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

François Gay-Balmaz  F. Gay-Balmaz: Laboratoire de Meteorologie Dynamique
Ecole Normale Superieure, CNRS, F-75231
Paris, France<EMAIL_ADDRESS>,  Juan C. Marrero  J. C. Marrero: ULL-CSIC
Geometría Diferencial y Mecánica Geométrica
Departamento de Matemáticas, Estadística e IO, Sección de Matemáticas,
Universidad de La Laguna
La Laguna, Tenerife, Canary Islands, Spain<EMAIL_ADDRESS>and  Nicolás
Martínez  N. Martínez: Departamento de Matematicas, Universidad Nacional de
Colombia
Bogotá, Colombia<EMAIL_ADDRESS>
###### Abstract.

It has been a long standing question how to extend, in the finite-dimensional
setting, 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.

J.C.M. thanks H. Bursztyn, A. Cabrera, S. Capriotti and D. Iglesias for
stimulating discussions on the topics of this paper. J.C.M. has been partially
supported by Ministerio de Economía y Competitividad (MINECO, Spain) under
grant PGC2018-098265-B-C32

##  1\. 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 $\sharp:T^{*}(T^{*}Q)\to T(T^{*}Q)$. Given the Hamiltonian $H\in
C^{\infty}(T^{*}Q)$ 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 $X_{H}=\sharp dH$, i.e., we have
the commutative diagram

$\textstyle{T^{*}(T^{*}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sharp}$$\textstyle{T(T^{*}Q)}$$\textstyle{T^{*}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{dH}$$\scriptstyle{X_{H}}$
(1.1)

The time evolution of an observable $F\in C^{\infty}(T^{*}Q)$ along a solution
$s:I\subseteq\mathbb{R}\to T^{*}Q$ of Hamilton’s equations, is given by

$\frac{d}{dt}(F\circ s)=\\{F,H\\}\circ s,$ (1.2)

where $\\{\cdot,\cdot\\}:C^{\infty}(T^{*}Q)\times C^{\infty}(T^{*}Q)\to
C^{\infty}(T^{*}Q)$ is the canonical Poisson bracket defined by

$\\{F,G\\}=\left\langle dF,X_{G}\right\rangle=\left\langle dF,\sharp
dG\right\rangle=-\left\langle\sharp dF,dG\right\rangle.$ (1.3)

Conversely, if a curve $s:I\subseteq\mathbb{R}\to T^{*}Q$ satisfies (1.2) for
any observables $F$, then $s$ is a solution of Hamilton’s equations (see, for
instance, [1, 23, 43]). Recall that (1.3) defines a Lie algebra structure on
$C^{\infty}(M)$, thereby satisfying the Jacobi identity

$\\{\\{F,G\\},H\\}=\\{F,\\{G,H\\}\\}-\\{G,\\{F,H\\}\\}.$ (1.4)

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
$\pi:E\to M$, whose sections are the fields of the theory. The case of time-
dependent mechanics corresponds to the special situation
$E=Q\times\mathbb{R}\rightarrow\mathbb{R}$. For classical field theories,
there are two useful generalisations of the notion of momentum phase space:
the restricted multi-momentum bundle ${\mathcal{M}}^{0}\pi$ and the extended
multi-momentum bundle ${\mathcal{M}}\pi$. Both are vector bundles on $E$ with
vector bundle projections denoted $\nu^{0}:{\mathcal{M}}^{0}\pi\to E$ and
$\nu:{\mathcal{M}}\pi\to E$, and there is a canonical line bundle projection
$\mu:{\mathcal{M}}\pi\to{\mathcal{M}}^{0}\pi$. The main property of the
extended multi-momentum bundle is that it admits a canonical multisymplectic
structure $\omega_{{\mathcal{M}}\pi}$ of degree $m+1$ ($m$ being the dimension
of $M$). The restricted multi-momentum bundle ${\mathcal{M}}^{0}\pi$, 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 $h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$ of the
canonical projection $\mu:{\mathcal{M}}\pi\to{\mathcal{M}}^{0}\pi$, see [11].
The corresponding evolution equations are the Hamilton-deDonder-Weyl
equations. They form a system of partial differential equations of first order
on ${\mathcal{M}}^{0}\pi$ with space of parameters $M$ and whose solutions are
sections of the projection $\pi\circ\nu^{0}:{\mathcal{M}}^{0}\pi\to M$. More
precisely, if we denote by

$h(x^{i},u^{\alpha},p_{\alpha}^{i})=(x^{i},u^{\alpha},-H(x,u,p),p_{\alpha}^{i}),$

the local expression of the Hamiltonian section $h$, then the Hamilton-
deDonder-Weyl equations are locally

$\displaystyle\frac{\partial u^{\alpha}}{\partial x^{i}}=\frac{\partial
H}{\partial p_{\alpha}^{i}},\;\;\mbox{ for all }i\mbox{ and }\alpha,$ (1.5)
$\displaystyle\sum_{i}\frac{\partial p_{\alpha}^{i}}{\partial
x^{i}}=-\frac{\partial H}{\partial u^{\alpha}},\;\;\mbox{ for all }\alpha.$

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
$\omega_{h}=h^{*}(\omega_{{\mathcal{M}}\pi})$ on ${\mathcal{M}}^{0}\pi$. Then,
using $\omega_{h}$, a special type of Ehresmann connections can be introduced
on the fibration $\pi\circ\nu^{0}:{\mathcal{M}}^{0}\pi\to M$ (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
$\omega_{h}$ can also be used to directly characterize the sections of the
projection $\pi\circ\nu^{0}:{\mathcal{M}}^{0}\pi\to M$ which are solutions of
the Hamilton-deDonder-Weyl equations for $h$ (see [11, 19, 21, 25, 26]). From
$\omega_{h}$, 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 $\nabla$ on the configuration bundle
$\pi:E\to M$ and the Hamiltonian section $h$, one may produce a Hamiltonian
energy $H^{\nabla}:{\mathcal{M}}^{0}\pi\to\mathbb{R}$ associated to $h$ and
$\nabla$ and a non-canonical multisymplectic structure on
${\mathcal{M}}^{0}\pi$ 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 ${\mathcal{M}}^{0}\pi$ which are horizontal with
respect to the projection $\pi\circ\nu^{0}:{\mathcal{M}}^{0}\pi\to M$) and one
may introduce a “Poisson bracket” of a current and an Hamiltonian energy
associated to $\nabla$. 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
${\mathcal{F}}_{h}$, which is a smooth $\pi^{*}(\Lambda^{m}T^{*}M)$-valued
function defined on ${\mathcal{M}}\pi$ see [10, 32]; see also [24] for the
particular case when a volume form on $M$ is fixed. Then, using
${\mathcal{F}}_{h}$ and the canonical multisymplectic structure
$\omega_{{\mathcal{M}}\pi}$ one write intrinsically a system of partial
differential equations on ${\mathcal{M}}\pi$ whose solutions are sections of
the fibration $\pi\circ\nu:{\mathcal{M}}\pi\to M$. The projection, via $\mu$,
of these sections are the solutions of the Hamilton-deDonder-Weyl equations
for $h$ (see [24]).

  * •

From the configuration bundle $\pi:E\to M$ one can construct the phase bundle
$\mathbb{P}(\pi)$, an affine bundle over ${\mathcal{M}}^{0}\pi$, and the
differential $dh:{\mathcal{M}}^{0}\pi\to\mathbb{P}(\pi)$ of the Hamiltonian
section $h$, as a section of $\mathbb{P}(\pi)$. In addition, an affine bundle
epimorphism $A:J^{1}(\pi\circ\nu^{0})\to\mathbb{P}(\pi)$ from the $1$-jet
bundle of the fibration $\pi\circ\nu^{0}:{\mathcal{M}}^{0}\pi\to M$ onto
$\mathbb{P}(\pi)$ may be also introduced. Then, the solutions of the Hamilton-
deDonder-Weyl equations are the sections $s^{0}:M\to{\mathcal{M}}^{0}\pi$
whose first prolongation $j^{1}s^{0}:M\to J^{1}(\pi\circ\nu^{0})$ is contained
in the submanifold $A^{-1}(dh({\mathcal{M}}^{0}\pi))$ (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 $\mathbb{P}(\pi)$ associated with the
configuration bundle $\pi:E\to M$ and the differential of a Hamiltonian
section $h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$ as a section of
$\mathbb{P}(\pi)$ (see [32]).

  * •

The affine bundle epimorphism $A:J^{1}(\pi\circ\nu^{0})\to\mathbb{P}(\pi)$
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

$\sharp^{\rm aff}:\mathbb{P}(\pi)\to
J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A$

where $\operatorname{Ker}A$ is the kernel of the affine bundle epimorphism
$A:J^{1}(\pi\circ\nu^{0})\to\mathbb{P}(\pi)$ and $\sharp^{\rm
aff}=\hat{A}^{-1}$, with
$\hat{A}:J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A\to\mathbb{P}(\pi)$ the
affine bundle isomorphism induced by $A$. Then, we introduce the section

$\Gamma_{h}:{\mathcal{M}}^{0}\pi\to
J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A$

of $J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A$, given by

$\Gamma_{h}=\sharp^{\rm aff}\circ dh$

and we prove the following result: the section $\Gamma_{h}$ is canonically
identified to the equivalence class of Ehresmann connections on the fibration
$\pi\circ\nu^{0}:{\mathcal{M}}^{0}\pi\to M$ that are Hamiltonian connections
for $h$, see Theorem 3.8.

So, the section $\Gamma_{h}$ associated to the Hamiltonian section $h$ is the
field-theoretic analogue to the Hamiltonian vector field $X_{H}$ associated to
a Hamiltonian function $H$ in Classical Mechanics. The following commutative
diagram illustrates the situation

$\textstyle{{\mathbb{P}(\pi)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sharp^{\rm
aff}}$$\textstyle{J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A}$$\textstyle{{\mathcal{M}}^{0}\pi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{dh}$$\scriptstyle{\Gamma_{h}}$

The analogy with the corresponding diagram given in 1.1 for classical
mechanics is evident.

The next step is to introduce a suitable space of currents ${\mathcal{O}}$ (a
vector subspace of $(m-1)$-forms on ${\mathcal{M}}^{0}\pi$ which are
horizontal with respect to the fibration
$\pi\circ\nu^{0}:{\mathcal{M}}^{0}\pi\to M$), in such a way that the
restriction of the standard exterior differential to ${\mathcal{O}}$ takes
values in the space of sections of the vector bundle
$(J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A)^{+}={\rm
Aff}(J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A,(\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))$,
that is, we have the linear map

$d:{\mathcal{O}}\to\Gamma(J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A)^{+}.$

The dual vector bundle $(J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A)^{+}$ is
chosen so that these differentials can be canonically paired with the
Hamiltonian connections $\Gamma_{h}$, thereby extending to the field-theoretic
context the pairing $\left\langle dF,X_{H}\right\rangle$ 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
${\mathcal{O}}$ and although it is different to the motivation in [12],
${\mathcal{O}}$ just coincides with the space of currents in [12] (see Remark
3.14).

Now, if $\Gamma(\mu)$ is the space of Hamiltonian sections, we can define the
linear-affine canonical bracket

$\\{\cdot,\cdot\\}:{\mathcal{O}}\times\Gamma(\mu)\to\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M)),$
(1.6)

given by

$\\{\alpha^{0},h\\}=\langle d\alpha^{0},\Gamma_{h}\rangle=\langle
d\alpha^{0},\sharp^{\rm aff}(dh)\rangle,\;\;\mbox{ for
}\alpha^{0}\in{\mathcal{O}}.$

Then, one may prove that the evolution of any current
$\alpha^{0}\in\mathcal{O}$ along a solution $s^{0}:M\to{\mathcal{M}}^{0}\pi$
of the Hamilton-deDonder-Weyl equations for $h$ is given by

$(s^{0})^{*}(d\alpha^{0})=\\{\alpha^{0},h\\}\circ s^{0}.$ (1.7)

Conversely, if $s^{0}:M\to{\mathcal{M}}^{0}\pi$ is such that (1.7) holds for
all $\alpha^{0}\in\mathcal{O}$, 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
$\\{\cdot,\cdot\\}$ in (1.6): the space $\Gamma(\mu)$ of Hamiltonian sections
is an affine space modelled over the vector space
$\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))$. Recalling that the
canonical Poisson bracket on $T^{*}Q$ induces a Lie algebra structure on
$C^{\infty}(T^{*}Q)$, a new question arises:

What are the algebraic properties of the bracket
$\\{\cdot,\cdot\\}:{\mathcal{O}}\times\Gamma(\mu)\to\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))$?

Related to this question, we will prove that ${\mathcal{O}}$ admits a
canonical Lie algebra structure $\\{\cdot,\cdot\\}_{\mathcal{O}}$ (see Theorem
4.2) and that the linear map

${\mathcal{R}}:{\mathcal{O}}\to{\rm
Aff}\left(\Gamma(\mu),\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))\right)$

defined by

${\mathcal{R}}(\alpha^{0})=\\{\alpha^{0},\cdot\\}$

is a representation of the Lie algebra
$({\mathcal{O}},\\{\cdot,\cdot\\}_{\mathcal{O}})$ on the affine space
$\Gamma(\mu)$ (see Theorem 4.3 and the property (4.13)).

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
$\\{\cdot,\cdot\\}:{\mathcal{O}}\times\Gamma(\mu)\to\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))$
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 ${\mathcal{O}}$ and we prove that $\\{\cdot,\cdot\\}$ induces a
representation of the Lie algebra ${\mathcal{O}}$ on the affine space
$\Gamma(\mu)$ 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.

##  2\. 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 $\pi:E\to
M$, that is, a surjective submersion from $E$ to $M$. We assume $dim\;M=m$ and
$dim\;E=m+n$.

The extended multimomentum bundle ${\mathcal{M}}\pi$ associated with the
configuration bundle $\pi:E\to M$ is the vector bundle over $E$ whose fiber at
the point $y\in E$ is

${\mathcal{M}}_{y}\pi=\\{\varphi:J^{1}_{y}\pi\to\Lambda^{m}T^{*}_{\pi(y)}M\mid\varphi\mbox{
is affine }\\}.$

Here, $J^{1}\pi=\cup_{y\in E}J^{1}_{y}\pi$ is the $1$-jet bundle of the
fibration $\pi:E\to M$ (see Appendix A).

It is well-known that ${\mathcal{M}}\pi$ may be identified with the vector
bundle $\Lambda_{2}^{m}(T^{*}E)$ over $E$, whose fiber at $y\in E$ is

$\Lambda^{m}_{2}(T_{y}^{*}E)=\\{\gamma\in\Lambda^{m}(T_{y}^{*}E)\mid
i_{u}i_{u^{\prime}}\gamma=0,\forall u,u^{\prime}\in V_{y}\pi\\}.$

In fact, if $\gamma\in\Lambda_{2}^{m}(T_{y}^{*}E)$ and $z:T_{\pi(y)}M\to
T_{y}E\in J^{1}_{y}\pi$ then

$\langle\gamma,z\rangle=\Lambda^{m}z^{*}(\gamma).$ (2.1)

If $(x^{i},u^{\alpha})$ are local coordinates on $E$ which are adapted with
the fibration $\pi$, then $\gamma\in\Lambda^{m}_{2}(T_{y}^{*}E)$ reads locally

$\gamma=p\,d^{m}x+p^{i}_{\alpha}du^{\alpha}\wedge d^{m-1}x_{i},$

where

$d^{m}x=dx^{1}\wedge\dots\wedge dx^{m}\;\;\mbox{ and
}\;\;d^{m-1}x_{j}=i(\frac{\partial}{\partial x^{j}})d^{m}x.$

So, $(x^{i},u^{\alpha},p,p_{\alpha}^{i})$ are local coordinates on
${\mathcal{M}}\pi$.

On ${\mathcal{M}}\pi\simeq\Lambda^{m}_{2}(T^{*}E)$ we can define a canonical
$m$-form $\lambda_{{\mathcal{M}}\pi}$ as follows

$\lambda_{{\mathcal{M}}\pi}(\gamma)(Y_{1},\dots,Y_{m})=\gamma((T_{\gamma}\nu)(Y_{1}),\dots,(T_{\gamma}\nu)(Y_{m})),$
(2.2)

for $\gamma\in\Lambda_{2}^{m}(T^{*}E)$ and $Y_{1},\dots,Y_{m}\in
T_{\gamma}\Lambda^{m}_{2}(T^{*}E)$, with $\nu:{\mathcal{M}}\pi\to E$ the
vector bundle projection.

From (2.2), $\lambda_{{\mathcal{M}}\pi}$ has the local expression

$\lambda_{{\mathcal{M}}\pi}=pd^{m}x+p_{\alpha}^{i}du^{\alpha}\wedge
d^{m-1}x_{i}.$

The canonical multisymplectic structure $\omega_{{\mathcal{M}}\pi}$ on
${\mathcal{M}}\pi$ is the $(m+1)$-form given by

$\omega_{{\mathcal{M}}\pi}=-d\lambda_{{\mathcal{M}}\pi}.$

Locally, we have

$\omega_{{\mathcal{M}}\pi}=-dp\wedge d^{m}x+du^{\alpha}\wedge
dp_{\alpha}^{i}\wedge d^{m-1}x_{i}.$ (2.3)

It is clear that $\omega_{{\mathcal{M}}\pi}$ is closed and non-degenerate,
that is, the vector bundle morphism

$\flat_{\omega_{{\mathcal{M}}\pi}}:T({\mathcal{M}}\pi)\to\Lambda^{m}(T^{*}({\mathcal{M}}\pi)),\;\;Y\in
T_{\gamma}({\mathcal{M}}\pi)\mapsto
i_{Y}\omega_{{\mathcal{M}}\pi}(\gamma)\in\Lambda^{m}T^{*}_{\gamma}({\mathcal{M}}\pi),$

is a linear monomorphism.

The restricted multimomentum bundle ${\mathcal{M}}^{0}\pi$ is the vector
bundle

${\mathcal{M}}^{0}\pi=(\pi^{*}(TM)\otimes
V^{*}\pi)\otimes\pi^{*}(\Lambda^{m}T^{*}M)\simeq\pi^{*}(\Lambda^{m-1}T^{*}M)\otimes
V^{*}\pi\simeq{\rm Lin}(\pi^{*}(T^{*}M)\otimes
V\pi,\pi^{*}(\Lambda^{m}T^{*}M)).$

Local coordinates on ${\mathcal{M}}^{0}\pi$ are
$(x^{i},u^{\alpha},p_{\alpha}^{i})$.

