# Multisymplectic Formalism for Cubic Horndeski Theories

Mauricio Doniz ,
Department of Mathematics, Universitat Politècnica de Catalunya
Jordi Gaset,
Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La
Rioja e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present the covariant multisymplectic formalism for the so-called cubic
Horndeski theories and discuss the geometrical and physical interpretation of
the constraints that arise in the unified Lagrangian-Hamiltonian approach. We
analyse in more detail the covariant Hamiltonian formalism of these theories
and we show that there are particular conditions that must be satisfied for
the Poincaré-Cartan form of the Lagrangian to project onto $J^{1}\pi$. From
this result, we study when a formulation using only multimomenta is possible.
We further discuss the implications of the general case, in which the
projection onto $J^{1}\pi$ conditions are not met.

Key words: Higher-order field theories, Covariant Hamiltonian field theory,
Multisymplectic structure, Horndeski theory.

MSC 2020 codes: Primary: 53D42,70S05, 83C05;
Secondary: 35R01, 53C15, 53C80, 53Z05.

###### Contents

  1. 1 Introduction
  2. 2 Setting up the problem
    1. 2.1 Geometry, manifolds and bundles of the theory
    2. 2.2 The higher-order jet multimomentum bundles
    3. 2.3 The Lagrangian-Hamiltonian problem
  3. 3 A suitable change of coordinates
  4. 4 Unified Lagrangian-Hamiltonian formalism and the Constraint Algorithm
    1. 4.1 Legendre maps
    2. 4.2 Field equations for multivector fields
    3. 4.3 Field equations for sections
  5. 5 Hamiltonian formalism
    1. 5.1 Hamiltonian formalism for a particular case
    2. 5.2 Hamiltonian formalism for the general case
  6. 6 Conclusion

##  1 Introduction

The multisymplectic formalism is a generalisation of symplectic geometry for
field theories. It provides a covariant framework of the Lagrangian,
Hamiltonian and the Hamilton-Jacobi formulations for field theories. The
geometrical aspects of the multisymplectic formalism for first order theories,
and its manifolds and forms have been studied in detail in [2, 3, 4, 6, 7, 8].
Throughout this work we go through the features and results of the formalism
that we will need. To understand the formalism at a deeper level, [9, 10, 11]
constitute a good starting point.

It is possible study field theories of up to second-order with this formalism.
In particular, [12] covers the most relevant features of the second-order
multisymplectic formalism. However, third and higher order field theories do
not have unique Poincaré-Cartan forms, although some efforts in that direction
have been discussed in [13, 14]. Besides this, more problems regarding the
non-uniqueness of the geometrical structures appear in the definition of the
Legendre maps associated with higher-order Lagrangians and also problems arise
while trying to impose a multimomentum phase space for the Hamiltonian
formalism of such theories.

The best way to overcome the aforementioned problems is to use the unified
Lagrangian-Hamiltonian formalism. It was first introduced by Skinner and Rusk
in [15, 16, 17], and the basic idea is to merge the Lagrangian and Hamiltonian
formalisms into one. Even though Skinner and Rusk’s idea tames some of the
problems, still some arbitrary parameters that appear in the solutions of the
higher-order field equations and in the definition of the Legendre maps must
be fixed to guarantee its consistency. A modification of this framework that
clarifies the choice of the jet and the multimomentum bundles and removes all
ambiguity for second-order field theories was developed in [18], and this is
the approach that we shall use throughout this work.

The unified Lagrangian-Hamiltonian formalism allows us to extract all the
relevant physical information of a given system. First, we identify the
geometry, manifolds and bundles of the theory and set the Lagrangian-
Hamiltonian problem. For regular Lagrangians, the field equations that arise
are compatible and have solutions on the jet-multimomentum bundle. This is not
the case for singular Lagrangians, where it is needed to implement a
constraint algorithm to be able to find the corresponding submanifolds of the
jet-multimomentum bundle on which the field equations are compatible and have
solutions. The constraint algorithm that we will apply was developed in [19].

In regular first-order theories, the holonomy condition is recovered from the
local coordinate expression of the field equations [20]. That is not the case
for second-order field theories (even for regular Lagrangians) and, therefore,
it is required to imposed it a priori. Moreover, in the unified formalism the
singularity of the Lagrangian appears also as constraints, in particular, as
the definition of the Legendre transform. This is convenient as the
implementation of the constraint algorithm is straightforward.

Another advantage of the unified Lagrangian-Hamiltonian formalism is that one
can derive a covariant Hamiltonian formulation, as long as some regularity
conditions are meet. Another common construction of a Hamiltonian formulation
for field theories consists on performing a space $+$ time decomposition of
the covariant Lagrangian formalism and then perform an instantaneous Legendre
transform. This was originally performed by Arnowitt, Deser and Misner for
General Relativity [21]. This ADM-like approach, also called the instantaneous
Hamiltonian formulation, has been studied from a geometric point of view [22,
23]. We shall delve into the relation and equivalence between these
Hamiltonian formalisms for theories of gravity in a future work.

For all these reasons the multisymplectic formalism, and particularly the
unified Lagrangian-Hamiltonian formalism, is suitable for studying singular
second-order field theories such as certain string theories, the Korteweg-de
Vries model and some of the most relevant theories of gravity, including
General Relativity.

It is known that General Relativity (GR) is one of the most successful
theories in the history of physics. For over a hundred years now, it has been
tested and shown to be the standard model of gravity. However, it is also
known that it is is a low-energy effective theory, incomplete in the sense
that it is non-renormalisable [24]. There are different motivations for
studying modified models of gravity. From the phenomenological perspective,
the relatively recent detection of the first gravitational waves opens a new
way of testing generalised models of gravity that predict something different
from what GR predict [25]. The problem of the fine-tuned cosmological constant
needed to explain the accelerated expansion of the universe in General
Relativity is a strong incentive for physicists to explore generalised models
of gravity as well. If we turn to a theoretical point of view, studying
modified gravity grants a deeper understanding of GR. Finally, mathematically
speaking, understanding the geometric structure of generalised models of
gravity could impose strong constraints on the theory that could be used by
physicists to discard models or, on the other hand, make them turn their
attention to a certain model.

That being said, Horndeski’s theory is an interesting candidate for being the
generalisation of GR since it is the most general diffeomorphism invariant,
scalar-tensor theory that leads to second order equations of motion [26]. It
is strongly hyperbolic, at least at weak coupling, and therefore admits a
well-posed initial value problem [27]. Most importantly, this type of theories
are causal and hence allow for the existence of dynamical black holes [28]
which could be potentially observed with the current techniques.

We will be focusing on the construction of the multisymplectic formalism for
the cubic subclass of Horndeski’s theory. This subclass leads to strongly
hyperbolic equations [29] and recently has gained attention among cosmologists
due to the fact that this model can describe a non-singular bouncing universe
[30]. Besides the importance of this subclass of theories as a physical model,
its covariant Hamiltonian formulation arises relevant geometric and physical
consequences, as we shall discuss in the last section.

The multisymplectic formalism of relevant theories of gravity can be found in
the literature. Some of the most relevant examples are General Relativity
[31], metric-affine gravity with [32, 33] and without vielbein [34], Lovelock
Gravity [35] and even Chern-Simons gravity and the bosonic string [36].

The aim of the present work is to present the multisymplectic framework for
the cubic subclass of Horndeski theories. A major feature of this formalism is
that it provides a recipe for obtaining the Hamiltonian formulation of
generalised theories of Gravity. Hence, we will first establish the geometric
framework of the theory and introduce a suited change of co-ordinates that
simplifies the calculations of the constraint algorithm. Finally we will show
how to obtain the Hamiltonian formulation of the theory and briefly discuss
its implications.

All the manifolds are real, second countable and of class $C^{\infty}$.
Manifolds and mappings are assumed to be smooth. Sum over crossed repeated
indices is understood. Comas denote partial derivatives and semicolons
covariant derivatives.

##  2 Setting up the problem

In this section, we will introduce the geometrical structures and the
manifolds and bundles that we need to construct the formalism for cubic
Horndeski’s theories. To do so, we will present the Horndeski’s Lagrangian,
explain its main features and justify the importance of the cubic subclass of
Horndeski’s theories. Finally we will set up the Lagrangian-Hamiltonian
problem for this theories.

###  2.1 Geometry, manifolds and bundles of the theory

Let $M$ be an oriented 4-dimensional spacetime with coordinates $x^{\mu}$,
$\mu=0,1,2,3$ and whose volume form is denoted by
$\eta\in{\mit\Omega}^{4}(M)$. A scalar field is a map
$\phi:M\rightarrow\mathbb{R}$ or, equivalently, its graph is a section of the
product bundle $M\times\mathbb{R}$ over $M$.

The covariant configuration bundle111Covariant configuration bundle will be
just referred to as configuration bundle from now on. In contrast, there is an
instantaneous configuration bundle that appears in the ADM-like formulation of
the theory. for this system is a fiber bundle $\pi\colon E\rightarrow M$, with
$E$ being the manifold
$\left(M\times\mathbb{R}\right)\times_{M}S_{2}^{3,1}(M)$ , where
$S_{2}^{3,1}(M)$ denotes the bundle of symmetric covariant two-tensors $g$ of
Lorentz signature $(-,+,+,+)$ acting on $T_{x}M$.

The adapted fiber coordinates in $E$ are $(x^{\mu},g_{\alpha\beta},\phi)$,
($\mu,\alpha,\beta=0,1,2,3$), where $g_{\alpha\beta}$ and $\phi$ are the
component functions of the metric and the scalar field respectively. The
volume form satisfies $\eta={\rm d}x^{0}\wedge{\rm d}x^{1}\wedge{\rm
d}x^{2}\wedge{\rm d}x^{3}\equiv{\rm d}^{4}x$. Provided that the metric is
symmetric, it only has ten independent components and hence $\dim E=15$. We
shall consider this when we sum over indices on the fiber, thus it is useful
to establish a convention and order the indices as $0\leq\alpha\leq\beta\leq
3$.

The $k$th-order jet bundles of the projection $\pi$, $J^{k}\pi$, ($k=1,2,3$);
which are the manifolds of the $k$-jets of local sections $\Psi\in\Gamma(\pi)$
are equivalence classes of local sections of $\pi$ [37]. Points in $J^{k}\pi$
are denoted by $j^{k}_{x}\psi$, with $x\in M$ and $\psi\in\Gamma(\pi)$ being a
representative of the equivalence class. If $\Psi\in\Gamma(\pi)$, we denote
the $k$th prolongation of $\Psi$ to $J^{k}\pi$ by
$j^{k}\Psi\in\Gamma(\bar{\pi}^{k})$. We have the following natural
projections: if $r\leqslant k$,

$\begin{array}[]{rcl}\pi^{k}_{r}\colon J^{k}\pi&\longrightarrow&J^{r}\pi\\\
j^{k}_{x}\Psi&\longmapsto&j^{r}_{x}\Psi\end{array}\quad;\quad\begin{array}[]{rcl}\pi^{k}\colon
J^{k}\pi&\longrightarrow&E\\\
j^{k}_{x}\Psi&\longmapsto&\Psi(x)\end{array}\quad;\quad\begin{array}[]{rcl}\bar{\pi}^{k}\colon
J^{k}\pi&\longrightarrow&M\\\ j^{k}_{x}\Psi&\longmapsto&x\end{array}\ .$

Observe that $\pi^{s}_{r}\circ\pi^{k}_{s}=\pi^{k}_{r}$, $\pi^{k}_{0}=\pi^{k}$,
$\pi^{k}_{k}=\textnormal{Id}_{J^{k}\pi}$, and $\bar{\pi}^{k}=\pi\circ\pi^{k}$.
The induced coordinates in $J^{3}\pi$ are
$(x^{\mu},\,g_{\alpha\beta},\phi,\,g_{\alpha\beta,\mu},\phi_{,\mu},\,g_{\alpha\beta,\mu\nu},\phi_{,\mu\nu},\,g_{\alpha\beta,\mu\nu\lambda},\phi_{,\mu\nu\lambda})$,
($0\leq\mu\leq\nu\leq\lambda\leq 3$). We shall use all the possible
permutations, although only the ordered ones are actual co-ordinates.

Now we can explicitly write the total derivative $D_{\tau}$ in these local co-
ordinates as

$\displaystyle
D_{\tau}=\displaystyle\frac{\partial{}}{\partial{x^{\tau}}}+\sum_{\begin{subarray}{c}\alpha\leq\beta\\\
\mu\leq\nu\leq\lambda\end{subarray}}\Big{(}g_{\alpha\beta,\tau}\displaystyle\frac{\partial{}}{\partial{g_{\alpha\beta}}}$
$\displaystyle+g_{\alpha\beta,\mu\tau}\displaystyle\frac{\partial{}}{\partial{g_{\alpha\beta,\mu}}}+g_{\alpha\beta,\mu\nu\tau}\displaystyle\frac{\partial{}}{\partial{g_{\alpha\beta,\mu\nu}}}+g_{\alpha\beta,\mu\nu\lambda\tau}\displaystyle\frac{\partial{}}{\partial{g_{\alpha\beta,\mu\nu\lambda}}}$
$\displaystyle+\phi_{;\tau}\displaystyle\frac{\partial{}}{\partial{\phi}}$
$\displaystyle+\phi_{,\mu\tau}\displaystyle\frac{\partial{}}{\partial{\phi_{,\mu}}}+\phi_{,\mu\nu\tau}\displaystyle\frac{\partial{}}{\partial{\phi_{,\mu\nu}}}+\phi_{,\mu\nu\lambda\tau}\displaystyle\frac{\partial{}}{\partial{\phi_{,\mu\nu\lambda}}}\Big{)}.$

Notice that, if $f\in{\rm C}^{\infty}(J^{k}\pi)$, then $D_{\tau}f\in{\rm
C}^{\infty}(J^{k+1}\pi)$.

Next, consider the bundle $J^{1}\pi$ and let ${\cal
M}\pi\equiv\Lambda_{2}^{4}(J^{1}\pi)$ be the bundle of $4$-forms over
$J^{1}\pi$ vanishing by the action of two $\bar{\pi}^{1}$-vertical vector
fields; with the canonical projections

$\pi_{J^{1}\pi}\colon\Lambda_{2}^{4}(J^{1}\pi)\to
J^{1}\pi\quad;\quad\bar{\pi}_{M}=\bar{\pi}^{1}\circ\pi_{J^{1}\pi}\colon\Lambda_{2}^{4}(J^{1}\pi)\to
M\,.$

The induced local coordinates in $\Lambda_{2}^{4}(J^{1}\pi)$ are
$(x^{\mu},g_{\alpha\beta},\phi,g_{\alpha\beta,\mu},\phi_{,\mu},p_{g},p_{\phi},p_{g}^{\alpha\beta,\mu},p_{\phi}^{,\mu},\overline{p}_{g}^{\alpha\beta,\mu\nu},\overline{p}_{\phi}^{,\mu\nu})$,
with $0\leq\alpha\leq\beta\leq 3$ and $\mu,\nu=0,1,2,3$.

This bundle is endowed with the tautological (or Liouville) $4$-form
$\Theta_{1}\in{\mit\Omega}^{4}(\Lambda_{2}^{4}(J^{1}\pi))$ and the canonical
(or Liouville) $5$-form
$\Omega_{1}=-d\Theta_{1}\in\Omega^{5}(\Lambda_{2}^{4}(J^{1}\pi))$, which is a
multisymplectic form; meaning it is closed and $1$-nondegenerate, with the
following local expressions

$\displaystyle\Theta_{1}$ $\displaystyle=$ $\displaystyle p{\rm
d}^{4}x+\sum_{\alpha\leq\beta}\left(p_{g}^{\alpha\beta,\mu}{\rm
d}g_{\alpha\beta}\wedge{\rm
d}^{3}x_{\mu}+\overline{p}_{g}^{\alpha\beta,\mu\nu}{\rm
d}g_{\alpha\beta,\mu}\wedge{\rm
d}^{3}x_{\nu}\right)+\overline{p}_{\phi}d\phi+\overline{p}_{\phi}^{,\mu}d\phi_{,\mu}\
,$ $\displaystyle\Omega_{1}$ $\displaystyle=$ $\displaystyle-{\rm
d}p\wedge{\rm d}^{4}x-\sum_{\alpha\leq\beta}\left({\rm
d}p_{g}^{\alpha\beta,\mu}\wedge{\rm d}g_{\alpha\beta}\wedge{\rm
d}^{3}x_{\mu}+{\rm d}\overline{p}_{g}^{\alpha\beta,\mu\nu}\wedge{\rm
d}g_{\alpha\beta,\mu}\wedge{\rm d}^{3}x_{\nu}\right)-{\rm
d}\overline{p}_{\phi}\wedge d\phi-{\rm d}\overline{p}_{\phi}^{,\mu}\wedge
d\phi_{,\mu}\ ;$

where $\displaystyle{\rm
d}^{3}x_{\nu}=\mathop{i}\nolimits\left(\displaystyle\frac{\partial{}}{\partial{x^{\nu}}}\right){\rm
d}^{4}x$.

