# Proof of the equivalence of the symplectic forms derived from the canonical
and the covariant phase space formalisms

Juan Margalef-Bentabol1,3 Eduardo J.S. Villaseñor2,3 1Department of
Mathematics and Statistics, Memorial University, St. John’s, Newfoundland and
Labrador A1C 5S7, Canada.
2Departamento de Matemáticas, Universidad Carlos III de Madrid. Avda. de la
Universidad 30, 28911 Leganés, Spain.
3Grupo de Teorías de Campos y Física Estadística. Instituto Gregorio Millán
(UC3M). Unidad Asociada al Instituto de Estructura de la Materia, CSIC,
Madrid, Spain.

###### Abstract

We prove that, for any theory defined over a space-time with boundary, the
symplectic form derived in the covariant phase space is equivalent to the one
derived from the canonical formalism.

CPS, symplectic geomety, Hamiltonian theory,boundaries, covariant methods,
canonical formalism

††preprint: APS/123-QED

##  I INTRODUCTION

In physics, we can identify, roughly speaking, two somewhat disjoint
frameworks to deal with a field theory: the canonical and the covariant. The
former breaks all the objects of the theory into spatial and normal
components. The main advantages are that we gain a dynamical perspective and
some universal structures like the symplectic form of a cotangent bundle. This
provides a way to approach the problem numerically [1] (essential in the study
of gravitational waves and LIGO observations) and a starting point for the
Hamiltonian quantization [2, 3, 4, 5]. A price to pay is the apparent loss of
some symmetries.

The covariant approach, on the other hand, considers fields over the whole
space-time [6]. Some noteworthy advantages are: symmetries are explicit, the
study of null infinity is easier, there are methods to compute conserved
quantities, and higher-derivative theories are treated on equal footing as
1st-order ones [7, 8, 9]. All this has important consequences in effective and
perturbation theory, both of which are relevant to the of study string
theories, edge modes, corner and BMS algebras, or the analysis of consistent
deformations [10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. However, one drawback
is that there are no known canonical structures on the spaces involved. In
particular, to have a symplectic structure, one has to rely on the covariant
phase space (CPS) formalism and fix a local action.

The question that arises naturally is if both approaches are equivalent. It is
possible to prove that in many aspects they are. However, the equivalence of
the symplectic forms was unknown in general. The problem has been studied in
concrete relevant examples, like GR without boundaries [20, 21]. Also in [22],
an analytical prove of the symplectic equivalence is provided over the reduced
phase space (using Poisson brackets) for 1st-order theories with no
boundaries. It is important to mention that adding boundaries complicates the
matter a great deal even in concrete examples. In that sense, our paper [23]
was a breakthrough because it provided a way to map any theory with boundary
to another one without boundary in the CPS formalism. Although this approach
would simplify the upcoming computations, we will not use it here as it
requires the introduction of a lot of notation and definitions. However, in
[23] we also provided a geometric language that bridges between the
mathematical formalism ($\infty$-jets framework) and the standard physics
notation. We will see in this paper that this language turns out to be
essential to prove, in full generality, the equivalence of both symplectic
structures. As a byproduct, the equivalence shows that our proposal of a CPS
symplectic structure in manifolds with boundaries (introduced also in [23]) is
the most natural one. Not only for its cohomological nature, as explained in
that paper, but also because it is equivalent to the one coming from the
canonical formalism.

##  II THE GEOMETRIC ARENA

###  II.1 The spacetime

Consider a globally hyperbolic $n$-manifold $M$ (up to diffeomorphism,
$M=\mathbb{R}\times\Sigma$) with boundary $\partial
M=\mathbb{R}\times\partial\Sigma$. We have the inclusions
$\overline{\jmath}:\partial\Sigma\hookrightarrow\Sigma$, $\jmath:\partial
M\hookrightarrow M$, and
$\imath_{t}:\Sigma\hookrightarrow\\{t\\}\times\Sigma\subset M$. As usual, we
have the exterior derivative $\mathrm{d}$, the wedge product $\wedge$, the
interior derivative $\iota_{\vec{V}}$, and the Lie derivative
$\mathcal{L}_{\vec{V}}$.

In this setting it is crucial to keep track of the orientations. We orient
$\mathbb{R}$ with the standard volume form $\mathrm{d}t$ and $\Sigma$ with
some volume form $\mathrm{vol}_{\Sigma}$ (which, up to easy to handle
technicalities, can be equally understood as an $(n-1)$-form on $\Sigma$ or on
$M$). We then orient $M$ with

$\mathrm{vol}_{M}:=\mathrm{d}t\wedge\mathrm{vol}_{\Sigma}$ (1)

Boundaries are oriented so that Stokes’ theorem holds

$\int_{M}\mathrm{d}\alpha=\int_{\partial
M}\jmath^{*}\alpha\qquad\qquad\int_{\Sigma}\mathrm{d}\beta=\int_{\partial\Sigma}\overline{\jmath}^{*}\beta$
(2)

For that, take any metric $\gamma_{ij}$ on $\Sigma$ and denote $\nu^{i}$ the
unit vector field $\gamma$-normal to $\partial\Sigma$. Consider the adapted
metric $g_{\alpha\beta}$ on $\mathbb{R}\times\Sigma$ such that
$g^{\alpha\beta}=-\partial_{t}^{\alpha}\partial_{t}^{\beta}+(\imath_{t})^{\alpha}_{i}(\imath_{t})^{\beta}_{j}\gamma^{ij}$,
then the unit vector field $g$-normal to $\partial M$ is (up to pushforward)
$\nu^{i}$, i.e., $\mathcal{V}^{\alpha}:=(\imath_{t})^{\alpha}_{i}\nu^{i}$.
Finally, we define