There is a canonical projection $\mu:{\mathcal{M}}\pi\to{\mathcal{M}}^{0}\pi$
given by

$\mu(\gamma)=\gamma^{l},$

for $\gamma:J^{1}_{y}\pi\to\Lambda^{m}T^{*}_{\pi(y)}M\in{\mathcal{M}}_{y}\pi$,
where $\gamma^{l}:T^{*}_{\pi(y)}M\otimes
V_{y}\pi\to\Lambda^{m}T^{*}_{\pi(y)}M$ is the linear map associated with
$\gamma$. The local expression of $\mu$ is

$\mu(x^{i},u^{\alpha},p,p_{\alpha}^{i})=(x^{i},u^{\alpha},p_{\alpha}^{i}).$

Note that if $\gamma,\gamma^{\prime}\in{\mathcal{M}}\pi=\Lambda_{2}^{m}T^{*}E$
then

$\mu(\gamma)=\mu(\gamma^{\prime})\Leftrightarrow\nu(\gamma)=\nu(\gamma^{\prime})\mbox{
and }\exists!\;\Omega\in\Lambda^{m}T^{*}_{\pi(\nu(\gamma))}M\mbox{ such that
}\gamma^{\prime}=\gamma+(\Lambda^{m}T^{*}_{\nu(\gamma)}\pi)(\Omega).$ (2.4)

######  Remark 2.1.

Note that this last statement implies that
$\mathcal{M}^{0}\pi\simeq{\mathcal{M}}\pi/\sim$. 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
$\Lambda^{m}T^{*}\mathcal{M}^{0}\pi$, 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 $\mathbb{R}$ on $\mathcal{M}^{0}\pi$, which
preserves the multisymplectic structure, and $\mathcal{M}\pi$ 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]. $\diamond$

###  2.2. Hamilton-deDonder-Weyl equations

Given a configuration bundle $\pi:E\to M$, a Hamiltonian section
$h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$ is a smooth section of the
canonical projection

$\mu:{\mathcal{M}}\pi\to{\mathcal{M}}^{0}\pi.$

The local expression of $h$ is

$h(x^{i},u^{\alpha},p_{\alpha}^{i})=(x^{i},u^{\alpha},-H(x,u,p),p_{\alpha}^{i}),$

where $H$ is a local real $C^{\infty}$-function on ${\mathcal{M}}^{0}\pi$.

Using the Hamiltonian section, we can define the $(m+1)$-form $\omega_{h}$ on
${\mathcal{M}}^{0}\pi$ given by

$\omega_{h}=h^{*}(\omega_{{\mathcal{M}}\pi}).$ (2.5)

The local expression of $\omega_{h}$ is

$\omega_{h}=du^{\alpha}\wedge dp_{\alpha}^{i}\wedge d^{m-1}x_{i}+dH\wedge
d^{m}x=du^{\alpha}\wedge dp_{\alpha}^{i}\wedge d^{m-1}x_{i}+\frac{\partial
H}{\partial u^{\alpha}}du^{\alpha}\wedge d^{m}x+\frac{\partial H}{\partial
p_{\alpha}^{i}}dp_{\alpha}^{i}\wedge d^{m}x.$ (2.6)

Note that if $m=1$, $\omega_{h}$ is degenerate and the rank of its kernel is
$1$. On the other hand, if $m\geq 2$, $\omega_{h}$ is non-degenerate which
implies that it is multisymplectic.

######  Proposition 2.2.

A (local) section $s^{0}:U\subseteq M\to{\mathcal{M}}^{0}\pi$ of the
projection $\pi\circ\nu^{0}:{\mathcal{M}}^{0}\pi\to M$ is a solution of the
Hamilton-deDonder-Weyl equations iff

$(s^{0})^{*}(i_{U}\omega_{h})=0,\;\;\;\forall\;U\in\Gamma(V(\pi\circ\nu^{0})),$

where $\Gamma(V(\pi\circ\nu^{0}))$ is the space of sections of the vertical
bundle to $\pi\circ\nu^{0}:{\mathcal{M}}^{0}\pi\to M$.

###### Proof.

Using the local expression
$s^{0}(x^{i})=(x^{i},u^{\alpha}(x),p_{\alpha}^{i}(x))$, it is routine to
verify that $(s^{0})^{*}(i_{U}\omega_{h})=0$ if and only if

$\frac{\partial u^{\alpha}}{\partial x^{i}}=\frac{\partial H}{\partial
p_{\alpha}^{i}},\;\;\mbox{ for all }i\mbox{ and }\alpha,$
$\sum_{i}\frac{\partial p_{\alpha}^{i}}{\partial x^{i}}=-\frac{\partial
H}{\partial u^{\alpha}},\;\;\mbox{ for all }\alpha.$

∎

##  3\. 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
$\flat:T(T^{*}Q)\to T^{*}(T^{*}Q)$ over the identity with inverse denoted
$\sharp:T^{*}(T^{*}Q)\to T(T^{*}Q)$. The Hamiltonian is a real
$C^{\infty}$-function on $T^{*}Q$ and the Hamiltonian vector field is given in
terms of the differential $dH$ and the vector bundle isomorphism $\sharp$ as
$X_{H}=\sharp dH$, see Diagram (1.1). For field theories, we don’t have a
Hamiltonian function, but a Hamiltonian section

$h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$

of the canonical projection $\mu:{\mathcal{M}}\pi\to{\mathcal{M}}^{0}\pi$. 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 $\mathbb{P}(\pi)$ associated with the
fibration $\mu:{\mathcal{M}}\pi\to{\mathcal{M}}^{0}\pi$. The bundle
$\mathbb{P}(\pi)$ 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
$\sharp$ in §3.2 and the field-theoretic analogue $\Gamma_{h}$ to the
Hamiltonian vector field $X_{H}$ in §3.3. In particular, we will show that
$\Gamma_{h}$ 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 $C^{\infty}(T^{*}Q)$ 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 $\Gamma_{h}$, 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. The phase bundle associated with a fibration and the differential of
a Hamiltonian section

Let $\pi:E\to M$ be the configuration bundle of the field theory and
$h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$ be the Hamiltonian section. Then,
although

$dim\;{\mathcal{M}}\pi=dim\;{\mathcal{M}}^{0}\pi+1,$

$h$ cannot be identified, in general, with a real $C^{\infty}$-function on
${\mathcal{M}}^{0}\pi$. However, to $h$ we can associate an extended
Hamiltonian density

${\mathcal{F}}_{h}:{\mathcal{M}}\pi\to\pi^{*}(\Lambda^{m}T^{*}M),$

defined as follows. If $\gamma\in{\mathcal{M}}\pi$ we have
$\mu(\gamma)=\mu(h(\mu(\gamma)))$ and hence using (2.4), we conclude that
there exists a unique $\Omega\in\Lambda^{m}T^{*}_{\pi(\nu(\gamma))}M$ such
that $\gamma=h(\mu(\gamma))+(\Lambda^{m}T^{*}_{\nu(\gamma)}\pi)(\Omega)$. We
thus define

${\mathcal{F}}_{h}(\gamma)=\Omega.$

####  3.1.1. The differential of $\mathcal{F}_{h}$ and the extended phase
bundle

Note that ${\mathcal{F}}_{h}$ may be considered, in a natural way, as a
$m$-form on ${\mathcal{M}}\pi$. Thus, we can take its exterior differential
and we obtain a $(m+1)$-form on ${\mathcal{M}}\pi$ which is a section of the
vector bundle

$\Lambda^{m+1}_{2}(T^{*}({\mathcal{M}}\pi))\to{\mathcal{M}}\pi.$

Now, it is easy to prove that the vector bundles
$\Lambda^{m+1}_{2}(T^{*}({\mathcal{M}}\pi))$ and
$V^{*}(\pi\circ\nu)\otimes(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)$ are
isomorphic. In fact, an isomorphism

$\Psi:\Lambda^{m+1}_{2}(T^{*}({\mathcal{M}}\pi))\to
V^{*}(\pi\circ\nu)\otimes(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)$

is given by

$\Psi(\tilde{\theta}):V_{\gamma}(\pi\circ\nu)\to\Lambda^{m}(T^{*}_{\pi(\nu(\gamma))}M),\;\;\tilde{U}\in
V_{\gamma}(\pi\circ\nu)\to
i_{\tilde{U}}\tilde{\theta}\in\Lambda^{m}(T^{*}_{\pi(\nu(\gamma))}M),$

for $\tilde{\theta}\in\Lambda^{m+1}_{2}(T^{*}_{\gamma}({\mathcal{M}}\pi))$ and
$\gamma\in{\mathcal{M}}\pi$. Note that
$i_{\tilde{U}}\tilde{\theta}\in\Lambda^{m}_{1}(T^{*}_{\gamma}({\mathcal{M}}\pi))$,
therefore, it induces an element of $\Lambda^{m}(T^{*}_{\pi(\nu(\gamma))}M)$.

We denote by $d^{v}{\mathcal{F}}_{h}$ the section of the vector bundle
$V^{*}(\pi\circ\nu)\otimes(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)\to{\mathcal{M}}\pi$
induced by the differential of ${\mathcal{F}}_{h}$. In local coordinates, if

$h(x^{i},u^{\alpha},p_{\alpha}^{i})=(x^{i},u^{\alpha},-H(x,u,p),p_{\alpha}^{i})$

then

${\mathcal{F}}_{h}(x^{i},u^{\alpha},p,p_{\alpha}^{i})=(p+H(x^{i},u^{\alpha},p_{\alpha}^{i}))d^{m}x$
(3.1)

and

$d^{v}{\mathcal{F}}_{h}(x^{i},u^{\alpha},p,p_{\alpha}^{i})=\Big{(}dp+\frac{\partial
H}{\partial u^{\alpha}}du^{\alpha}+\frac{\partial H}{\partial
p_{\alpha}^{i}}dp_{\alpha}^{i}\Big{)}\otimes d^{m}x.$ (3.2)

Note that if $\mu:{\mathcal{M}}\pi\to{\mathcal{M}}^{0}\pi$ is the canonical
projection, $\Phi$ is a $m$-form on $M$ and $\Phi^{\bf v}\in\Gamma(V\mu)$ is
the vertical lift to ${\mathcal{M}}\pi$ (see Appendix B) then, using (3.2) and
(B.1), we deduce that

$(d^{v}{\mathcal{F}}_{h})(\Phi^{\bf v})=(\pi\circ\nu)^{*}(\Phi).$ (3.3)

This property of $d^{v}{\mathcal{F}}_{h}$ motivates the definition of the
following affine subbundle of
$V^{*}(\pi\circ\nu)\otimes(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)$.

######  Definition 3.1.

The extended phase bundle of the configuration bundle $\pi:E\to M$ is the
affine subbundle $\widetilde{\mathbb{P}(\pi)}$ of
$V^{*}(\pi\circ\nu)\otimes(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)$ whose fiber at
the point $\gamma\in{\mathcal{M}}\pi$ is

$\widetilde{\mathbb{P}(\pi)}(\gamma)=\big{\\{}\tilde{\mathcal{A}}\in\operatorname{Lin}(V_{\gamma}(\pi\circ\nu),\Lambda^{m}T^{*}_{\pi(\nu(\gamma))}M)\mid\tilde{\mathcal{A}}(\Omega^{\bf
v}(\gamma))=\Omega,\forall\;\Omega\in\Lambda^{m}T^{*}_{\pi(\nu(\gamma))}M\big{\\}}.$
(3.4)

From (3.3) we have

$d^{v}{\mathcal{F}}_{h}\in\Gamma(\widetilde{\mathbb{P}(\pi)}).$

Note that $\widetilde{\mathbb{P}(\pi)}$ is modelled over the vector bundle
$V(\widetilde{\mathbb{P}(\pi)})$ whose fiber at the point
$\gamma\in{\mathcal{M}}\pi$ is

$V(\widetilde{\mathbb{P}(\pi)})(\gamma)=\\{\tilde{\nu}\in\operatorname{Lin}(V_{\gamma}(\pi\circ\nu),\Lambda^{m}T^{*}_{\pi(\nu(\gamma))}M)\mid\tilde{\nu}(\Omega^{\bf
v}(\gamma))=0,\forall\;\Omega\in\Lambda^{m}T^{*}_{\pi(\nu(\gamma))}M\\}.$

We remark that an element $\tilde{\mathcal{A}}$ of
$\widetilde{\mathbb{P}(\pi)}$ has the following local form

$\tilde{\mathcal{A}}=(dp+\tilde{\mathcal{A}}_{\alpha}du^{\alpha}+\tilde{\mathcal{A}}_{i}^{\alpha}dp_{\alpha}^{i})\otimes
d^{m}x$

and a generic element $\tilde{\nu}$ of $V(\widetilde{\mathbb{P}(\pi)})$ has
the local form

$\tilde{\nu}=(\tilde{\mathcal{A}}_{\alpha}du^{\alpha}+\tilde{\mathcal{A}}_{i}^{\alpha}dp_{\alpha}^{i})\otimes
d^{m}x.$

Therefore, the local coordinates on $\widetilde{\mathbb{P}(\pi)}$ and
$V(\widetilde{\mathbb{P}(\pi)})$ are
$(x^{i},u^{\alpha},p,p_{\alpha}^{i};\tilde{\mathcal{A}}_{\alpha},\tilde{\mathcal{A}}_{i}^{\alpha})$.
In addition,

$(d^{v}{\mathcal{F}}_{h})(x^{i},u^{\alpha},p,p_{\alpha}^{i})=\Big{(}x^{i},u^{\alpha},p,p_{\alpha}^{i};\frac{\partial
H}{\partial u^{\alpha}},\frac{\partial H}{\partial p_{\alpha}^{i}}\Big{)}.$
(3.5)

####  3.1.2. The differential of a Hamiltonian section and the phase bundle

Now, given a point $x\in M$, we can consider an action of the abelian group
$\Lambda^{m}T^{*}_{x}M$ on the fiber $(\pi\circ\nu)^{-1}(x)$ defined as
follows. If $\Phi\in\Lambda^{m}(T^{*}_{x}M)$ then we define
$\Phi\;\cdot:(\pi\circ\nu)^{-1}(x)\to(\pi\circ\nu)^{-1}(x)$ by

$\Phi\cdot\gamma=\gamma+(\Lambda^{m}T^{*}_{\nu(\gamma)}\pi)(\Phi),\;\;\mbox{
for }\gamma\in(\pi\circ\nu)^{-1}(x).$ (3.6)

In local coordinates, we get

$(\lambda
d^{m}x)\cdot(x^{i},u^{\alpha},p,p_{\alpha}^{i})=(x^{i},u^{\alpha},p+\lambda,p_{\alpha}^{i})$

and thus the quotient space ${\mathcal{M}}\pi/\pi^{*}(\Lambda^{m}T^{*}M)$ may
be identified with the reduced multimomentum bundle ${\mathcal{M}}^{0}\pi$.

The tangent and cotangent lift of the previous action induces a fibred action
of the vector bundle $(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)$ on the vector
bundles $V(\pi\circ\nu)$ and
$V^{*}(\pi\circ\nu)\otimes(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)$. In fact, if
$\gamma\in\Lambda^{m}_{2}(T^{*}_{y}E)$, $\tilde{U}\in V_{\gamma}(\pi\circ\nu)$
and $\Phi\in\Lambda^{m}T^{*}_{\pi(y)}M$ then the tangent lift is

$\Phi\;\cdot\tilde{U}=(T_{\gamma}(\Phi\cdot))(\tilde{U})\in
V_{\Phi\cdot\gamma}(\pi\circ\nu).$ (3.7)

Note that, using (3.6), (3.7) and (B.1), it follows that

$\Phi\cdot\Omega^{\bf v}_{\gamma}=\Omega^{\bf v}_{\Phi\cdot\gamma},$ (3.8)

for $\Omega\in\Lambda^{m}T^{*}_{\pi(y)}M$. If
$(x^{i},u^{\alpha},p,p_{\alpha}^{i};\dot{u}^{\alpha},\dot{p},\dot{p}_{\alpha}^{i})$
are local coordinates on $V(\pi\circ\nu)$, we have

$(\lambda
d^{m}x)\cdot(x^{i},u^{\alpha},p,p_{\alpha}^{i};\dot{u}^{\alpha},\dot{p},\dot{p}_{\alpha}^{i})=(x^{i},u^{\alpha},p+\lambda,p_{\alpha}^{i};\dot{u}^{\alpha},\dot{p},\dot{p}_{\alpha}^{i}).$
(3.9)

In a similar way, if $\tilde{\theta}\in
V^{*}_{\gamma}(\pi\circ\nu)\otimes\Lambda^{m}T^{*}_{\pi(\nu(\gamma))}M$ then
the cotangent lift is

$(\Phi\cdot\tilde{\theta})(\tilde{U}^{\prime})=\tilde{\theta}(-\Phi\cdot\tilde{U}^{\prime}),$
(3.10)

for $\tilde{U}^{\prime}\in V_{\Phi\cdot\gamma}(\pi\circ\nu)$. From (3.4) and
(3.8), we deduce that this action restricts to the extended phase bundle
$\widetilde{\mathbb{P}(\pi)}$, and to the vector bundle
$V(\widetilde{\mathbb{P}(\pi)})$. In local coordinates, we have

$(\lambda
d^{m}x)\cdot(x^{i},u^{\alpha},p,p_{\alpha}^{i};\tilde{\mathcal{A}}_{\alpha},\tilde{\mathcal{A}}^{\alpha}_{i})=(x^{i},u^{\alpha},p+\lambda,p_{\alpha}^{i};\tilde{\mathcal{A}}_{\alpha},\tilde{\mathcal{A}}^{\alpha}_{i}).$

Taking the quotient with respect to the action, we can introduce the following
definition.

######  Definition 3.2.

The phase bundle of the configuration bundle $\pi:E\to M$ is defined by

$\displaystyle\mathbb{P}(\pi)=\frac{\widetilde{\mathbb{P}(\pi)}}{(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)}.$

We note that $\mathbb{P}(\pi)$ is an affine bundle over
${\mathcal{M}}^{0}\pi={\mathcal{M}}\pi/\pi^{*}(\Lambda^{m}T^{*}M)$ modelled
over the vector bundle $\displaystyle
V(\mathbb{P}(\pi))=\frac{V(\widetilde{\mathbb{P}(\pi))}}{(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)}$.
This bundle is isomorphic to the vector bundle

$V^{*}(\pi\circ\nu^{0})\otimes(\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M)=\cup_{\gamma^{0}\in{\mathcal{M}}^{0}\pi}\operatorname{Lin}(V_{\gamma^{0}}(\pi\circ\nu^{0}),\Lambda^{m}T^{*}_{\pi(\nu^{0}(\gamma^{0}))}M),$

an isomorphism being given by

${\mathcal{A}}\in\operatorname{Lin}(V_{\gamma^{0}}(\pi\circ\nu^{0}),\Lambda^{m}T^{*}_{\pi(\nu^{0}(\gamma^{0}))}M)\to[\tilde{\mathcal{A}}]\in
V(\mathbb{P}(\pi))_{\gamma^{0}},$

where
$\tilde{\mathcal{A}}:V_{\gamma}(\pi\circ\nu)\to\Lambda^{m}T^{*}_{\pi(\nu(\gamma))}M$
is defined by

$\tilde{\mathcal{A}}(\tilde{U})={\mathcal{A}}((T_{\gamma}\mu)(\tilde{U})),\;\;\;\mbox{
for }\tilde{U}\in V_{\gamma}(\pi\circ\nu)$

with $\gamma\in{\mathcal{M}}\pi$ and $\mu(\gamma)=\gamma^{0}$. Local
coordinates on $\mathbb{P}(\pi)$ and $V({\mathbb{P}(\pi))}$ are

$(x^{i},u^{\alpha},p_{\alpha}^{i};{\mathcal{A}}_{\alpha},{\mathcal{A}}_{i}^{\alpha}).$
(3.11)

It is clear that there exists a one-to-one correspondence between the space of
sections of the affine bundle $\mathbb{P}(\pi)\to{\mathcal{M}}^{0}\pi$ and the
set of sections of the extended phase bundle $\widetilde{\mathbb{P}(\pi)}$
associated with $\pi$, which are
$(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)$-equivariant. So, if
$h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$ is a Hamiltonian section then,
using (3.5), it is easy to see that the vertical differential
$d^{v}{\mathcal{F}}_{h}$ is $(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)$-equivariant
and, therefore, it induces a section

$dh:{\mathcal{M}}^{0}\pi\to\mathbb{P}(\pi)$

of the phase bundle $\mathbb{P}(\pi)$. We can thus write the following
definition.