Consider the $\pi_{J^{1}\pi}$-transverse submanifold $\jmath_{s}\colon
J^{2}\pi^{\dagger}\hookrightarrow\Lambda_{2}^{4}(J^{1}\pi)$ defined locally by
the constraints
$\overline{p}_{g}^{\alpha\beta,\mu\nu}=\overline{p}_{g}^{\alpha\beta,\nu\mu}$
and $\overline{p}_{\phi}^{,\mu\nu}=\overline{p}_{\phi}^{,\nu\mu}$. This
submanifold is called the extended $2$-symmetric multimomentum bundle and its
construction is canonical even though it is defined in local coordinates [38].

Let

$\pi_{J^{1}\pi}^{\dagger}\colon J^{2}\pi^{\dagger}\to
J^{1}\pi\quad;\quad\bar{\pi}_{M}^{\dagger}=\bar{\pi}^{1}\circ\pi_{J^{1}\pi}^{\dagger}\colon
J^{2}\pi^{\dagger}\to M$

be the canonical projections.

The coordinates in $J^{2}\pi^{\dagger}$ are

$(x^{\mu},g_{\alpha\beta},\phi,g_{\alpha\beta,\mu},\phi_{,\mu},p,p_{g}^{\alpha\beta,\mu},p_{\phi}^{,\mu},p_{g}^{\alpha\beta,\mu\nu},p_{\phi}^{,\mu\nu}),$

with ($0\leq\alpha\leq\beta\leq 3$; $0\leq\mu\leq\nu\leq 3$), and
$\displaystyle\jmath_{s}^{*}\overline{p}_{g}^{\alpha\beta,\mu\nu}=\frac{1}{n(\mu\nu)}p_{g}^{\alpha\beta,\mu\nu}$,
$\displaystyle\jmath_{s}^{*}\overline{p}_{\phi}^{,\mu\nu}=\frac{1}{n(\mu\nu)}p_{\phi}^{,\mu\nu}$,
where $n(\mu\nu)$ is a combinatorial factor defined by $n(\mu\nu)=1$ for
$\mu=\nu$, and $n(\mu\nu)=2$ for $\mu\neq\nu$.

Denote
$\Theta_{1}^{s}=\jmath_{s}^{*}\Theta_{1}\in\Omega^{4}(J^{2}\pi^{\dagger})$ and
the multisymplectic form
$\Omega_{1}^{s}=\jmath_{s}^{*}\Omega_{1}=-d\Theta_{1}^{s}\in\Omega^{5}(J^{2}\pi^{\dagger})$,
which are called symmetrised Liouville $m$ and $(m+1)$-forms. In this case,
the local expressions are

$\displaystyle\Theta_{1}^{s}=$ $\displaystyle p\,{\rm
d}^{4}x+\sum_{\alpha\leq\beta}p_{g}^{\alpha\beta,\mu}{\rm
d}g_{\alpha\beta}\wedge{\rm
d}^{3}x_{\mu}+\sum_{\alpha\leq\beta}\frac{1}{n(\mu\nu)}p_{g}^{\alpha\beta,\mu\nu}{\rm
d}g_{\alpha\beta,\mu}\wedge{\rm d}^{3}x_{\nu}$
$\displaystyle+p_{\phi}^{,\mu}{\rm d}\phi\wedge{\rm
d}^{3}x_{\mu}+\frac{1}{n(\mu\nu)}p_{\phi}^{,\mu\nu}{\rm d}\phi_{\mu}\wedge{\rm
d}^{3}x_{\nu}\ ,$ $\displaystyle\Omega_{1}^{s}=$ $\displaystyle-{\rm
d}p\wedge{\rm d}^{4}x-\sum_{\alpha\leq\beta}{\rm
d}p_{g}^{\alpha\beta,\mu}\wedge{\rm d}g_{\alpha\beta}\wedge{\rm
d}^{3}x_{\mu}-\sum_{\alpha\leq\beta}\frac{1}{n(\mu\nu)}{\rm
d}p_{g}^{\alpha\beta,\mu\nu}\wedge{\rm d}g_{\alpha\beta,\mu}\wedge{\rm
d}^{3}x_{\nu}$ $\displaystyle-{\rm d}p_{\phi}^{,\mu}\wedge{\rm
d}\phi\wedge{\rm d}^{3}x_{\mu}-\frac{1}{n(\mu\nu)}{\rm
d}p_{\phi}^{,\mu\nu}\wedge{\rm d}\phi_{\mu}\wedge{\rm d}^{3}x_{\nu}\ .$

Last, consider the quotient bundle
$J^{2}\pi^{\ddagger}=J^{2}\pi^{\dagger}/\Lambda^{4}_{1}(J^{1}\pi)$, called the
restricted $2$-symmetric multimomentum bundle, endowed with the following
projections

$\mu\colon J^{2}\pi^{\dagger}\to
J^{2}\pi^{\ddagger}\quad;\quad\pi_{J^{1}\pi}^{\ddagger}\colon
J^{2}\pi^{\ddagger}\to J^{1}\pi\quad;\quad\bar{\pi}_{M}^{\ddagger}\colon
J^{2}\pi^{\ddagger}\to M.$

Observe that $J^{2}\pi^{\ddagger}$ is also the submanifold of
$\Lambda^{4}_{2}(J^{1}\pi)/\Lambda^{4}_{1}(J^{1}\pi)$ defined by the local
constraints
$\overline{p}^{\alpha\beta,\mu\nu}=\overline{p}^{\alpha\beta,\nu\mu}$ and
$\overline{p}_{\phi}^{;\mu\nu}=\overline{p}_{\phi}^{;\nu\mu}$ Thus, the
coordinates in $J^{2}\pi^{\ddagger}$ are

$(x^{\mu},g_{\alpha\beta},\phi,g_{\alpha\beta,\mu},\phi_{,\mu},p_{g}^{\alpha\beta,\mu},p_{\phi}^{,\mu},p_{g}^{\alpha\beta,\mu\nu},p_{\phi}^{,\mu\nu}),$

with ($0\leq\alpha\leq\beta\leq 3$; $0\leq\mu\leq\nu\leq 3$). The dimension of
this submanifold is $\dim J^{2}\pi^{\ddagger}=\dim J^{2}\pi^{\dagger}-1$.

The Horndeski Lagrangian density is a $\overline{\pi}^{2}$-semibasic form
$\mathcal{L}_{\mathfrak{V}}\in\Omega^{4}(J^{2}\pi)$. Hence
$\mathcal{L}_{\mathfrak{V}}=L_{\mathfrak{V}}\,(\overline{\pi}^{2})^{*}\eta$,
where $L_{\mathfrak{V}}\in{\rm C}^{\infty}(J^{2}\pi)$ is the Horndeski
Lagrangian function

$L_{\mathfrak{V}}=\frac{1}{16\pi
G}\sqrt{|g|}\left(\sum_{i=1}^{3}L_{i}\right),$

where, in the coordinates of the manifold:

$\displaystyle L_{1}=$ $\displaystyle R+X\,;\quad L_{2}=G_{2}(\phi,X)\,;\quad
L_{3}=G_{3}(\phi,X)\square\phi\,;$ $\displaystyle L_{4}=$ $\displaystyle
G_{4}(\phi,X)R+G_{4,X}(\phi,X)\left[(\square\phi)^{2}-g^{\mu\alpha}g^{\nu\beta}\phi_{;\mu\nu}\phi_{;\alpha\beta}\right]$
$\displaystyle L_{5}=$ $\displaystyle
G_{5}(\phi,X)G_{\mu\nu}g^{\mu\alpha}g^{\nu\beta}\phi_{;\alpha\beta}$
$\displaystyle-\frac{1}{6}G_{5,X}(\phi,X)\left[(\square\phi)^{3}+2g^{\nu\alpha}g^{\beta\gamma}g^{\sigma\nu}\phi_{;\mu\nu}\phi_{;\alpha\beta}\phi_{;\gamma\sigma}-3g^{\alpha\mu}g^{\beta\nu}g\phi_{;\mu\nu}\phi_{;\alpha\beta}\square\phi\right].$

Here $g^{\mu\nu}$ are the components of the inverse of the metric tensor, that
is, $g_{\alpha\mu}g^{\mu\nu}=\delta_{\alpha}^{\nu}$; $g$ the determinant of
the metric tensor; $X=-\frac{1}{2}g^{\mu\nu}\phi_{;\mu}\phi_{;\nu}$, and
$\square\phi=g^{\mu\nu}\phi_{;\mu\nu}$. The Ricci tensor is given by:
$R_{\alpha\beta}=D_{\gamma}\Gamma^{\gamma}_{\alpha\beta}-D_{\alpha}\Gamma^{\gamma}_{\gamma\beta}+\Gamma^{\gamma}_{\alpha\beta}\Gamma^{\delta}_{\delta\gamma}-\Gamma^{\gamma}_{\delta\beta}\Gamma^{\delta}_{\alpha\gamma}$.
Hence, the Ricci scalar has the local expression:

$R=g^{\alpha\beta}R_{\alpha\beta}=g^{\alpha\beta}\left(D_{\gamma}\Gamma^{\gamma}_{\alpha\beta}-D_{\alpha}\Gamma^{\gamma}_{\gamma\beta}+\Gamma^{\gamma}_{\alpha\beta}\Gamma^{\delta}_{\delta\gamma}-\Gamma^{\gamma}_{\delta\beta}\Gamma^{\delta}_{\alpha\gamma}\right),$

As we mentioned in the introduction, throughout this work we shall consider
the cubic subclass of Horndeski’s theories, which means we will only preserve
up to the $L_{3}$ term in the Lagrangian, i.e. we will set $G_{4}(\phi,X)=0$
and $G_{5}(\phi,X)=0$.

###  2.2 The higher-order jet multimomentum bundles

The unified Lagrangian-Hamiltornian formalism is set in a bundle that
encompasses the jets and bundles described in the previous section and hence
the manifolds $M$ and $E$. First, we construct the symmetric higher-order jet
multimomentum bundle $\mathcal{W}$ and the restricted symmetric higher-order
jet multimomentum bundle $\mathcal{W}_{r}$ as described in [12, 18]

$\mathcal{W}=J^{3}\pi\times_{J^{1}\pi}J^{2}\pi^{\dagger},$

$\mathcal{W}_{r}=J^{3}\pi\times_{J^{1}\pi}\,J^{2}\pi^{\ddagger}.$

Here $J^{2}\pi^{\dagger}$ $J^{2}\pi^{\ddagger}$ are the extended and the
restricted 2-symmetric multimomentum bundle respectively, as discussed in the
previous section. The symmetric higher-order jet multimomentum bundles have
the following natural local coordinates

$(x^{\mu},\,g_{\alpha\beta},\phi,\,g_{\alpha\beta,\mu},\phi_{,\mu},\,g_{\alpha\beta,\mu\nu},\phi_{,\mu\nu},\,g_{\alpha\beta,\mu\nu\lambda},\phi_{,\mu\nu\lambda},p,p_{g}^{\alpha\beta,\mu},p_{\phi}^{,\mu},p_{g}^{\alpha\beta,\mu\nu},p_{\phi}^{,\mu\nu}),$

and

$(x^{\mu},\,g_{\alpha\beta},\phi,\,g_{\alpha\beta,\mu},\phi_{,\mu},\,g_{\alpha\beta,\mu\nu},\phi_{,\mu\nu},\,g_{\alpha\beta,\mu\nu\lambda},\phi_{,\mu\nu\lambda},p_{g}^{\alpha\beta,\mu},p_{\phi}^{,\mu},p_{g}^{\alpha\beta,\mu\nu},p_{\phi}^{,\mu\nu}),$

with ($0\leq\alpha\leq\beta\leq 3$; $0\leq\mu\leq\nu\leq 3$), and are endowed
with the following projections

$\displaystyle\rho_{1}\colon\mathcal{W}\to J^{3}\pi\ ,\ $
$\displaystyle\rho_{2}\colon\mathcal{W}\to J^{2}\pi^{\dagger}\ ,\ $
$\displaystyle\rho_{M}\colon\mathcal{W}\to M$
$\displaystyle\rho^{r}_{1}\colon\mathcal{W}_{r}\to J^{3}\pi\ ,\ $
$\displaystyle\rho^{r}_{2}\colon\mathcal{W}_{r}\to J^{2}\pi^{\ddagger}\ ,\ $
$\displaystyle\rho_{M}^{r}\colon\mathcal{W}_{r}\to M\ .$

Moreover, the quotient map $\mu\colon J^{2}\pi^{\dagger}\to
J^{2}\pi^{\ddagger}$ induces a natural submersion
$\mu_{\mathcal{W}}\colon\mathcal{W}\to\mathcal{W}_{r}$.

Now, we define the canonical pairing which will help us determine the
Hamiltonian function.

$\begin{array}[]{rcl}\mathcal{C}\colon
J^{2}\pi\times_{J^{1}\pi}\Lambda_{2}^{4}(J^{1}\pi)&\longrightarrow&\Lambda_{1}^{4}(J^{1}\pi)\\\
(j^{2}_{x}\phi,\omega)&\longmapsto&(j^{1}\phi)^{*}_{j^{1}_{x}\phi}\omega\end{array}\
,$

hence we have can define a new pairing $\mathcal{C}^{s}\colon
J^{2}\pi\times_{J^{1}\pi}J^{2}\pi^{\dagger}\to\Lambda_{1}^{4}(J^{1}\pi)$ as

$\mathcal{C}^{s}(j^{2}_{x}\phi,\omega)=\mathcal{C}(j^{2}_{x}\phi,j_{s}(\omega))=(j^{1}\phi)_{j^{1}_{x}\phi}^{*}\
j_{s}(\omega)\,.$

From here we get the second-order coupling $4$-form in $\mathcal{W}$, which is
the $\rho_{M}$-semibasic $4$-form
$\hat{\mathcal{C}}\in\Omega^{4}(\mathcal{W})$ defined by

$\hat{\mathcal{C}}(j^{3}_{x}\phi,\omega)=\mathcal{C}^{s}(\pi^{3}_{2}(j^{3}_{x}\phi),\omega)\quad,\quad(j^{3}_{x}\phi,\omega)\in\mathcal{W}\
.$

As $\hat{\mathcal{C}}$ is a $\rho_{M}$-semibasic $4$-form, there exists a
function $\hat{C}\in C^{\infty}(\mathcal{W})$ such that
$\hat{\mathcal{C}}=\hat{C}\rho_{M}^{*}\eta$. In co-ordinates this is written
as

$\hat{\mathcal{C}}=\left(p+\sum_{\alpha\leq\beta}p_{g}^{\alpha\beta,\mu}g_{\alpha\beta,\mu}+\sum_{\begin{subarray}{c}\alpha\leq\beta\\\
\mu\leq\nu\end{subarray}}p_{g}^{\alpha\beta,\mu\nu}g_{\alpha\beta,\mu\nu}+p_{\phi}^{,\mu}\phi_{,\mu}+\sum_{\mu\leq\nu}p_{\phi}^{,\mu\nu}\phi_{,\mu\nu}\right){\rm
d}^{4}x\ .$

A $4-$form
$\hat{\mathcal{L}}=(\pi^{3}_{2}\circ\rho_{1})^{*}\mathcal{L}_{\mathfrak{V}}\in\Omega^{4}(\mathcal{W})$,
which can be written as $\hat{\mathcal{L}}=\hat{L}\,\rho_{M}^{*}\eta$, where
$\hat{L}=(\pi^{3}_{2}\circ\rho_{1})^{*}L_{\mathfrak{V}}\in
C^{\infty}(\mathcal{W})$, can be used to define the Hamiltonian submanifold
$\mathcal{W}_{o}$