$\mathrm{vol}_{\partial\Sigma}:=\iota_{\vec{\nu}}\mathrm{vol}_{\Sigma}\qquad\qquad\mathrm{vol}_{\partial
M}:=\iota_{\vec{\mathcal{V}}}\mathrm{vol}_{M}$ (3)

It is important to notice that

$\mathrm{vol}_{\partial
M}=\iota_{\vec{\mathcal{V}}}(\mathrm{d}t\wedge\mathrm{vol}_{\Sigma})=-\mathrm{d}t\wedge\mathrm{vol}_{\partial\Sigma}$
(4)

###  II.2 The space of fields

Let $\mathcal{F}$ be a space of tensor fields (of any tensorial character)
over $M$. This space is $\infty$-dimensional and non-linear in general.
Although it can rigorously be described with the $\infty$-jets formalism, for
our purposes it is enough to think of $\mathcal{F}$ as a standard smooth
manifold with the usual operators such as the exterior derivative
$\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}$, the wedge product $\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}$ , the
interior derivative
$\raisebox{-0.17218pt}{\includegraphics[height=4.73611pt]{./ii.pdf}}_{\mathbb{V}}$,
or the Lie derivative
$\includegraphics[height=7.70004pt]{./LL.pdf}_{\mathbb{V}}$. Here,
$\mathbb{V}$ is a vector field of $\mathcal{F}$ (see [23] for a careful
discussion). Of course, $\mathcal{F}$ may consist of different types of tensor
fields, hence $\mathcal{F}=\mathcal{F}^{1}\times\cdots\times\mathcal{F}^{N}$
with the fields labeled as $(\phi^{I})_{I=1\cdots N}\in\mathcal{F}$.

##  III THE COVARIANT PHASE SPACE FORMALISM IN A NUTSHELL

Roughly speaking, the CPS method studies the space of solutions of a theory
over the whole space-time. The equations of motion are not dynamic but,
rather, give conditions for the fields to be solutions. We devote this section
to summarizing how to define a presymplectic form canonically associated with
a local action. For a detailed discussion and some applications see [23, 24,
25, 26].

Consider a local action $\SS:\mathcal{F}\to\mathbb{R}$ given by

$\SS=\int_{M}L-\int_{\partial M}\overline{\ell}$ (5)

where
$(L,\overline{\ell})\in\Omega^{(n,0)}(M\times\mathcal{F})\times\Omega^{(n-1,0)}(\partial
M\times\mathcal{F})$, known as CPS bulk and boundary Lagrangians, are top-
forms on $(M,\partial M)$ and $0$-forms on $\mathcal{F}$ (they are bigraded
forms). We assume that they are locally constructed, i.e., when evaluating
$(L(\phi),\overline{\ell}(\phi))\in\Omega^{n}(M)\times\Omega^{n-1}(\partial
M)$ at $p\in M$, they only depend on $p$, $\phi(p)$, and finitely many of its
derivatives at $p$.

It is a standard result [23, 27] that the $(n,1)$-form
$\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}L$ can be split as

$\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}L=E_{I}\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\phi^{I}+\mathrm{d}\Theta$ (6)

for some $(n,0)$-forms $E_{I}$ (Euler-Lagrange forms) and some $(n-1,1)$-form
$\Theta$ (bulk symplectic potential, uniquely defined up to an exact form). In
practice, this is achieved by using Leibniz’s rule to remove all the
derivatives from
$\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\phi^{I}$. Taking the
$\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}$-exterior derivative of $\SS$, using (6), and
Stokes’ theorem, we have

$\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\SS=\int_{M}E^{I}\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\phi^{I}-\int_{\partial
M}(\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\overline{\ell}-\jmath^{*}\Theta)$ (7)

We can split
$\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\overline{\ell}$ as in (6), but we have the
additional term $\jmath^{*}\Theta$ so we need it to be “splittable” as well.
This condition has to be imposed, in which case we say that $\SS$ defines a
good variational principle. This leads to

$\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}\overline{\ell}-\jmath^{*}\Theta=\overline{b}_{I}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\phi^{I}-\mathrm{d}\overline{\theta}$ (8)

for some $(n-1,0)$-forms $\overline{b}_{I}$ (boundary Euler-Lagrange forms)
and some $(n-2,1)$-form $\overline{\theta}$ (boundary symplectic potential,
uniquely defined up to an exact form). With these ingredients, we define the
space of solutions

$\mathrm{Sol}(\SS)=\\{\phi\in\mathcal{F}\ /\
(E^{I}(\phi),\overline{b}^{I}(\phi))=(0,0)\\}\overset{\raisebox{-1.59302pt}{\includegraphics[height=6.14993pt]{./jj.pdf}}_{\SS}}{\hookrightarrow}\mathcal{F}$

and the symplectic structure associated with an embedding
$\imath:\Sigma\hookrightarrow M$

$\includegraphics[height=6.83331pt]{./OOmega.pdf}_{\SS}^{\imath}=\int_{\Sigma}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}\imath^{*}\Theta-\int_{\partial\Sigma}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\imath^{*}\overline{\theta}$ (9)

$\includegraphics[height=6.83331pt]{./OOmega.pdf}_{\SS}^{\imath}$ is
independent of the choice of Lagrangians (as long as they define the same
$\SS$) and of the symplectic potentials chosen in (6) and (8). Moreover, if we
denote
$\includegraphics[height=6.83331pt]{./OOmega.pdf}_{\SS}:=\raisebox{-1.59302pt}{\includegraphics[height=6.14993pt]{./jj.pdf}}_{\SS}^{*}\Omega_{\SS}^{\imath}$
the pullback of the symplectic form to $\mathrm{Sol}(\SS)$, then it can be
proved that $\includegraphics[height=6.83331pt]{./OOmega.pdf}_{\SS}$ does not
depend on the embedding either. Thus, we have constructed a presymplectic form
on $\mathrm{Sol}(\SS)$ canonically associated with $\SS$.