######  Definition 3.3.

The differential of a Hamiltonian section
$h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$ is the section

$dh:{\mathcal{M}}^{0}\pi\to\mathbb{P}(\pi)$

defined by the following commutative diagram

$\textstyle{{\mathcal{M}}\pi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{d^{v}{\mathcal{F}}_{h}}$$\textstyle{\widetilde{\mathbb{P}(\pi)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\mu}}$$\textstyle{{\mathcal{M}}^{0}\pi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{dh}$$\textstyle{\mathbb{P}(\pi)}$

where $\mu:{\mathcal{M}}\pi\to{\mathcal{M}}^{0}\pi$ and
$\tilde{\mu}:\widetilde{\mathbb{P}(\pi)}\to\mathbb{P}(\pi)$ are the canonical
projections.

The local expression of $dh$ is

$dh(x^{i},u^{\alpha},p_{\alpha}^{i})=\Big{(}x^{i},u^{\alpha},p_{\alpha}^{i};\frac{\partial
H}{\partial u^{\alpha}},\frac{\partial H}{\partial p_{\alpha}^{i}}\Big{)}.$
(3.12)

So, we have given an answer to Questions 1 and 2 stated above.

From the previous definition, we get the map

$d:\Gamma(\mu)\to\Gamma(\mathbb{P}(\pi)),\quad h\in\Gamma(\mu)\to
dh\in\Gamma(\mathbb{P}(\pi)).$

Note that $\Gamma(\mu)$ and $\Gamma(\mathbb{P}(\pi))$ are affine spaces
modelled over the vector spaces
$\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))$ and
$\Gamma(V(\mathbb{P}(\pi)))$, respectively, and $d$ is an affine map. Later in
the paper, we shall use the corresponding linear map
$d^{l}:\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))\to\Gamma(V(\mathbb{P}(\pi)))$
defined as follows. If
${\mathcal{F}}^{0}\in\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))$ is a
$(\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M)$-valued function on
${\mathcal{M}}^{0}\pi$ then it may be considered as a section of the vector
bundle

$\Lambda^{m}_{1}(T^{*}({\mathcal{M}}^{0}\pi))\to{\mathcal{M}}^{0}\pi.$

So, we can take the standard differential $d{\mathcal{F}}^{0}$ and we obtain a
section of the vector bundle

$\Lambda^{m+1}_{2}(T^{*}({\mathcal{M}}^{0}\pi))\to{\mathcal{M}}^{0}\pi.$

This vector bundle is isomorphic to
$V^{*}(\pi\circ\nu^{0})\otimes(\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M)\to{\mathcal{M}}^{0}\pi$,
an isomorphism

$\Psi^{0}:\Lambda^{m+1}_{2}(T^{*}({\mathcal{M}}^{0}\pi))\to
V^{*}(\pi\circ\nu^{0})\otimes(\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M)$

is given by

$\Psi^{0}(\theta):V_{{\gamma}^{0}}(\pi\circ\nu^{0})\to\Lambda^{m}(T^{*}_{\pi(\nu^{0}(\gamma^{0}))}M),\;\;U\to
i_{U}\theta,$

for $\theta\in\Lambda^{m+1}_{2}(T^{*}_{\gamma^{0}}({\mathcal{M}}^{0}\pi))$ and
$\gamma^{0}\in{\mathcal{M}}^{0}\pi$. Note that
$i_{U}\theta\in\Lambda^{m}_{1}(T^{*}_{\gamma^{0}}({\mathcal{M}}^{0}\pi))$ and,
therefore, it induces an element of
$\Lambda^{m}(T^{*}_{\pi(\nu^{0}(\gamma^{0}))}M)$. We denote by
$d^{l}{\mathcal{F}}^{0}$ the section of the vector bundle
$V^{*}(\pi\circ\nu^{0})\otimes(\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M)\to{\mathcal{M}}^{0}\pi$
induced by the differential $d{\mathcal{F}}^{0}$ via the isomorphism
$\Psi^{0}$. If locally

${\mathcal{F}}^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})=F^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})\otimes
d^{m}x,$

the local expression of $d^{l}{\mathcal{F}}^{0}$ is

$d^{l}{\mathcal{F}}^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})=\Big{(}x^{i},u^{\alpha},p_{\alpha}^{i};\frac{\partial
F^{0}}{\partial u^{\alpha}},\frac{\partial F^{0}}{\partial
p_{\alpha}^{i}}\Big{)}.$ (3.13)

####  3.1.3. Comments on the next steps

The differential $dh:{\mathcal{M}}^{0}\pi\to\mathbb{P}(\pi)$ of a Hamiltonian
section is the field theoretic analogue to the differential
$dH:T^{*}Q\rightarrow T^{*}(T^{*}Q)$ of a Hamiltonian function in classical
mechanics. In the next section, we will introduce a quotient affine bundle
$J^{1}(\pi\circ\nu^{0})/\operatorname{KerA}\to{\mathcal{M}}^{0}\pi$ 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

$\sharp:T^{*}(T^{*}Q)\to T(T^{*}Q)$

and the Hamiltonian vector field $X_{H}$ on $T^{*}Q$ associated with a
Hamiltonian function $H\in C^{\infty}(T^{*}Q)$ is given by $X_{H}=\sharp\circ
dH$. So, a natural question arises:

Question 3: Does there exist an affine bundle isomorphism $\sharp^{\rm
aff}:\mathbb{P}(\pi)\rightarrow J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A$
which, in the presence of a Hamiltonian section
$h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$, allows us to introduce a
distinguished section $\Gamma_{h}$ of the affine bundle
$J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A\to{\mathcal{M}}^{0}\pi$?

In the next Section 3.2, we will give an affirmative answer to Question 3 and
we will discuss the relation between $\Gamma_{h}$ and the solutions of the
Hamilton-deDonder-Weyl equations for $h$. The section $\Gamma_{h}$ will play
the role of $X_{H}$ in Hamiltonian Mechanics.

###  3.2. The field-theoretic analogue to the canonical isomorphism
$\sharp:T^{*}(T^{*}Q)\to T(T^{*}Q)$

We will show that it is given by an affine bundle isomorphism $\sharp^{\rm
aff}:\mathbb{P}(\pi)\rightarrow J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A$.

Let $J^{1}(\pi\circ\nu^{0})$ be the $1$-jet bundle associated with the
fibration $\pi\circ\nu^{0}:{\mathcal{M}}^{0}\pi\to M$ (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

$A:J^{1}(\pi\circ\nu^{0})\to\displaystyle\mathbb{P}(\pi)=\frac{\widetilde{\mathbb{P}(\pi)}}{(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)}$

over the identity of ${\mathcal{M}}^{0}\pi$. This epimorphism is constructed
in several steps.

First, we consider the vector bundle monomorphism

$\tilde{\flat}:V(\pi\circ\nu)\to{\mathcal{M}}(\pi\circ\nu)\simeq\Lambda^{m}_{2}(T^{*}{\mathcal{M}}\pi)\subseteq\Lambda^{m}(T^{*}{\mathcal{M}}\pi)$

induced by the canonical multisymplectic structure $\omega_{{\mathcal{M}}\pi}$
as follows

$\tilde{\flat}(\tilde{U})=i_{\tilde{U}}\omega_{{\mathcal{M}}\pi}(\gamma),\;\;\mbox{
for }\tilde{U}\in V_{\gamma}(\pi\circ\nu).$ (3.14)

Note that, from (2.3), there exists a unique $m$-form at $\mu(\gamma)$ on
${\mathcal{M}}^{0}\pi$, which we denote by $\bar{\flat}(\tilde{U})$, such that

$i_{\tilde{U}}\omega_{{\mathcal{M}}\pi}(\gamma)=(\Lambda^{m}T^{*}_{\gamma}\mu)(\bar{\flat}(\tilde{U})).$
(3.15)

This defines a vector bundle morphism

$\bar{\flat}:V(\pi\circ\nu)\to{\mathcal{M}}(\pi\circ\nu^{0})\simeq\Lambda^{m}_{2}(T^{*}{\mathcal{M}}^{0}\pi)\subseteq\Lambda^{m}(T^{*}{\mathcal{M}}^{0}\pi)$

over the canonical projection $\mu:{\mathcal{M}}\pi\to{\mathcal{M}}^{0}\pi$.
Using local coordinates
$(x^{i},u^{\alpha},p,p_{\alpha}^{i};\dot{u^{\alpha}},\dot{p},\dot{p}_{\alpha}^{i})$
and
$(x^{i},u^{\alpha},p_{\alpha}^{i};\bar{p},\bar{p}_{\alpha}^{i},\bar{p}_{i}^{\alpha
j})$ on $V(\pi\circ\nu)$ and ${\mathcal{M}}(\pi\circ\nu^{0})$, respectively,
the local expression of $\bar{\flat}$ is

$\bar{\flat}(x^{i},u^{\alpha},p,p_{\alpha}^{i};\dot{u^{\alpha}},\dot{p},\dot{p}_{\alpha}^{i})=(x^{i},u^{\alpha},p_{\alpha}^{i};-\dot{p},-\dot{p}_{\alpha}^{i},\bar{p}_{i}^{\alpha
j}=\dot{u}^{\alpha}\delta^{j}_{i}).$ (3.16)

If $\gamma\in{\mathcal{M}}\pi$, the following commutative diagram

$\textstyle{V_{\gamma}(\pi\circ\nu)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\flat}_{\gamma}}$$\scriptstyle{\bar{\flat}_{\gamma}}$$\textstyle{{\mathcal{M}}_{\gamma}(\pi\circ\nu)=\Lambda_{2}^{m}T_{\gamma}^{*}({\mathcal{M}}\pi)}$$\textstyle{{\mathcal{M}}_{\mu(\gamma)}(\pi\circ\nu^{0})=\Lambda_{2}^{m}T_{\mu(\gamma)}^{*}({\mathcal{M}}^{0}\pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Lambda^{m}T^{*}_{\gamma}\mu}$
(3.17)

illustrates the relation between $\tilde{\flat}$ and $\bar{\flat}$.

We shall now use the vector bundle morphism $\bar{\flat}$ to construct $A$.
For $\gamma^{0}\in{\mathcal{M}}^{0}\pi$ and $Z^{0}\in
J^{1}_{\gamma^{0}}(\pi\circ\nu^{0})$ one first defines

$\widetilde{A(Z^{0})}\in\widetilde{\mathbb{P}(\pi)}(\gamma)\subseteq
V_{\gamma}^{*}(\pi\circ\nu)\otimes\Lambda^{m}T^{*}_{\pi(\nu(\gamma))}M,$

with $\gamma\in{\mathcal{M}}\pi$ such that $\mu(\gamma)=\gamma^{0}$, as
follows:

$\big{\langle}\widetilde{A(Z^{0})},\tilde{U}\big{\rangle}=-\Lambda^{m}((Z^{0})^{*})(\bar{\flat}(\tilde{U})),$
(3.18)

for $\tilde{U}\in V_{\gamma}(\pi\circ\nu)$. Then, if
$\tilde{\mu}:\widetilde{\mathbb{P}(\pi)}\to\mathbb{P}(\pi)$ is the canonical
projection, we set

$A(Z^{0})=\tilde{\mu}(\widetilde{A(Z^{0})}).$ (3.19)

Note that $A$ is well-defined and its local expression is

$A(x^{i},u^{\alpha},p_{\alpha}^{i};u^{\alpha}_{j},p_{\alpha
j}^{i})=\Big{(}x^{i},u^{\alpha},p_{\alpha}^{i};-\sum_{i}p^{i}_{\alpha
i},u^{\alpha}_{j}\Big{)}.$ (3.20)

This proves that $A$ is an affine bundle epimorphism over the identity of
${\mathcal{M}}^{0}\pi$ (for more details, see [32]).

Recall that the affine bundle $J^{1}(\pi\circ\nu^{0})\to{\mathcal{M}}^{0}\pi$
is modelled over the vector bundle
$V(J^{1}(\pi\circ\nu^{0}))=(\pi\circ\nu^{0})^{*}(T^{*}M)\otimes
V(\pi\circ\nu^{0})\to{\mathcal{M}}^{0}\pi$. We denote by
$(x^{i},u^{\alpha},p_{\alpha}^{i};u^{\alpha}_{j},p^{i}_{\alpha j})$ the
standard coordinates on $J^{1}(\pi\circ\nu^{0})$ and
$V(J^{1}(\pi\circ\nu^{0}))$ (see Appendix A). From the local expression
(3.20), it follows that the kernel of $A$ is a vector subbundle of
$V(J^{1}(\pi\circ\nu^{0}))$ which is locally characterized by

$\operatorname{Ker}A=\big{\\{}(x^{i},u^{\alpha},p_{\alpha}^{i};u^{\alpha}_{j},p^{i}_{\alpha
j})\in V(J^{1}(\pi\circ\nu^{0}))\mid u^{\alpha}_{j}=0,\sum_{i}p_{\alpha
i}^{i}=0,\forall\alpha,i\big{\\}}.$ (3.21)

We can thus consider the quotient affine bundle
$J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A\to{\mathcal{M}}^{0}\pi$ which is
modelled over the quotient vector bundle $(\pi\circ\nu^{0})^{*}(T^{*}M)\otimes
V(\pi\circ\nu^{0})/\operatorname{Ker}A\to{\mathcal{M}}^{0}\pi$. From (3.21),
we have that a local basis of sections for this vector bundle is

$\big{\\{}\nu_{\alpha}^{i}=\big{[}dx^{i}\otimes\frac{\partial}{\partial
u^{\alpha}}\big{]},\nu^{\alpha}=1/m\big{[}dx^{1}\otimes\frac{\partial}{\partial
p_{\alpha}^{1}}+\dots+dx^{m}\otimes\frac{\partial}{\partial
p_{\alpha}^{m}}\big{]}\big{\\}}$

for $i\in\\{1,\dots,m\\}$ and $\alpha\in\\{1,\dots,n\\}$. Note that in the
quotient vector bundle

$\big{[}dx^{i}\otimes\frac{\partial}{\partial
p_{\alpha}^{i}}\big{]}=\big{[}dx^{j}\otimes\frac{\partial}{\partial
p_{\alpha}^{j}}\big{]},\;\;\forall\;i,j\mbox{ and }\alpha.$

Local coordinates associated to this basis of sections on the quotient vector
bundle $(\pi\circ\nu^{0})^{*}(T^{*}M)\otimes
V(\pi\circ\nu^{0})/\operatorname{Ker}A\to{\mathcal{M}}^{0}\pi$ (and also on
the quotient affine bundle $J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A$) are
denoted
$(x^{i},u^{\alpha},p_{\alpha}^{i};\hat{u}^{\alpha}_{j},\hat{p}_{\alpha})$

The affine bundle epimorphism $A:J^{1}(\pi\circ\nu^{0})\to\mathbb{P}(\pi)$
induces an affine bundle isomorphism

$\hat{A}:J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A\to\mathbb{P}(\pi)$

and, from (3.20), we deduce that the local expression of $\hat{A}$ is

$\hat{A}(x^{i},u^{\alpha},p_{\alpha}^{i};\hat{u}^{\alpha}_{i},\hat{p}_{\alpha})=(x^{i},u^{\alpha},p_{\alpha}^{i};-\hat{p}_{\alpha},\hat{u}^{\alpha}_{i}).$
(3.22)

By definition, the affine bundle isomorphism

$\sharp^{\rm aff}:\mathbb{P}(\pi)\to
J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A$

is the inverse isomorphism to
$\hat{A}:J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A\to\mathbb{P}(\pi)$. If we
consider the local coordinates
$(x^{i},u^{\alpha},p_{\alpha}^{i};{\mathcal{A}}_{\alpha},{\mathcal{A}}^{\alpha}_{i})$
on the phase bundle $\mathbb{P}(\pi)$ then, using (3.22), it follows that

$\sharp^{\rm
aff}(x^{i},u^{\alpha},p_{\alpha}^{i};{\mathcal{A}}_{\alpha},{\mathcal{A}}^{\alpha}_{i})=(x^{i},u^{\alpha},p_{\alpha}^{i};{\mathcal{A}}^{\alpha}_{i},-{\mathcal{A}}_{\alpha}).$
(3.23)

###  3.3. The field-theoretic analogue to the Hamiltonian vector field
$X_{H}:TQ\rightarrow T(T^{*}Q)$

Let $h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$ be a Hamiltonian section. We
have seen that the differential of $h$

$dh:{\mathcal{M}}^{0}\pi\to\mathbb{P}(\pi)$

is a section of the phase bundle $\mathbb{P}(\pi)$. So, we can define the
section

$\Gamma_{h}:{\mathcal{M}}^{0}\pi\to
J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A$

of the quotient affine bundle
$J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A\to{\mathcal{M}}^{0}\pi$ by

$\Gamma_{h}=\sharp^{\rm aff}\circ dh.$ (3.24)

Using (3.12), (3.23) and the local basis of sections
$\\{\nu_{\alpha}^{i},\nu^{\alpha}\\}$ introduced above, we obtain that the
local expression of $\Gamma_{h}$ is

$\Gamma_{h}-[dx^{i}\otimes\frac{\partial}{\partial x^{i}}]=\frac{\partial
H}{\partial p_{\alpha}^{i}}\nu^{i}_{\alpha}-\frac{\partial H}{\partial
u^{\alpha}}\nu^{\alpha}.$ (3.25)

Now, we will show that $\Gamma_{h}$ 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 $\Gamma_{h}$ and the
solutions of the Hamilton-deDonder-Weyl equations for $h$. This uses the
notion of a Hamiltonian connection (see [19, 21]).

Let $\tau:N\to B$ be an arbitrary fibration and $H$ an Ehresmann connection on
$\tau:N\to B$. Denote by ${}^{H}:N\times_{B}TB\to H\subseteq TN$ the
horizontal lift induced by $H$ (see Appendix A). It is clear that if $1\leq
r\leq b=\operatorname{dim}B$ the previous map induces a vector bundle
isomorphism between $N\times_{B}\Lambda^{r}(TB)$ and $\Lambda^{r}H$ which we
also denote by

${}^{H}:N\times_{B}\Lambda^{r}(TB)\to\Lambda^{r}H\subseteq\Lambda^{r}(TN).$

So, if $\chi\in\Gamma(N\times_{B}\Lambda^{r}(TB))$, the image $\chi^{H}$ of
$\chi$ by the previous map is called the horizontal lift of $\chi$. If
$(b^{i},n^{\alpha})$ are local coordinates on $N$ which are adapted to the
fibration $\tau$, then the horizontal lift reads locally

$\left(\frac{\partial}{\partial b^{i}}\right)^{H}=\frac{\partial}{\partial
b^{i}}+H^{\alpha}_{i}\frac{\partial}{\partial n^{\alpha}},\;\;\forall
i\in\\{1,\dots,b\\}$

and if

$\chi=\chi_{i_{1}\dots i_{r}}\frac{\partial}{\partial
b^{i_{1}}}\wedge\dots\wedge\frac{\partial}{\partial
b^{i_{r}}}\in\Gamma(\Lambda^{r}TB)$

then its horizontal lift is

$\chi^{H}=\chi_{i_{1}\dots i_{r}}\left(\frac{\partial}{\partial
b^{i_{1}}}\right)^{H}\wedge\dots\wedge\left(\frac{\partial}{\partial
b^{i_{r}}}\right)^{H}=\chi_{i_{1}\dots i_{r}}\left(\frac{\partial}{\partial
b^{i_{1}}}+H_{i_{1}}^{\alpha_{1}}\frac{\partial}{\partial
n^{\alpha_{1}}}\right)\wedge\dots\wedge\left(\frac{\partial}{\partial
b^{i_{r}}}+H_{i_{r}}^{\alpha_{r}}\frac{\partial}{\partial
n^{\alpha_{r}}}\right).$

Now, suppose that $\pi:E\to M$ is the configuration bundle of a Hamiltonian
Classical Field Theory of first order, with $m=\operatorname{dim}M$, and
consider the fibration $\pi\circ\nu^{0}:{\mathcal{M}}^{0}\pi\to M$. For a
Hamiltonian section $h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$, we denote by
$\omega_{h}$ the $(m+1)$-form on ${\mathcal{M}}^{0}\pi$ given by (2.5).