$\mathcal{W}_{o}=\left\\{w\in\mathcal{W}\colon\hat{\mathcal{L}}(w)=\hat{\mathcal{C}}(w)\right\\}\stackrel{{\scriptstyle\jmath_{o}}}{{\hookrightarrow}}\mathcal{W}\
,$

which is ultimately defined by the following constraint

$\hat{\mathcal{C}}-\hat{L}\equiv
p+\sum_{\alpha\leq\beta}p_{g}^{\alpha\beta,\mu}g_{\alpha\beta,\mu}+\sum_{\begin{subarray}{c}\alpha\leq\beta\\\
\mu\leq\nu\end{subarray}}p_{g}^{\alpha\beta,\mu\nu}g_{\alpha\beta,\mu\nu}+p_{\phi}^{,\mu}\phi_{,\mu}+\sum_{\mu\leq\nu}p_{\phi}^{,\mu\nu}\phi_{,\mu\nu}-\hat{L}=0\
.$

This submanifold is $\mu_{\mathcal{W}}$-transverse and diffeomorphic to
$\mathcal{W}_{r}$, $\Phi_{o}\colon\mathcal{W}_{o}\to\mathcal{W}_{r}$.
$\mathcal{W}_{o}$ induces a Hamiltonian section
$\hat{h}\in\Gamma(\mu_{\mathcal{W}})$ by
$\hat{h}=\jmath_{o}\circ\Phi_{o}^{-1}\colon\mathcal{W}_{r}\to\mathcal{W}$,
specified by the local Hamiltonian function

$\hat{H}=\sum_{\alpha\leq\beta}p_{g}^{\alpha\beta,\mu}g_{\alpha\beta,\mu}+\sum_{\begin{subarray}{c}\alpha\leq\beta\\\
\mu\leq\nu\end{subarray}}p_{g}^{\alpha\beta,\mu\nu}g_{\alpha\beta,\mu\nu}+p_{\phi}^{,\mu}\phi_{,\mu}+\sum_{\mu\leq\nu}p_{\phi}^{,\mu\nu}\phi_{,\mu\nu}-\hat{L}\
.$

that is,

$\displaystyle\hat{h}(x^{\mu},\,g_{\alpha\beta},\phi,\,g_{\alpha\beta,\mu},\phi_{,\mu},\,g_{\alpha\beta,\mu\nu},\phi_{,\mu\nu},\,g_{\alpha\beta,\mu\nu\lambda},\phi_{,\mu\nu\lambda},p_{g}^{\alpha\beta,\mu},p_{\phi}^{,\mu},p_{g}^{\alpha\beta,\mu\nu},p_{\phi}^{,\mu\nu})$
$\displaystyle=(x^{\mu},\,g_{\alpha\beta},\phi,\,g_{\alpha\beta,\mu},\phi_{,\mu},\,g_{\alpha\beta,\mu\nu},\phi_{,\mu\nu},\,g_{\alpha\beta,\mu\nu\lambda},\phi_{,\mu\nu\lambda},-\hat{H},p_{g}^{\alpha\beta,\mu},p_{\phi}^{,\mu},p_{g}^{\alpha\beta,\mu\nu},p_{\phi}^{,\mu\nu}).$

This is all summarised in the following commutative diagram

$\textstyle{\mathcal{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{1}}$$\scriptstyle{\mu_{\mathcal{W}}}$$\scriptstyle{\rho_{2}}$
$\textstyle{\mathcal{W}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{h}}$$\scriptstyle{\rho_{1}^{r}}$$\scriptstyle{\rho_{2}^{r}}$$\scriptstyle{\rho_{J^{1}\pi}^{r}}$$\scriptstyle{\rho_{M}^{r}}$
$\textstyle{J^{2}\pi^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{\pi_{J^{1}\pi}^{\dagger}}$$\textstyle{J^{3}\pi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{3}_{1}}$$\scriptstyle{\bar{\pi}^{3}}$$\scriptstyle{\pi^{3}}$
$\textstyle{J^{2}\pi^{\ddagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{J^{1}\pi}^{\ddagger}}$
$\textstyle{J^{1}\pi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\pi}^{1}}$$\scriptstyle{\pi^{1}}$
$\textstyle{M}$
$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$

The Liouville forms  in $\mathcal{W}_{r}$,
$\Theta_{r}=(\rho_{2}\circ\hat{h})^{*}\Theta_{1}^{s}\in{\mit\Omega}^{4}(\mathcal{W}_{r})$
and $\Omega_{r}=-{\rm
d}\Theta_{r}=(\rho_{2}\circ\hat{h})^{*}\Omega_{1}^{s}\in{\mit\Omega}^{5}(\mathcal{W}_{r})$,
for second order field theories [18], in these specific co-ordinates, are

$\displaystyle\Theta_{r}=$ $\displaystyle-\hat{H}{\rm
d}^{4}x+\sum_{\alpha\leq\beta}p^{\alpha\beta,\mu}{\rm
d}g_{\alpha\beta}\wedge{\rm
d}^{3}x_{\mu}+\sum_{\alpha\leq\beta}\frac{1}{n(\mu\nu)}p^{\alpha\beta,\mu\nu}{\rm
d}g_{\alpha\beta,\mu}\wedge{\rm d}^{3}x_{\nu}$ $\displaystyle+p^{,\mu\nu}{\rm
d}\phi\wedge{\rm d}^{3}x_{\mu}+\frac{1}{n(\mu\nu)}p^{,\mu\nu}{\rm
d}\phi_{,\mu}\wedge{\rm d}^{3}x_{\nu}$

$\displaystyle\Omega_{r}=$ $\displaystyle{\rm d}\hat{H}\wedge{\rm
d}^{4}x-\sum_{\alpha\leq\beta}{\rm d}p_{g}^{\alpha\beta,\mu}\wedge{\rm
d}g_{\alpha\beta}\wedge{\rm
d}^{3}x_{\mu}-\sum_{\alpha\leq\beta}\frac{1}{n(\mu\nu)}{\rm
d}p_{g}^{\alpha\beta,\mu\nu}\wedge{\rm d}g_{\alpha\beta,\mu}\wedge{\rm
d}^{3}x_{\nu}$ $\displaystyle-{\rm d}p_{g}^{,\mu\nu}\wedge{\rm
d}\phi\wedge{\rm d}^{3}x_{\mu}-\frac{1}{n(\mu\nu)}{\rm
d}p_{g}^{,\mu\nu}\wedge{\rm d}\phi_{,\mu}\wedge{\rm d}^{3}x_{\nu};$ (1)

To obtain the form (2.2) explicitly, we need to calculate the exterior
derivative of the Hamiltonian function. The exterior derivative of a function
is the differential of the function. Specifically we have

$\displaystyle{\rm d}\hat{H}=$
$\displaystyle\sum_{\alpha\leq\beta}\left(g_{\alpha\beta,\mu}{\rm
d}p_{g}^{\alpha\beta,\mu}+p_{g}^{\alpha\beta,\mu}{\rm
d}g_{\alpha\beta,\mu}\right)+\sum_{\begin{subarray}{c}\alpha\leq\beta\\\
\mu\leq\nu\end{subarray}}\left(g_{\alpha\beta,\mu\nu}{\rm
d}p_{g}^{\alpha\beta,\mu\nu}+p_{g}^{\alpha\beta,\mu\nu}{\rm
d}g_{\alpha\beta,\mu\nu}\right)$ $\displaystyle+\phi_{,\mu}{\rm
d}p_{\phi}^{,\mu}+p_{\phi}^{,\mu}{\rm
d}\phi_{,\mu}+\sum_{\mu\leq\nu}\left(\phi_{,\mu\nu}{\rm
d}p_{\phi}^{,\mu\nu}+p_{\phi}^{,\mu\nu}{\rm d}\phi_{,\mu\nu}\right)-{\rm
d}\hat{L}.$

The differential of the Lagrangian, provided its dependency upon the metric,
the first and second order derivatives of the metric, the scalar field and the
first and second order derivatives of the scalar field, is

$\displaystyle{\rm d}\hat{L}=$
$\displaystyle\sum_{\alpha\leq\beta}\displaystyle\frac{\partial{\hat{L}}}{\partial{g_{\alpha\beta}}}{\rm
d}g_{\alpha\beta}+\sum_{\alpha\leq\beta}\displaystyle\frac{\partial{\hat{L}}}{\partial{g_{\alpha\beta,\mu}}}{\rm
d}g_{\alpha\beta,\mu}+\sum_{\begin{subarray}{c}\alpha\leq\beta\\\
\mu\leq\nu\end{subarray}}\displaystyle\frac{\partial{\hat{L}}}{\partial{g_{\alpha\beta,\mu\nu}}}{\rm
d}g_{\alpha\beta,\mu\nu}$
$\displaystyle+\displaystyle\frac{\partial{\hat{L}}}{\partial{\phi}}{\rm
d}\phi+\displaystyle\frac{\partial{\hat{L}}}{\partial{\phi_{,\mu}}}{\rm
d}\phi_{,\mu}+\displaystyle\frac{\partial{\hat{L}}}{\partial{\phi_{,\mu\nu}}}{\rm
d}\phi_{,\mu\nu}.$

The Liouville forms are degenerate; this is

$\ker\,\Theta_{r}=\ker\,\Omega_{r}\supset{\left<\frac{\partial}{\partial
g_{\alpha\beta,\mu\nu}},\frac{\partial}{\partial
g_{\alpha\beta,\mu\nu\lambda}},\frac{\partial}{\partial\phi_{,\mu\nu\lambda}}\right>}_{0\leq\alpha\leq\beta\leq
3;\,0\leq\mu\leq\nu\leq\lambda\leq 3}\ .$ (2)

For a premultisymplectic form $\Omega$, we call (geometric) gauge vector
fields to those vector fields belonging to $\ker\,\Omega$ (see [8, 43] for
more details). Furthermore, $\Theta_{r}$ is
$(\pi^{3}_{1}\circ\rho^{r}_{2})$-projectable.

###  2.3 The Lagrangian-Hamiltonian problem

Consider the system $(\mathcal{W}_{r},\Omega_{r})$.

######  Definition 1.

A section $\psi\in\Gamma(\bar{\pi}^{k})$ is holonomic if
$j^{k}(\pi^{k}\circ\psi)=\psi$; that is, $\psi$ is the $k$th prolongation of a
section $\phi=\pi^{k}\circ\psi\in\Gamma(\pi)$, and an integrable and
$\bar{\pi}_{M}$-transverse multivector field ${\bf
X}\in\mathfrak{X}^{4}(J^{k}\pi)$ is holonomic if its integral sections are
holonomic.

A section $\psi\in\Gamma(\bar{\pi}_{M}^{\ddagger})$ is holonomic in
$J^{2}\pi^{\ddagger}$ if
$\bar{\pi}_{J^{1}\pi}^{\ddagger}\circ\psi\in\Gamma(\bar{\pi}^{1})$ is
holonomic in $J^{1}\pi$, and an integrable and
$\bar{\pi}_{M}^{\ddagger}$-transverse multivector field ${\bf
X}\in\mathfrak{X}^{4}(J^{2}\pi^{\ddagger})$ is holonomic if its integral
sections are holonomic.

Finally, a section $\psi\in\Gamma(\rho_{M}^{r})$ is holonomic in
$\mathcal{W}_{r}$ if $\rho_{1}^{r}\circ\psi\in\Gamma(\bar{\pi}^{3})$ is
holonomic in $J^{3}\pi$, and an integrable and $\rho_{M}^{r}$-transverse
multivector field ${\bf X}\in\mathfrak{X}^{4}(\mathcal{W}_{r})$ is holonomic
if its integral sections are holonomic.

It is important to point out that the fact that a multivector field in
$\mathcal{W}_{r}$ has the local expression (5) (and then being locally
decomposable and $\rho^{r}_{M}$-transverse) is just a necessary condition to
be holonomic, since it may not be integrable. However, if such a multivector
field admits integral sections, then its integral sections are holonomic. In
general, a locally decomposable and $\rho^{r}_{M}$-transverse multivector
field which has (5) as coordinate expression, is said to be semiholonomic in
$\mathcal{W}_{r}$.

The Lagrangian-Hamiltonian problem associated with the system
$(\mathcal{W}_{r},\Omega_{r})$ consists in finding holonomic sections
$\psi\in\Gamma(\rho_{M}^{r})$ satisfying any of the following equivalent
conditions:

  1. 1.

$\psi$ is a solution to the equation

$\psi^{*}\mathop{i}\nolimits(X)\Omega_{r}=0\,,\quad\mbox{for every
}X\in\mathfrak{X}(\mathcal{W}_{r})\ .$ (3)

  2. 2.

$\psi$ is an integral section of a multivector field contained in a class of
holonomic multivector fields $\\{{\bf
X}\\}\subset\mathfrak{X}^{4}(\mathcal{W}_{r})$ satisfying the equation

$\mathop{i}\nolimits({\bf X})\Omega_{r}=0\ .$ (4)

As the form $\Omega_{r}$ is $1$-degenerate we have that
$(\mathcal{W}_{r},\Omega_{r})$ is a premultisymplectic system, and solutions
to (3) or (4) do not exist everywhere in $\mathcal{W}_{r}$.

##  3 A suitable change of coordinates

At this point, for the sake of facilitating calculations, it is useful to
introduce a new set of co-ordinates on $\mathcal{W}_{r}$. The argument is
simple; most interesting Lagrangians contain covariant derivative terms,
making it necessary to transform them to terms with partial derivatives and
components of the Levi-Civita connection. However, if we choose the velocity,
acceleration and jerk co-ordinates of the scalar field on $\mathcal{W}_{r}$ to
be the covariant derivatives of first, second and third order respectively,
instead of the partial derivatives, we will reduce significantly the
difficulty in further computations. The cost of this co-ordinate
transformation is that the Poincaré-Cartan forms must be transformed to the
new co-ordinates.