##  IV THE CANONICAL FORMALISM IN A NUTSHELL

The canonical (CAN) formalism deals, roughly speaking, with “instant fields”
that evolve according to some dynamical equations. By evolving specific
initial data, we obtain a curve in the space of “instantaneous fields” which
corresponds (up to integrability issues) to a solution over the whole space-
time. We devote this section to summarize the results necessary for the
present work. For now, we will focus on 1st-order theories and delay the
generalization to higher order ones to section VI.

Consider the space
$\mathcal{Q}=\mathcal{Q}^{1}\times\cdots\times\mathcal{Q}^{M}$ of fields over
$\Sigma$ and its tangent bundle $T\mathcal{Q}$ (its standard geometric
operators like
$\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}$ or $\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}$ are
denoted as the ones of $\mathcal{F}$). It consists of elements of the form
$(q^{1},\ldots,q^{M};v^{1},\ldots,v^{M})$. Consider a local CAN Lagrangian-
action $\mathcal{L}:T\mathcal{Q}\to\mathbb{R}$ given by

$\mathcal{L}=\int_{\Sigma}\mathrm{L}-\int_{\partial\Sigma}l$ (10)

where $(\mathrm{L},l)\in\Omega^{(n-1,0)}(\Sigma\times
T\mathcal{Q})\times\Omega^{(n-2,0)}(\partial\Sigma\times T\mathcal{Q})$ are
some (locally constructed and possibly time-dependent) CAN Lagrangians.
$\mathcal{L}$ is historically called Lagrangian but it plays a role similar to
the action (5) although not entirely equal since the time integration is
missing. So we define the CAN action
$\mathcal{S}:\mathcal{C}^{\infty}(\mathbb{R},\mathcal{Q})\to\mathbb{R}$

$\mathcal{S}(q)=\int{R}\mathcal{L}\big{(}q(t),\dot{q}(t)\big{)}\mathrm{d}t$
(11)

The analogues to equations (6) and (8) are

$\displaystyle\begin{split}&\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\mathrm{L}=A_{I}^{(0)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}q^{I}+A_{I}^{(1)}\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}v^{I}+\mathrm{d}\widetilde{\Theta}\\\
&\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}l-\overline{\jmath}^{*}\widetilde{\Theta}=B_{I}^{(0)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}q^{I}+B_{I}^{(1)}\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}v^{I}-\mathrm{d}\widetilde{\theta}\end{split}$ (12)

Taking the
$\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}$-exterior derivative of (11), using Stokes’
theorem, and integrating by parts with respect to time (notice that in
$\mathcal{S}$ every $v^{I}$ is replaced by $\dot{q}^{I}$), we obtain

$\displaystyle\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}\mathcal{S}\\!=\\!\\!\int{R}\\!\mathrm{d}t\Big{(}\\!\int_{\Sigma}(A_{I}^{(0)}-\dot{A}_{I}^{(1)})\\!\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}q^{I}\\!\\!-\\!\\!\int_{\partial\Sigma}(B_{I}^{(0)}-\dot{B}_{I}^{(1)})\\!\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}q^{I}\\!\Big{)}$

$(A_{I}^{(0)}-\dot{A}_{I}^{(1)},B_{I}^{(0)}-\dot{B}_{I}^{(1)})$ are the bulk
and boundary dynamical equations (if some of them do not involve time
derivatives, we obtain bulk or boundary constraints).

Once we have the Lagrangian formalism, we proceed to introduce the Hamiltonian
formalism which, instead of living in the tangent bundle $T\mathcal{Q}$, lives
on the cotangent bundle $T^{*}\\!\mathcal{Q}$. The advantage of the latter is
its canonical symplectic form, which plays an essential role on the
Hamiltonian formulation. Indeed, denoting the elements of
$T^{*}\\!\mathcal{Q}$ as $(q;p)=(q^{1},\ldots,q^{M};p_{1},\ldots,p_{M})$, the
canonical symplectic structure is given by

$\includegraphics[height=6.83331pt]{./OOmega.pdf}_{T^{*}\\!\mathcal{Q}}:=\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}q^{I}\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}p_{I}$ (13)

The usual pairing between position and momenta applies when evaluated over
fields. In order to go from the Lagrangian (tangent bundle) to the Hamiltonian
(cotangent bundle), we use the fiber derivative $F\mathcal{L}:T\mathcal{Q}\to
T^{*}\\!\mathcal{Q}$. For each $(q;v)\in T_{q}\mathcal{Q}$ we define
$F\mathcal{L}_{(q;v)}\in T_{q}^{*}\mathcal{Q}$ as

$F\mathcal{L}_{(q;v)}\big{(}q;w\big{)}=\left.\frac{\mathrm{d}}{\mathrm{d}\tau}\right|_{\tau=0}\mathcal{L}\big{(}q;v_{\tau}\big{)}$
(14)

where $v_{\tau}$ is a curve in $T_{q}\mathcal{Q}$ with $v_{0}=v$ and
$\left.\frac{\mathrm{d}}{\mathrm{d}\tau}\right|_{0}v_{\tau}=w$. In general,
this map is not surjective and the relevant symplectic structure is actually
the one induced on its image (the primary constraint submanifold), i.e.,
induced by the inclusion
$\raisebox{-1.59302pt}{\includegraphics[height=6.14993pt]{./jj.pdf}}_{\mathcal{L}}:F\mathcal{L}(T\mathcal{Q})\hookrightarrow
T^{*}\\!\mathcal{Q}$. Since over $F\mathcal{L}(T\mathcal{Q})$ we have
$p=F\mathcal{L}_{(q;v)}$, we schematically obtain

