is trivial, we have J 1 π (t,x,u) ≃ T * (t,x) M ⊗ T u Q. We denote by (t, x i , u α , V α , F α i ) the local coordinates. Writing locally the Lagrangian density as L =L(t, x i , u α , V α , F α i )dt ∧ d N x, the Euler-Lagrange equations are given by ∂ ∂t In the infinite dimensional classical Lagrangian description, the map ϕ is interpreted as a curve ϕ(t) in the infinite dimensional manifold Emb(B, Q) of smooth embeddings of B into Q. The equations are given by the (classical) Euler-Lagrange equations for the Lagrangian function L : where we assumed that the Lagrangian density L does not depend explicitly on the time t, and T X ϕ : T X B → T ϕ(X) Q denotes the tangent map to ϕ ∈ Emb(B, M ), i.e. locally T x ϕ = ϕ α ,i ∂ ∂u α ⊗dx i . When L is hyperregular, to this classical Lagrangian description is formally associated a classical Hamiltonian description with respect to the Hamiltonian H : T * Emb(B, Q) → R defined on the (regular) cotangent bundle of Emb(B, Q). The Hamiltonian is defined by Q). In this case, the associated equations can formally be writtenḞ = {F, H} can with respect to the canonical Poisson bracket on T * Emb(B, Q).
The Hamiltonian formulation that we present below is different from this one, since it is associated to the field theoretic Lagrangian formulation. Roughly speaking, while the canonical Hamiltonian formulation recalled above is based on a Legendre transform with respect to the time direction only, the canonical Hamiltonian field theoretic description that we will describe below is based on a Legendre transform with respect to all the variables in the base manifold M .
We warn the reader that the coordinates x i that were used in the previous sections for the base manifold M are here given by (t, x i ) for M = [0, T ] × B. The coordinates u α i used earlier on the fiber of J 1 π are here given by (V α , F α i ) and represent the material velocity and the deformation gradient of the continuum.

5.3.2.
Lagrangian density and Legendre transform. The Lagrangian density of continuum mechanics is defined with the help of given tensor fields on B and Q. In order to treat both fluid dynamics and elasticity from a unified perspective, we shall consider here a Riemannian metric G on B, two volume forms ̺ and ς on B, and a Riemannian metric g on Q. Additional tensor fields can be introduced to describe electromagnetic effects or microstructures. The volume forms ̺ and ς are the mass density and the entropy density in the reference configuration and are locally written as ̺ =̺d N X and ς =ςd N X. The potential energy density is a bundle map In local coordinates, it reads E(x i , u α , F α i ,̺,ς, G ij , g αβ )d N X. This is a general form of potential energy density for continua, including fluid and elasticity, which may describe both internal and stored energies.
The associated Lagrangian density L : J 1 π → Λ m T * M is given by the kinetic minus the potential energy, and reads in local coordinates. Note that the Lagrangian is defined with the help of the given tensor fields ∂ ∂x j , and g = g αβ dx α dx β . We chose to work with the cometric G ij associated to G ij , in order to directly get the Finger deformation (or left Cauchy-Green) tensor b αβ , rather than its inverse, later.
The restricted multimomentum bundle for continuum mechanics is given by with coordinates (t, x i , u α , M α , P i α ). The restricted Legendre transform of the Lagrangian density is with M α and P i α given by with M α the momentum density (in the Lagrangian description) and P i α is the Piola-Kirchoff stress tensor density. Note that the coordinates (M α , P i α ) on the fiber of M 0 π correspond to the coordinates denoted p i α earlier. The Eulerian versions of these tensor densities are the Eulerian momentum density m α and the Cauchy stress tensor density σ αβ given by the Piola transformation From the second relation, we have Note that the first relation in (5.7) is always invertible, but the invertibility of the second relation depends in the potential energy density E. As we shall illustrate below, relation (5.9) is extremely useful to check if the restricted Legendre transform (5.6) is an isomorphism, in which case we say that the Lagrangian density is hyperregular.

5.3.3.