The suited coordinates on $\mathcal{W}_{r}$ in this case are:

$(\tilde{x}^{\mu},\,\tilde{g}_{\alpha\beta},\tilde{\phi},\,\tilde{g}_{\alpha\beta,\mu},\tilde{\phi}_{,\mu},\,\tilde{g}_{\alpha\beta,\mu\nu},\tilde{\phi}_{,\mu\nu},\,\tilde{g}_{\alpha\beta,\mu\nu\lambda},\tilde{\phi}_{,\mu\nu\lambda},\tilde{p}_{g}^{\alpha\beta,\mu},\tilde{p}_{\phi}^{,\mu},\tilde{p}_{g}^{\alpha\beta,\mu\nu},\tilde{p}_{\phi}^{,\mu\nu}).$

They are related to the previous chart by a set of maps. For the coordinates
of spacetime we just set $\tilde{x}^{\mu}=x^{\mu}$. The metric is left
unchanged related, and the scalar field is changed as:

$\displaystyle\tilde{\phi}=$
$\displaystyle\phi\,;\qquad\tilde{\phi}_{;\mu}=\phi_{,\mu}\,;\qquad\tilde{\phi}_{;\mu\nu}=\phi_{,\mu\nu}-\phi_{,\gamma}\Gamma_{\mu\nu}^{\gamma}\,;$
$\displaystyle\tilde{\phi}_{;\mu\nu\lambda}=$
$\displaystyle\phi_{,\mu\nu\lambda}-\phi_{,\gamma\lambda}\Gamma^{\gamma}_{\nu\mu}-\phi_{,\gamma}\Gamma^{\gamma}_{\nu\mu,\lambda}-\phi_{,\gamma\nu}\Gamma^{\gamma}_{\mu\lambda}+\phi_{,\sigma}\Gamma^{\sigma}_{\gamma\nu}\Gamma^{\gamma}_{\mu\lambda}-\phi_{,\mu\gamma}\Gamma^{\gamma}_{\lambda\nu}+\phi_{,\sigma}\Gamma^{\sigma}_{\mu\gamma}\Gamma^{\gamma}_{\lambda\nu}.$

The multimomenta coordinates are mapped via the identity, and they are also
unchanged. These expressions are a mere change of coordinates and hence the
manifold and all the geometric structure of the theory remains intact. From
these relations we get

$\displaystyle\frac{\partial}{\partial g_{\alpha\beta}}=$
$\displaystyle\frac{\partial}{\partial\tilde{g}_{\alpha\beta}}+\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}}}{\partial
g_{\alpha\beta}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}}}+\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}{\partial
g_{\alpha\beta}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}},$
$\displaystyle\frac{\partial}{\partial g_{\alpha\beta,\mu}}=$
$\displaystyle\frac{\partial}{\partial\tilde{g}_{\alpha\beta,\mu}}+\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}}}{\partial
g_{\alpha\beta,\mu}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}}}+\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}{\partial
g_{\alpha\beta,\mu}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}},$
$\displaystyle\frac{\partial}{\partial g_{\alpha\beta,\mu\nu}}=$
$\displaystyle\frac{\partial}{\partial\tilde{g}_{\alpha\beta,\mu\nu}}+\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}{\partial
g_{\alpha\beta,\mu\nu}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}},$
$\displaystyle\frac{\partial}{\partial\phi}=$
$\displaystyle\frac{\partial}{\partial\tilde{\phi}}$
$\displaystyle\frac{\partial}{\partial\phi_{,\mu}}=$
$\displaystyle\frac{\partial}{\partial\tilde{\phi}_{;\mu}}+\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}}}{\partial\phi_{,\mu}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}}}+\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}{\partial\phi_{,\mu}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}},$
$\displaystyle\frac{\partial}{\partial\phi_{,\mu\nu}}=$
$\displaystyle\frac{\partial}{\partial\tilde{\phi}_{;\mu\nu}}+\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}{\partial\phi_{,\mu\nu}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}},$

where

$\displaystyle\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}}}{\partial
g_{\alpha\beta}}=$ $\displaystyle
n(\alpha\beta)\tilde{\phi}_{;\gamma}g^{\gamma(\alpha}\Gamma_{\mu^{\prime}\nu^{\prime}}^{\beta)}$
$\displaystyle\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}}}{\partial
g_{\alpha\beta,\mu}}=$
$\displaystyle\frac{n(\alpha\beta)}{2}\tilde{\phi}_{;\gamma}g^{\gamma\rho}\left[\delta_{\mu^{\prime}}^{\mu}\delta_{\nu^{\prime}}^{(\alpha}\delta_{\rho}^{\beta)}+\delta_{\nu^{\prime}}^{\mu}\delta_{\rho}^{(\alpha}\delta_{\mu^{\prime}}^{\beta)}-\delta_{\rho}^{\mu}\delta_{\mu^{\prime}}^{(\alpha}\delta_{\nu^{\prime}}^{\beta)}\right]$
$\displaystyle\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}}}{\partial\phi_{,\mu}}=$
$\displaystyle-\Gamma_{\mu^{\prime}\nu^{\prime}}^{\mu}$
$\displaystyle\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}{\partial\phi_{,\mu}}=$
$\displaystyle-\Gamma_{\mu^{\prime}\nu^{\prime},\lambda^{\prime}}^{\mu}+\Gamma_{\gamma\nu^{\prime}}^{\mu}\Gamma_{\mu^{\prime}\lambda^{\prime}}^{\gamma}+\Gamma_{\mu^{\prime}\gamma}^{\mu}\Gamma_{\lambda^{\prime}\nu^{\prime}}^{\gamma}$
$\displaystyle\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}{\partial\phi_{,\mu\nu}}=$
$\displaystyle-3n(\mu\nu)\Gamma_{[\nu^{\prime}\mu^{\prime}}^{(\mu}\delta^{\nu)}_{\lambda^{\prime}]}$
$\displaystyle\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}{\partial
g_{\alpha\beta}}=$ $\displaystyle
n(\alpha\beta)\Bigg{\\{}-\left(\phi_{;\gamma\lambda^{\prime}}+\phi_{;\tau}\Gamma_{\gamma\lambda^{\prime}}^{\tau}\right)g^{\gamma(\alpha}\Gamma^{\beta)}_{\nu^{\prime}\mu^{\prime}}$
$\displaystyle+\frac{1}{2}\phi_{;\gamma}\Big{[}g^{\gamma(\alpha}g^{\beta)\rho}(g_{\mu^{\prime}\rho,\nu^{\prime}\lambda^{\prime}}+g_{\rho\nu^{\prime},\mu^{\prime}\lambda^{\prime}}-g_{\mu^{\prime}\nu^{\prime},\rho\lambda^{\prime}})$
$\displaystyle-g^{\gamma(\alpha}g^{\beta)\sigma}g^{\rho\delta}g_{\sigma\delta,\lambda^{\prime}}(g_{\mu^{\prime}\rho,\nu^{\prime}}+g_{\rho\nu^{\prime},\mu^{\prime}}-g_{\nu^{\prime}\mu^{\prime},\rho})$
$\displaystyle-g^{\gamma\sigma}g^{\rho(\alpha}g^{\beta)\delta}g_{\sigma\delta,\lambda^{\prime}}(g_{\mu^{\prime}\rho,\nu^{\prime}}+g_{\rho\nu^{\prime},\mu^{\prime}}-g_{\nu^{\prime}\mu^{\prime},\rho})\Big{]}$
$\displaystyle+(\phi_{;\gamma\nu^{\prime}}+\phi_{;\tau}\Gamma_{\gamma\nu^{\prime}}^{\tau})g^{\gamma(}\Gamma^{\beta)}_{\mu^{\prime}\lambda^{\prime}}-\phi_{;\sigma}g^{\sigma(\alpha}\Gamma^{\beta)}_{\gamma\nu^{\prime}}\Gamma^{\gamma}_{\mu^{\prime}\sigma^{\prime}}$
$\displaystyle-\phi_{;\sigma}\Gamma^{\sigma}_{\gamma\nu^{\prime}}g^{\gamma(\alpha}\Gamma^{\beta)}_{\mu^{\prime}\sigma^{\prime}}+(\phi_{;\mu^{\prime}\gamma}+\phi_{;\tau}\Gamma^{\tau}_{\mu^{\prime}\gamma})g^{\gamma(\alpha}\Gamma^{\beta)}_{\gamma^{\prime}\nu^{\prime}}$
$\displaystyle-\phi_{;\sigma}g^{\sigma(\alpha}\Gamma^{\beta)}_{\mu^{\prime}\gamma}\Gamma^{\gamma}_{\lambda^{\prime}\nu^{\prime}}-\phi_{;\sigma}\Gamma^{\sigma}_{\mu^{\prime}\gamma}g^{\gamma(\alpha}\Gamma^{\beta)}_{\nu^{\prime}\lambda^{\prime}}\Bigg{\\}}$
$\displaystyle\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}{\partial
g_{\alpha\beta,\mu}}=$
$\displaystyle-\frac{1}{2}n(\alpha\beta)\Big{[}g^{\gamma\sigma}g^{\rho\delta}\delta^{\mu}_{\lambda^{\prime}}\delta_{\lambda^{\prime}}^{(\alpha}\delta^{\beta)}_{\delta}(g_{\mu^{\prime}\rho,\nu^{\prime}}+g_{\rho\nu^{\prime},\mu^{\prime}}-g_{\nu^{\prime}\mu^{\prime},\rho})$
$\displaystyle+g^{\gamma\sigma}g^{\rho\delta}g_{\sigma\delta,\lambda^{\prime}}(\delta_{\nu^{\prime}}^{\mu}\delta_{\mu^{\prime}}^{(\alpha}\delta_{\rho}^{\beta)}+\delta_{\mu^{\prime}}^{\mu}\delta_{\rho}^{(\alpha}\delta_{\nu^{\prime}}^{\beta)}-\delta_{\rho}^{\mu}\delta_{\nu^{\prime}}^{(\alpha}\delta_{\mu^{\prime}}^{\beta)})\Big{]}$
$\displaystyle\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}{\partial
g_{\alpha\beta,\mu\nu}}=$
$\displaystyle\frac{1}{2}n(\alpha\beta)n(\mu\nu)g^{\gamma\rho}\left(\delta_{\mu^{\prime}}^{(\alpha}\delta_{\rho}^{\beta)}\delta_{\nu^{\prime}}^{(\mu}\delta_{\sigma^{\prime}}^{\nu)}+\delta_{\rho}^{(\alpha}\delta_{\nu^{\prime}}^{\beta)}\delta_{\mu^{\prime}}^{(\mu}\delta_{\sigma^{\prime}}^{\nu)}-\delta_{\nu^{\prime}}^{(\alpha}\delta_{\mu^{\prime}}^{\beta)}\delta_{\rho}^{(\mu}\delta_{\sigma^{\prime}}^{\nu)}\right)$

It is important to clarify that we have used the standard notation for
symmetrization and antisymmetrization of indices. To symmetrize on $n$
indices, we sum over all possible permutations of these indices and divide the
result by $n!$ . To antisymmetrize, we go through the same procedure, but
weighting each term in the sum by the sign of the permutation. For instance

$\displaystyle T^{(\mu\nu\rho)\lambda}=$
$\displaystyle\frac{1}{3!}\left(T^{\mu\nu\rho\lambda}+T^{\nu\rho\mu\lambda}+T^{\rho\mu\nu\lambda}+T^{\nu\mu\rho\lambda}+T^{\rho\nu\mu\lambda}+T^{\mu\rho\nu\lambda}\right),$
$\displaystyle T^{\mu}_{\;\;\;[\nu cd]}=$
$\displaystyle\frac{1}{3!}\left(T^{\mu}_{\;\;\;\nu\rho\sigma}-T^{\mu}_{\;\;\;\rho\nu\sigma}+T^{\mu}_{\;\;\;\rho\sigma\nu}-T^{\mu}_{\;\;\;\sigma\rho\nu}+T^{\mu}_{\;\;\;\sigma\nu\rho}-T^{\mu}_{\;\;\;\nu\sigma\rho}\right).$

Sometimes it is convenient to (anti)symmetrize over indices which are not
adjacent. In this case, we use vertical bars to denote that some indices will
be excluded. For instance,

$T_{(\mu|\rho\sigma|\nu)}=\frac{1}{2}\left(T_{\mu\rho\sigma\nu}+T_{\nu\rho\sigma\mu}\right).$

From now on, we shall drop the tildes over the names of the co-ordinates, but
it must be understood that the following calculations are performed in the new
chart.

The Liouville forms in $\mathcal{W}_{r}$, in these new coordinates, become

$\displaystyle\Theta_{r}=$ $\displaystyle-\hat{H}{\rm
d}^{4}x+\sum_{\alpha\leq\beta}p^{\alpha\beta,\mu}{\rm
d}g_{\alpha\beta}\wedge{\rm
d}^{3}x_{\mu}+\sum_{\alpha\leq\beta}\frac{1}{n(\mu\nu)}p^{\alpha\beta,\mu\nu}{\rm
d}g_{\alpha\beta,\mu}\wedge{\rm d}^{3}x_{\nu}$ $\displaystyle+p^{,\mu\nu}{\rm
d}\phi\wedge{\rm d}^{3}x_{\mu}+\frac{1}{n(\mu\nu)}p^{,\mu\nu}_{\phi}{\rm
d}\phi_{;\mu}\wedge{\rm d}^{3}x_{\nu}$

$\displaystyle\Omega_{r}=$ $\displaystyle{\rm d}\hat{H}\wedge{\rm
d}^{4}x-\sum_{\alpha\leq\beta}{\rm d}p_{g}^{\alpha\beta,\mu}\wedge{\rm
d}g_{\alpha\beta}\wedge{\rm
d}^{3}x_{\mu}-\sum_{\alpha\leq\beta}\frac{1}{n(\mu\nu)}{\rm
d}p_{g}^{\alpha\beta,\mu\nu}\wedge{\rm d}g_{\alpha\beta,\mu}\wedge{\rm
d}^{3}x_{\nu}$ $\displaystyle-{\rm d}p_{\phi}^{,\mu\nu}\wedge{\rm
d}\phi\wedge{\rm d}^{3}x_{\mu}-\frac{1}{n(\mu\nu)}{\rm
d}p_{\phi}^{,\mu\nu}\wedge{\rm d}\phi_{;\mu}\wedge{\rm d}^{3}x_{\nu};$

where

$\displaystyle\hat{H}=$
$\displaystyle\sum_{\alpha\leq\beta}p_{g}^{\alpha\beta,\mu}g_{\alpha\beta,\mu}+\sum_{\begin{subarray}{c}\alpha\leq\beta\\\
\mu\leq\nu\end{subarray}}p_{g}^{\alpha\beta,\mu\nu}g_{\alpha\beta,\mu\nu}+p_{\phi}^{,\mu}\phi_{;\mu}+\sum_{\mu\leq\nu}p_{\phi}^{,\mu\nu}\phi_{;\mu\nu}+\sum_{\mu\leq\nu}p_{\phi}^{,\mu\nu}\phi_{;\gamma}\Gamma^{\gamma}_{\mu\nu}-{\rm
d}\hat{L}$

$\displaystyle{\rm d}\hat{H}=$
$\displaystyle\sum_{\alpha\leq\beta}p_{g}^{\alpha\beta,\mu}{\rm
d}g_{\alpha\beta,\mu}+\sum_{\alpha\leq\beta}g_{\alpha\beta,\mu}{\rm
d}p_{g}^{\alpha\beta,\mu}+\sum_{\alpha\leq\beta}p_{g}^{\alpha\beta,\mu\nu}{\rm
d}g_{\alpha\beta,\mu\nu}+\sum_{\alpha\leq\beta}g_{\alpha\beta,\mu\nu}{\rm
d}p_{g}^{\alpha\beta,\mu\nu}$ $\displaystyle+p_{\phi}^{,\mu}{\rm
d}\phi_{;\mu}+\phi_{;\mu}{\rm
d}p_{\phi}^{,\mu}+\sum_{\mu\leq\nu}p_{\phi}^{,\mu\nu}{\rm
d}\phi_{;\mu\nu}+\sum_{\mu\leq\nu}p_{\phi}^{,\mu\nu}\Gamma^{\gamma}_{\mu\nu}{\rm
d}\phi_{;\gamma}+\sum_{\mu\leq\nu}p_{\phi}^{,\mu\nu}\phi_{;\gamma}{\rm
d}\Gamma^{\gamma}_{\mu\nu}$
$\displaystyle+\sum_{\mu\leq\nu}\left(\phi_{;\mu\nu}+\phi_{;\gamma}\Gamma^{\gamma}_{\mu\nu}\right){\rm
d}p_{\phi}^{,\mu\nu}-{\rm d}\hat{L},$

with

${\rm
d}\Gamma_{\mu\nu}^{\gamma}=-\frac{1}{2}g^{\gamma\lambda}g^{\rho\sigma}\left(g_{\nu\rho,\mu}+g_{\rho\mu,\nu}-g_{\mu\nu,\rho}\right){\rm
d}g_{\lambda\sigma}+\frac{1}{2}g^{\gamma\rho}\left({\rm d}g_{\nu\rho,\mu}+{\rm
d}g_{\rho\mu,\nu}-{\rm d}g_{\mu\nu,\rho}\right).$

Provided that we are interested in second order theories, we have

$\displaystyle{\rm d}\hat{L}=$ $\displaystyle\frac{\partial\hat{L}}{\partial
x^{\mu}}{\rm d}x^{\mu}+\frac{\partial\hat{L}}{\partial g_{\alpha\beta}}{\rm
d}g_{\alpha\beta}+\frac{\partial\hat{L}}{\partial\tilde{g}_{\alpha\beta,\mu}}{\rm
d}g_{\alpha\beta,\mu}+\frac{\partial\hat{L}}{\partial\tilde{g}_{\alpha\beta,\mu\nu}}{\rm
d}g_{\alpha\beta,\mu\nu}$
$\displaystyle+\frac{\partial\hat{L}}{\partial\phi}{\rm
d}\phi+\frac{\partial\hat{L}}{\partial\phi_{;\mu}}{\rm
d}\phi_{;\mu}+\frac{\partial\hat{L}}{\partial\phi_{;\mu\nu}}{\rm
d}\phi_{;\mu\nu}.$