$\includegraphics[height=6.83331pt]{./OOmega.pdf}_{\mathcal{L}}:=\raisebox{-1.59302pt}{\includegraphics[height=6.14993pt]{./jj.pdf}}_{\mathcal{L}}^{*}\includegraphics[height=6.83331pt]{./OOmega.pdf}_{T^{*}\\!\mathcal{Q}}=\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}q^{I}\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}\raisebox{-1.59302pt}{\includegraphics[height=6.14993pt]{./jj.pdf}}_{\mathcal{L}}^{*}p_{I}=\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}q^{I}\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}(F\mathcal{L}_{(q;v^{I})})$ (15)

where by $v^{I}$ we mean $(0,\ldots,0,v^{I},0,\ldots,0)$. In order to give a
concrete sense to (15), we rely on the geometric language introduced before.
The key observation is that the momenta can be rewritten as

$\displaystyle\begin{split}&p_{I}(q;w^{I}):=\includegraphics[height=7.70004pt]{./LL.pdf}_{(0,w^{I})}\mathcal{L}=\raisebox{-0.17218pt}{\includegraphics[height=4.73611pt]{./ii.pdf}}_{(0,w^{I})}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\mathcal{L}=\\\
&=\int_{\Sigma}\raisebox{-0.17218pt}{\includegraphics[height=4.73611pt]{./ii.pdf}}_{(0,w^{I})}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}\mathrm{L}-\int_{\partial\Sigma}\raisebox{-0.17218pt}{\includegraphics[height=4.73611pt]{./ii.pdf}}_{(0,w^{I})}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}l\end{split}$ (16)

The Lie derivative acts following Cartan’s rule while the interior derivative
acts as follows:
$\raisebox{-0.17218pt}{\includegraphics[height=4.73611pt]{./ii.pdf}}_{(\alpha^{I},\beta^{J})}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}q^{K}=\delta^{K}_{I}\alpha^{I}$ and
$\raisebox{-0.17218pt}{\includegraphics[height=4.73611pt]{./ii.pdf}}_{(\alpha^{I},\beta^{J})}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}v^{K}=\delta^{K}_{J}\beta^{J}$. Using (LABEL:def_p),
equation (LABEL:eq:_dd_L_canonic), and Stokes’ theorem leads to (removing the
argument)

$\displaystyle p_{I}(\cdot{})=\int_{\Sigma}A_{I}^{(1)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\cdot{}-\int_{\partial\Sigma}B_{I}^{(1)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\cdot{}$

Taking its
$\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}$-exterior derivative and using the $p-q$ pairing
finally allows us to rewrite (15) as

$\includegraphics[height=6.83331pt]{./OOmega.pdf}_{\mathcal{L}}=\int_{\Sigma}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}A_{I}^{(1)}\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}q^{I}-\int_{\partial\Sigma}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}B_{I}^{(1)}\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}q^{I}$ (17)

##  V PROVING THE EQUIVALENCE FOR FIRST ORDER THEORIES

In the previous sections we have seen that, on the one hand, we can derive a
symplectic structure $\includegraphics[height=6.83331pt]{./OOmega.pdf}_{\SS}$
canonically associated with a CPS action $\SS$ (defined over the space of
fields of $M$). On the other hand, from a CAN Lagrangian-action $\mathcal{L}$
defined over some $T\mathcal{Q}$, we get the symplectic structure
$\includegraphics[height=6.83331pt]{./OOmega.pdf}_{\mathcal{L}}$ canonically
associated with $\mathcal{L}$. It is well known that we can go from the CPS
formalism to the CAN one by performing a $(1,n-1)$-decomposition (there are
many equivalent ways of doing it). Therefore, for a given theory we end up
with two symplectic structures canonically associated with it. The issue of
understanding the relation between them has been a long-lasting open question
that we answer now.

###  V.1 CANonicalizing the Lagrangian

In order to go from the CPS Lagrangians $(L,\overline{\ell})$ to the CAN
Lagrangians $(\mathrm{L},l)$, we need Fubini’s theorem.

Theorem
Let $A$ and $B$ be manifolds oriented with $\mathrm{vol}_{A}$ and
$\mathrm{vol}_{B}$ respectively. We orient $A\times B$ with
$\mathrm{vol}_{A}\wedge\mathrm{vol}_{B}$ and consider an integrable function
$f:A\times B\to\mathbb{R}$, then

$\int_{A\times
B}f\mathrm{vol}_{A}\wedge\mathrm{vol}_{B}=\int_{A}\left(\int_{B}f\mathrm{vol}_{B}\right)\mathrm{vol}_{A}$

Let us now use the $(1,n-1)$-decomposition induced from
$M=\mathbb{R}\times\Sigma$ to decompose the CPS Lagrangians. In general, given
$\alpha\in\Omega^{k}(M)$, we have
$\alpha=\mathrm{d}t\wedge\alpha_{\perp}+\alpha^{\top}$ with
$\alpha_{\perp}:=\iota_{\partial_{t}}\alpha$ and
$\alpha^{\top}:=\iota_{\partial_{t}}(\mathrm{d}t\wedge\alpha)$. Since the CPS
Lagrangians are top-forms, we simply have $L=\mathrm{d}t\wedge L_{\perp}$ and
$\overline{\ell}=\mathrm{d}t\wedge\overline{\ell}_{\perp}$. Besides, there
exist $F,f$ such that