Then, we may prove the following result.

######  Lemma 3.4.

Let $h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$ be a Hamiltonian section, let
$H$ be an Ehresmann connection on the fibration
$\pi\circ\nu^{0}:{\mathcal{M}}^{0}\pi\to M$, and let $s^{0}:U\subseteq
M\to{\mathcal{M}}^{0}\pi$ be a section horizontal with respect to $H$. Then,
$s^{0}$ is a solution of the Hamilton-deDonder-Weyl equations for $h$ if and
only if

$(s^{0})^{*}\left(i_{\chi^{H}}\omega_{h}\right)=0,\;\;\forall\;\chi\in\Gamma(\Lambda^{m}(TM)).$
(3.26)

###### Proof.

Using that $s^{0}$ is horizontal with respect to $H$ we deduce that

$(s^{0})^{*}(i_{U}\omega_{h})=0,\;\;\forall\;U\in\Gamma(V(\pi\circ\nu^{0}))$

if and only if

$(s^{0})^{*}\left(i_{\chi^{H}}i_{U}\omega_{h}\right)=0,\;\;\forall\;\chi\in\Gamma(\Lambda^{m}(TM)).$
(3.27)

So, it is clear that if (3.26) holds then $s^{0}$ is a solution of the
Hamilton-deDonder-Weyl equations for $h$ by Proposition 2.2.

Conversely, assume that $s^{0}$ is a solution of the Hamilton-deDonder-Weyl
equations for $h$. Then (3.27) holds. Thus, using that

$T_{s_{0}(x)}{\mathcal{M}}^{0}\pi=\kern-3.0pt({s_{0}(x)}\times
T_{x}M)^{H}\oplus V_{s_{0}(x)}(\pi\circ\nu^{0}),\;\;\forall x\in U\subseteq
M,$

we deduce that

$(s^{0})^{*}\left(i_{\chi^{H}}\omega_{h}\right)(x)=0,\;\;\forall\chi\in\Gamma(\Lambda^{m}TM),\;\;\forall
x\in U\subseteq M.$

This proves the result. ∎

The previous result suggests the introduction of the following definition.

######  Definition 3.5.

Let $h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$ be a Hamiltonian section. An
Ehresmann connection ${}^{H}:{\mathcal{M}}^{0}\pi\times_{M}TM\to H\subseteq
T({\mathcal{M}}^{0}\pi)$ on the fibration
$\pi\circ\nu^{0}:{\mathcal{M}}^{0}\pi\to M$ is said to be a Hamiltonian
connection for $h$ if

$i_{\chi^{H}}\omega_{h}=0,\;\;\forall\chi\in\Gamma(\Lambda^{m}TM).$

As a direct consequence of the definition, if locally the Hamiltonian section
$h$ is

$h(x^{i},u^{\alpha},p_{\alpha}^{i})=(x^{i},u^{\alpha},-H(x,u,p),p_{\alpha}^{i})$

and the Ehresmann connection is

$\left(\frac{\partial}{\partial x^{i}}\right)^{H}=\frac{\partial}{\partial
x^{i}}+H_{i}^{\alpha}\frac{\partial}{\partial u^{\alpha}}+H_{\alpha
i}^{j}\frac{\partial}{\partial p_{\alpha}^{j}},$

then from (2.6), we have the equivalence

$^{H}:{\mathcal{M}}^{0}\pi\times_{M}TM\to T({\mathcal{M}}^{0}\pi)\mbox{
Hamiltonian connection for $h$ }\Leftrightarrow H^{\alpha}_{i}=\frac{\partial
H}{\partial p_{\alpha}^{i}},\;\;H^{i}_{\alpha i}=-\frac{\partial H}{\partial
u^{\alpha}}.$ (3.28)

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 $s^{H}:{\mathcal{M}}^{0}\pi\to
J^{1}(\pi\circ\nu^{0})$ of the affine bundle
$J^{1}(\pi\circ\nu^{0})\to{\mathcal{M}}^{0}\pi$ (see Appendix A). Moreover, if
$\Gamma_{h}$ is the section of the quotient affine bundle
$J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A\to{\mathcal{M}}^{0}\pi$ defined in
(3.24) then, using (3.25) and (3.28), we obtain the following result.

######  Proposition 3.6.

Let $s^{H}:{\mathcal{M}}^{0}\pi\to J^{1}(\pi\circ\nu^{0})$ be the section of
the affine bundle $J^{1}(\pi\circ\nu^{0})\to{\mathcal{M}}^{0}\pi$ induced by a
Hamiltonian connection $H$ for $h$ and $p:J^{1}(\pi\circ\nu^{0})\to
J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A$ be the canonical projection. Then

$\Gamma_{h}=p\circ s^{H}.$

From (3.21) and (3.28), we also get the following result.

######  Proposition 3.7.

If $H,\bar{H}:{\mathcal{M}}^{0}\pi\to J^{1}(\pi\circ\nu^{0})$ are Hamiltonian
connections for the same Hamiltonian section
$h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$, then they satisfy

$\bar{H}-H\in\Gamma(\operatorname{Ker}A).$

Finally, from Propositions 3.6 and 3.7, it follows the following
characterization of Hamiltonian connections for $h$.

######  Theorem 3.8.

Let $h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$ be a Hamiltonian section and
let $H$ be an Ehresmann connection for the fibration
$\pi\circ\nu^{0}:{\mathcal{M}}^{0}\pi\to M$. Then, $H$ is a Hamiltonian
connection for $h$ if and only if

$\Gamma_{h}=p\circ s^{H}.$

Theorem 3.8 suggests the introduction of the following definition.

######  Definition 3.9.

Let $h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$ be a Hamiltonian section.
Then, the section $\Gamma_{h}:{\mathcal{M}}^{0}\pi\to
J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A$ is called the equivalence class of
the Hamiltonian connections for $h$.

The following commutative diagram illustrates the results obtained in Sections
3.1, 3.2, 3.3

$\textstyle{{\mathbb{P}(\pi)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sharp^{\rm
aff}}$$\textstyle{J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A}$$\textstyle{{\mathcal{M}}^{0}\pi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{dh}$$\scriptstyle{\Gamma_{h}}$

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 $T(T^{*}Q)\to T^{*}Q$ and the space of
observables is the set $C^{\infty}(T^{*}Q)$ of real $C^{\infty}$-functions on
$T^{*}Q$. Given an observable $F\in C^{\infty}(T^{*}Q)$, we can consider a
section $dF$ (the differential of $F$) of the dual bundle $T^{*}(T^{*}Q)\to
T^{*}Q$ to $T(T^{*}Q)\to T^{*}Q$ and the evolution of the observable $F$ along
a solution $s:I\subseteq\mathbb{R}\to T^{*}Q$ of Hamilton’s equation is given
as

$\frac{d}{dt}(F\circ s)=\langle dF,X_{H}\rangle\circ s.$

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
$h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$, the object corresponding to the
Hamiltonian vector field $X_{H}$ is the section $\Gamma_{h}$ of the quotient
affine bundle
$J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A\to{\mathcal{M}}^{0}\pi$. So, we
need to overcome the following two steps:

First step: Describe the dual vector bundle
$(J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A)^{+}$ to the affine bundle
$J^{1}(\pi\circ\nu^{0})/\operatorname{KerA}\to{\mathcal{M}}^{0}\pi$.

Second step: Introduce a space ${\mathcal{O}}$ of currents and a differential
operator

$d:{\mathcal{O}}\to\Gamma\big{(}(J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A)^{+}\big{)}$
(3.29)

on this space, such that the evolution of a current
$\alpha^{0}\in{\mathcal{O}}$ along a solution $s^{0}:U\subseteq
M\to{\mathcal{M}}^{0}\pi$ is given by

$(s^{0})^{*}(d\alpha^{0})=\langle d\alpha^{0},\Gamma_{h}\rangle\circ s^{0}.$

We will show that $s^{0}$ satisfies these equations for any
$\alpha\in\mathcal{O}$ if and only if $s^{0}$ is a solution of the Hamilton-
deDonder-Weyl equations.

First step: Let $A:J^{1}(\pi\circ\nu^{0})\to\mathbb{P}(\pi)$ (respectively,
$\hat{A}:J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A\to\mathbb{P}(\pi)$) be the
affine bundle epimorphism (respectively, isomorphism) considered in Section
3.2. Denote by $\mathbb{P}(\pi)^{+}$ and $J^{1}(\pi\circ\nu^{0})^{+}$ the
vector bundles over ${\mathcal{M}}^{0}\pi$ defined by

$\displaystyle\mathbb{P}(\pi)^{+}$ $\displaystyle={\rm
Aff}(\mathbb{P}(\pi),(\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))$ $\displaystyle
J^{1}(\pi\circ\nu^{0})^{+}$ $\displaystyle={\rm
Aff}(J^{1}(\pi\circ\nu^{0}),(\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M)).$

It is clear that $A$ induces the vector bundle morphism

$A^{+}:\mathbb{P}(\pi)^{+}\to
J^{1}(\pi\circ\nu^{0})^{+}={\mathcal{M}}(\pi\circ\nu^{0}),\qquad\langle
A^{+}(U^{0}),Z^{0}\rangle=\langle U^{0},A(Z^{0})\rangle$

for $U^{0}\in\mathbb{P}(\pi)^{+}_{\gamma^{0}}$ and $Z^{0}\in
J^{1}_{\gamma^{0}}(\pi\circ\nu^{0})$, with
$\gamma^{0}\in{\mathcal{M}}^{0}\pi$. Since $A$ is an epimorphism, we deduce
that $A^{+}$ is a vector bundle monomorphism. In addition, the image of
$A^{+}$ is the vector subbundle of $J^{1}(\pi\circ\nu^{0})^{+}$ whose fiber at
the point $\gamma^{0}\in{\mathcal{M}}^{0}\pi$ is

$A^{+}(\mathbb{P}(\pi)^{+})_{\gamma^{0}}=\big{\\{}\theta^{0}\in{\rm
Aff}(J^{1}_{\gamma^{0}}(\pi\circ\nu^{0}),\Lambda^{m}T^{*}_{\pi(\nu^{0}(\gamma^{0}))}M)\mid\big{\langle}(\theta^{0})^{l},\operatorname{Ker}A\cap
V_{\gamma^{0}}(J^{1}(\pi\circ\nu^{0}))\big{\rangle}=0\big{\\}}.$

Here,
$(\theta^{0})^{l}:V_{\gamma^{0}}(J^{1}(\pi\circ\nu^{0}))\to\Lambda^{m}T^{*}_{\pi(\nu^{0}(\gamma^{0}))}M$
denotes the linear map associated with the affine map
$\theta^{0}:J^{1}_{\gamma^{0}}(\pi\circ\nu^{0})\to\Lambda^{m}T^{*}_{\pi(\nu^{0}(\gamma^{0}))}M$.
So, we have a vector bundle isomorphism

$A^{+}:\mathbb{P}(\pi)^{+}\to
A^{+}(\mathbb{P}(\pi)^{+})\subseteq{\mathcal{M}}(\pi\circ\nu^{0})$

over the identity of ${\mathcal{M}}^{0}\pi$.

Now, denote by
$\hat{A}^{+}:\mathbb{P}(\pi)^{+}\to(J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A)^{+}$
the vector bundle isomorphism induced by
$\hat{A}:J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A\to\mathbb{P}(\pi)$. Then,
it is clear that the vector bundles $A^{+}(\mathbb{P}(\pi)^{+})$ and
$(J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A)^{+}$ can be identified and,
under this identification, $\hat{A}^{+}$ is just the vector bundle isomorphism
$A^{+}:\mathbb{P}(\pi)^{+}\to
A^{+}(\mathbb{P}(\pi)^{+})\subseteq{\mathcal{M}}(\pi\circ\nu^{0})$.

The following commutative diagram

$\textstyle{{\mathbb{P}(\pi)^{+}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{A}^{+}}$$\scriptstyle{A^{+}}$$\textstyle{(J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A)^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{A^{+}(\mathbb{P}(\pi)^{+})}$

illustrates the situation.

It is desirable to have an explicit realisation of dual vector bundle
$(J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A)^{+}$. As we know

$J^{1}(\pi\circ\nu^{0})^{+}={\mathcal{M}}(\pi\circ\nu^{0})\simeq\Lambda^{m}_{2}(T^{*}({\mathcal{M}}^{0}\pi))$

(see Section 2.1). So it is possible to describe $A^{+}(\mathbb{P}(\pi)^{+})$
and, therefore, $(J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A)^{+}$, as a
certain vector subbundle $L$ of
$\Lambda^{m}_{2}(T^{*}({\mathcal{M}}^{0}\pi))$. We shall now give such a
description.

First of all, using (3.4), it follows that the vector bundle

$\widetilde{\mathbb{P}(\pi)}^{+}={\rm
Aff}\big{(}\widetilde{\mathbb{P}(\pi)},(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)\big{)}$

is isomorphic to the vertical bundle $V(\pi\circ\nu)$ of the fibration
$\pi\circ\nu:{\mathcal{M}}\pi\to M$. An isomorphism

$\tilde{\mathcal{I}}:V(\pi\circ\nu)\to\widetilde{\mathbb{P}(\pi)}^{+}$

is given by

$\langle\tilde{\mathcal{I}}(\tilde{U}),\tilde{\mathcal{A}}\rangle=\tilde{\mathcal{A}}(\tilde{U}),\;\;\mbox{
for }\tilde{U}\in V_{\gamma}(\pi\circ\nu)\mbox{ and
}\tilde{\mathcal{A}}\in\widetilde{\mathbb{P}(\pi)}_{\gamma},$ (3.30)

with $\gamma\in{\mathcal{M}}\pi$.

Now, if we consider the standard fibred actions of
$(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)$ on $V(\pi\circ\nu)$ and
$\widetilde{\mathbb{P}(\pi)}^{+}$ then, it is clear that $\tilde{\mathcal{I}}$
is $(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)$-equivariant and, thus, it induces a
vector bundle isomorphism

${\mathcal{I}}:V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)\to\widetilde{\mathbb{P}(\pi)}^{+}/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)$

over the identity of ${\mathcal{M}}^{0}\pi$. Then, from Definition 3.2, we
deduce that the quotient vector bundle
$\widetilde{\mathbb{P}(\pi)}^{+}/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)$ is
isomorphic to $\mathbb{P}(\pi)^{+}$. So, we have a vector bundle isomorphism

${\mathcal{I}}:V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)\to\mathbb{P}(\pi)^{+}$

which is characterized by the following condition

$\langle{\mathcal{I}}[\tilde{U}],\tilde{\mu}(\tilde{\mathcal{A}})\rangle=\langle\tilde{\mathcal{I}}(\tilde{U}),\tilde{\mathcal{A}}\rangle$
(3.31)

for $\tilde{U}\in V_{\gamma}(\pi\circ\nu)$ and
$\tilde{\mathcal{A}}\in\widetilde{\mathbb{P}(\pi)}_{\gamma}$, with
$\gamma\in{\mathcal{M}}\pi$ and
$\tilde{\mu}:\widetilde{\mathbb{P}(\pi)}\to\mathbb{P}(\pi)$ the canonical
projection.

We now consider the composition

$A^{+}\circ{\mathcal{I}}:V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)\to
A^{+}(\mathbb{P}(\pi)^{+})\subseteq{\mathcal{M}}(\pi\circ\nu^{0})=\Lambda^{m}_{2}(T^{*}{\mathcal{M}}^{0}\pi)$

of the two vector bundles isomorphisms $A^{+}$ and $\mathcal{I}$ defined above
and show that $A^{+}\circ{\mathcal{I}}$ can be expressed in a simple way,
which allows to describe its image $L$ explicitly.

Consider the vector bundle morphism
$\bar{\flat}:V(\pi\circ\nu)\to{\mathcal{M}}(\pi\circ\nu^{0})\simeq\Lambda^{m}_{2}(T^{*}{\mathcal{M}}^{0}\pi)$
defined in Section 3.2 which is characterized by Eq. (3.15) and has the local
expression (3.16). Using (3.9), we deduce that $\bar{\flat}$ induces the
vector bundle morphism

$\flat:V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)\to{\mathcal{M}}(\pi\circ\nu^{0})\simeq\Lambda^{m}_{2}(T^{*}({\mathcal{M}}^{0}\pi))$

over the identity of ${\mathcal{M}}^{0}\pi$ given by

$\flat[\tilde{U}]=\bar{\flat}(\tilde{U}),\;\;\mbox{ for }\tilde{U}\in
V(\pi\circ\nu).$ (3.32)

So, if $\gamma\in{\mathcal{M}}\pi$ and
${\mathcal{J}}_{\gamma}:V_{\gamma}(\pi\circ\nu)\to(V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M))_{\mu(\gamma)}$
is the canonical isomorphism between the fibers by $\gamma$ and $\mu(\gamma)$
of the vector bundles $V(\pi\circ\nu)\to{\mathcal{M}}\pi$ and
$V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)\to{\mathcal{M}}^{0}\pi$,
then the following diagram

$\textstyle{V_{\gamma}(\pi\circ\nu)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\flat}_{\gamma}}$$\scriptstyle{{\mathcal{J}}_{\gamma}}$$\textstyle{{\mathcal{M}}_{\mu(\gamma)}(\pi\circ\nu^{0})=\Lambda_{2}^{m}T_{\mu(\gamma)}^{*}({\mathcal{M}}^{0}\pi)}$$\textstyle{(V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M))_{\mu(\gamma)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\flat_{\gamma}}$
(3.33)

is commutative

######  Proposition 3.10.

We have the equality

$-A^{+}\circ{\mathcal{I}}=\flat.$ (3.34)

This implies that the vector bundle
$(J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A)^{+}$ is isomorphic to the vector
subbundle $L$ of
${\mathcal{M}}(\pi\circ\nu^{0})\simeq\Lambda^{m}_{2}(T^{*}({\mathcal{M}}^{0}\pi))$
given by

$L=\bar{\flat}(V(\pi\circ\nu)).$ (3.35)

In particular, a local basis of sections of the vector subbundle $L$ is

$\big{\\{}d^{m}x,du^{\alpha}\wedge d^{m-1}x_{i},\sum_{i}dp_{\alpha}^{i}\wedge
d^{m-1}x_{i}\big{\\}}.$ (3.36)

###### Proof.

If $\gamma\in{\mathcal{M}}\pi$, $\tilde{U}\in V_{\gamma}(\pi\circ\nu)$ and
$Z^{0}\in J^{1}_{\mu(\gamma)}(\pi\circ\nu^{0})$ then, from (3.19), we obtain

$\langle
A^{+}({\mathcal{I}}[\tilde{U}]),Z^{0}\rangle=\langle{\mathcal{I}}[\tilde{U}],A(Z^{0})\rangle=\langle{\mathcal{I}}[\tilde{U}],\tilde{\mu}(\widetilde{A(Z^{0})})\rangle.$

Therefore, using (3.30) and (3.31), it follows that

$\langle
A^{+}({\mathcal{I}}[\tilde{U}]),Z^{0}\rangle=\langle\tilde{\mathcal{I}}(\tilde{U}),\widetilde{A(Z^{0})}\rangle=\langle\widetilde{A(Z^{0})},\tilde{U}\rangle.$

So, from (2.1), (3.18) and (3.32), we conclude that

$\langle
A^{+}({\mathcal{I}}[\tilde{U}]),Z^{0}\rangle=-(\Lambda^{m}(Z^{0})^{*})(\bar{\flat}(\tilde{U}))=-(\Lambda^{m}(Z^{0})^{*})(\flat[\tilde{U}])=-\langle\flat[\tilde{U}],Z^{0}\rangle.$

This proves (3.34).