The Hamiltonian density and the linear-affine bracket for Continuum Mechanics. By assuming that L is hyperregular, we get the Hamiltonian H ∈ C ∞ (M 0 π) where F α i is expressed in terms of the variables in M 0 π by inverting the second relation in (5.7). A section of the restricted momentum bundle M 0 π is locally given by x i ) and, in the hyperregular case, the Euler-Lagrange equation are equivalent to the Hamilton-deDonder-Weyl equations given by These equations admit the canonical linear-affine bracket formulation, that is, a section (t, where the currents α 0 ∈ O for Continuum Mechanics are of the form This formulation assumes that the Legendre transform is invertible. Except in some simple situations, this invertibility is a priori difficult to check. We shall show below how to facilitate the approach by using two symmetries of the potential energy density E. The first one, the material covariance, is related to the isotropy of the continuum, while the second, the material frame indifference, is a general covariance assumption of continuum theories, see [48] and [30].
We assume that the potential energy density is of the form is the potential energy density in the Eulerian description. This assumption is compatible with the assumption of material covariance. Here ǫ =ǭd N u is a bundle map From this expression, we compute the momenta from the second equality in (5.7) as where we introduced the notations These are the local expressions of the mass density and entropy density in Eulerian description, and of the Finger deformation (or left Cauchy-Green) tensor.
The associated Cauchy stress tensor density σ, see the second equation in (5.8), is If in addition E satisfies the material frame indifference, then ǫ(ψ * ρ, ψ * s, ψ * b, ψ * g) = ψ * ǫ(ρ, s, b, g) , By inserting these relations into (5.10), we get the following result which is a step towards a more explicit expression of the Hamiltonian density, because in practice ǫ, rather than E, is given. Proposition 5.3. Assume that the Lagrangian is hyperregular and that E satisfies the two invariance mentioned above, and consider the associated Eulerian potential energy density ǫ. Then, the Hamiltonian density of continuum mechanics H ∈ C ∞ (M 0 π) is given by For a continuum moving in a fixed domain B ′ ⊂ Q diffeomorphic to B, we have the boundary condition ϕ(t, ∂B) = ∂B ′ on the motion and, in addition, the boundary condition on the Piola-Kirchhoff stress tensor P given by P i α (t, x)N ♭ i (x)| T B ′ = 0, for all x ∈ ∂B, with N the normal vector field to B with respect to G. This corresponds to zero tangential traction on the boundary, a condition that vanishes for fluids. In this case, the subbundle B 0 → [0, T ] × ∂B is given by For a free boundary continuum we take T ] × ∂B, which corresponds to zero traction on the boundary. This reduces to zero pressure at the boundary for fluids. We have B 0 5.3.5. Fluid dynamics. In this case the energy density ǫ only depends on the mass density and entropy density ρ =ρd N u and s =sd N u, so the Cauchy stress density is given by see (5.14), where p is the pressure of the fluid. In this case (5.9) yields This relation is of the form In this case the Lagrangian density is hyperregular. Note that the function f, and hence the hyperregularity, depends on the state function of the fluid, i.e., the relation ǫ = ǫ(ρ, s, g).
Hyperregularity is satisfied for a large class of state equations, including the important case of a perfect gas for which ǫ(ρ, s, g) = ǫ 0 e 1 Cv s ρ − s 0 ρ 0 ρ ρ0µ(g) γ µ(g), where γ = C p /C v is the adiabatic index and µ(g) is the volume form associated to g, i.e. µ(g) = √ det gd N u. In this case, we compute the pressure as p √ det g = (γ − 1)ǭ. Note that, as it should, ǫ satisfies (5.14). Computing the derivative ofǭ with respect to the Riemannian metric, we get ∂ǭ ∂g αβ = 1 2 (1 − γ)ǭg αβ , so one directly checks that the Doyle-Ericksen formula (5.16) is verified.
The fluid equations can thus be written in the canonical linear-affine bracket form (5.12). 5.3.6. Nonlinear elasticity. In general, the Hamiltonian density in nonlinear elasticity takes a complicate expression due to the dependence of ǫ on the Finger deformation tensor b. For example, for the compressible neo-Hookean material (see [51], [3]), with N = 3, the energy density is where κ is the bulk modulus, µ is the Lamé constant, and µ(b ♭ ) is the volume form associated to the Riemannian metric b ♭ , obtained by lowering the indices of b. One observes that (5.14) is satisfied. The Doyle-Ericksen formula yields the expression of the stress tensor density We thus get 2 ∂ǫ ∂g : g = 3κ ln Jρ which can then be inserted in (5.17) to yield the Hamiltonian density.