$\displaystyle\mathrm{d}t\wedge
L_{\perp}=L=F\mathrm{vol}_{M}\\!\overset{\eqref{eq: vol
M}}{=}\\!\mathrm{d}t\wedge(F\mathrm{vol}_{\Sigma})$
$\displaystyle\\!\\!\\!\\!\\!\\!\to\
F=\frac{L_{\perp}}{\mathrm{vol}_{\Sigma}}$
$\displaystyle\mathrm{d}t\wedge\overline{\ell}_{\perp}=\overline{\ell}=f\mathrm{vol}_{\partial
M}\\!\overset{\eqref{eq: vol partial
M}}{=}\\!(-\mathrm{d}t)\wedge(f\mathrm{vol}_{\partial\Sigma})$
$\displaystyle\\!\\!\\!\\!\\!\\!\to\
f=\frac{-\overline{\ell}_{\perp}}{\mathrm{vol}_{\partial\Sigma}}$

where $\frac{\omega}{\mathrm{vol}}$ denotes the function that relates the top-
form $\omega$ with the volume form vol. Plugging these expressions of
$(L,\overline{\ell})$ into (5) and using Fubini’s theorem we get

$\displaystyle\SS$
$\displaystyle=\int{R}\left(\int_{\Sigma}L_{\perp}-\int_{\partial\Sigma}(-\overline{\ell}_{\perp})\right)\mathrm{d}t$

A minus sign appears in the boundary because, from (4), we have
$(A,\mathrm{vol}_{A})=(\mathbb{R},-\mathrm{d}t)$ in Fubini’s theorem.

All this allows us to identify $\mathrm{L}$ with $\iota_{\partial_{t}}L$ and
$l$ with $-\iota_{\partial_{t}}\overline{\ell}$. To formalize this
identification, we perform the $(1,n-1)$-decomposition on the fields
$(\phi^{I})_{I}$ and their derivatives. Since the theory is of 1st order, we
end up with some tangential fields $(\hat{q}^{J})_{J}$ and their velocities
$\hat{v}^{J}:=\mathcal{L}_{\partial_{t}}\hat{q}^{J}$. We now express
$(\iota_{\partial_{t}}L,-\iota_{\partial_{t}}\overline{\ell})$ in terms of
$(\hat{q}^{J};\hat{v}^{J})_{J}$ and pull everything back to $\Sigma$ (where
the positions and velocities are denoted as $(q^{J};v^{J})_{J}$) to obtain
$(\mathrm{L},l)$. Plugging them into equation (10) leads to a CAN Lagrangian-
action associated with $\SS$ that we denote $\mathcal{L}_{\SS}$.

###  V.2 CANonicalizing the symplectic potentials

Now we want to rewrite the symplectic potentials derived from the CPS
formalism in a way suitable for the comparison with the CAN formalism. For
that, we compute

$\displaystyle E_{I}\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}\phi^{I}+\mathrm{d}\Theta=\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}L=\mathrm{d}t\wedge\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\iota_{\partial_{t}}L\overset{\dagger}{=}$
$\displaystyle=\mathrm{d}t\wedge(\hat{A}_{I}^{(0)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\hat{q}^{I}+\hat{A}_{I}^{(1)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}\mathcal{L}_{\partial_{t}}\hat{q}^{I}+\mathrm{d}\hat{\Theta})=$
$\displaystyle=\mathrm{d}t\wedge(\hat{A}_{I}^{(0)}-\mathcal{L}_{\partial_{t}}\hat{A}_{I}^{(1)})\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\hat{q}^{I}+$
$\displaystyle\qquad+\mathrm{d}\Big{(}\iota_{\partial_{t}}(\mathrm{d}t\wedge\hat{A}_{I}^{(1)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\hat{q}^{I})-\mathrm{d}t\wedge\hat{\Theta}\Big{)}$

In the $\dagger$ equality we have “lifted” (LABEL:eq:_dd_L_canonic) from
$\Sigma$ to $M$ (the corresponding objects are denoted with a hat) taking into
account that, although additional terms might seem to appear, they have a
$\mathrm{d}t$ in them so they vanish in our previous computation. We know from
[28, 23, 29], that if $r<n$ and $s>0$, then any $\mathrm{d}$-closed
$(r,s)$-form is also $\mathrm{d}$-exact. So there exists some
$Z\in\Omega^{(n-2,1)}(M)$ such that

$\Theta=\iota_{\partial_{t}}(\mathrm{d}t\wedge\hat{A}_{I}^{(1)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}\hat{q}^{I})-\mathrm{d}t\wedge\hat{\Theta}+\mathrm{d}Z$ (18)

Since $\jmath^{*}$ and $\iota_{\partial_{t}}$ commute ($\partial_{t}$ is
tangent to the boundary), if we compute $\jmath^{*}\Theta$, the first term
vanishes. Hence

$\displaystyle b_{I}\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}\phi^{I}-\mathrm{d}\overline{\theta}=\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\overline{\ell}-\jmath^{*}\Theta\overset{\eqref{eq:
Theta CPC to canonical}}{=}$
$\displaystyle=-\mathrm{d}t\wedge(\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}(-\iota_{\partial_{t}}\overline{\ell})-\jmath^{*}\hat{\Theta})-\mathrm{d}\jmath^{*}Z\overset{\eqref{eq:
dd L canonic}}{=}$
$\displaystyle=-\mathrm{d}t\wedge(\hat{B}_{I}^{(0)}-\mathcal{L}_{\partial_{t}}\hat{B}_{I}^{(1)})\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\hat{q}^{I}-$
$\displaystyle\qquad-\mathrm{d}\Big{(}\iota_{\partial_{t}}(\mathrm{d}t\wedge\hat{B}_{I}^{(1)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}\hat{q}^{I})-\mathrm{d}t\wedge\hat{\theta}+\jmath^{*}Z\Big{)}$