From this, (3.35) follows using (3.32) and (3.34).

Finally, the local expression is obtained by using (3.16) and (3.35). ∎

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
$\pi:E\to M$ is

${\mathcal{O}}=\big{\\{}\alpha^{0}\in\Gamma(\Lambda^{m-1}_{1}(T^{*}({\mathcal{M}}^{0}\pi)))\mid
d\alpha^{0}\in\Gamma(L)\big{\\}}.$

######  Example 3.12.

i) Let $\alpha$ be a $(m-1)$-form on $E$ which is semi-basic with respect to
the projection $\pi:E\to M$. Then,

$\alpha^{0}=(\nu^{0})^{*}(\alpha)\in{\mathcal{O}}.$

Indeed, for $\alpha=\alpha^{i}(x,u)d^{m-1}x_{i}$, we have

$d\alpha=\left(\frac{\partial\alpha^{i}}{\partial
x^{i}}\right)d^{m}x+\left(\frac{\partial\alpha^{i}}{\partial
u^{\alpha}}\right)du^{\alpha}\wedge d^{m-1}x_{i}$

which implies that $d\alpha\in\Gamma(L)$, see (3.36).

ii) Let $Y$ be a section of the vector bundle $V\pi\to E$, that is, $Y$ is a
vector field on $E$ and

$(T_{y}\pi)(Y(y))=0,\;\;\forall y\in E.$

Define the $(m-1)$-form $\hat{Y}$ on
${\mathcal{M}}^{0}\pi=\operatorname{Lin}(V\pi,\pi^{*}(\Lambda^{m-1}T^{*}M))$
as follows

$\hat{Y}(\gamma^{0})(Z_{1},\dots,Z_{m-1})=\gamma^{0}(Y(\nu^{0}(\gamma^{0})))\left(T_{\gamma^{0}}(\pi\circ\nu^{0})(Z_{1}),\dots,T_{\gamma^{0}}(\pi\circ\nu^{0})(Z_{m-1})\right),$

for $\gamma^{0}\in{\mathcal{M}}^{0}\pi$ and $Z_{1},\dots,Z_{m-1}\in
T_{\gamma^{0}}{\mathcal{M}}^{0}\pi$. If the local expression of $Y$ is

$Y=Y^{\alpha}(x,u)\frac{\partial}{\partial u^{\alpha}}$

it follows that
$\hat{Y}=\left(Y^{\alpha}(x,u)p_{\alpha}^{i}\right)d^{m-1}x_{i}$. Thus,

$d\hat{Y}=\left(\frac{\partial Y^{\alpha}}{\partial
x^{i}}p_{\alpha}^{i}\right)d^{m}x+\left(\frac{\partial Y^{\alpha}}{\partial
u^{\beta}}p_{\alpha}^{i}\right)du^{\beta}\wedge
d^{m-1}x_{i}+Y^{\alpha}dp_{\alpha}^{i}\wedge d^{m-1}x_{i},$

which implies that $d\hat{Y}\in\Gamma(L)$ and $\hat{Y}\in{\mathcal{O}}$.

In the following theorem, we give the explicit description of the currents for
the case when $m\geq 2$. Note that if $m=1$ then
${\mathcal{O}}=C^{\infty}({\mathcal{M}}^{0}\pi)$.

######  Theorem 3.13.

If $m\geq 2$ then a section $\alpha^{0}$ of the vector bundle
$\Lambda^{m-1}_{1}(T^{*}({\mathcal{M}}^{0}\pi))\to{\mathcal{M}}^{0}\pi$ is a
current if and only if there exists a unique $\pi$-semibasic $(m-1)$-form
$\alpha$ on $E$ and a unique $\pi$-vertical vector field $Y$ on $E$ such that

$\alpha^{0}=\hat{Y}+(\nu^{0})^{*}(\alpha).$

###### Proof.

It is clear that if $Y$ is a $\pi$-vertical vector field and $\alpha$ is a
$\pi$-semibasic $(m-1)$-form on $E$ then
$\alpha^{0}=\hat{Y}+(\nu^{0})^{*}(\alpha)$ is a current (see Examples 3.12).

Conversely, suppose that $\alpha^{0}$ is a current. The local expression of
$\alpha^{0}$ is
$\alpha^{0}=\alpha^{0i}(x^{j},u^{\beta},p_{\beta}^{j})d^{m-1}x_{i}$ and

$d\alpha^{0}=\left(\frac{\partial\alpha^{0i}}{\partial
x^{i}}\right)d^{m}x+\left(\frac{\partial\alpha^{0i}}{\partial
u^{\beta}}\right)du^{\beta}\wedge
d^{m-1}x_{i}+\left(\frac{\partial\alpha^{0i}}{\partial
p_{\beta}^{j}}\right)dp_{\beta}^{j}\wedge d^{m-1}x_{i}.$

Thus, using (3.36), we deduce that

$\displaystyle\frac{\partial\alpha^{0i}}{\partial p_{\beta}^{j}}=0\mbox{ and
}\frac{\partial\alpha^{0i}}{\partial
p_{\beta}^{i}}=\frac{\partial\alpha^{0j}}{\partial p_{\beta}^{j}}\mbox{ if
}i\neq j.$ (3.37)

This implies that

$\displaystyle\frac{\partial^{2}\alpha^{0i}}{\partial(p_{\alpha}^{i})^{2}}=\frac{\partial^{2}\alpha^{0j}}{\partial
p_{\alpha}^{i}\partial p_{\alpha}^{j}}=0,\mbox{ with }i\neq j$

so if $m\geq 2$ we conclude that

$\displaystyle\frac{\partial^{2}\alpha^{0i}}{\partial(p_{\alpha}^{i})^{2}}=0,\mbox{
for all }i.$ (3.38)

Therefore, from (3.37) and (3.38), it follows that

$\alpha^{0i}(x^{j},u^{\beta},p_{\beta}^{j})=Y^{\alpha}(x,u)p_{\alpha}^{i}+\alpha^{i}(x,u),\mbox{
for all }i.$

Consequently, we have proved that there exists a local $\pi$-vertical vector
field $Y=Y^{\alpha}(x,u)\displaystyle\frac{\partial}{\partial u^{\alpha}}$ and
a local $\pi$-semibasic $(m-1)$-form $\alpha=\alpha^{i}(x,u)d^{m-1}x_{i}$ on
$E$ such that

$\alpha^{0}=\hat{Y}+(\nu^{0})^{*}(\alpha).$

Note that $Y$ and $\alpha$ are unique. Then, this last fact also proves the
global result. ∎

######  Remark 3.14.

(i) Note that ${\mathcal{O}}$ is a $C^{\infty}(E)$-module.
(ii) In [12], the authors consider as a space of currents the set of
horizontal Poisson $(m-1)$-forms on ${\mathcal{M}}^{0}\pi$. Moreover, they
prove that a $(m-1)$-form $F$ of this type may be described as

$F=\hat{Y}+(\nu^{0})^{*}(\alpha^{\prime})+\beta^{\prime},$

where $X$ is a vertical vector field on $E$, $\alpha^{\prime}$ is a
$\pi$-semibasic $(m-1)$-form on $E$ and $\beta^{\prime}$ is a closed
$(\pi\circ\nu^{0})$-semibasic $(m-1)$-form on ${\mathcal{M}}^{0}\pi$. Now, it
is easy to prove that, under the previous conditions, there exists a unique
closed $(m-1)$-form $\beta$ on $M$ such that
$\beta^{\prime}=(\pi\circ\nu^{0})^{*}(\beta)=(\nu^{0})^{*}(\pi^{*}(\beta))$.
So, if we take $\alpha=\alpha^{\prime}+\pi^{*}\beta$, we conclude that

$F=\hat{Y}+(\nu^{0})^{*}(\alpha).$

The previous discussion shows that ${\mathcal{O}}$ is just the space of
currents which was considered in [12]. $\diamond$

###  3.5. A suitable linear-affine bracket and the Hamilton-deDonder-Weyl
equations

We consider the linear-affine bracket

$\\{\cdot,\cdot\\}:{\mathcal{O}}\times\Gamma(\mu)\to\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))$

defined by

$\\{\alpha^{0},h\\}=\langle d\alpha^{0},\Gamma_{h}\rangle=\langle
d\alpha^{0},\sharp^{\rm aff}(dh)\rangle.$ (3.39)

Assume that $m\geq 2$, that the local expression of the Hamiltonian section
$h\in\Gamma(\mu)$ is

$h(x^{i},u^{\alpha},p_{\alpha}^{i})=(x^{i},u^{\alpha},-H(x,u,p),p_{\alpha}^{i})$

and that the local expression of the current $\alpha^{0}\in{\mathcal{O}}$ is

$\alpha^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})=(Y^{\alpha}(x,u)p_{\alpha}^{i}+\beta^{i}(x,u))d^{m-1}x_{i}$

with $Y^{\alpha}$ and $\beta^{i}$ local real $C^{\infty}$-functions on $E$.
Then, using (3.25) and (3.39), we obtain the local expression of the linear-
affine bracket (3.39) as

$\begin{array}[]{l}\\{\left(Y^{\alpha}(x,u)p_{\alpha}^{i}+\beta^{i}(x,u)\right)d^{m-1}x_{i},h\\}\\\\[5.0pt]
\;=\left(\displaystyle\frac{\partial\beta^{i}}{\partial x^{i}}+\frac{\partial
Y^{\alpha}}{\partial
x^{i}}p_{\alpha}^{i}+\left(\frac{\partial\beta^{i}}{\partial
u^{\alpha}}+\frac{\partial Y^{\beta}}{\partial
u^{\alpha}}p_{\beta}^{i}\right)\frac{\partial H}{\partial
p_{\alpha}^{i}}-\frac{\partial H}{\partial u^{\alpha}}Y^{\alpha}\right)\otimes
d^{m}x.\end{array}$ (3.40)

Note that if we write the current as
$\alpha^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})=\alpha^{i}d^{m-1}x_{i}$, with
$\alpha^{i}(x,u,p)=Y^{\alpha}(x,u)p_{\alpha}^{i}+\beta^{i}(x,u)$, the bracket
takes the elegant form

$\\{\alpha^{i}d^{m-1}x_{i},h\\}=\left(\frac{\partial\alpha^{i}}{\partial
x^{i}}+\frac{\partial\alpha^{i}}{\partial u^{\alpha}}\frac{\partial
H}{\partial p_{\alpha}^{i}}-\frac{1}{m}\frac{\partial H}{\partial
u^{\alpha}}\frac{\partial\alpha^{i}}{\partial p^{i}_{\alpha}}\right)\otimes
d^{m}x.$

As we know, $\Gamma(\mu)$ is an affine space which is modelled over the vector
space $\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))$. Therefore, using
(3.39), it follows that the bilinear bracket

$\\{\cdot,\cdot\\}_{l}:{\mathcal{O}}\times\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))\to\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))$

associated with the linear-affine bracket $\\{\cdot,\cdot\\}$ is given by

$\\{\alpha^{0},{\mathcal{F}}^{0}\\}_{l}=\langle\mu^{0}(d\alpha^{0}),\sharp^{\rm
lin}(d^{l}{\mathcal{F}}^{0})\rangle.$

Here, $d^{l}{\mathcal{F}}^{0}$ is the vertical differential of
${\mathcal{F}}^{0}$ (see (3.13)),

$\sharp^{\rm
lin}:V(\mathbb{P}(\pi))\to\displaystyle\frac{(\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M)\otimes
V(\pi\circ\nu^{0})}{\operatorname{Ker}A}$

is the vector bundle isomorphism associated with the affine bundle isomorphism
$\sharp^{\rm aff}:\mathbb{P}(\pi)\to
J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A$, and

$\mu^{0}:{\rm
Aff}\Big{(}\frac{J^{1}(\pi\circ\nu^{0})}{\operatorname{Ker}A},(\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M)\Big{)}\to{\rm
Lin}\Big{(}\displaystyle\frac{(\pi\circ\nu^{0})^{*}(T^{*}M)\otimes
V(\pi\circ\nu^{0})}{\operatorname{Ker}A},(\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M)\Big{)}$

is the canonical projection. In local coordinates, we have

$\begin{array}[]{l}\\{\left(Y^{\alpha}(x,u)p_{\alpha}^{i}+\beta^{i}(x,u)\right)d^{m-1}x_{i},{\mathcal{F}}^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})\otimes
d^{m}x\\}_{l}\\\\[5.0pt]
\;=\left(\left(\displaystyle\frac{\partial\beta^{i}}{\partial
u^{\alpha}}+\frac{\partial Y^{\beta}}{\partial
u^{\alpha}}p_{\beta}^{i}\right)\displaystyle\frac{\partial{\mathcal{F}}^{0}}{\partial
p_{\alpha}^{i}}-\frac{\partial{\mathcal{F}}^{0}}{\partial
u^{\alpha}}Y^{\alpha}\right)\otimes d^{m}x.\end{array}$ (3.41)

Again, if we write the current as
$\alpha^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})=\alpha^{i}d^{m-1}x_{i}$, with
$\alpha^{i}(x,u,p)=(Y^{\alpha}(x,u)p_{\alpha}^{i}+\beta^{i}(x,u))$, the
bilinear bracket takes the form

$\\{\alpha^{i}d^{m-1}x_{i},{\mathcal{F}}^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})\otimes
d^{m}x\\}_{l}=\left(\frac{\partial\alpha^{i}}{\partial
u^{\alpha}}\frac{\partial{\mathcal{F}}^{0}}{\partial
p_{\alpha}^{i}}-\frac{1}{m}\frac{\partial{\mathcal{F}}^{0}}{\partial
u^{\alpha}}\frac{\partial\alpha^{i}}{\partial p^{i}_{\alpha}}\right)\otimes
d^{m}x.$

On the other hand, if $m=1$ then the space of currents is

${\mathcal{O}}=\Gamma((\pi\circ\nu^{0})^{*}(T^{*}M))=C^{\infty}({\mathcal{M}}^{0}\pi)$

and, using (3.25) and (3.39), we deduce that the linear-affine bracket

$\\{\cdot,\cdot\\}:C^{\infty}({\mathcal{M}}^{0}\pi)\times\Gamma(\mu)\to
C^{\infty}({\mathcal{M}}^{0}\pi)$

and the bilinear bracket

$\\{\cdot,\cdot\\}_{l}:C^{\infty}({\mathcal{M}}^{0}\pi)\times
C^{\infty}({\mathcal{M}}^{0}\pi)\to C^{\infty}({\mathcal{M}}^{0}\pi)$

are locally given by

$\\{f^{0},h\\}=\displaystyle\frac{\partial f^{0}}{\partial x}+\frac{\partial
f^{0}}{\partial u^{\alpha}}\frac{\partial H}{\partial
p_{\alpha}}-\frac{\partial f^{0}}{\partial p_{\alpha}}\frac{\partial
H}{\partial u^{\alpha}}$ (3.42)

and

$\\{f^{0},g^{0}\\}_{l}=\displaystyle\frac{\partial f^{0}}{\partial
u^{\alpha}}\frac{\partial g^{0}}{\partial p_{\alpha}}-\frac{\partial
f^{0}}{\partial p_{\alpha}}\frac{\partial g^{0}}{\partial u^{\alpha}}$ (3.43)

for $f^{0},g^{0}\in C^{\infty}({\mathcal{M}}^{0}\pi)$ and $h\in\Gamma(\mu)$.

The following result extends to the field-theoretic context the canonical
Poisson bracket formulation of Hamilton’s equations.

######  Theorem 3.15.

Let $h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$ be a Hamiltonian section and
$s^{0}:M\to{\mathcal{M}}^{0}\pi$ a (local) section of the projection
$\pi\circ\nu^{0}:{\mathcal{M}}^{0}\pi\to M$. Then, $s^{0}$ is a solution of
the Hamilton-deDonder-Weyl equations for $h$ if and only if

$(s^{0})^{*}(d\alpha^{0})=\\{\alpha^{0},h\\}\circ
s^{0},\;\;\;\forall\;\alpha^{0}\in{\mathcal{O}}.$ (3.44)

###### Proof.

Suppose that $m\geq 2$ and that

$s^{0}(x^{i})=(x^{i},u^{\alpha}(x),p_{\alpha}^{i}(x)),\quad\alpha^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})=\left(Y^{\alpha}(x,u)p_{\alpha}^{i}+\beta^{i}(x,u)\right)d^{m-1}x_{i},$

with $Y^{\alpha}$, $\beta^{i}$ local real $C^{\infty}$-functions on $E$. Then,

$(s^{0})^{*}(d\alpha^{0})=\left(\frac{\partial\beta^{i}}{\partial
x^{i}}+\frac{\partial Y^{\alpha}}{\partial
x^{i}}p_{\alpha}^{i}+\left(\frac{\partial\beta^{i}}{\partial
u^{\alpha}}+\frac{\partial Y^{\beta}}{\partial
u^{\alpha}}p_{\beta}^{i}\right)\frac{\partial u^{\alpha}}{\partial
x^{i}}+\frac{\partial p_{\alpha}^{i}}{\partial x^{i}}Y^{\alpha}\right)\otimes
d^{m}x.$

Thus, using (3.40), we conclude that (3.44) hold if and only if

$\displaystyle\frac{\partial u^{\alpha}}{\partial x^{i}}=\frac{\partial
H}{\partial p_{\alpha}^{i}},\;\;\;\;\frac{\partial p_{\alpha}^{i}}{\partial
x^{i}}=-\frac{\partial H}{\partial u^{\alpha}},$

or, equivalently, $s^{0}$ is a solution of the Hamilton-deDonder-Weyl
equations for $h$.

If $m=1$ the result is proved in a similar way using (3.42). ∎

###  3.6. A remark on boundary conditions

If the boundary $\partial M$ of the base space $M$ of the configuration bundle
$\pi:E\to M$ is not empty, then the Hamilton-deDonder-Weyl equations for a
Hamiltonian section can be supplemented by boundary conditions.

The boundary of the configuration space $E$ is just

$\partial E=\pi^{-1}(\partial M)$

in such a way that

$\pi_{|\partial E}:\partial E\to\partial M$

is again a fibration.