The local expression of a holonomic multivector field ${\bf
X}\in\mathfrak{X}^{4}(\mathcal{W}_{r})$ in the new coordinates is

$\displaystyle\mathbf{X}$ $\displaystyle=$
$\displaystyle\bigwedge_{\lambda=0}^{3}\sum_{\begin{subarray}{c}\alpha\leq\beta\\\
\mu\leq\nu\leq\tau\end{subarray}}\left[\displaystyle\frac{\partial{}}{\partial{x^{\lambda}}}+g_{\alpha\beta,\lambda}\left(\frac{\partial}{\partial\tilde{g}_{\alpha\beta}}+\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}}}{\partial
g_{\alpha\beta}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}}}+\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}{\partial
g_{\alpha\beta}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}\right)\right.$
(5)
$\displaystyle\qquad\qquad\qquad\left.+g_{\alpha\beta,\mu\lambda}\left(\frac{\partial}{\partial\tilde{g}_{\alpha\beta,\mu}}+\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}}}{\partial
g_{\alpha\beta,\mu}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}}}+\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}{\partial
g_{\alpha\beta,\mu}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}\right)\right.$
$\displaystyle\qquad\qquad\qquad\left.+g_{\alpha\beta,\mu\nu\lambda}\left(\frac{\partial}{\partial\tilde{g}_{\alpha\beta,\mu\nu}}+\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}{\partial
g_{\alpha\beta,\mu\nu}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}\right)+\phi_{,\lambda}\displaystyle\frac{\partial{}}{\partial{\phi}}\right.$
$\displaystyle\qquad\qquad\qquad\left.+\left(\phi_{;\mu\lambda}+\phi_{;\gamma}\Gamma_{\mu\lambda}^{\gamma}\right)\left(\frac{\partial}{\partial\tilde{\phi}_{;\mu}}+\frac{\partial\tilde{\phi}_{\mu^{\prime}\nu^{\prime}}}{\partial\phi_{,\mu}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}}}+\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}{\partial\phi_{,\mu}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}\right)\right.$
$\displaystyle\left.+\left(\phi_{;\mu\nu\lambda}+\phi_{;\gamma\lambda}\Gamma_{\nu\mu}^{\gamma}-\phi_{;\rho}\Gamma_{\gamma\lambda}^{\rho}\Gamma_{\mu\nu}^{\gamma}+\phi_{;\gamma}\Gamma_{\nu\mu,\lambda}^{\gamma}+\phi_{;\gamma\nu}\Gamma_{\mu\lambda}^{\gamma}+\phi_{;\mu\gamma}\Gamma_{\lambda\nu}^{\gamma}\right)\left(\frac{\partial}{\partial\tilde{\phi}_{;\mu\nu}}+\frac{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}{\partial\phi_{,\mu\nu}}\frac{\partial}{\partial\tilde{\phi}_{;\mu^{\prime}\nu^{\prime}\lambda^{\prime}}}\right)\right.$
$\displaystyle\qquad\qquad\qquad\left.+F_{g\;\alpha\beta,\mu\nu\tau\lambda}\displaystyle\frac{\partial{}}{\partial{g_{\alpha\beta,\mu\nu\tau}}}+G^{\alpha\beta,\mu}_{g\;\lambda}\displaystyle\frac{\partial{}}{\partial{p_{g}^{\alpha\beta,\mu}}}+G^{\alpha\beta,\mu\nu}_{g\;\lambda}\displaystyle\frac{\partial{}}{\partial{p_{g}^{\alpha\beta,\mu\nu}}}\right.$
$\displaystyle\qquad\qquad\qquad\left.+F_{\phi\;,\mu\nu\tau\lambda}\displaystyle\frac{\partial{}}{\partial{\phi_{,\mu\nu\tau}}}+G^{,\mu}_{\phi\;\lambda}\displaystyle\frac{\partial{}}{\partial{p_{\phi}^{,\mu}}}+G^{,\mu\nu}_{\phi\;\lambda}\displaystyle\frac{\partial{}}{\partial{p_{\phi}^{,\mu\nu}}}\right]\
.$

and, if

$\displaystyle\psi(x^{\lambda})=(x^{\lambda},\,$
$\displaystyle\psi_{g\;\alpha\beta}(x^{\lambda}),\,\psi_{g\;\alpha\beta,\mu}(x^{\lambda}),\,\psi_{g\;\alpha\beta,\mu\nu}(x^{\lambda}),\,\psi_{\phi}(x^{\lambda}),\,\psi_{\phi\;;\mu}(x^{\lambda}),\,\psi_{\phi\;;\mu\nu}(x^{\lambda}),$
$\displaystyle\psi_{g\;\alpha\beta,\mu\nu\tau}(x^{\lambda}),\,\psi_{g}^{\alpha\beta,\mu}(x^{\lambda}),\psi_{g}^{\alpha\beta,\mu\nu}(x^{\lambda})\,\psi_{\phi\;,\mu\nu\tau}(x^{\lambda}),\,\psi_{\phi}^{,\mu}(x^{\lambda}),\psi_{\phi}^{,\mu\nu}(x^{\lambda}))$

is an integral section of ${\bf X}$, its component functions satisfy the
following system of partial differential equations

$\displaystyle\displaystyle\frac{\partial{\psi_{g\;\alpha\beta}}}{\partial{x^{\lambda}}}=g_{\alpha\beta,\lambda}\circ\psi\
,\
\displaystyle\frac{\partial{\psi_{g\;\alpha\beta,\mu}}}{\partial{x^{\lambda}}}=g_{\alpha\beta,\mu\lambda}\circ\psi\
,\
\displaystyle\frac{\partial{\psi_{g\;\alpha\beta,\mu\nu}}}{\partial{x^{\lambda}}}=g_{\alpha\beta,\mu\nu\lambda}\circ\psi\
,$
$\displaystyle\displaystyle\frac{\partial{\psi_{\phi}}}{\partial{x^{\lambda}}}=\phi_{;\lambda}\circ\psi\
,\
\displaystyle\frac{\partial{\psi_{\phi\;;\mu}}}{\partial{x^{\lambda}}}=\phi_{;\mu\lambda}\circ\psi\
,\
\displaystyle\frac{\partial{\psi_{\phi\;;\mu\nu}}}{\partial{x^{\lambda}}}=\phi_{;\mu\nu\lambda}\circ\psi\
,\ $
$\displaystyle\displaystyle\frac{\partial{\psi_{g\;\alpha\beta,\mu\nu\tau}}}{\partial{x^{\lambda}}}=F_{g\;\alpha\beta,\mu\nu\tau\lambda}\circ\psi\
,\
\displaystyle\frac{\partial{\psi_{g}^{\alpha\beta,\mu}}}{\partial{x^{\lambda}}}=G^{\alpha\beta,\mu}_{g\;\lambda}\circ\psi\
,\
\displaystyle\frac{\partial{\psi_{g}^{g\;\alpha\beta,\mu\nu}}}{\partial{x^{\lambda}}}=G^{\alpha\beta,\mu\nu}_{g\;\lambda}\circ\psi$
$\displaystyle\displaystyle\frac{\partial{\psi_{\phi\;,\mu\nu\tau}}}{\partial{x^{\lambda}}}=F_{\phi\;\alpha\beta,\mu\nu\tau\lambda}\circ\psi\
,\
\displaystyle\frac{\partial{\psi_{\phi}^{,\mu}}}{\partial{x^{\lambda}}}=G^{,\mu}_{\phi\;\lambda}\circ\psi\
,\
\displaystyle\frac{\partial{\psi_{\phi}^{,\mu\nu}}}{\partial{x^{\lambda}}}=G^{,\mu\nu}_{\phi\;\lambda}\circ\psi\
.$

##  4 Unified Lagrangian-Hamiltonian formalism and the Constraint Algorithm

Now we explicitly calculate the Legendre maps and the corresponding field
equations for the multivector fields in the new coordinates.

###  4.1 Legendre maps

######  Proposition 1.

A section $\psi\in\Gamma(\rho_{M}^{r})$ solution to the equation (3) takes
values in a $140$-codimensional submanifold $\jmath_{{\cal
L}_{\mathfrak{V}}}\colon\mathcal{W}_{\mathcal{L}_{\mathfrak{V}}}\hookrightarrow\mathcal{W}_{r}$
which is identified with the graph of a bundle map
$\mathcal{FL}_{\mathfrak{V}}\colon J^{3}\pi\to J^{2}\pi^{\ddagger}$, over
$J^{1}\pi$, defined locally by

$\displaystyle\mathcal{FL}_{\mathfrak{V}}^{\ \ *}\,p_{g}^{\alpha\beta,\mu}=$
$\displaystyle\frac{\partial\hat{L}}{\partial
g_{\alpha\beta,\mu}}-\sum_{\nu=0}^{3}\frac{1}{n(\mu\nu)}X_{\nu}\left(\frac{\partial\hat{L}}{\partial
g_{\alpha\beta,\mu\nu}}\right)-\frac{1}{2}\left(\frac{\partial\hat{L}}{\partial\phi_{;\mu\alpha}}\phi_{;\gamma}g^{\gamma\beta}+\frac{\partial\hat{L}}{\partial\phi_{;\beta\mu}}\phi_{;\gamma}g^{\gamma\alpha}-\frac{\partial\hat{L}}{\partial\phi_{;\alpha\beta}}\phi_{;\gamma}g^{\gamma\mu}\right)$
$\displaystyle=$ $\displaystyle\hat{L}_{g}^{\alpha\beta,\mu},$
$\displaystyle\mathcal{FL}_{\mathfrak{V}}^{\ \
*}\,p_{g}^{\alpha\beta,\mu\nu}=$ $\displaystyle\frac{\partial\hat{L}}{\partial
g_{\alpha\beta,\mu\nu}},$ $\displaystyle\mathcal{FL}_{\mathfrak{V}}^{\ \
*}\,p_{\phi}^{,\mu}=$
$\displaystyle\frac{\partial\hat{L}}{\partial\phi_{;\mu}}-\sum_{\nu=0}^{3}\frac{1}{n(\mu\nu)}X_{\nu}\left(\frac{\partial\hat{L}}{\partial\phi_{;\mu\nu}}\right)-\frac{\partial\hat{L}}{\partial\phi_{;\gamma\nu}}\Gamma_{\gamma\nu}^{\mu}=\hat{L}_{\phi}^{,\mu},$
$\displaystyle\mathcal{FL}_{\mathfrak{V}}^{\ \ *}\,p_{\phi}^{,\mu\nu}=$
$\displaystyle\frac{\partial\hat{L}}{\partial\phi_{;\mu\nu}}.$

The submanifold $\mathcal{W}_{\mathcal{L}_{\mathfrak{V}}}$ is the graph of a
bundle morphism $\widetilde{\mathcal{FL}}_{\mathfrak{V}}\colon J^{3}\pi\to
J^{2}\pi^{\dagger}$ over $J^{1}\pi$ defined locally by

$\displaystyle\displaystyle\widetilde{\mathcal{FL}}_{\mathfrak{V}}^{\ \
*}\,p_{g}^{\alpha\beta,\mu}=$ $\displaystyle\frac{\partial\hat{L}}{\partial
g_{\alpha\beta,\mu}}-\sum_{\nu=0}^{3}\frac{1}{n(\mu\nu)}X_{\nu}\left(\frac{\partial\hat{L}}{\partial
g_{\alpha\beta,\mu\nu}}\right)$
$\displaystyle-\frac{1}{2}\left(\frac{\partial\hat{L}}{\partial\phi_{;\mu\alpha}}\phi_{;\gamma}g^{\gamma\beta}+\frac{\partial\hat{L}}{\partial\phi_{;\beta\mu}}\phi_{;\gamma}g^{\gamma\alpha}-\frac{\partial\hat{L}}{\partial\phi_{;\alpha\beta}}\phi_{;\gamma}g^{\gamma\mu}\right)$
$\displaystyle=$ $\displaystyle\hat{L}_{g}^{\alpha\beta,\mu},$
$\displaystyle\displaystyle\widetilde{\mathcal{FL}}_{\mathfrak{V}}^{\ \
*}\,p_{g}^{\alpha\beta,\mu\nu}=$ $\displaystyle\frac{\partial\hat{L}}{\partial
g_{\alpha\beta,\mu\nu}},$
$\displaystyle\displaystyle\widetilde{\mathcal{FL}}_{\mathfrak{V}}^{\ \
*}\,p_{\phi}^{,\mu}=$
$\displaystyle\frac{\partial\hat{L}}{\partial\phi_{;\mu}}-\sum_{\nu=0}^{3}\frac{1}{n(\mu\nu)}X_{\nu}\left(\frac{\partial\hat{L}}{\partial\phi_{;\mu\nu}}\right)-\frac{\partial\hat{L}}{\partial\phi_{;\gamma\nu}}\Gamma_{\gamma\nu}^{\mu}=\hat{L}_{\phi}^{,\mu},$
$\displaystyle\displaystyle\widetilde{\mathcal{FL}}_{\mathfrak{V}}^{\ \
*}\,p_{\phi}^{,\mu\nu}=$
$\displaystyle\frac{\partial\hat{L}}{\partial\phi_{;\mu\nu}},$
$\displaystyle\displaystyle\widetilde{\mathcal{FL}}_{\mathfrak{V}}^{\ \
*}\,p=$ $\displaystyle\hat{L}-\sum_{\begin{subarray}{c}\alpha\leq\beta\\\
\mu\leq\nu\end{subarray}}g_{\alpha\beta,\mu\nu}\frac{\partial\hat{L}}{\partial
g_{\alpha\beta,\mu\nu}}-\sum_{\begin{subarray}{c}\mu\leq\nu\end{subarray}}\left(\phi_{;\mu\nu}+\phi_{;\gamma}\Gamma^{\gamma}_{\mu\nu}\right)\frac{\partial\hat{L}}{\partial\phi_{;\mu\nu}}$
$\displaystyle-\phi_{;\mu}\left(\frac{\partial\hat{L}}{\partial\phi_{;\mu}}-\sum_{\nu=0}^{3}\frac{1}{n(\mu\nu)}X_{\nu}\left(\frac{\partial\hat{L}}{\partial\phi_{;\mu\nu}}\right)-\frac{\partial\hat{L}}{\partial\phi_{;\gamma\nu}}\Gamma_{\gamma\nu}^{\mu}\right)$
$\displaystyle-\sum_{\begin{subarray}{c}\alpha\leq\beta\end{subarray}}g_{\alpha\beta,\mu}\Bigg{[}\frac{\partial\hat{L}}{\partial
g_{\alpha\beta,\mu}}-\sum_{\nu=0}^{3}\frac{1}{n(\mu\nu)}X_{\nu}\left(\frac{\partial\hat{L}}{\partial
g_{\alpha\beta,\mu\nu}}\right)$
$\displaystyle-\frac{1}{2}\left(\frac{\partial\hat{L}}{\partial\phi_{;\mu\alpha}}\phi_{;\gamma}g^{\gamma\beta}+\frac{\partial\hat{L}}{\partial\phi_{;\beta\mu}}\phi_{;\gamma}g^{\gamma\alpha}-\frac{\partial\hat{L}}{\partial\phi_{;\alpha\beta}}\phi_{;\gamma}g^{\gamma\mu}\right)\Bigg{]}.$

The maps $\mathcal{FL}_{\mathfrak{V}}$ and
$\widetilde{\mathcal{FL}}_{\mathfrak{V}}$ are the restricted and the extended
Legendre maps (associated with the Lagrangian density ${\cal
L}_{\mathfrak{V}}$), and they satisfy that
$\mathcal{FL}_{\mathfrak{V}}=\mu\circ\widetilde{\mathcal{FL}}_{\mathfrak{V}}$.
For every $j^{3}_{x}\phi\in J^{3}\pi$, we have that
$\operatorname{rank}(\widetilde{\mathcal{FL}}_{\mathfrak{V}}(j^{3}_{x}\phi))=\operatorname{rank}(\mathcal{FL}_{\mathfrak{V}}(j^{3}_{x}\phi))$.