We shall illustrate the derivation of the Hamiltonian density by considering the simplified situation ǫ(ρ, b, g) = 1 2 Tr g (b)ρ. In this case σ = bρ, so we get the momenta P i α = −G ij F β j g αβ̺ . Using this and σ : g = Tr g (b)ρ, we get the Hamiltonian density The nonlinear elasticity equations can thus be written in the canonical linear-affine bracket form (5.12).

5.4.
Yang-Mills theory. Yang-Mills theory may be considered as a singular Lagrangian field theory of first order associated with a principal G-bundle over an oriented Riemannian (or a Lorentzian manifold) space M (possibly with boundary) of dimension m and where G is a compact Lie group of dimension n (we will follow [39]). We will denote by g the metric on M . For simplicity, we will assume that the principal bundle is trivial, g is a Riemannian metric and ∂M = φ.
Under the previous conditions, the configuration bundle of the theory is the vector bundle where g is the Lie algebra of G.
Then, we will proceed as follows. We will introduce a Lagrangian density on the 1-jet bundle of the fibration π M,g : T * M ⊗ g → M . This Lagrangian density is singular. In fact, the image of the corresponding Legendre transformation is a proper submanifold M 1 of the restricted multimomentum bundle M 0 π M,g . Using the restricted and the extended Legendre transformation, we will construct a constrained Hamiltonian section h 1 : M 1 → µ −1 (M 1 ) ⊆ Mπ M,g of the fibration µ |µ −1 (M 1 ) : µ −1 (M 1 ) → M 1 . Now, if we consider an (arbitrary) hamiltonian section h : M 0 π M,g → Mπ M,g , whose restriction to M 1 coincides with h 1 , we will obtain a Hamiltonian field theory in such a way that the solutions of the Hamilton-deDonder-Weyl equations for h which are contained in M 1 are just the solutions of the corresponding Yang-Mills theory. 5.4.1. The Lagrangian formalism. Note that the sections of the vector bundle π M,g : T * M ⊗g → M are the principal connections on the trivial principal bundle pr 1 : M × G → M . As we know, the 1-jet bundle J 1 π M,g is an affine bundle over T * M ⊗ g. The key point is that there is a canonical epimorphism off affine bundles (over the vector bundle projection π M,g : T * M ⊗ g → M ), F : J 1 π M,g → Λ 2 T * M ⊗ g, which is characterized by the condition for all principal connections Θ. In other words, the image by F of the 1-jet bundle of a principal connection is just the curvature of the connection.
If (x i ) are local coordinates on M and {e α } is a basis of g, we have the corresponding local coordinates (x i , u α i ) on E and (x i , u α i , u α ij ) on J 1 π M,g . Moreover, Here, c γ αβ are the structure constants of the Lie algebra g with respect to the basis {e γ }. Next, we will introduce the Lagrangian density First of all, since the manifold M is oriented, the vector bundle π * M,g (Λ m T * M ) → E is the trivial line bundle E × R → E. So, the Lagrangian density L is, in fact, a real C ∞ -function L : J 1 π M,g → R.
In addition, we will fix an Ad-invariant scalar product ·, · on g (which is possible, since G is compact). Then, the scalar product on g and the Riemannian metric on M induce a bundle metric on the vector bundle So, we can consider the real function L : J 1 π M,g → R given by where the norm is taken with respect to the bundle metric on the vector bundle pr 1 : The local expression of L is with F kl γ = F β mn g km g ln ·, · βγ and (g ij ) the matrix of the coefficients of g, (g ij ) the inverse matrix and ·, · βγ = e β , e γ . Note that, since M is oriented, the restricted multimomentum bundle M 0 π M,g may be identified with the dual bundle V * (J 1 π M,g ) of V (J 1 π M,g ). So, The transformation leg L is given by for z ∈ J 1 y π M,g , v ∈ M 0 y π M,g and y ∈ E. The local expression of leg l is This implies that the image of leg L is the vector subbundle M 1 (over E) of M 0 π M,g M 1 ≃ E × M (Λ 2 T M ⊗ g * ).