In a similar way, the restricted multimomentum bundle ${\mathcal{M}}^{0}\pi$
is a manifold with boundary,

$\partial({\mathcal{M}}^{0}\pi)=(\nu^{0})^{-1}(\partial
E)=(\pi\circ\nu^{0})^{-1}(\partial M),$

and we have fibrations

$(\nu^{0})_{|\partial({\mathcal{M}}^{0}\pi)}:\partial({\mathcal{M}}^{0}\pi)\to\partial
E,\;\;\;(\pi\circ\nu^{0})_{|\partial({\mathcal{M}}^{0}\pi)}:\partial({\mathcal{M}}^{0}\pi)\to\partial
M.$

A boundary condition for the Hamiltonian Classical Field theory is given by
specifying a subbundle $B^{0}\rightarrow\partial M$ of
$(\pi\circ\nu^{0})_{|\partial({\mathcal{M}}^{0}\pi)}:\partial({\mathcal{M}}^{0}\pi)\to\partial
M$, such that $B^{0}\rightarrow
B_{E}^{0}:=(\nu^{0})_{|\partial({\mathcal{M}}^{0}\pi)}(B^{0})$ is a subbundle
of
$(\nu^{0})_{|\partial({\mathcal{M}}^{0}\pi)}:\partial({\mathcal{M}}^{0}\pi)\to\partial
E$. In such a case, we will consider only sections
$s^{0}:M\to{\mathcal{M}}^{0}\pi$ such that

$s^{0}(\partial M)\subseteq B^{0}.$ (3.45)

A standard assumption in the literature for the subbundle $B^{0}$ is

$i_{B^{0}}^{*}\omega_{h}=0,$

where $i_{B^{0}}:B^{0}\to{\mathcal{M}}^{0}\pi$ is the canonical inclusion and
$h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$ 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

${}^{H}:{\mathcal{M}}^{0}\pi\times_{M}TM\to T({\mathcal{M}}^{0}\pi)$

we should only consider those whose restriction to
$\partial({\mathcal{M}}^{0}\pi)\times_{\partial M}T(\partial M)$ takes values
in the tangent bundle $TB^{0}$, that is, H should induce a monomorphism of
vector bundles

${}^{H}:\partial({\mathcal{M}}^{0}\pi)\times_{\partial M}T(\partial M)\to
TB^{0}.$

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 $\Gamma_{h}$ of
the quotient affine bundle $J^{1}(\pi\circ\nu^{0})/\operatorname{Ker}A$ 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).

##  4\. 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 ${\mathcal{O}}$ of a
Hamiltonian Field theory of first order admits a Lie algebra structure and we
show that the linear affine bracket $\\{\cdot,\cdot\\}$ introduced in Section
3.5 (see (3.39)) induces an affine representation of ${\mathcal{O}}$ 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 $V$, ${\rm Aff}(A,V)$, is a Lie algebra and the
Lie bracket on ${\rm Aff}(A,V)$ is given by

$[\varphi,\psi]=\varphi^{l}\circ\psi-\psi^{l}\circ\varphi,$

for $\varphi,\psi\in{\rm Aff}(A,V)$, where $\varphi^{l},\psi^{l}:V\to V$ are
the linear maps associated with $\varphi,\psi$, respectively.

An affine representation of a real Lie algebra $\mathfrak{g}$ on $A$ is a Lie
algebra morphism

$\mathcal{R}:\mathfrak{g}\to{\rm Aff(A,V)}.$

We first note that if $\pi:E\to M$ is a configuration bundle with ${\rm
dim}M=1$ then it is easy to prove the following facts (see Section 5.1 for the
particular case when $M$ is the real line $\mathbb{R}$ and $E=\mathbb{R}\times
Q$):

  * •

$J^{1}\pi$ is an affine subbundle of corank $1$ of the tangent bundle $TE\to
E$ which is modelled over the vertical bundle $V\pi\to E$ to the fibration
$\pi:E\to M$.

  * •

The restricted multimomentum bundle is just the dual bundle $V^{*}\pi\to E$ to
$V\pi\to E$.

  * •

The extended multimomentum bundle is the cotangent bundle $T^{*}E\to E$ of
$E$.

  * •

$\Gamma(\mu)$ is an affine space which is modelled over the vector space
${\mathcal{O}}=C^{\infty}(V^{*}\pi)$ of the currents.

So, in this case, we have a Lie algebra structure on
${\mathcal{O}}=C^{\infty}(V^{*}\pi)$. In fact, the Lie bracket on
${\mathcal{O}}$ is just the Poisson bracket $\\{\cdot,\cdot\\}_{l}$ on
$V^{*}\pi$ given by (3.43). Moreover, using (3.42) and (3.43), we deduce that
the linear-affine bracket

$\\{\cdot,\cdot\\}:C^{\infty}(V^{*}\pi)\times\Gamma(\mu)\to
C^{\infty}(V^{*}\pi)$

induces an affine representation of the Lie algebra
$(C^{\infty}(V^{*}\pi),\\{\cdot,\cdot\\}_{l})$ on the affine space
$\Gamma(\mu)$. More explicitly, we have

$\\{\\{\alpha_{0},\beta_{0}\\}_{l},h\\}=\\{\alpha_{0},\\{\beta_{0},h\\}\\}_{l}-\\{\beta_{0},\\{\alpha_{0},h\\}\\}_{l},$

for $\alpha_{0},\beta_{0}\in C^{\infty}(V^{*}\pi)$ and $h\in\Gamma(\mu)$.

Therefore, in the rest of this section, we will assume the following
hypothesis:

Assumption: In what follows, we will suppose that ${\rm dim}M\geq 2$.

First we introduce a Lie algebra structure on ${\mathcal{O}}$, then we show
that the linear affine bracket
$\\{\cdot,\cdot\\}:{\mathcal{O}}\times\Gamma(\mu)\to\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))$
induces an affine representation of the Lie algebra
$({\mathcal{O}},\\{\cdot,\cdot\\}_{\mathcal{O}})$ on the affine space
$\Gamma(\mu)$.

###  4.1. The Lie algebra structure on the space of currents ${\mathcal{O}}$

The construction of the Lie bracket is made in several steps which involve the
definition of a vertical vector field on ${\mathcal{M}}^{0}\pi$ associated to
a current.

####  4.1.1. Definition of the vertical vector field on ${\mathcal{M}}^{0}\pi$
associated to a current

Let $\flat:V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)\to
L\subseteq{\mathcal{M}}(\pi\circ\nu^{0})=\Lambda^{m}_{2}(T^{*}({\mathcal{M}}^{0}\pi))$
be the vector bundle isomorphism over the identity of ${\mathcal{M}}^{0}\pi$
given by (3.32) and denote by $\sharp:L\to
V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)$ the inverse morphism. If
$\alpha^{0}\in{\mathcal{O}}$ then, from Definition 3.11, we have
$\alpha^{0}\in\Gamma(\Lambda^{m-1}_{1}(T^{*}{\mathcal{M}}^{0}\pi))$ and
$d\alpha^{0}\in\Gamma(L)$. So, we can consider the section
$\sharp(d\alpha^{0})$ of the vector bundle
$V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)\to{\mathcal{M}}^{0}\pi={\mathcal{M}}\pi/\pi^{*}(\Lambda^{m}T^{*}M)$.

######  Remark 4.1.

The notation $\sharp(d\alpha^{0})$ is justified by the following fact. The
vector bundle $V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)$ is
canonically isomorphic to $\mathbb{P}(\pi)^{+}$ (an isomorphism
${\mathcal{I}}$ between these vector bundles is characterized by condition
(3.31)). So, if $h:{\mathcal{M}}^{0}\pi\to{\mathcal{M}}\pi$ is a Hamiltonian
section then $dh\in\mathbb{P}(\pi)$ and

$\langle\sharp(d\alpha^{0}),dh\rangle\in\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M)).$

Moreover, as will be proved later (see the next Lemma 4.5), we can write the
linear-affine bracket as

$\\{\alpha^{0},h\\}=-\langle\sharp(d\alpha^{0}),dh\rangle.$

The reader can compare the previous expression with Eq. (1.3) for the
definition of the canonical Poisson bracket on $T^{*}Q$. $\diamond$

Note that a section of the vector bundle
$V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)\to{\mathcal{M}}^{0}\pi={\mathcal{M}}\pi/\pi^{*}(\Lambda^{m}T^{*}M)$
can be identified with a section $U:{\mathcal{M}}\pi\to V(\pi\circ\nu)$ of
$V(\pi\circ\nu)\to{\mathcal{M}}\pi$ which is equivariant with respect to the
fibred actions of $\pi^{*}(\Lambda^{m}T^{*}M)$ on ${\mathcal{M}}\pi$ and on
$V(\pi\circ\nu)$.

By applying this observation to $\sharp(d\alpha^{0})$, we denote by

$\tilde{\mathcal{H}}_{\alpha^{0}}:{\mathcal{M}}\pi\to V(\pi\circ\nu)$

the equivariant vector field on ${\mathcal{M}}\pi$ associated with the section
$\sharp(d\alpha^{0})$. Since $\tilde{\mathcal{H}}_{\alpha^{0}}$ is
equivariant, it follows that it is $\mu$-projectable to a vertical vector
field

${\mathcal{H}}_{\alpha^{0}}:{\mathcal{M}}^{0}\pi\to V(\pi\circ\nu^{0})$

on ${\mathcal{M}}^{0}\pi$.

We now present a description of $\tilde{\mathcal{H}}_{\alpha^{0}}$ in terms of
$\alpha^{0}$. Let
$\tilde{\flat}:V(\pi\circ\nu)\to{\mathcal{M}}(\pi\circ\nu)=\Lambda^{m}_{2}(T^{*}({\mathcal{M}}\pi))$
be the vector bundle monomorphism given by (3.14). Denote by $\tilde{L}$ the
image of $V(\pi\circ\nu)$ by $\tilde{\flat}$, so that

$\tilde{\flat}:V(\pi\circ\nu)\to\tilde{L}\subseteq{\mathcal{M}}(\pi\circ\nu)=\Lambda^{m}_{2}(T^{*}{\mathcal{M}}\pi)$

is a vector bundle isomorphism over the identity of ${\mathcal{M}}\pi$. From
(2.3), it follows that a local basis of $\Gamma(\tilde{L})$ is

$\\{d^{m}x,du^{\alpha}\wedge d^{m-1}x_{i},dp_{\alpha}^{i}\wedge
d^{m-1}x_{i}\\}.$

Note that if $\gamma\in{\mathcal{M}}\pi$ then

$(\Lambda^{m}T^{*}_{\gamma}\mu)(L_{\mu(\gamma)})=\tilde{L}_{\gamma}.$

In fact, if $\theta$ is a section of
${\mathcal{M}}(\pi\circ\nu^{0})=\Lambda^{m}_{2}(T^{*}{\mathcal{M}}^{0}\pi)$
then

$\theta\in\Gamma(L)\Leftrightarrow\mu^{*}\theta\in\Gamma(\tilde{L}).$ (4.1)

Let $\bar{\flat}:V(\pi\circ\nu)\to
L\subseteq{\mathcal{M}}(\pi\circ\nu^{0})=\Lambda_{2}^{m}(T^{*}{\mathcal{M}}^{0}\pi)$
be the vector bundle morphism over the canonical projection
$\mu:{\mathcal{M}}\pi\to{\mathcal{M}}^{0}\pi$ which is characterized by Eq.
(3.15). If $\gamma\in{\mathcal{M}}\pi$ and
${\mathcal{J}}_{\gamma}:V_{\gamma}(\pi\circ\nu)\to(V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M))_{\mu(\gamma)}$
is the canonical isomorphism between the fibers by $\gamma$ and $\mu(\gamma)$
of the vector bundles $V(\pi\circ\nu)\to{\mathcal{M}}\pi$ and
$V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)\to{\mathcal{M}}^{0}\pi$
then, using (3.17) and (3.33), it follows that the following diagram

$\textstyle{V_{\gamma}(\pi\circ\nu)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\flat}_{\gamma}}$$\scriptstyle{\bar{\flat}_{\gamma}}$$\scriptstyle{{\mathcal{J}}_{\gamma}}$$\textstyle{\tilde{L}_{\gamma}}$$\textstyle{(V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M))_{\mu(\gamma)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
42.67912pt\flat_{\mu(\gamma)}}$$\textstyle{L_{\mu(\gamma)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Lambda^{m}T^{*}_{\gamma}\mu}$
(4.2)

is commutative. This implies that

$\tilde{\flat}_{\gamma}(\tilde{\mathcal{H}}_{\alpha^{0}}(\gamma))=(\Lambda^{m}T^{*}_{\gamma}\mu)\big{(}\flat_{\mu(\gamma)}(\sharp(d\alpha^{0})(\mu(\gamma)))\big{)}=(\Lambda^{m}T^{*}_{\gamma}\mu)((d\alpha^{0})(\mu(\gamma))),$

or, in other words, $\tilde{\mathcal{H}}_{\alpha^{0}}$ satisfies the following
condition

$i_{\tilde{\mathcal{H}}_{\alpha^{0}}}\omega_{{\mathcal{M}}\pi}=\mu^{*}(d\alpha^{0}).$
(4.3)

Note that, since $\omega_{{\mathcal{M}}\pi}$ is non-degenerate, (4.3) may be
considered as a definition of the equivariant vector field
$\tilde{\mathcal{H}}_{\alpha^{0}}$.

So, in conclusion, for a current $\alpha^{0}\in{\mathcal{O}}$ we have the
following objects:

  * •

An equivariant vertical vector field
$\tilde{\mathcal{H}}_{\alpha^{0}}:{\mathcal{M}}\pi\to V(\pi\circ\nu)$ on
${\mathcal{M}}\pi$, which is characterized by condition (4.3).

  * •

The induced section $\sharp(d\alpha^{0})$ of the vector bundle

$\mathbb{P}(\pi)^{+}\simeq
V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)\to{\mathcal{M}}^{0}\pi={\mathcal{M}}\pi/\pi^{*}(\Lambda^{m}T^{*}M).$

  * •

The vertical vector field on ${\mathcal{M}}^{0}\pi$

${\mathcal{H}}_{\alpha^{0}}:{\mathcal{M}}^{0}\pi\to V(\pi\circ\nu^{0})$

which is the projection, via $\mu:{\mathcal{M}}\pi\to{\mathcal{M}}^{0}\pi$, of
$\tilde{\mathcal{H}}_{\alpha^{0}}$.

We now present the local expressions of the vector fields
$\tilde{\mathcal{H}}_{\alpha^{0}}$ and ${\mathcal{H}}_{\alpha^{0}}$. As we
have seen (see Section 3.4), the local expression of an element
$\alpha^{0}\in{\mathcal{O}}$ is

$\alpha^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})=\left(Y^{\alpha}(x^{j},u^{\beta})p_{\alpha}^{i}+\alpha^{i}(x^{j},u^{\beta})\right)d^{m-1}x_{i}$

with $Y^{\alpha}$ and $\alpha^{i}$ local real $C^{\infty}$-functions on $E$.
We have

$d\alpha^{0}=\left(\frac{\partial\alpha^{i}}{\partial x^{i}}+\frac{\partial
Y^{\alpha}}{\partial
x^{i}}p_{\alpha}^{i}\right)d^{m}x+\left(\frac{\partial\alpha^{i}}{\partial
u^{\beta}}+\frac{\partial Y^{\alpha}}{\partial
u^{\beta}}p_{\alpha}^{i}\right)du^{\beta}\wedge
d^{m-1}x_{i}+Y^{\alpha}dp_{\alpha}^{i}\wedge d^{m-1}x_{i}.$ (4.4)

Thus, using (2.3) and (4.3), we deduce that

$\tilde{\mathcal{H}}_{\alpha^{0}}=Y^{\alpha}\frac{\partial}{\partial
u^{\alpha}}-\left(\frac{\partial\alpha^{i}}{\partial x^{i}}+\frac{\partial
Y^{\alpha}}{\partial x^{i}}p_{\alpha}^{i}\right)\frac{\partial}{\partial
p}-\left(\frac{\partial\alpha^{i}}{\partial u^{\beta}}+\frac{\partial
Y^{\alpha}}{\partial u^{\beta}}p_{\alpha}^{i}\right)\frac{\partial}{\partial
p_{\beta}^{i}}.$ (4.5)

Therefore, it follows that

${\mathcal{H}}_{\alpha^{0}}=Y^{\alpha}\frac{\partial}{\partial
u^{\alpha}}-\left(\frac{\partial\alpha^{i}}{\partial u^{\beta}}+\frac{\partial
Y^{\alpha}}{\partial u^{\beta}}p_{\alpha}^{i}\right)\frac{\partial}{\partial
p_{\beta}^{i}}.$ (4.6)

Following the terminology in [11] (see also [31]), Eq. (4.3) implies that
$\mu^{*}\alpha^{0}$ is a Hamiltonian $(m-1)$-form and that the vector field
$\tilde{\mathcal{H}}_{\alpha^{0}}$ is a Hamiltonian vector field on the
multisymplectic manifold $({\mathcal{M}}\pi,\omega_{{\mathcal{M}}\pi})$.

####  4.1.2. Definition of the Lie bracket $\\{\cdot,\cdot\\}_{\mathcal{O}}$

As in [11], we consider the $(m-1)$-form
$\\{\mu^{*}(\alpha^{0}),\mu^{*}(\beta^{0})\\}^{\tilde{}}_{\mathcal{O}}$ on the
multisymplectic manifold $({\mathcal{M}}\pi,\omega_{{\mathcal{M}}\pi})$
defined by

$\\{\mu^{*}(\alpha^{0}),\mu^{*}(\beta^{0})\\}^{\tilde{}}_{\mathcal{O}}=-i_{\tilde{\mathcal{H}}_{\alpha^{0}}}i_{\tilde{\mathcal{H}}_{\beta^{0}}}\omega_{{\mathcal{M}}\pi}=-i_{\tilde{\mathcal{H}}_{\alpha^{0}}}(\mu^{*}(d\beta^{0})).$

Using the fact that $\tilde{\mathcal{H}}_{\alpha^{0}}$ is $\mu$-projectable,
it follows that such a $(m-1)$-form is basic with respect to $\mu$. In fact,

$\\{\mu^{*}(\alpha^{0}),\mu^{*}(\beta^{0})\\}_{\mathcal{O}}^{\tilde{}}=-\mu^{*}(i_{{\mathcal{H}}_{\alpha^{0}}}d\beta^{0}).$

This equation suggests the introduction of the the $(m-1)$-form
$\\{\alpha^{0},\beta^{0}\\}_{\mathcal{O}}$ on ${\mathcal{M}}^{0}\pi$ given by

$\\{\alpha^{0},\beta^{0}\\}_{\mathcal{O}}=-i_{{\mathcal{H}}_{\alpha^{0}}}d\beta^{0}.$
(4.7)

Using the local expressions

$\alpha^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})=\left(Y^{\alpha}(x,u)p_{\alpha}^{i}+\alpha^{i}(x,u)\right)d^{m-1}x_{i},\;\;\beta^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})=\left(Z^{\alpha}(x,u)p_{\alpha}^{i}+\beta^{i}(x,u)\right)d^{m-1}x_{i}$

we deduce from (4.4) and (4.6) that

$\begin{array}[]{l}\\{\left(Y^{\alpha}(x,u)p_{\alpha}^{i}+\alpha^{i}(x,u)\right)d^{m-1}x_{i},\left(Z^{\alpha}(x,u)p_{\alpha}^{i}+\beta^{i}(x,u)\right)d^{m-1}x_{i}\\}_{\mathcal{O}}\\\\[5.0pt]
\;=-\left(\big{(}\displaystyle Y^{\beta}\frac{\partial Z^{\alpha}}{\partial
u^{\beta}}-Z^{\beta}\frac{\partial Y^{\alpha}}{\partial
u^{\beta}}\big{)}p_{\alpha}^{i}+\big{(}\displaystyle
Y^{\beta}\frac{\partial\beta^{i}}{\partial
u^{\beta}}-Z^{\beta}\frac{\partial\alpha^{i}}{\partial
u^{\beta}}\big{)}\right)d^{m-1}x_{i}\end{array}$ (4.8)

So, it is clear that $\\{\alpha,\beta\\}_{\mathcal{O}}\in{\mathcal{O}}$.