Thus, there exists some $\overline{z}\in\Omega^{(n-3,1)}(\partial M)$ such
that

$\overline{\theta}=\iota_{\partial_{t}}(\mathrm{d}t\wedge\hat{B}_{I}^{(1)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}\hat{q}^{I})-\mathrm{d}t\wedge\hat{\theta}+\jmath^{*}Z-\mathrm{d}\overline{z}$
(19)

Finally, we rewrite the CPS symplectic potentials in a more suitable way:

$\displaystyle\begin{split}&\Theta=\hat{A}_{I}^{(1)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\hat{q}^{I}+\mathrm{d}t\wedge(\cdots)+\mathrm{d}Z\\\
&\overline{\theta}=\hat{B}_{I}^{(1)}\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}\hat{q}^{I}+\mathrm{d}t\wedge(\cdots)-\mathrm{d}t\wedge\hat{\theta}+\jmath^{*}Z-\mathrm{d}\overline{z}\end{split}$
(20)

###  V.3 CANonicalizing the symplectic form

Since the symplectic structure over $\mathrm{Sol}(\SS)$ does not depend on the
embedding, we consider $\imath_{t_{0}}$ (where $\mathrm{d}t=0$, so in
particular $\imath_{t_{0}}^{*}\hat{A}_{I}=A_{I}$ and
$\imath_{t_{0}}^{*}\hat{B}_{I}=B_{I}$) and we finally obtain the desired
equivalence

$\displaystyle\begin{split}\includegraphics[height=6.83331pt]{./OOmega.pdf}_{\SS}\overset{\eqref{eq:
OOmega SS}}{\underset{\eqref{eq: Theta
CPS=canonical}}{=}}\int_{\Sigma}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}A_{I}^{(1)}\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}q^{I}-\int_{\partial\Sigma}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}B_{I}^{(1)}\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}q^{I}\overset{\eqref{eq: OOmega third
try}}{=}\includegraphics[height=6.83331pt]{./OOmega.pdf}_{\mathcal{L}_{\SS}}\end{split}$
(21)

##  VI PROVING THE EQUIVALENCE FOR HIGHER ORDER THEORIES

In section V.1 we obtained the 1st-order CAN Lagrangians from the CPS ones by
breaking $(\phi^{I})_{I}$ into
$(\hat{q}^{J},\mathcal{L}_{\partial_{t}}\hat{q}^{J})_{J}$. However, for
general theories higher order “velocities”
$\\{(\mathcal{L}_{\partial_{t}})^{\mu}\hat{q}^{J}\\}_{\mu=1\cdots K}$ will
appear. While the CPS formalism does not change, the CAN one changes
drastically. We devote this section to briefly summarizing how to prove the
equivalence for higher order theories.

###  VI.1 The higher order canonical formalism in a nutshell

For a detailed description, see [30]. For our purposes, we only need to
generalize some of the equations of section V.1. First, equation
(LABEL:eq:_dd_L_canonic) has to be changed to

$\displaystyle\begin{split}&\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\mathrm{L}=\sum_{\mu}A_{I}^{(\mu)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}q_{(\mu)}^{I}+\mathrm{d}\widetilde{\Theta}\\\
&\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}l-\overline{\jmath}^{*}\widetilde{\Theta}=\sum_{\mu}B_{I}^{(\mu)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}q_{(\mu)}^{I}-\mathrm{d}\widetilde{\theta}\end{split}$ (22)

where $q^{I}_{(0)}$ are the positions, $q^{I}_{(1)}$ their velocities,
$q^{I}_{(2)}$ their accelerations, and so on up to $\mu=K$. Thus, we are
working on the $K$-th tangent bundle $T^{K}\\!\mathcal{Q}$. The Hamiltonian
formalism takes place in $T^{*}\\!(T^{K-1}\\!\mathcal{Q})$ where we have
$\\{q^{I}_{(\mu)}\\}_{\mu=0\cdots K-1}$ and their momenta
$\\{p_{I}^{(\mu)}\\}_{\mu=0\cdots K-1}$. The canonical symplectic structure
generalizing (13) is

$\includegraphics[height=6.83331pt]{./OOmega.pdf}_{T^{*}\\!(T^{K-1}\mathcal{Q})}:=\sum_{\mu=0}^{K-1}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}q^{I}_{(\mu)}\mathrel{\\!{\hbox to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}p_{I}^{(\mu)}$ (23)

In order to go from the Lagrangian to the Hamiltonian formulation, we have to
generalize the fiber derivative. For our purposes it is enough to generalize
(LABEL:def_p). The standard way is

$\displaystyle
p^{(K)}_{I}(q;w^{I}):=\includegraphics[height=7.70004pt]{./LL.pdf}_{(0,w^{I})}\mathcal{L}$
$\displaystyle
p^{(\mu)}_{I}(q;w^{I}):=\includegraphics[height=7.70004pt]{./LL.pdf}_{(0,w^{I})}\mathcal{L}-\frac{\mathrm{d}}{\mathrm{d}t}p^{(\mu+1)}_{I}(q;w^{I})$

for $\mu=1,\ldots,K-1$. It is not hard to prove by (backwards) induction that

$\displaystyle
p_{I}^{(\mu)}(\cdot{})=\sum_{k=\mu}^{K}\left(\\!-\frac{\mathrm{d}}{\mathrm{d}t}\right)^{\\!k-\mu}\left\\{\int_{\Sigma}A_{I}^{(k)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\cdot{}-\int_{\partial\Sigma}B_{I}^{(k)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\cdot{}\right\\}$

Taking the
$\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}$-exterior derivative, plugging the result into
(23), and performing some standard manipulations with double finite sums, we
obtain

$\displaystyle\begin{split}&\includegraphics[height=6.83331pt]{./OOmega.pdf}_{\mathcal{L}}=\sum_{k=1}^{K}\sum_{\mu=0}^{k-1}\left\\{\int_{\Sigma}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}\\!\left(\\!-\frac{\mathrm{d}}{\mathrm{d}t}\right)^{\\!k-1-\mu}\\!A_{I}^{(k)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}q^{I}_{(\mu)}-\right.\\\
&\qquad\qquad\left.-\int_{\partial\Sigma}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}\\!\left(\\!-\frac{\mathrm{d}}{\mathrm{d}t}\right)^{\\!k-1-\mu}\\!B_{I}^{(k)}\mathrel{\\!{\hbox
to0.0pt{\hbox
to6.66669pt{\hss\,$\wedge$\hss}\hss}\hbox{$\mathrel{\wedge}$}}\\!}\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}q^{I}_{(\mu)}\right\\}\end{split}$ (24)

Taking $K=1$ leads, as expected, to (17).