Remember that, according to [38], a second-order Lagrangian density
$\mathcal{L}\in\Omega^{4}(J^{2}\pi)$ is regular if

$\operatorname{rank}(\widetilde{\mathcal{FL}}(j^{3}_{x}\phi))=\operatorname{rank}(\mathcal{FL}(j^{3}_{x}\phi))=\dim
J^{2}\pi+\dim J^{1}\pi-\dim E=\dim J^{2}\pi^{\ddagger}\,,$

otherwise, the Lagrangian density is singular. Regularity is equivalent to
demand that $\mathcal{FL}\colon J^{3}\pi\to J^{2}\pi^{\ddagger}$ is a
submersion onto $J^{2}\pi^{\ddagger}$ and this implies that there exist local
sections of $\mathcal{FL}$. If $\mathcal{FL}$ admits a global section
$\Upsilon\colon J^{2}\pi^{\ddagger}\to J^{3}\pi$, then the Lagrangian density
is said to be hyperregular. Recall that the regularity of $\mathcal{L}$
determines if the section $\psi\in\Gamma(\rho_{M}^{r})$ solution to the
equation (3) lies in $\mathcal{W}_{\mathcal{L}}$ or in a submanifold
$\mathcal{W}_{f}\hookrightarrow\mathcal{W}_{\mathcal{L}}$ where the section
$\psi$ takes values. In order to obtain this final constraint submanifold, the
best way is to work with the equation (4) instead of (3).

###  4.2 Field equations for multivector fields

For a generic cubic Horndenski theory we have, at least, the following
constraints:

######  Theorem 1.

Consider the Horndeski Lagrangian with $G_{4}(\phi,X)=0$ and
$G_{5}(\phi,X)=0$, and let ${\cal W}_{f}\hookrightarrow\mathcal{W}_{r}$ be the
submanifold defined locally by the constraints

$p_{g}^{\alpha\beta,\mu\nu}-\frac{\partial\hat{L}}{\partial
g_{\alpha\beta,\mu\nu}}=0\quad,\quad
p_{g}^{\alpha\beta,\mu}-\hat{L}_{g}^{\alpha\beta,\mu}=0\quad,\quad\hat{L}_{g}^{\alpha\beta}=0\quad,\quad
X_{\tau}\hat{L}_{g}^{\alpha\beta}=0\ ,$
$p_{\phi}^{,\mu\nu}-\frac{\partial\hat{L}}{\partial\phi_{;\mu\nu}}=0\quad,\quad
p_{\phi}^{,\mu}-\hat{L}_{\phi}^{,\mu}=0\quad,\quad\hat{L}_{\phi}=0\quad,\quad
X_{\tau}\hat{L}_{\phi}=0\ ;$

for $0\leq\alpha\leq\beta\leq 3$, $0\leq\mu\leq\nu\leq 3$ and $0\leq\tau\leq
3$. Then, there exist classes of semiholonomic multivector fields $\\{{\bf
X}\\}\subset\mathfrak{X}^{4}({\cal W}_{r})$ which are tangent to
$\mathcal{W}_{f}$ and such that

${}\mathop{i}\nolimits{({\bf
X})}\Omega_{r}|_{\mathcal{W}_{f}}=0\quad,\quad\forall{\bf X}\in\\{{\bf
X}\\}\subset\mathfrak{X}^{4}({\cal W}_{r})\ .$ (6)

###### Proof.

In order to find the final submanifold $\mathcal{W}_{f}$ we use a local
coordinate procedure which is equivalent to the constraint algorithm for
premultisymplectic field theories. Bearing in mind (5), the local expression
of a representative of a class of a semiholonomic multivector fields, not
necessarily integrable, is, in this case, 5

then, equation (4) leads to

$\displaystyle G^{\alpha\beta,\mu}_{g\;\mu}-\frac{\partial\hat{L}}{\partial
g_{\alpha\beta}}-\sum_{\mu\leq\nu}p_{\phi}^{,\mu\nu}\phi_{;\gamma}g^{\gamma\alpha}\Gamma_{\mu\nu}^{\beta}=0\
,$ (7) $\displaystyle
G^{,\mu}_{\phi\;\mu}-\frac{\partial\hat{L}}{\partial\phi}=0\ ,$ (8)
$\displaystyle\sum_{\nu=0}^{3}\frac{1}{n(\mu\nu)}G^{\alpha\beta,\mu\nu}_{g\;\nu}-\frac{\partial\hat{L}}{\partial\tilde{g}_{\alpha\beta,\mu}}+p_{g}^{\alpha\beta,\mu}+\sum_{\rho\leq\sigma}\phi_{;\gamma}p^{,\rho\sigma}_{\phi}\frac{\partial\Gamma^{\gamma}_{\rho\sigma}}{\partial\tilde{g}_{\alpha\beta,\mu}}=0\
,$ (9)
$\displaystyle\sum_{\nu=0}^{3}\frac{1}{n(\mu\nu)}G^{,\mu\nu}_{\phi\;\nu}-\frac{\partial\hat{L}}{\partial\phi_{;\mu}}+p_{\phi}^{,\gamma\nu}\Gamma_{\gamma\nu}^{\mu}+p_{\phi}^{,\mu}=0\
,$ (10) $\displaystyle
p_{g}^{\alpha\beta,\mu\nu}-\frac{\partial\hat{L}}{\partial\tilde{g}_{\alpha\beta,\mu\nu}}=0\
,$ (11) $\displaystyle
p_{\phi}^{,\mu\nu}-\frac{\partial\hat{L}}{\partial\phi_{;\mu\nu}}=0\ .$ (12)

Expressions (11) and (12) are constraints that define the compatibility
submanifold $\mathcal{W}_{c}\hookrightarrow\mathcal{W}_{r}$. If we require
tangency of the multivector field to $\mathcal{W}_{c}$,

$\mathop{\rm
L}\nolimits(X_{\tau})\left(p_{g}^{\alpha\beta,\mu\nu}-\frac{\partial\hat{L}}{\partial
g_{\alpha\beta,\mu\nu}}\right)|_{\mathcal{W}_{c}}=0\ ,$ $\mathop{\rm
L}\nolimits(X_{\tau})\left(p_{\phi}^{,\mu\nu}-\frac{\partial\hat{L}}{\partial\phi_{;\mu\nu}}\right)|_{\mathcal{W}_{c}}=0\
,$

we get

$G^{\alpha\beta,\mu\nu}_{g\;\tau}=X_{\tau}\left(\frac{\partial\hat{L}}{\partial\tilde{g}_{\alpha\beta,\mu\nu}}\right)\quad;\quad\mbox{\rm(on
$\mathcal{W}_{c}$)}\ ,$ (13)

$G^{,\mu\nu}_{\phi\;\tau}=X_{\tau}\left(\frac{\partial\hat{L}}{\partial\phi_{;\mu\nu}}\right)\quad;\quad\mbox{\rm(on
$\mathcal{W}_{c}$)}\ .$ (14)

Contracting $\tau$ and $\nu$ and combining these expressions with (9) and (10)
leads to

$0=\sum_{\nu=0}^{3}\frac{1}{n(\mu\nu)}X_{\nu}\left(\frac{\partial\hat{L}}{\partial\tilde{g}_{\alpha\beta,\mu\nu}}\right)-\frac{\partial\hat{L}}{\partial\tilde{g}_{\alpha\beta,\mu}}+p_{g}^{\alpha\beta,\mu}+\sum_{\rho\leq\sigma}\phi_{;\gamma}p^{,\rho\sigma}_{\phi}\frac{\partial\Gamma^{\gamma}_{\rho\sigma}}{\partial\tilde{g}_{\alpha\beta,\mu}}\quad;\quad\mbox{\rm(on
$\mathcal{W}_{c}$)}\ ,$ (15)

$0=\sum_{\nu=0}^{3}\frac{1}{n(\mu\nu)}X_{\nu}\left(\frac{\partial\hat{L}}{\partial\phi_{;\mu\nu}}\right)-\frac{\partial\hat{L}}{\partial\phi_{;\mu}}+p_{\phi}^{,\gamma\nu}\Gamma_{\gamma\nu}^{\mu}+p_{\phi}^{,\mu}\quad;\quad\mbox{\rm(on
$\mathcal{W}_{c}$)}\ ,$ (16)

which can be rewritten as
$p_{g}^{\alpha\beta,\mu}=\hat{L}^{\alpha\beta,\mu}_{g}\,;\quad
p_{\phi}^{,\mu}=\hat{L}_{\phi}^{\mu}\,$, respectively. These constraints
define the submanifold
$\mathcal{W}_{\mathcal{L}}\hookrightarrow\mathcal{W}_{c}$. Imposing tangency
conditions on these, gives

$G^{\alpha\beta,\mu}_{g\;\tau}=X_{\tau}\left(\hat{L}^{\alpha\beta,\mu}_{g}\right)\,;\quad
G^{,\mu}_{\phi\;\tau}=X_{\tau}\left(\hat{L}_{\phi}^{\mu}\right)\,;\quad\mbox{\rm(on
$\mathcal{W}_{{\cal L}}$)}$

If we contract $\mu$ and $\tau$, and use (7), (8), (11), (12), (13) and (14),
we get

$\displaystyle 0=$ $\displaystyle\frac{\partial\hat{L}}{\partial
g_{\alpha\beta}}-X_{\mu}\left(\hat{L}^{\alpha\beta,\mu}_{g}\right)+\sum_{\mu\leq\nu}p_{\phi}^{,\mu\nu}\phi_{;\gamma}g^{\gamma\alpha}\Gamma_{\mu\nu}^{\beta}\quad;\quad\mbox{\rm(on
$\mathcal{W}_{{\cal L}}$)}\ ,$ (17) $\displaystyle 0=$
$\displaystyle\frac{\partial\hat{L}}{\partial\phi}-X_{\tau}\left(\hat{L}_{\phi}^{\mu}\right)\quad;\quad\mbox{\rm(on
$\mathcal{W}_{{\cal L}}$)}\ .$ (18)

These results hold for the full Horndeski’s theory since up to this point we
have only assumed that our Lagrangian is constructed out of the metric tensor,
a scalar field and its first and second order derivatives. To provide a better
insight of the physical meaning of expressions (17) and (18) we shall consider
the Horndeski Lagrangian with $G_{4}(\phi,X)=0$ and $G_{5}(\phi,X)=0$ and
after carefully computing these expressions explicitly, we get

$\displaystyle\hat{L}_{\phi}^{,\mu}=$
$\displaystyle-\sqrt{-g}\Bigg{\\{}g^{\mu\nu}\phi_{;\nu}\left[1+\frac{\partial
G_{2}{\phi,X}}{\partial X}+\frac{\partial
G_{3}(\phi,X)}{\partial\phi}+\frac{\partial G_{3}(\phi,X)}{\partial
X}\square\phi\right]$
$\displaystyle+\sum_{\nu=0}^{3}\frac{1}{n(\mu\nu)}g^{\mu\nu}g^{\alpha\beta}\phi_{;\alpha}\left(\phi_{;\beta\nu}+\phi_{;\gamma}\Gamma^{\gamma}_{\beta\nu}\right)\frac{\partial
G_{3}(\phi,X)}{\partial X}\Bigg{\\}}$ (19)

$\displaystyle\hat{L}_{g}^{\alpha\beta,\mu}=\frac{1}{2}\sqrt{-g}\Big{\\{}g_{\rho\lambda,\sigma}\Big{[}$
$\displaystyle-3g^{\rho\lambda}g^{\mu(\alpha}g^{\beta)\lambda}+2g^{\rho\lambda}g^{\mu(\alpha}g^{\beta)\sigma}+2g^{\alpha\beta}g^{\rho\sigma}g^{\mu\lambda}$
$\displaystyle+3g^{\mu\sigma}g^{\lambda(\alpha}g^{\beta)\rho}-2g^{\rho\mu}g^{\lambda(\alpha}g^{\beta)\sigma}-g^{\alpha\beta}g^{\rho\lambda}g^{\mu\sigma}$
$\displaystyle-\frac{1}{2}g^{\rho
b}g^{\mu\sigma}g^{\alpha\beta}+g^{\mu(\rho}g^{\lambda)c}g^{\alpha\beta}+g^{\mu\sigma}g^{\alpha(\rho}g^{\lambda)\beta}\Big{]}$
$\displaystyle-
G_{3}(\phi,X)\phi_{;\gamma}(g^{\gamma\beta}g^{\alpha\mu}+g^{\gamma\alpha}g^{\beta\mu}-g^{\gamma\mu}g^{\alpha\beta})\Big{\\}}\,.$
(20)

Therefore:

$\displaystyle\hat{L}^{\alpha\beta}_{g}=$
$\displaystyle-\sqrt{g}n(\alpha\beta)\Bigg{[}R^{\alpha\beta}-\frac{1}{2}\left(R+X+G_{2}+2X\frac{\partial
G_{3}}{\partial\phi}\right)g^{\alpha\beta}-\frac{1}{2}\left(1+\frac{\partial
G_{2}}{\partial X}+2\frac{\partial
G_{3}}{\partial\phi}\right)g^{\alpha\rho}g^{\beta\sigma}\phi_{;\rho}\phi_{;\sigma}$
$\displaystyle+\frac{1}{2}\frac{\partial G_{3}}{\partial
X}\left(-g^{\rho\sigma}g^{\alpha\delta}g^{\beta\sigma}\phi_{;\rho\sigma}\phi_{;\delta}\phi_{;\sigma}+2g^{\alpha\rho}g^{\beta\sigma}g^{\gamma\delta}\phi_{;(\rho}\phi_{;|\gamma|\sigma)}\phi_{;\delta}\right)\Bigg{]}=0\quad;\quad\mbox{\rm(on
$\mathcal{W}_{{\cal L}_{\mathfrak{V}}}$)}\ ,$ (21)

$\displaystyle\hat{L}_{\phi}=$
$\displaystyle-\sqrt{-g}n(\alpha\beta)\Bigg{[}-\left(1+\frac{\partial
G_{2}}{\partial X}+2X\frac{\partial^{2}G_{2}}{\partial X^{2}}+2\frac{\partial
G_{3}}{\partial\phi}+2X\frac{\partial^{2}G_{3}}{\partial
X\partial\phi}\right)g^{\mu\nu}\phi_{;\mu\nu}$
$\displaystyle+g^{\mu\rho}g^{\nu\sigma}R_{\mu\nu}\frac{\partial
G_{3}}{\partial
X}\phi_{;\rho}\phi_{;\sigma}-\left(\frac{\partial^{2}G_{2}}{\partial
X^{2}}+2\frac{\partial^{2}G_{3}}{\partial
X\partial\phi}\right)\left(g^{\mu\nu}\phi_{;\mu\nu}g^{\rho\sigma}\phi_{;\rho}\phi_{;\sigma}-g^{\rho\nu}g^{\sigma\mu}\phi_{;\rho}\phi_{;\sigma}\phi_{;\nu\mu}\right)$
$\displaystyle-\frac{\partial G_{3}}{\partial
X}\left(g^{\mu\nu}\phi_{;\mu\nu}g^{\rho\sigma}\phi_{;\rho\sigma}-g^{\mu\rho}g^{\nu\sigma}\phi_{;\mu\nu}\phi_{;\rho\sigma}\right)+\left(2X\frac{\partial^{2}G_{3}}{\partial\phi^{2}}+\frac{\partial^{2}G_{2}}{\partial
X\partial\phi}\right)-\frac{\partial G_{2}}{\partial\phi}$
$\displaystyle+\frac{\partial^{2}G_{3}}{\partial
X^{2}}g^{\mu\sigma}g^{\nu\rho}\phi_{;\mu}\phi_{;\nu}\left(g^{\gamma\delta}\phi_{;\gamma\delta}\phi_{;\rho\sigma}-g^{\gamma\delta}\phi_{;\gamma\sigma}\phi_{;\rho\delta}\right)\Bigg{]}=0\quad,$
(22)

Expresions (4.2) and (4.2) are the Euler-Lagrange equations, and when they are
evaluated on sections in $\mathcal{W}_{{\cal L}_{\mathfrak{V}}}$ we recover
the cubic Horndeski equations of motion.