Moreover, we can prove the following result.

######  Theorem 4.2.

The bracket $\\{\cdot,\cdot\\}_{\mathcal{O}}$ given by

$\\{\alpha^{0},\beta^{0}\\}_{\mathcal{O}}=-i_{{\mathcal{H}}_{\alpha^{0}}}(d\beta^{0}),\;\;\;\mbox{
for }\alpha^{0},\beta^{0}\in{\mathcal{O}},$

defines a Lie algebra structure on the space of currents ${\mathcal{O}}$.

###### Proof.

Using Theorem 3.13, we deduce that there exists and isomorphism between the
$C^{\infty}(E)$-modules ${\mathcal{O}}$ and
$\Gamma(V\pi)\times\Gamma(\Lambda^{m-1}_{1}T^{*}E)$. In addition, from (4.8),
it follows that under the previous isomorphism, the bracket
$\\{\cdot,\cdot\\}_{\mathcal{O}}$ is given by

$\\{(Y,\alpha),(Z,\beta)\\}_{\mathcal{O}}=-([Y,Z],{\mathcal{L}}_{Y}\beta-{\mathcal{L}}_{Z}\alpha)=-([Y,Z],i_{Y}d\beta-
i_{Z}d\alpha),$

for
$(Y,\alpha),(Z,\beta)\in\Gamma(V\pi)\times\Gamma(\Lambda^{m-1}_{1}T^{*}E)$,
where $[\cdot,\cdot]$ is the Lie bracket of vector fields in $E$ and
${\mathcal{L}}$ is the Lie derivative operator. Using this definition of
$\\{\cdot,\cdot\\}_{\mathcal{O}}$, it is easy to prove that
$\\{\cdot,\cdot\\}_{\mathcal{O}}$ induces a Lie algebra structure on
${\mathcal{O}}$. ∎

Note that if we write the observables locally as
$\alpha^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})=\alpha^{0i}d^{m-1}x_{i}$ and
$\beta^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})=\beta^{0i}d^{m-1}x_{i}$ with
$\alpha^{0i}(x,u,p)=Y^{\alpha}(x,u)p_{\alpha}^{i}+\alpha^{i}(x,u)$ and
$\beta^{0i}(x,u,p)=Z^{\alpha}(x,u)p_{\alpha}^{i}+\beta^{i}(x,u)$, then the Lie
bracket $\\{\cdot,\cdot\\}_{\mathcal{O}}$ has the local expression

$\\{\alpha^{0i}d^{m-1}x_{i},\beta^{0i}d^{m-1}x_{i}\\}_{\mathcal{O}}=\left(\frac{\partial\alpha^{0j}}{\partial
u^{\gamma}}\frac{\partial\beta^{0i}}{\partial
p_{\gamma}^{j}}-\frac{\partial\beta^{0j}}{\partial
u^{\gamma}}\frac{\partial\alpha^{0i}}{\partial
p^{j}_{\gamma}}\right)d^{m-1}x_{i}$

which is reminiscent of the local expression of the canonical Poisson bracket.

As a final remark on the definition of the morphism $\flat$ and the bracket
$\\{\cdot,\cdot\\}_{\mathcal{O}}$ 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
$M^{0}\pi$.

Indeed, it is clear that the vector bundle $V(\pi\circ\nu)\to\mathcal{M}\pi$
admits a Lie algebroid structure. The Lie bracket in the space
$\Gamma(V(\pi\circ\nu))$ of sections is just the restriction of the standard
Lie bracket to $(\pi\circ\nu)$-vertical vector fields and the anchor map is
the inclusion
$\Gamma(V(\pi\circ\nu))\hookrightarrow\mathfrak{X}(\mathcal{M}\pi)$. So, using
the vector bundle isomorphism
$\tilde{\flat}:V(\pi\circ\nu)\to\tilde{L}\subseteq\mathcal{M}(\pi\circ\nu)$,
we can induce a Lie algebroid structure on the vector bundle $\tilde{L}$. In
fact, a direct computation proves that the Lie bracket in the space of
sections of $\tilde{L}$, $\Gamma(\tilde{L})$, is given by

$[\tilde{\theta},\tilde{\theta}^{\prime}]_{\tilde{L}}=\mathcal{L}_{\tilde{\rho}(\tilde{\theta})}\tilde{\theta}^{\prime}-\mathcal{L}_{\tilde{\rho}(\tilde{\theta^{\prime}})}\tilde{\theta}+d(i_{\tilde{\rho}(\tilde{\theta}^{\prime})}\tilde{\theta}),\;\;\mbox{
for }\tilde{\theta},\tilde{\theta}^{\prime}\in\Gamma(\tilde{L}),$ (4.9)

where
$\tilde{\rho}=\tilde{\flat}^{-1}:\Gamma(\tilde{L})\to\Gamma(V(\pi\circ\nu))$
is the anchor map.

On the other hand, the space of sections of the vector bundle
$V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}(T^{*}M))\to\mathcal{M}^{0}\pi$
may be identified with the $(\pi\circ\nu)$-vertical and
$(\pi\circ\nu)^{*}(\Lambda^{m}(T^{*}M))$-equivariant vector fields on
$\mathcal{M}\pi$. Thus, using that the standard Lie bracket of vector fields
is closed for this subspace, we may induce a Lie algebroid structure on the
vector bundle
$V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}(T^{*}M))\to\mathcal{M}^{0}\pi$.
Note that every $(\pi\circ\nu)$-vertical and
$(\pi\circ\nu)^{*}(\Lambda^{m}(T^{*}M))$-equivariant vector field on
$\mathcal{M}\pi$ is $\mu$-projectable a vector field on $\mathcal{M}^{0}\pi$
and this fact determines the anchor map of the Lie algebroid
$V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}(T^{*}M))\to\mathcal{M}^{0}\pi$.

Now, using the vector bundle isomorphism

$\flat:V(\pi\circ\nu)/(\pi\circ\nu)^{*}(\Lambda^{m}(T^{*}M))\to
L\subseteq\mathcal{M}(\pi\circ\nu^{0})$

we can translate the previous Lie algebroid structure to a Lie algebroid
structure on $L$. In fact, if $\theta\in\Gamma(L)$ then, from the
commutativity of the diagram (4.2), it follows that vector field
$\tilde{\rho}(\mu^{*}(\theta))$ on $\mathcal{M}\pi$ is $\mu$-projectable to a
vector field $\rho(\theta)\in\Gamma(V(\pi\circ\nu^{0}))$. This fact determines
the anchor map $\rho:\Gamma(L)\to\Gamma(V(\pi\circ\nu^{0}))$ of the Lie
algebroid $L$. Moreover, from (4.9), we deduce that the Lie bracket
$[\cdot,\cdot]$ on $\Gamma(L)$ is given by

$[\theta,\theta^{\prime}]_{L}=\mathcal{L}_{\rho(\theta)}\theta^{\prime}-\mathcal{L}_{\rho(\theta^{\prime})}\theta+d(i_{\rho(\theta^{\prime})}\theta),\;\;\mbox{
for }\theta,\theta^{\prime}\in\Gamma(L).$ (4.10)

Finally, if $\alpha^{0},\beta^{0}\in\mathcal{O}$ then, using (4.7), (4.10) and
the fact that

$\rho(d\alpha^{0})=\mathcal{H}_{\alpha^{0}},\;\;\rho(d\beta^{0})=\mathcal{H}_{\beta^{0}},$

we conclude that

$d\\{\alpha^{0},\beta^{0}\\}_{\mathcal{O}}=-[d\alpha^{0},d\beta^{0}]_{L},\;\;\mbox{
for }\alpha^{0},\beta^{0}\in{\mathcal{O}}.$ (4.11)

Eqs (4.10) and (4.11) are reminiscent of the properties of the Lie algebroid
structure on the cotangent bundle of a Poisson manifold (in particular, the
cotangent bundle of an arbitrary manifold).

###  4.2. The linear-affine bracket is an affine representation

In this subsection, we show that the linear-affine bracket
$\\{\cdot,\cdot\\}:{\mathcal{O}}\times\Gamma(\mu)\to\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))$
induces an affine representation of the Lie algebra of currents on the affine
space of Hamiltonian sections.

######  Theorem 4.3.

The map

${\mathcal{R}}:{\mathcal{O}}\to{\rm
Aff}\left(\Gamma(\mu),(\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M)\right),\quad{\mathcal{R}}(\alpha^{0})=\\{\alpha^{0},\cdot\\}$
(4.12)

is an affine representation of the Lie algebra
$({\mathcal{O}},\\{\cdot,\cdot\\}_{\mathcal{O}})$ on the affine space
$\Gamma(\mu)$.

More explicitly, we have

$\\{\\{\alpha^{0},\beta^{0}\\}_{\mathcal{O}},h\\}=\\{\alpha^{0},\\{\beta^{0},h\\}\\}_{l}-\\{\beta^{0},\\{\alpha^{0},h\\}\\}_{l},\;\;\mbox{
for }\alpha^{0},\beta^{0}\in{\mathcal{O}}\mbox{ and }h\in\Gamma(\mu).$ (4.13)

In order to prove this theorem, we will use the following results.

######  Lemma 4.4.

The linear map

$\tilde{\mathcal{H}}:{\mathcal{O}}\to\Gamma(V(\pi\circ\nu)),\;\;\alpha^{0}\to\tilde{\mathcal{H}}(\alpha^{0}):=\tilde{\mathcal{H}}_{\alpha^{0}}$

is a Lie algebra anti-morphism between the Lie algebras
$({\mathcal{O}},\\{\cdot,\cdot\\}_{\mathcal{O}})$ and
$(\Gamma(V(\pi\circ\nu)),[\cdot,\cdot])$, where $[\cdot,\cdot]$ is the
standard Lie bracket of vector fields.

######  Lemma 4.5.

Let $\alpha^{0}$ be an element of ${\mathcal{O}}$.

  1. (i)

If $h\in\Gamma(\mu)$ is a Hamiltonian section then

$\mu^{*}\\{\alpha^{0},h\\}=-(d^{v}{\mathcal{F}}_{h})(\tilde{\mathcal{H}}_{\alpha^{0}}),$
(4.14)

where ${\mathcal{F}}_{h}\in
C^{\infty}\big{(}{\mathcal{M}}\pi,(\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M)\big{)}$
is the extended Hamiltonian density associated with $h$ and
$d^{v}{\mathcal{F}}_{h}$ is the vertical differential of ${\mathcal{F}}_{h}$.
So, we have

$\\{\alpha^{0},h\\}=-\langle\sharp(d\alpha^{0}),dh\rangle.$ (4.15)

  2. (ii)

If ${\mathcal{F}}^{0}\in\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))$ then

$\\{\alpha^{0},{\mathcal{F}}^{0}\\}_{l}=-(d^{l}{\mathcal{F}}^{0})({\mathcal{H}}_{\alpha^{0}}),$
(4.16)

where $d^{l}{\mathcal{F}}^{0}$ is the linear part of the differential of
${\mathcal{F}}^{0}$.

######  Remark 4.6.

Note that
$\\{\alpha^{0},h\\}\in\Gamma((\pi\circ\nu^{0})^{*}(\Lambda^{m}T^{*}M))$ and,
thus,
$\mu^{*}\\{\alpha^{0},h\\}\in\Gamma((\pi\circ\nu)^{*}(\Lambda^{m}T^{*}M))$. In
fact,

$\mu^{*}\\{\alpha^{0},h\\}(\gamma)=\\{\alpha^{0},h\\}(\mu(\gamma)),\;\;\;\mbox{
for }\gamma\in{\mathcal{M}}\pi.$

$\diamond$

###### Proof.

(of Theorem 4.3) It is clear that ${\mathcal{R}}$ is a linear map. So, we must
prove that ${\mathcal{R}}$ is a Lie algebra morphism. For this purpose, we
will use Lemmas 4.4 and 4.5.

If $\alpha^{0},\beta^{0}$ are currents and $h\in\Gamma(\mu)$ is a Hamiltonian
section then, using (4.12) and (4.14), we obtain

$\mu^{*}\left({\mathcal{R}}(\\{\alpha^{0},\beta^{0}\\}_{\mathcal{O}})h\right)=-(d^{v}{\mathcal{F}}_{h})(\tilde{\mathcal{H}}_{\\{\alpha^{0},\beta^{0}\\}_{\mathcal{O}}}).$

Therefore, from Lemma 4.4, it follows that

$\mu^{*}\left({\mathcal{R}}(\\{\alpha^{0},\beta^{0}\\}_{\mathcal{O}})h\right)=\tilde{\mathcal{H}}_{\alpha^{0}}\left((d^{v}{\mathcal{F}}_{h})(\tilde{\mathcal{H}}_{\beta^{0}})\right)-\tilde{\mathcal{H}}_{\beta^{0}}\left((d^{v}{\mathcal{F}}_{h})(\tilde{\mathcal{H}}_{\alpha^{0}})\right).$

Now, using (4.14) and the fact the vector fields
$\tilde{\mathcal{H}}_{\alpha^{0}}$ and $\tilde{\mathcal{H}}_{\beta^{0}}$ are
$\mu$-projectable on the vector fields ${\mathcal{H}}_{\alpha^{0}}$ and
${\mathcal{H}}_{\beta^{0}}$, respectively, we deduce

$\displaystyle\mu^{*}\left({\mathcal{R}}(\\{\alpha^{0},\beta^{0}\\}_{\mathcal{O}})h\right)$
$\displaystyle=\mu^{*}\left(-{\mathcal{H}}_{\alpha^{0}}(\\{\beta^{0},h\\})+{\mathcal{H}}_{\beta^{0}}(\\{\alpha^{0},h\\})\right)$
$\displaystyle=\mu^{*}\left(-d^{l}\\{\beta^{0},h\\}({\mathcal{H}}_{\alpha^{0}})+d^{l}\\{\alpha^{0},h\\}({\mathcal{H}}_{\beta^{0}})\right).$

So, from (4.16), we obtain that

$\mu^{*}\left({\mathcal{R}}(\\{\alpha^{0},\beta^{0}\\}_{\mathcal{O}})h\right)=\mu^{*}\left(\\{\alpha^{0},\\{\beta^{0},h\\}\\}_{l}-\\{\beta^{0},\\{\alpha^{0},h\\}\\}_{l}\right).$

Finally, using (4.12) and the fact that

${\mathcal{R}}(\alpha^{0})^{l}=\\{\alpha^{0},\cdot\\}_{l},\;\;\;{\mathcal{R}}(\beta^{0})^{l}=\\{\beta^{0},\cdot\\}_{l},$

we conclude that

$\displaystyle{\mathcal{R}}(\\{\alpha^{0},\beta^{0}\\}_{\mathcal{O}})h$
$\displaystyle=\left({\mathcal{R}}(\alpha^{0})^{l}({\mathcal{R}}(\beta^{0})h)-{\mathcal{R}}(\beta^{0})^{l}({\mathcal{R}}(\alpha^{0})h)\right)$
$\displaystyle=\big{(}{\mathcal{R}}(\alpha^{0})^{l}\circ{\mathcal{R}}(\beta^{0})-{\mathcal{R}}(\beta^{0})^{l}\circ{\mathcal{R}}(\alpha^{0})\big{)}h,$

which proves the result. ∎

###### Proof.

(of Lemma 4.4) It is clear that the map
$\tilde{\mathcal{H}}:{\mathcal{O}}\to\Gamma(V(\pi\circ\nu))$ is linear. So, we
will prove that

$[\tilde{\mathcal{H}}_{\alpha^{0}},\tilde{\mathcal{H}}_{\beta^{0}}]=-\tilde{\mathcal{H}}_{\\{\alpha^{0},\beta^{0}\\}_{\mathcal{O}}},$

for $\beta^{0}\in{\mathcal{O}}$.

Now, we have that

$i_{[\tilde{\mathcal{H}}_{\alpha^{0}},\tilde{\mathcal{H}}_{\beta^{0}}]}\omega_{{\mathcal{M}}\pi}=\left({\mathcal{L}}_{\tilde{\mathcal{H}}_{\alpha^{0}}}\circ
i_{\tilde{\mathcal{H}}_{\beta^{0}}}-i_{\tilde{\mathcal{H}}_{\beta^{0}}}\circ{\mathcal{L}}_{\tilde{\mathcal{H}}_{\alpha^{0}}}\right)\omega_{{\mathcal{M}}\pi}.$

Thus, using (4.3) and the fact that $\omega_{{\mathcal{M}}\pi}$ is closed, we
deduce that

$i_{[\tilde{\mathcal{H}}_{\alpha^{0}},\tilde{\mathcal{H}}_{\beta^{0}}]}\omega_{{\mathcal{M}}\pi}={\mathcal{L}}_{\tilde{\mathcal{H}}_{\alpha^{0}}}(\mu^{*}(d\beta^{0}))-i_{\tilde{\mathcal{H}}_{\beta^{0}}}d(\mu^{*}(d\alpha^{0}))={\mathcal{L}}_{\tilde{\mathcal{H}}_{\alpha^{0}}}(\mu^{*}(d\beta^{0}))=d(i_{\tilde{\mathcal{H}}_{\alpha^{0}}}(\mu^{*}(d\beta^{0}))).$

But, since the vector field $\tilde{\mathcal{H}}_{\alpha^{0}}$ on
${\mathcal{M}}\pi$ is $\mu$-projectable over the vector field
${\mathcal{H}}_{\alpha^{0}}$ on ${\mathcal{M}}^{0}\pi$, it follows that

$i_{[\tilde{\mathcal{H}}_{\alpha^{0}},\tilde{\mathcal{H}}_{\beta^{0}}]}\omega_{{\mathcal{M}}\pi}=d(\mu^{*}(i_{{\mathcal{H}}_{\alpha^{0}}}d\beta^{0})).$

Therefore, using (4.7), we obtain that

$i_{[\tilde{\mathcal{H}}_{\alpha^{0}},\tilde{\mathcal{H}}_{\beta^{0}}]}\omega_{{\mathcal{M}}\pi}=-\mu^{*}(d\\{\alpha^{0},\beta^{0}\\}_{\mathcal{O}}).$

So, from (4.3), it follows that

$i_{[\tilde{\mathcal{H}}_{\alpha^{0}},\tilde{\mathcal{H}}_{\beta^{0}}]}\omega_{{\mathcal{M}}\pi}=-i_{\tilde{\mathcal{H}}_{\\{\alpha^{0},\beta^{0}\\}_{\mathcal{O}}}}\omega_{{\mathcal{M}}\pi}$

and, since $\omega_{{\mathcal{M}}\pi}$ is non-degenerate, this implies the
result. ∎

###### Proof.

(of Lemma 4.5) Suppose that

$\alpha^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})=\left(Y^{\alpha}(x,u)p_{\alpha}^{i}+\alpha^{i}(x,u)\right)d^{m-1}x_{i}.$

(i) If

$h(x^{i},u^{\alpha},p_{\alpha}^{i})=(x^{i},u^{\alpha},-H(x^{j},u^{\beta},p_{\beta}^{j}),p_{\alpha}^{i})$

then, from (3.2) and (4.5), we have

$(d^{v}{\mathcal{F}}_{h})(\tilde{\mathcal{H}}_{\alpha^{0}})=\left(Y^{\alpha}\frac{\partial
H}{\partial u^{\alpha}}-\left(\frac{\partial Y^{\beta}}{\partial
u^{\alpha}}p_{\beta}^{i}+\frac{\partial\alpha^{i}}{\partial
u^{\alpha}}\right)\frac{\partial H}{\partial
p_{\alpha}^{i}}-\left(\frac{\partial\alpha^{i}}{\partial x^{i}}+\frac{\partial
Y^{\alpha}}{\partial x^{i}}p_{\alpha}^{i}\right)\right)d^{m}x.$

This, by (3.40), implies

$\mu^{*}\\{\alpha^{0},h\\}=-(d^{v}{\mathcal{F}}_{h})(\tilde{\mathcal{H}}_{\alpha^{0}}).$

Finally, using (4.14), Definition 3.3, and the fact that
$\tilde{\mathcal{H}}_{\alpha^{0}}$ is $\mu$-projectable on the vector field
${\mathcal{H}}_{\alpha^{0}}$, we deduce that (4.15) also holds.