###  VI.2 Equivalence for higher order theories

Reasoning by induction, using computations similar to those of section V.2,
and equation (LABEL:eq:_dd_L_canonic_higher) (where $q^{I}_{(\mu)}$ has to be
replaced by $(\mathcal{L}_{\partial_{t}})^{\mu}\hat{q}^{I}$ since we are in
the CPS formalism), we obtain

$\displaystyle\begin{split}&\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}L=\mathrm{d}t\wedge\sum_{\mu}(-\mathcal{L}_{\partial_{t}})^{\mu}\hat{A}_{I}^{(\mu)}\wedge\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\hat{q}^{I}+\mathrm{d}\Theta\\\
&\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}\overline{\ell}-\jmath^{*}\Theta=-\mathrm{d}t\wedge\sum_{\mu}(-\mathcal{L}_{\partial_{t}})^{\mu}\hat{B}_{I}^{(\mu)}\wedge\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip 0.6458pt}\hat{q}^{I}-\mathrm{d}\overline{\theta}\end{split}$
(25)

where (up to exact forms)

$\displaystyle\Theta=\sum_{k=1}^{K}\sum_{\mu=0}^{k-1}(-\mathcal{L}_{\partial_{t}})^{k-1-\mu}\hat{A}_{I}^{(k)}\wedge\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}(\mathcal{L}_{\partial_{t}})^{\mu}\hat{q}^{I}-\mathrm{d}t\wedge(\cdots)$
$\displaystyle\overline{\theta}=\sum_{k=1}^{K}\sum_{\mu=0}^{k-1}(-\mathcal{L}_{\partial_{t}})^{k-1-\mu}\hat{B}_{I}^{(k)}\wedge\mathchoice{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{6.91037pt}\raisebox{0.0pt}{\kern
0.1873pt\rotatebox{90.0}{\kern-0.1873pt\makebox[6.91037pt][l]{}}\kern
0.1873pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{4.84373pt}\raisebox{0.0pt}{\kern
0.17221pt\rotatebox{90.0}{\kern-0.17221pt\makebox[4.84373pt][l]{}}\kern
0.17221pt}\hskip
0.6458pt}{\mathbbm{d}\rule[0.0pt]{0.0pt}{3.44444pt}\raisebox{0.0pt}{\kern
0.12915pt\rotatebox{90.0}{\kern-0.12915pt\makebox[3.44444pt][l]{}}\kern
0.12915pt}\hskip
0.6458pt}(\mathcal{L}_{\partial_{t}})^{\mu}\hat{q}^{I}-\mathrm{d}t\wedge(\cdots)$

Plugging them into (9), considering an embedding $\imath_{t_{0}}$, and taking
into account that in CAN $(\mathcal{L}_{\partial_{t}})^{\mu}\hat{q}^{I}$ is
identified with $q^{I}_{(\mu)}$, we obtain the general equivalence

$\includegraphics[height=6.83331pt]{./OOmega.pdf}_{\SS}\overset{\eqref{eq:
OOmega canonical
higher}}{=}\includegraphics[height=6.83331pt]{./OOmega.pdf}_{\mathcal{L}_{\SS}}$

##  VII CONCLUSIONS AND COMMENTS

We have proved the equivalence of the symplectic form induced by the
Hamiltonian formalism and the one derived in the covariant phase space. The
equivalence, which has been an open question for several decades, holds for
theories of any order and even when boundaries are present. The proof relies
strongly on the geometric formalism introduced in [23], where we also checked
the equivalence in some concrete examples. Finally, this work also proves that
the symplectic form introduced in [23] for manifolds with boundaries is the
natural one.

###### Acknowledgements.

This work has been supported by the Spanish Ministerio de Ciencia e
Innovación-AEI PID2020-116567GB-C22 grants. J.M.B. is supported by the AARMS
postdoctoral fellowship and by the NSERC Grants Discovery-2018-04873 and
RGPIN-2018-04887. E.J.S.V. is supported by the Madrid Government (Comunidad de
Madrid-Spain) under the Multiannual Agreement with UC3M in the line of
Excellence of University Professors (EPUC3M23), and in the context of the V
PRICIT (Regional Programme of Research and Technological Innovation). We thank
F. Barbero for his very useful comments and discussions.