It is important to remark that $\hat{L}_{g}^{\alpha\beta}$ and
$\hat{L}_{\phi}$ do not depend on any of the momenta. There is a dependence on
the velocities and accelerations of both the metric and the scalar field, but
not on higher-order velocities of any of them. Hence,
$\hat{L}_{g}^{\alpha\beta}$ and $\hat{L}_{\phi}$ project onto ${J^{2}\pi}$ and
they can be regarded as new constraints defining locally a submanifold
$\mathcal{W}_{1}\hookrightarrow\mathcal{W}_{\mathcal{L}_{\mathfrak{V}}}\hookrightarrow\mathcal{W}_{r}$.
Once again, demanding tangency of the multivector field to this new manifold
we get

$\displaystyle\mathop{\rm
L}\nolimits(X_{\tau})\hat{L}_{g}^{\alpha\beta}|_{\mathcal{W}_{1}}=0\ ,$
$\displaystyle\mathop{\rm
L}\nolimits(X_{\tau})\hat{L}_{\phi}|_{\mathcal{W}_{1}}=0\ ,$

which are just

$\displaystyle X_{\tau}\left(\hat{L}_{g}^{\alpha\beta}\right)$
$\displaystyle=0;\quad\mbox{\rm(on $\mathcal{W}_{1}$)}\ ,$ $\displaystyle
X_{\tau}\left(\hat{L}_{\phi}\right)$ $\displaystyle=0;\quad\mbox{\rm(on
$\mathcal{W}_{1}$)}\ .$

These are new constraints again that project onto $J^{3}\pi$. They define
locally the submanifold
$\mathcal{W}_{f}\hookrightarrow\mathcal{W}_{1}\hookrightarrow\mathcal{W}_{\mathcal{L}}\hookrightarrow\mathcal{W}_{r}$.
This manifold $\mathcal{W}_{f}$ is the final constraint submanifold because
there exist holonomic multivector fields, solutions to (6). Finally, the new
tangency conditions,

$\displaystyle\mathop{\rm
L}\nolimits(X_{\sigma})X_{\tau}\left(\hat{L}_{g}^{\alpha\beta}\right)|_{\mathcal{W}_{1}}=0\
,$ $\displaystyle\mathop{\rm
L}\nolimits(X_{\sigma})X_{\tau}\left(\hat{L}_{\phi}\right)|_{\mathcal{W}_{1}}=0\
,$

which are explicitly

$\displaystyle
X_{\sigma}\left(X_{\tau}\left(\hat{L}_{g}^{\alpha\beta}\right)\right)=0;\quad\mbox{\rm(on
$\mathcal{W}_{1}$)}\ ,$ $\displaystyle
X_{\sigma}\left(X_{\tau}\left(\hat{L}_{\phi}\right)\right)=0\quad\mbox{\rm(on
$\mathcal{W}_{1}$)}\ .$

These allow us to determine the remaining components of (5),
$F_{g\;\alpha\beta,\mu\nu\tau\lambda}$ and $F_{\phi\;,\mu\nu\tau\lambda}$.
Finally, the complete set of constraints that define the final constraint
submanifold $\mathcal{W}_{f}\hookrightarrow\mathcal{W}_{r}$ are

$p_{g}^{\alpha\beta,\mu\nu}-\frac{\partial\hat{L}}{\partial
g_{\alpha\beta,\mu\nu}}=0\quad,\quad
p_{g}^{\alpha\beta,\mu}-\hat{L}_{g}^{\alpha\beta,\mu}=0\quad,\quad\hat{L}_{g}^{\alpha\beta}=0\quad,\quad
X_{\tau}\left(\hat{L}_{g}^{\alpha\beta}\right)=0\ ,$
$p_{\phi}^{,\mu\nu}-\frac{\partial\hat{L}}{\partial\phi_{;\mu\nu}}=0\quad,\quad
p_{\phi}^{,\mu}-\hat{L}_{\phi}^{,\mu}=0\quad,\quad\hat{L}_{\phi}=0\quad,\quad
X_{\tau}\left(\hat{L}_{\phi}\right)=0\ ;$

∎

In contrasts with the Hilbert-Einstein case, we cannot assume that there
exists a holonomic solution (that is, integrable) in $\mathcal{W}_{f}$.
Depending on the particular choice of $G_{1}$, $G_{2}$ and $G_{3}$, new
constraints may appear when demanding integrability of the multivector field
in the constraint algorithm.

###  4.3 Field equations for sections

Provided that we now know the solution for the holonomic multivector fields,
we can evaluate equation (3) to recover the field equations for sections.

$\displaystyle\displaystyle\frac{\partial{\psi_{g\;\alpha\beta}}}{\partial{x^{\mu}}}$
$\displaystyle=\psi_{g\;\alpha\beta,\mu}\ ,\ $ (23)
$\displaystyle\displaystyle\frac{\partial{\psi_{g\;\alpha\beta,\mu}}}{\partial{x^{\nu}}}$
$\displaystyle=\psi_{g\;\alpha\beta,\mu\nu}\ ,\ $ (24)
$\displaystyle\nabla_{\mu}\psi_{\phi}$ $\displaystyle=\psi_{\phi\;;\mu}\ ,\ $
(25) $\displaystyle\nabla_{\nu}\psi_{\phi\;;\mu}$
$\displaystyle=\psi_{\phi\;;\mu\nu}\ ,\ $ (26)
$\displaystyle\displaystyle\frac{\partial{\psi_{g}^{\alpha\beta,\mu}}}{\partial{x^{\mu}}}$
$\displaystyle=\displaystyle\frac{\partial{\hat{L}}}{\partial{g_{\alpha\beta}}}+\psi_{\phi}^{,\mu\nu}\psi_{\phi\;;\gamma}\psi_{g}^{\gamma\alpha}\Gamma^{\beta}_{\mu\nu}\
,\ $ (27)
$\displaystyle\displaystyle\frac{\partial{\psi_{\phi}^{,\mu}}}{\partial{x^{\mu}}}$
$\displaystyle=\displaystyle\frac{\partial{\hat{L}}}{\partial{\phi}}\ ,$ (28)
$\displaystyle\displaystyle\frac{\partial{\psi_{g}^{g\;\alpha\beta,\mu\nu}}}{\partial{x^{\mu}}}$
$\displaystyle=\displaystyle\frac{\partial{\hat{L}}}{\partial{g_{\alpha\beta,\mu}}}-\psi_{g}^{\alpha\beta,\mu}-\frac{1}{2}\psi_{\phi\;;\gamma}\left(\psi_{\phi}^{,\mu\alpha}\psi_{g}^{\gamma\beta}+\psi_{\phi}^{,\beta\mu}\psi_{g}^{\gamma\alpha}-\psi_{\phi}^{,\alpha\beta}\psi_{g}^{\gamma\mu}\right)$
(29)
$\displaystyle\displaystyle\frac{\partial{\psi_{\phi}^{,\mu\nu}}}{\partial{x^{\nu}}}$
$\displaystyle=\displaystyle\frac{\partial{\hat{L}}}{\partial{\phi_{;\mu}}}-\psi_{\phi}^{,\gamma\nu}\Gamma^{\mu}_{\gamma\nu}-\psi_{\phi}^{,\mu}\
,$ (30) $\displaystyle\psi_{g}^{\alpha\beta,\mu\nu}$
$\displaystyle=\displaystyle\frac{\partial{\hat{L}}}{\partial{g_{\alpha\beta,\mu\nu}}}\
,$ (31) $\displaystyle\psi_{\phi}^{,\mu\nu}$
$\displaystyle=\displaystyle\frac{\partial{\hat{L}}}{\partial{\phi_{;\mu\nu}}}\
.$ (32)

Equations (23)-(26) are the holonomy conditions for the multivector field.
Equations (29)-(32) define the Legendre transformations.

##  5 Hamiltonian formalism

The covariant Hamiltonian formalism takes place in the image of the Legendre
Transformation. For singular Lagrangian this space could be highly degenerate.
The Legendre maps in our case are given by Proposition 1. Then

${\rm
T}_{{j^{3}_{x}}\phi}\mathcal{FL}_{\mathfrak{V}}=\left(\begin{array}[]{ccccccccc}1&0&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0\\\
0&0&0&0&1&0&0&0&0\\\
0&\displaystyle\frac{\partial\hat{L}_{g}^{\alpha\beta,\mu}}{\partial
g_{\gamma\delta}}&\displaystyle\frac{\partial\hat{L}_{g}^{\alpha\beta,\mu}}{\partial\phi}&\displaystyle\frac{\partial\hat{L}_{g}^{\alpha\beta,\mu}}{\partial
g_{\gamma\delta,\tau}}&\displaystyle\frac{\partial\hat{L}_{g}^{\alpha\beta,\mu}}{\partial\phi_{;\tau}}&0&0&0&0\\\
0&\displaystyle\frac{\partial\hat{L}_{\phi}^{,\mu}}{\partial
g_{\gamma\delta}}&\displaystyle\frac{\partial\hat{L}_{\phi}^{,\mu}}{\partial\phi}&\displaystyle\frac{\partial\hat{L}_{\phi}^{,\mu}}{\partial
g_{\gamma\delta,\tau}}&\displaystyle\frac{\partial\hat{L}_{\phi}^{,\mu}}{\partial\phi_{;\tau}}&\displaystyle\frac{\partial\hat{L}_{\phi}^{,\mu}}{\partial\phi_{;\tau\lambda}}&0&0&0\\\
0&\displaystyle\frac{\partial^{2}\hat{L}}{\partial g_{\gamma\delta}\partial
g_{\alpha\beta,\mu\nu}}&\displaystyle\frac{\partial^{2}\hat{L}}{\partial\phi\partial
g_{\alpha\beta,\mu\nu}}&\displaystyle\frac{\partial^{2}\hat{L}}{\partial
g_{\gamma\delta,\tau}\partial
g_{\alpha\beta,\mu\nu}}&\displaystyle\frac{\partial^{2}\hat{L}}{\partial\phi_{;\tau}\partial
g_{\alpha\beta,\mu\nu}}&0&0&0&0\\\
0&\displaystyle\frac{\partial^{2}\hat{L}}{\partial
g_{\gamma\delta}\partial\phi_{;\mu\nu}}&\displaystyle\frac{\partial^{2}\hat{L}}{\partial\phi\partial\phi_{;\mu\nu}}&\displaystyle\frac{\partial^{2}\hat{L}}{\partial
g_{\gamma\delta,\tau}\partial\phi_{;\mu\nu}}&\displaystyle\frac{\partial^{2}\hat{L}}{\partial\phi_{;\tau}\partial\phi_{;\mu\nu}}&0&0&0&0\end{array}\right)\
$

Notice that, in general, ${\rm rank}({\rm
T}_{{j^{3}_{x}}\phi}\mathcal{FL}_{\mathfrak{V}})\geq 59$, depending on the
arbitrary function $G_{3}(\phi,X)$. Also, locally

$\ker\,(\mathcal{FL}_{{\mathfrak{V}}})_{*}=\ker\,\Omega_{{\cal
L}_{\mathfrak{V}}}\supset{\left<\frac{\partial}{\partial
g_{\alpha\beta,\mu\nu}},\frac{\partial}{\partial
g_{\alpha\beta,\mu\nu\lambda}},\frac{\partial}{\partial\phi_{;\mu\nu\lambda}}\right>}_{0\leq\alpha\leq\beta\leq
3;\,0\leq\mu\leq\nu\leq\lambda\leq 3}\ ,$ (33)

hence $\mathcal{FL}_{\mathfrak{V}}$ is highly degerated.

We denote
$\widetilde{\mathcal{P}}=\widetilde{\mathcal{FL}}_{\mathfrak{V}}(J^{3}\pi)\stackrel{{\scriptstyle\tilde{\jmath}}}{{\hookrightarrow}}J^{2}\pi^{\dagger}$
and
$\mathcal{P}=\mathcal{FL}_{\mathfrak{V}}(J^{3}\pi)\stackrel{{\scriptstyle\jmath}}{{\hookrightarrow}}J^{2}\pi^{\ddagger}$,
and let $\mathcal{FL}_{\mathfrak{V}}^{o}$ be the map defined by
$\mathcal{FL}_{\mathfrak{V}}=\jmath\circ\mathcal{FL}_{\mathfrak{V}}^{o}$ and
$\bar{\pi}_{\mathcal{P}}\colon\mathcal{P}\to M$ the natural projection. In
order to assure the existence of the Hamiltonian formalism it is needed that
the Lagrangian density $\mathcal{L}_{\mathfrak{V}}\in\Omega^{4}(J^{2}\pi)$ is,
at least, almost-regular; i.e, $\mathcal{P}$ is a closed submanifold of
$J^{2}\pi^{\ddagger}$, $\mathcal{FL}_{\mathfrak{V}}$ is a submersion onto its
image and, for every $j^{3}_{x}\phi\in J^{3}\pi$, the fibers
$\mathcal{FL}_{\mathfrak{V}}^{-1}(\mathcal{FL}_{\mathfrak{V}}(j^{3}_{x}\phi))$
are connected submanifolds of $J^{3}\pi$. For more details in almost-regular
Lagrangians and how to recover the Hamiltonian formalism from the unified
Lagrangian-Hamiltonian formalism, we recommend consulting references [12, 39].

The proof that the Hilbert-Einstein Lagrangian is almost-regular is based on
the fact that $\mathcal{P}$ is diffeomorphic to the first jet of the
corresponding fiber bundle[31]. This property is closely related to the fact
that the Euler-Lagrange equations (Einstein’s Field Equations) are second-
order, although one expects fourth-order equations for a second-order
Lagrangian. This topic is called order-reduction (or projectability) of a
theory [40, 41, 42]. Horndenski Lagrangians are constructed such that the
corresponding field equations are second-order, therefore, one hopes to
proceed in a similar way as in the Hilbert-Einstein case. Nevertheless, the
fact that the Euler-Lagrangians equations projects to lower order doesn’t
implies that the geometric structures also project to a lower order. The
Horndenski theories that have this property are characterised by proposition
2.

A form $\alpha\in\Omega^{*}(J^{3}\pi)$ projects to $J^{s}\pi$, $s=1,2$, if it
is $\pi^{3}_{s}$-basic, that is, $\mathop{\rm L}\nolimits_{Y}\alpha=0$ for all
vector fields $Y$ vertical with respect to $\pi^{3}_{s}$. The multisymplectic
Lagrangian system is $(J^{3}\pi,\Omega_{\mathcal{L}})$, where
$\Omega_{\mathcal{L}_{\mathcal{V}}}=\widetilde{\mathcal{FL}_{\mathcal{V}}}^{\
*}\Omega_{1}^{s}$ is the Poincaré-Cartan form.

######  Proposition 2.

The Poincaré-Cartan form $\Omega_{\mathcal{L}_{\mathcal{V}}}$ of a cubic
Horndeski Lagrangian projects to $J^{1}\pi$ if, and only if,

$\frac{\partial G_{3}(\phi,X)}{\partial X}=0\,.$

###### Proof.

The necessary and sufficient conditions for the associated Poincaré-Cartan
form of a second-order theory to project on $J^{1}\pi$, according to [40] and
[41], are that $L\in C^{\infty}(J^{2}\pi)$ is an affine function with respect
to the affine structure of $p_{1}^{2}:J^{2}\pi\rightarrow J^{1}\pi$, i.e.,

$L=L^{ij}_{\alpha}y^{\alpha}_{(ij)}+L_{0},L^{ij}_{\alpha}=L^{ji}_{\alpha}\in
C^{\infty}(J^{1}\pi),L_{0}\in C^{\infty}(J^{1}\pi),$ (34)

and the following equations hold:

$2\frac{\partial L_{\beta}^{hi}}{\partial y_{a}^{\alpha}}-\frac{\partial
L_{\alpha}^{ai}}{\partial y_{h}^{\beta}}-\frac{\partial
L_{\alpha}^{ah}}{\partial y_{i}^{\beta}}=0,a,h,i=1,...n,\alpha,\beta=1,...,m,$
(35)

which in cubic Horndeski’s theory translate into

$L=\sum_{\alpha\leq\beta}L_{g}^{\alpha\beta,\mu\nu}g_{\alpha\beta,\mu\nu}+L_{\phi}^{,\mu\nu}\phi_{,\mu\nu}+L_{0},$
(36)

where

$\displaystyle L_{0}=$
$\displaystyle\sqrt{-g}\Bigg{\\{}g^{\alpha\beta}\Big{[}g^{\gamma\delta}(g_{\delta\mu,\beta}\Gamma^{\mu}_{\alpha\gamma}-g_{\delta\mu,\gamma}\Gamma^{\mu}_{\alpha\beta})+\Gamma^{\delta}_{\alpha\beta}\Gamma^{\gamma}_{\gamma\delta}-\Gamma^{\delta}_{\alpha\gamma}\Gamma^{\gamma}_{\beta\delta}\Big{]}$
$\displaystyle+X+G_{2}(\phi,X)-\phi_{,\gamma}\Gamma^{\gamma}_{\nu\mu}g^{\mu\nu}G_{3}(\phi,X)\Bigg{\\}},$
(37)

$L_{g}^{\alpha\beta,\mu\nu}=\frac{n(\alpha\beta)}{2}\sqrt{-g}\left(g^{\alpha\mu}g^{\beta\nu}+g^{\alpha\nu}g^{\beta\mu}-2g^{\alpha\beta}g^{\mu\nu}\right),$
(38)

$L_{\phi}^{,\mu\nu}=\sqrt{-g}g^{\mu\nu}G_{3}(\phi,X),$ (39)

and the equations (35) hold if, and only, if

$\sqrt{-g}\phi_{,\delta}\frac{\partial G_{3}(\phi,X)}{\partial
X}\left(-2g^{\mu\nu}g^{\gamma\delta}+g^{\gamma\nu}g^{\mu\delta}+g^{\gamma\mu}g^{\nu\delta}\right)=0.$
(40)

Equation 40 only holds if $\frac{\partial G_{3}(\phi,X)}{\partial X}=0$. ∎

If $\frac{\partial G_{3}}{\partial X}=0$, then we should expect that the
systems behaves like a first-order system. We will study this particular case
first, and then we present the general case.