Thus, the map leg 1 : J 1 π M,g → M 1 is a submersion with connected fibers and L is almost regular.
On the other hand, we can consider the extended Legendre transformation Leg L : J 1 π M,g → Mπ M,g ≃ Aff(J 1 π M,g , R) associated with L defined by Leg L (z)(z ′ ) = d dt |t=0 L(z + t(z ′ − z)), for z, z ′ ∈ J 1 y π M,g and y ∈ E.
The local expression of Leg L is Note that if µ : Mπ M,g → M 0 π M,g is the canonical projection then the image of Leg L is a submanifold M of µ −1 (M 1 ) ⊆ Mπ M,g which is diffeomorphic to M 1 , via the restriction of µ to M.
It is clear that On the other hand, if h : M 0 π M,g → Mπ M,g is a Hamiltonian section which extends h 1 (that is, h |M 1 = h 1 ), then we may consider the corresponding Hamiltonian field theory associated with h. Furthermore, using the classical results on singular Lagrangian field theories (see [21]), if s 0 : U ⊆ M → M 0 π M,g is a solution of the Hamilton-deDonder-Weyl equations for h which is contained in M 1 then s 0 is just a solution of the Yang-Mills equations.
The following diagram illustrates the situation The Lie algebra of currents and the linear-affine bracket for the extended Hamiltonian field theory. After the previous subsections, we could apply all the machinery in this paper for the extended Hamiltonian field theory and, as a consequence, we could deduce results on the Yang-Mills theory. This will be the subject of a future research. Anyway, we will remark a couple of general facts on the Lie algebra of currents, the linear-affine bracket (in Section 3.5) and the Yang-Mills equations as constrained Hamilton-deDonder-Weyl equations: • First of all, following the proof of Theorem 4.2, we have that the space of currents, as a C ∞ (E)-module, may be identified with the product Γ(V π M,g ) × Γ(Λ m−1 1 T * E). But, since the configuration bundle π M,g : T * M ⊗ g → M is a vector bundle, we have that Γ(V π M,g ) is generated by vertical lifts of sections of the projection π M,g (see Appendix B). In fact, if θ = θ i (x)dx i is a 1-form on M and ξ ∈ g then the local expression of the vertical lift of the section s = θ ⊗ ξ is

Conclusions and future work
In this paper, we have developed a completely canonical geometric formulation of Hamiltonian Classical Field Theories of first order which is analogous to the canonical Poisson formulation of time-independent Hamiltonian Mechanics. This formulation is valid for any configuration bundle and is independent of any external structures such as connections or volume forms. We have defined a space of currents and endowed it with a Lie algebra structure, and we have shown that the bracket induces an affine representation of the Lie algebra of currents on the affine space of Hamiltonian sections. An important difference with the case of time-independent Hamiltonian Mechanics is the linear-affine character of our bracket, which is consistent with the fact that the set of currents and the set of Hamiltonian sections are linear and affine spaces, respectively. We have applied our results to several examples and we have proved their effectiveness.
The results of this paper open some interesting future directions of research: • Develop appropriate processes of reduction by symmetry for Hamiltonian Classical Field Theories of first order by exploiting the canonical linear-affine bracket formulation proposed in this paper. • Include boundary conditions in the geometric formulation (see [44]) and discuss the relation between the resultant construction and the theory of Peierls brackets [50] in the space of solutions (see [27,28]; see also the recent papers [13,14] and the references therein).
• Discuss a canonical affine formulation of Lagrangian Classical Field Theories of first oder and obtain the equivalence with the Hamiltonian formulation for the case when Lagrangian density is almost regular (this is, for instance, the case of Yang-Mills theories discussed in this paper).
then s is a horizontal section of H if and only if it satisfies the following system of partial differential equations ∂u α ∂x i = H α i (x, u(x)), ∀ i, α. The Ehresmann connection H is said to be integrable if the distribution H is completely integrable. In such a case, for every point y ∈ E there exists a unique horizontal (local) section s : U ⊆ M → E of H such that π(y) ∈ U and s(π(y)) = y.