## References

  * Gourgoulhon [2012] E. Gourgoulhon.  _3+1 formalism in general relativity: bases of numerical relativity_ , volume 846.  Springer Science & Business Media, 2012.
  * Woodhouse [1997] N.M.J. Woodhouse.  _Geometric quantization_.  Oxford university press, 1997.
  * Ashtekar and Lewandowski [2004] A. Ashtekar and J. Lewandowski.  _Background independent quantum gravity: A status report_.  Class. Quant. Grav., 21(15):R53, 2004.  [arXiv:gr-qc/0404018].
  * Margalef-Bentabol [2018] J. Margalef-Bentabol.  _Towards general relativity through parametrized theories_.  PhD thesis, Universidad Carlos III de Madrid, 2018.  [arXiv:1807.05534].
  * Juárez-Aubry et al. [2015] B.A. Juárez-Aubry, J.F. Barbero G., J. Margalef-Bentabol, and E.J. Villaseñor.  _Quantization of scalar fields coupled to point masses_.  Classical and Quantum Gravity, 32(24):245009, 2015\.
  * Gotay et al. [1998] M.J. Gotay, J. Isenberg, J.E. Marsden, and R. Montgomery.  _Momentum maps and classical relativistic fields. part I: Covariant field theory_.  Preprint, 1998.  [arXiv:physics/9801019].
  * Stelle [1977] K.S. Stelle.  _Renormalization of higher derivative quantum gravity_.  Phys. Rev. D, 16:953, 1977.
  * Stelle [1978] .  _Classical gravity with higher derivatives_.  Gen. Rel. Grav., 9:353, 1978.
  * Lu et al. [2015] H. Lu, A. Perkins, C.N. Pope, and K.S. Stelle.  _Black holes in higher-derivative gravity_.  Phys. Rev. Lett., 114(17):171601, 2015.  [arXiv:1502.01028].
  * Barnich and Henneaux [1993] G. Barnich and M. Henneaux.  _Consistent couplings between fields with a gauge freedom and deformations of the master equation_.  Phys. Lett. B, 311:123, 1993.  [arXiv:hep-th/9304057].
  * Barnich et al. [1995a] G. Barnich, F. Brandt, and M. Henneaux.  _Local BRST cohomology in the antifield formalism: I. General theorems_.  Commun. Math. Phys., 174:57, 1995a.  [arXiv:hep-th/9405109].
  * Barnich et al. [1995b] .  _Local BRST cohomology in the antifield formalism: II. Application to Yang-Mills theory_.  Commun. Math. Phys., 174:93, 1995b.  [arXiv:hep-th/9405109].
  * Odak and Speziale [2021] G. Odak and S. Speziale.  _Brown-York charges with mixed boundary conditions_.  JHEP, 11(11):224, 2021.  [arXiv:2109.02883].
  * Freidel et al. [2021] L. Freidel, R. Oliveri, D. Pranzetti, and S. Speziale.  _Extended corner symmetry, charge bracket and Einstein’s equations_.  JHEP, 09(083):083, 2021.  [arXiv:2104.12881].
  * Freidel et al. [2020] L. Freidel, M. Geiller, and D. Pranzetti.  _Edge modes of gravity. Part I. Corner potentials and charges_.  JHEP, 11(11):026, 2020.  [arXiv:2006.12527].
  * Wieland [2021a] W. Wieland.  _Gravitational SL(2,R) algebra on the light cone_.  JHEP, 07(057):057, 2021a.  [arXiv:2104.05803].
  * Wieland [2021b] .  _Null infinity as an open Hamiltonian system_.  JHEP, 04(095):095, 2021b.  [arXiv:2012.01889].
  * Barbero G. et al. [2021a] J.F. Barbero G., B. Díaz, J. Margalef-Bentabol, and E.J.S. Villaseñor.  _Concise symplectic formulation for tetrad gravity_.  Phys. Rev. D, 103(2):024051, 2021a.  [arXiv:2011.00661].
  * Barbero G. et al. [2021b] .  _Hamiltonian Gotay-Nester-Hinds analysis of the parametrized unimodular extension of the Holst action_.  Phys. Rev. D, 103(6):064062, 2021b.  [arXiv:2101.12311].
  * Frauendiener and Sparling [1992] J. Frauendiener and G.A.J. Sparling.  _On the symplectic formalism for general relativity_.  Proc. R. Soc. Lond. A, 436:141–153, 1992.
  * Ashtekar and Magnon-Ashtekar [1982] A. Ashtekar and A. Magnon-Ashtekar.  _On the symplectic structure of general relativity_.  Commun. Math. Phys., 86:55, 1982.
  * Barnich et al. [1991] G. Barnich, M. Henneaux, and C. Schomblond.  _Covariant description of the canonical formalism_.  Phys. Rev. D, 44(4):R939, 1991.
  * Margalef-Bentabol and Villaseñor [2021] J. Margalef-Bentabol and E.J.S. Villaseñor.  _Geometric formulation of the covariant phase space methods with boundaries_.  Phys. Rev. D, 103:025011, 2021.  [arXiv:2008.01842].
  * Barbero G. et al. [2021c] J.F. Barbero G., J. Margalef-Bentabol, V. Varo, and E.J.S. Villaseñor.  _Covariant phase space for gravity with boundaries: Metric versus tetrad formulations_.  Phys. Rev. D, 104:044048, 2021c.  [arXiv:2103.06362].
  * Barbero G. et al. [2021d] .  _Palatini gravity with nonmetricity, torsion, and boundaries in metric and connection variables_.  Phys. Rev. D, 104:044046, 2021d.  [arXiv:2105.07053].
  * Barbero G. et al. [2022] .  _On the on-shell equivalence of general relativity and Holst theories with nonmetricity, torsion, and boundaries_.  Preprint, 2022.  [arXiv:2201.12141].
  * Anderson [1989] I.M. Anderson.  _The variational bicomplex_.  Technical report, Utah State Technical Report, 1989.
  * Wald [1990] R.M. Wald.  _On identically closed forms locally constructed from a field_.  J. Math. Phys., 31(10):2378, 1990.
  * Takens [1979] F. Takens.  _A global version of the inverse problem of the calculus of variations_.  J. Differ. Geom., 14(4):543, 1979.
  * De León and Rodrigues [2011] M. De León and P.R. Rodrigues.  _Generalized Classical Mechanics and Field Theory_.  Elsevier, 2011\.