(ii) Suppose that

${\mathcal{F}}^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})=F^{0}(x^{i},u^{\alpha},p_{\alpha}^{i})d^{m}x.$

Then, we have

$d^{l}{\mathcal{F}}^{0}=\left(\frac{\partial F^{0}}{\partial
u^{\alpha}}du^{\alpha}+\frac{\partial F^{0}}{\partial
p_{\alpha}^{i}}dp_{\alpha}^{i}\right)\otimes d^{m}x.$

Thus, using (4.6), it follows

$(d^{l}{\mathcal{F}}^{0})({\mathcal{H}}_{\alpha^{0}})=\left(Y^{\alpha}\frac{\partial
F^{0}}{\partial u^{\alpha}}-\left(\frac{\partial Y^{\beta}}{\partial
u^{\alpha}}p_{\beta}^{i}+\frac{\partial\alpha^{i}}{\partial
u^{\alpha}}\right)\frac{\partial F^{0}}{\partial
p_{\alpha}^{i}}\right)d^{m}x.$

Therefore, from (3.41), we deduce the result. ∎

##  5\. Examples

###  5.1. Time-dependent Hamiltonian Mechanics

In this section, we will use the following terminology. Let $\tau:V\to P$ be a
vector bundle. Then, we can consider the vector bundle

$id_{\mathbb{R}}\times\tau:\mathbb{R}\times V\to\mathbb{R}\times P.$

The sections of this vector bundle are just the time-dependent sections of
$\tau:V\to P$. For this reason, the vector bundle
$id_{\mathbb{R}}\times\tau:\mathbb{R}\times V\to\mathbb{R}\times P$ will be
called the time-dependent vector bundle associated with $\tau:V\to P$.

For time-dependent Mechanics, the base space of the configuration bundle is
the real line $\mathbb{R}$, that is, we have a fibration $\pi:E\to\mathbb{R}$.
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 $\mathbb{R}\times Q$ and, under
this identification, $\pi:E\simeq\mathbb{R}\times Q\to\mathbb{R}$ 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
$J^{1}\pi=J^{1}(pr_{1})\to E=\mathbb{R}\times Q$ may be identified with the
affine subbundle of $T(\mathbb{R}\times Q)=\mathbb{R}\times\mathbb{R}\times
TQ$

$J^{1}(pr_{1})=\big{\\{}\frac{\partial}{\partial t}_{|t}+v_{q}\in
T_{t}\mathbb{R}\times T_{q}Q\mid(t,q)\in\mathbb{R}\times Q\mbox{ and }v_{q}\in
T_{q}Q\big{\\}}.$

Thus, $J^{1}(pr_{1})$ is isomorphic to the vector bundle $\mathbb{R}\times
TQ\to\mathbb{R}\times Q$.

An Ehresmann connection ${}^{H}:(\mathbb{R}\times
Q)\times_{\mathbb{R}}T\mathbb{R}\to H\subseteq T(\mathbb{R}\times Q)$ on the
fibration $pr_{1}:\mathbb{R}\times Q\to\mathbb{R}$ is completely determined by
a vector field $\Gamma$ on $\mathbb{R}\times Q$ satisfying

$\left\langle dt,\Gamma\right\rangle=1.$

In fact, the horizontal subbundle associated with the connection is of rank
$1$ and generated by the vector field $\Gamma$.

The extended (resp. restricted) multimomentum bundle may be identified with
the cotangent bundle $T^{*}(\mathbb{R}\times
Q)=\mathbb{R}\times\mathbb{R}\times T^{*}Q$ (resp. the vector bundle
$\mathbb{R}\times T^{*}Q$) and, under this identification, the multisymplectic
structure on $\mathbb{R}\times\mathbb{R}\times T^{*}Q$ is just the canonical
symplectic structure $\omega_{(\mathbb{R}\times Q)}$ on
$\mathbb{R}\times\mathbb{R}\times T^{*}Q$.

We have a principal $\mathbb{R}$-action on $T^{*}(\mathbb{R}\times
Q)=\mathbb{R}\times\mathbb{R}\times T^{*}Q$ given by

$p^{\prime}\cdot(t,p,\alpha_{q})=(t,p+p^{\prime},\alpha_{q})$

for $(t,p,\alpha_{q})\in\mathbb{R}\times T^{*}_{q}Q$ and
$p^{\prime}\in\mathbb{R}$. The principal bundle projection is just the
canonical projection

$\mu:T^{*}(\mathbb{R}\times Q)=\mathbb{R}\times\mathbb{R}\times T^{*}Q\to
V^{*}(pr_{1})=\mathbb{R}\times T^{*}Q,\;\;(t,p,\alpha_{q})\to(t,\alpha_{q}).$

Note that $\mathbb{R}\times T^{*}Q$ admits a Poisson structure of corank $1$
which is induced by the canonical symplectic structure on $T^{*}Q$. In fact,
the $\mathbb{R}$-action on the extended multimomentum bundle preserves the
symplectic form and the canonical projection $\mu$ is a Poisson map.

In this case, a Hamiltonian section $h:V^{*}(pr_{1})=\mathbb{R}\times
T^{*}Q\to T^{*}(\mathbb{R}\times Q)=\mathbb{R}\times\mathbb{R}\times T^{*}Q$
may be identified with a global time-dependent Hamiltonian function
$H:\mathbb{R}\times T^{*}Q\to\mathbb{R}$. In addition, the couple

$(\omega_{h}=h^{*}\omega_{(\mathbb{R}\times
Q)},\eta=(\pi\circ\nu^{0})^{*}(dt))$

is a cosymplectic structure on $V^{*}(pr_{1})=\mathbb{R}\times T^{*}Q$ and,
thus, we can consider the Reeb vector field $\Gamma_{H}$ of
$(\omega_{H}=\omega_{h},\eta)$.

######  Remark 5.1.

We recall that a cosymplectic structure on a manifold $P$ of odd dimension
$2p+1$ is a couple $(\omega,\theta)$, where $\omega$ is a closed $2$-form,
$\theta$ is a closed $1$-form and $\theta\wedge\omega^{p}$ is a volume form on
$P$. The Reeb vector field $\Gamma$ associated with the structure
$(\omega,\theta)$ is the vector field on $P$ which is completely characterized
by the conditions

$i_{\Gamma}\omega=0,\quad i_{\Gamma}\eta=1.$

$\diamond$

Note that the $2$-form $\omega_{H}$ on $\mathbb{R}\times T^{*}Q$ is given by

$\omega_{H}=\omega_{Q}+dH\wedge dt,$

where $\omega_{Q}$ is the canonical symplectic structure on $T^{*}Q$. So, the
Reeb vector field $\Gamma_{H}$ of the cosymplectic structure
$(\omega_{H},\eta)$ on $\mathbb{R}\times T^{*}Q$ is

$(t,\alpha)\to\Gamma_{H}(t,\alpha)=\frac{\partial}{\partial
t}_{|t}+X_{H(t,\cdot)}(\alpha),$

with $X_{H(t,\cdot)}$ the Hamiltonian vector field on $T^{*}Q$ associated with
the function

$H(t,\cdot):T^{*}Q\to\mathbb{R},\;\;\beta\to H(t,\beta).$

Thus, if $(t,q^{i},p_{i})$ are canonical coordinates on $\mathbb{R}\times
T^{*}Q$ then

$\Gamma_{H}=\frac{\partial}{\partial t}+\sum_{i}\left(\frac{\partial
H}{\partial p_{i}}\frac{\partial}{\partial q^{i}}-\frac{\partial H}{\partial
q^{i}}\frac{\partial}{\partial p_{i}}\right)$

and the integral curves $t\to(t,q^{i}(t),p_{i}(t))$ of $\Gamma_{H}$ are just
of the solution of the Hamilton equations for $H$, that is,

$\displaystyle\frac{dq^{i}}{dt}=\frac{\partial H}{\partial
p_{i}},\;\;\;\frac{dp_{i}}{dt}=-\frac{\partial H}{\partial q^{i}}.$

Therefore, $\Gamma_{H}$ is the evolution vector field associated with the
time-dependent Hamiltonian function $H:\mathbb{R}\times T^{*}Q\to\mathbb{R}$
(for more details see, for instance, [15, 20]).

In addition, the vector bundle $V^{*}(\pi\circ\nu)$ may be identified with the
time-dependent cotangent bundle to $\mathbb{R}\times T^{*}Q$,

$id_{\mathbb{R}}\times\pi_{\mathbb{R}\times T^{*}Q}:\mathbb{R}\times
T^{*}(\mathbb{R}\times T^{*}Q)\to\mathbb{R}\times(\mathbb{R}\times T^{*}Q),$

where $\pi_{\mathbb{R}\times T^{*}Q}:T^{*}(\mathbb{R}\times
T^{*}Q)\to\mathbb{R}\times T^{*}Q$ is the canonical projection.

Under the previous identification, the principal $\mathbb{R}$-action on
$V^{*}(\pi\circ\nu)\simeq\mathbb{R}\times T^{*}(\mathbb{R}\times
T^{*}Q)\simeq\mathbb{R}\times(\mathbb{R}\times\mathbb{R}\times T^{*}T^{*}Q)$
is given by

$p^{\prime}\cdot(t,(p,p_{q},\tilde{\gamma}))=(t,(p+p^{\prime},p_{q},\tilde{\gamma})),$

for $p^{\prime}\in\mathbb{R}$ and
$(t,(p,p_{q},\tilde{\gamma}))\in\mathbb{R}\times(\mathbb{R}\times\mathbb{R}\times
T^{*}T^{*}Q)$.

Moreover, the extended phase bundle $\widetilde{\mathbb{P}(\pi)}$ is

$\widetilde{\mathbb{P}(\pi)}=\\{(t,dp_{|p}+\gamma)\mid(t,p)\in\mathbb{R}\mbox{
and }\gamma\in T^{*}(T^{*}Q)\\}.$

In other words, $\widetilde{\mathbb{P}(\pi)}$ may be identified with the time-
dependent vector bundle associated with the vector bundle
$id_{\mathbb{R}}\times\pi_{T^{*}Q}:\mathbb{R}\times
T^{*}T^{*}Q\to\mathbb{R}\times T^{*}Q$, that is,

$id_{\mathbb{R}}\times(id_{\mathbb{R}}\times\pi_{T^{*}Q}):\widetilde{\mathbb{P}(\pi)}\simeq\mathbb{R}\times(\mathbb{R}\times
T^{*}T^{*}Q)\to T^{*}(\mathbb{R}\times
Q)\simeq\mathbb{R}\times\mathbb{R}\times T^{*}Q$

and the principal $\mathbb{R}$-action on
$\widetilde{\mathbb{P}(\pi)}\simeq\mathbb{R}\times(\mathbb{R}\times
T^{*}T^{*}Q)$ is given by

$p^{\prime}\cdot(t,(p,\tilde{\gamma}))=(t,p+p^{\prime}\tilde{\gamma})$

for $p^{\prime},t,p\in\mathbb{R}$ and $\tilde{\gamma}\in T^{*}(T^{*}Q)$.

Thus, the phase bundle $\mathbb{P}(\pi)$ may be identified with the time-
dependent cotangent bundle to $T^{*}Q$

$id_{\mathbb{R}}\times\pi_{T^{*}Q}:\mathbb{P}(\pi)\simeq\mathbb{R}\times
T^{*}T^{*}Q\to\mathbb{R}\times T^{*}Q.$

Then, the differential $dh$ of $h$ is just the vertical differential $d^{v}H$
(with respect to the projection $pr_{1}:\mathbb{R}\times T^{*}Q\to\mathbb{R}$)
of the Hamiltonian function $H$, that is,

$d^{v}H:\mathbb{R}\times T^{*}Q\to\mathbb{R}\times
T^{*}T^{*}Q,\;\;(t,\alpha)\to d^{v}H(t,\alpha)=(t,dH_{(t,\cdot)}(\alpha)).$

On the other hand, the vector bundle $L$, which was introduced in Proposition
3.10, is isomorphic to the cotangent bundle to $\mathbb{R}\times T^{*}Q$.

Moreover, as we know, the $1$-jet bundle to the projection

$\pi\circ\nu^{0}:\mathbb{R}\times T^{*}Q\to\mathbb{R}$

may be identified with the affine subbundle of $T(\mathbb{R}\times T^{*}Q)$
given by

$\big{\\{}\frac{\partial}{\partial t}_{|t}+X_{\alpha}\in T_{t}\mathbb{R}\times
T_{\alpha}T^{*}Q\mid(t,\alpha)\in\mathbb{R}\times T^{*}Q\big{\\}}$ (5.1)

or, equivalently, with the time-dependent tangent bundle to $T^{*}Q$

$id_{\mathbb{R}}\times\tau_{T^{*}Q}:\mathbb{R}\times
TT^{*}Q\to\mathbb{R}\times T^{*}Q.$

Under all the previous identifications, the affine bundle isomorphism
$A:J^{1}(\pi\circ\nu^{0})\simeq\mathbb{R}\times
TT^{*}Q\to\mathbb{P}(\pi)\simeq\mathbb{R}\times T^{*}T^{*}Q$ is given by

$A(t,X)=(t,i_{X}\omega_{Q}),\;\;\mbox{ for }(t,X)\in\mathbb{R}\times TT^{*}Q,$

with $\omega_{Q}$ the canonical symplectic structure of $T^{*}Q$. Thus, in
this case, the vector subbundle $\operatorname{Ker}A$ 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 $\Gamma_{H}$.

In addition, if $\sharp^{\rm aff}:\mathbb{P}(\pi)\to J^{1}(\pi\circ\nu^{0})$
is the inverse morphism of $A:J^{1}(\pi\circ\nu^{0})\to\mathbb{P}(\pi)$ it is
clear that, under the identification of $J^{1}(\pi\circ\nu^{0})$ with the
affine subbundle of $T(\mathbb{R}\times T^{*}Q)$ given by (5.1), the image of
the section $dh$ of $\mathbb{P}(\pi)$ is just the evolution vector field
$\Gamma_{H}$.

On the other hand, the space ${\mathcal{O}}$ of currents is the set of smooth
real functions on $\mathbb{R}\times T^{*}Q$ (the space of observables in
Classical Mechanics)

${\mathcal{O}}\simeq C^{\infty}(\mathbb{R}\times T^{*}Q)$

and it is clear that the space $\Gamma(\mu)$ of sections of the projection

$\mu:T^{*}(\mathbb{R}\times Q)\to\mathbb{R}\times T^{*}Q$

may be also identified with $C^{\infty}(\mathbb{R}\times T^{*}Q)$. Then, the
linear-affine bracket

$\\{\cdot,\cdot\\}:{\mathcal{O}}\times\Gamma(\mu)\to
C^{\infty}(\mathbb{R}\times T^{*}Q)$

is given by

$\\{F^{0},H\\}=\langle dF^{0},\Gamma_{H}\rangle=\frac{\partial F^{0}}{\partial
t}+\sum_{i}\left(\frac{\partial F^{0}}{\partial q^{i}}\frac{\partial
H}{\partial p_{i}}-\frac{\partial F^{0}}{\partial p_{i}}\frac{\partial
H}{\partial q^{i}}\right),$

for $(F^{0},H)\in{\mathcal{O}}\times\Gamma(\mu)\simeq
C^{\infty}(\mathbb{R}\times T^{*}Q)\times C^{\infty}(\mathbb{R}\times
T^{*}Q)$.

This bracket was considered in [34, 35, 47].

In addition, the bracket $\\{\cdot,\cdot\\}_{\mathcal{O}}$ on the space of
observables ${\mathcal{O}}\simeq C^{\infty}(\mathbb{R}\times T^{*}Q)$ is just
the standard Poisson bracket $\\{\cdot,\cdot\\}_{\mathbb{R}\times T^{*}Q}$
induced by the canonical symplectic structure of $T^{*}Q$. In other words,

$\\{F^{0},G^{0}\\}_{\mathbb{R}\times T^{*}Q}=\sum_{i}\left(\frac{\partial
F^{0}}{\partial q^{i}}\frac{\partial G^{0}}{\partial p_{i}}-\frac{\partial
F^{0}}{\partial p_{i}}\frac{\partial G^{0}}{\partial q^{i}}\right)$

for $F^{0},G^{0}\in C^{\infty}(\mathbb{R}\times T^{*}Q)$.

Finally, the affine representation ${\mathcal{R}}$ of the Lie algebra
$(C^{\infty}(\mathbb{R}\times T^{*}Q),\\{\cdot,\cdot\\}_{\mathbb{R}\times
T^{*}Q})$ on the affine space $\Gamma(\mu)\simeq C^{\infty}(\mathbb{R}\times
T^{*}Q)$ is given by

${\mathcal{R}}:C^{\infty}(\mathbb{R}\times T^{*}Q)\to{\rm
Aff}(C^{\infty}(\mathbb{R}\times T^{*}Q),C^{\infty}(\mathbb{R}\times
T^{*}Q)),\quad{\mathcal{R}}(F^{0})H=\langle dF^{0},\Gamma_{H}\rangle.$

Concretely, we have

$\\{\\{F_{0},G_{0}\\}_{\mathbb{R}\times
T^{*}Q},H\\}=\\{F_{0},\\{G_{0},H\\}\\}_{\mathbb{R}\times
T^{*}Q}-\\{G_{0},\\{F_{0},H\\}\\}_{\mathbb{R}\times T^{*}Q},$

###  5.2. A particular case: the configuration bundle is trivial and the base
space is orientable

In this section we will assume that $E=M\times Q$, $\pi:E\to M$ is the
canonical projection $pr_{1}:M\times Q\to M$ on the first factor. In this
case, the affine bundle

$J^{1}(pr_{1})\to M\times Q$

can be identified with the vector bundle