###  5.1 Hamiltonian formalism for a particular case

Throughout this subsection we shall consider that $\frac{\partial
G_{3}(\phi,X)}{\partial X}=0$, i.e. $G_{3}(\phi,X)=G_{3}(\phi)$. Hence

$L_{\mathfrak{V}}=\frac{1}{16\pi
G}\sqrt{|g|}\left[R+X+G_{2}(\phi,X)+G_{3}(\phi)\square\phi\right].$ (41)

######  Proposition 3.

$L_{\mathfrak{V}}$ is an almost-regular Lagrangian and $\mathcal{P}$ is
diffeomorphic to $J^{1}\pi$.

###### Proof.

$\mathcal{P}$ is a closed submanifold of $J^{2}\pi^{\ddagger}$ since it is
defined by the constraints

$\displaystyle p_{g}^{\alpha\beta,\mu\nu}-\frac{\partial\hat{L}}{\partial
g_{\alpha\beta,\mu\nu}}$ $\displaystyle=0;\quad
p_{g}^{\alpha\beta,\mu}-\hat{L}_{g}^{\alpha\beta,\mu}=0\ ,$ $\displaystyle
p_{\phi}^{,\mu\nu}-\frac{\partial\hat{L}}{\partial\phi_{;\mu\nu}}$
$\displaystyle=0;\quad\quad\quad p_{\phi}^{,\mu}-\hat{L}_{\phi}^{,\mu}=0\ .$

The dimension of $\mathcal{P}$ is $4+10+1+40+4=59$ and, as ${\rm rank}({\rm
T}\mathcal{FL}_{\mathfrak{V}})=59$ in every point, ${\rm
T}\mathcal{FL}_{\mathfrak{V}}$ is surjective and $\mathcal{FL}_{\mathfrak{V}}$
is a submersion. Moreover,

$\ker\,(\mathcal{FL}_{{\mathfrak{V}}})_{*}={\left<\frac{\partial}{\partial
g_{\alpha\beta,\mu\nu}},\frac{\partial}{\partial
g_{\alpha\beta,\mu\nu\lambda}},\frac{\partial}{\partial\phi_{;\mu\nu\lambda}}\right>}_{0\leq\alpha\leq\beta\leq
3;\,0\leq\mu\leq\nu\leq\lambda\leq 3}\ ,$ (42)

therefore, the fibers of the Legendre map are the fibers of the projection
$\bar{\pi}^{3}_{1}$. As we consider metric with fixed signature, they are
connected submanifolds of $J^{3}\pi$.

Taking any local section $\phi$ of the projection $\pi^{3}_{1}$, the map
$\Phi=\mathcal{FL}_{\mathfrak{V}}\circ\phi\colon J^{1}\pi\to{\cal P}$ is a
local diffeomorphism and it does not depend on the chosen section. Therefore,
${\cal P}$ and $J^{1}\pi$ are diffeomorphic.

|
---|---
$\textstyle{\
J^{3}\pi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\mathcal{FL}_{\mathfrak{V}}}$$\scriptstyle{\pi^{3}_{1}}$
$\textstyle{\ \mathcal{P}\subset J^{2}\pi^{\ddagger}}$ $\textstyle{\ J^{1}\pi\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\Phi}$

∎

Then, the $\mu$-transverse submanifolds $\widetilde{\cal P}$ and ${\cal P}$
are diffeomorphic and the diffeomorphism, denoted
$\tilde{\mu}\colon\tilde{\cal P}\to{\cal P}$, is just the restriction of the
projection $\mu$ to $\tilde{\cal P}$. Therefore we can define a Hamiltonian
$\mu$-section as $h_{\mathfrak{V}}=\tilde{\jmath}\circ\widetilde{\mu}^{-1}$,
which is specified by a local Hamiltonian function $H_{\mathcal{P}}\in
C^{\infty}(\mathcal{P})$; that is,

$h_{\mathfrak{V}}(x^{\mu},g_{\alpha\beta},\phi,g_{\alpha\beta,\mu},\phi_{\mu},p_{g}^{\alpha\beta,\mu},p_{\phi}^{,\mu},p_{g}^{\alpha\beta,\mu\nu},p_{\phi}^{,\mu\nu})=(x^{\mu},g_{\alpha\beta},\phi,g_{\alpha\beta,\mu},\phi_{\mu},-H_{\mathcal{P}},p_{g}^{\alpha\beta,\mu},p_{\phi}^{,\mu},p_{g}^{\alpha\beta,\mu\nu},p_{\phi}^{,\mu\nu}).$

This function $H_{\mathfrak{V}}$ is the Hamiltonian function defined on
$\mathcal{P}$ and is given by
$H=(\mathcal{FL}_{\mathfrak{V}}^{o})^{*}\,H_{\mathfrak{V}}$; where $H$, which
is $\mathcal{FL}_{\mathfrak{V}}^{o}$-projectable, is

$\displaystyle
H=\sum_{\alpha\leq\beta}L_{g}^{\alpha\beta,\mu\nu}g_{\alpha\beta,\mu\nu}+L_{\phi}^{,\mu\nu}\left(\phi_{;\mu\nu}+\phi_{;\gamma}\Gamma^{\gamma}_{\mu\nu}\right)+\sum_{\alpha\leq\beta}L_{g}^{\alpha\beta,\mu}g_{\alpha\beta,\mu}+L_{\phi}^{,\mu}\phi_{;\mu}-L,$
(43)

This is summarised in the following diagram:

$\textstyle{\widetilde{\mathcal{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\jmath}}$$\scriptstyle{\tilde{\mu}}$
$\textstyle{J^{2}\pi^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$
$\textstyle{\mathcal{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\mathcal{W}}}$$\scriptstyle{\rho_{2}}$$\textstyle{\mathcal{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\jmath}$$\scriptstyle{h_{\mathfrak{V}}}$
$\textstyle{J^{2}\pi^{\ddagger}}$
$\textstyle{\mathcal{W}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{h}}$$\scriptstyle{\rho_{2}^{r}}$

Now, it is possible to define the Hamiltonian forms

$\Theta_{h_{\mathfrak{V}}}:=h_{\mathfrak{V}}^{*}\Theta_{1}^{s}\in{\mit\Omega}^{4}({\cal
P})\quad,\quad\Omega_{h_{\mathfrak{V}}}:=-{\rm
d}\Theta_{h_{\mathfrak{V}}}=h_{\mathfrak{V}}^{*}\Omega_{1}^{s}\in{\mit\Omega}^{5}({\cal
P})\ ,$

and thus we have the Hamiltonian system $({\cal
P},\Omega_{h_{\mathfrak{V}}})$. Then, the Hamiltonian problem associated with
this system consists in finding holonomic sections $\psi_{h}\colon
M\rightarrow\mathcal{P}$ satisfying any of the following equivalent
conditions:

  1. 1.

$\psi_{h}$ is a solution to the equation

${}\psi_{h}^{*}i(X)\Omega_{h_{\mathfrak{V}}}=0\quad,\quad\mbox{\rm for every
$X\in\mathfrak{X}(\mathcal{P})$}\ .$ (44)

  2. 2.

$\psi_{h}$ is an integral section of a multivector field contained in a class
of holonomic multivector fields $\\{{\bf
X}_{h}\\}\subset\mathfrak{X}^{4}(\mathcal{P})$ satisfying the equation

${}\mathop{i}\nolimits{({\bf
X}_{h})}\Omega_{h_{\mathfrak{V}}}=0\quad,\quad\forall{\bf X}_{h}\in\\{{\bf
X}_{h}\\}\subset\mathfrak{X}^{4}(\mathcal{P})\ .$ (45)

(Here, holonomic sections and multivector fields are defined as in
$J^{2}\pi^{\dagger}$). The solutions of the Hamiltonian formalism can be
recovered geometrically from the unified formalism using the adequate
projections (see [18] for more details). Nevertheless, we will continue by
presenting the local expression of equations 45.

Formulation using multimomentum coordinates.

The natural coordinates of $J^{2}\pi^{\ddagger}$ are

$(x^{\mu},g_{\alpha\beta},\phi,g_{\alpha\beta,\mu},\phi_{;\mu},p_{g}^{\alpha\beta,\mu},p_{\phi}^{,\mu},p_{g}^{\alpha\beta,\mu\nu},p_{\phi}^{,\mu\nu}),$

which contain the multimomenta and the velocities. These are the expected
coordinates for a Hamiltonian formulation of a second-order regular
Lagrangian. Nevertheless, our Lagrangian is singular and the Hamiltonian
formulation takes place in the submanifold $\mathcal{P}$. Since it is
diffeomorphic to $J^{1}\pi$ by proposition 3, a natural set of coordinates is

$(x^{\mu},g_{\alpha\beta},\phi,g_{\alpha\beta,\mu},\phi_{,\mu})\,.$

This is an uninteresting coordinate system, as the resulting equations are
identical than the Lagrangian ones. It is customary to write the Hamiltonian
in terms of the positions and multimomenta only, so we need to isolate the
velocities to be able to write the Hamiltonian in these terms. The relation
between momenta and velocities is given by

$\displaystyle p_{g}^{\alpha\beta,\mu}=$
$\displaystyle-\frac{1}{2}\sqrt{-g}\Big{\\{}G_{3}(\phi)\phi_{;\gamma}\left(g^{\gamma\beta}g^{\alpha\mu}+g^{\gamma\alpha}g^{\beta\mu}-g^{\gamma\mu}g^{\alpha\beta}\right)$
$\displaystyle-
g_{\rho\lambda,\sigma}\Big{[}-3g^{\rho\sigma}g^{\mu(\alpha}g^{\beta)\lambda}+2g^{\rho\lambda}g^{\mu(\alpha}g^{\beta)\sigma}+2g^{\alpha\beta}g^{\rho\sigma}g^{\mu\lambda}$
$\displaystyle+3g^{\mu\sigma}g^{\lambda(\alpha}g^{\beta)\rho}-2g^{\rho\mu}g^{\lambda(\alpha}g^{\beta)\sigma}-g^{\alpha\beta}g^{\rho\lambda}g^{\mu\sigma}$
$\displaystyle-\frac{1}{2}g^{\rho\lambda}g^{\mu\sigma}g^{\alpha\beta}+g^{\mu(\rho}g^{\lambda)\sigma}g^{\alpha\beta}+g^{\mu\sigma}g^{\alpha(\rho}g^{\lambda)\beta}\Big{]}\Big{\\}}\,,$
(46)

$\displaystyle p_{\phi}^{,\mu}=$
$\displaystyle-\sqrt{-g}g^{\mu\nu}\Bigg{\\{}\phi_{;\nu}\left[1+\frac{\partial
G_{2}(\phi,X)}{\partial X}+\frac{\partial
G_{3}(\phi)}{\partial\phi}\right]\Bigg{\\}}$ (47)

Unlike in General Relativity, where the $p_{g}^{\alpha\beta,\mu}$ and
$g_{\alpha\beta,\mu}$ are in one to one correspondence [31], this is not
generally true even in the particular case where $\frac{\partial
G_{3}}{\partial X}=0$. To isolate the velocities we would need to fully
specify $G_{2}(\phi,X)$ and in some cases it would not even be possible to do
so.

To illustrate this procedure, we will consider the case
$\frac{\partial^{2}G_{2}}{\partial X^{2}}=0$ and $1+\frac{\partial
G_{2}(\phi,X)}{\partial X}+\frac{\partial G_{3}(\phi)}{\partial\phi}\neq 0$.
With this in mind, we isolate the velocities in terms of the positions and
multimomenta only:

$\displaystyle\phi_{;\nu}=$
$\displaystyle-\frac{1}{\sqrt{-g}}\left[\frac{p_{\phi}^{,\mu}g_{\mu\nu}}{1+\frac{\partial
G_{2}(\phi,X)}{\partial X}+\frac{\partial
G_{3}(\phi)}{\partial\phi}}\right]=U_{,\nu}$ (48) $\displaystyle
g_{\alpha\beta,\mu}=$
$\displaystyle\frac{1}{3\sqrt{-g}}\frac{1}{n(\alpha\beta)}\Bigg{\\{}p_{g}^{\lambda\sigma,\nu}-\frac{1}{2}p_{\phi}^{,\delta}g_{\delta\epsilon}\left[\frac{G_{3}(\phi)}{1+\frac{\partial
G_{2}(\phi,X)}{\partial X}+\frac{\partial
G_{3}(\phi)}{\partial\phi}}\right]\left[g^{\nu\lambda}g^{\epsilon\sigma}+g^{\nu\sigma}g^{\epsilon\lambda}-g^{\lambda\sigma}g^{\epsilon\nu}\right]\Bigg{\\}}$
$\displaystyle\Bigg{\\{}-2g_{\alpha\lambda}g_{\beta\mu}g_{\sigma\nu}-2g_{\alpha\mu}g_{\beta\lambda}g_{\sigma\nu}+6g_{\alpha\lambda}g_{\beta\sigma}g_{\mu\nu}+g_{\alpha\nu}g_{\beta\mu}g_{\lambda\sigma}+g_{\alpha\mu}g_{\beta\nu}g_{\lambda\sigma}\Bigg{\\}}=V_{\alpha\beta,\mu}$
(49)

Notice that we require that if $\frac{\partial G_{2}(\phi,X)}{\partial
X}+\frac{\partial G_{3}(\phi)}{\partial\phi}=-1\,,$ then $p^{,\mu}_{\phi}=0$
and there is no hope to use $p^{,\mu}_{\phi}$ as a coordinate instead of
$\phi;_{\mu}$.

Now we can set
$(x^{\mu},g_{\alpha\beta},\phi,p_{g}^{\alpha\beta,\mu},p_{\phi}^{,\mu})$ as
coordinates of $\mathcal{P}$ and then rewrite the Hamiltonian function

$H_{\mathfrak{V}}(x^{\mu},g_{\alpha\beta},\phi,p_{g}^{\alpha\beta,\mu},p_{\phi}^{,\mu})=H_{\mathfrak{V}}(x^{\mu},g_{\alpha\beta},\phi,V_{\alpha\beta,\mu}(p_{g}^{\alpha\beta,\mu},p_{\phi}^{,\mu},g_{\alpha\beta},\phi),U_{,\mu}(p_{\phi}^{,\mu},g_{\alpha\beta},\phi))\
.$

The Hamiltonian function is hence

$\displaystyle H_{\mathfrak{V}}=$
$\displaystyle\sum_{\alpha\leq\beta}p_{g}^{\alpha\beta,\mu}V_{\alpha\beta,\mu}+\sqrt{-g}G_{3}(\phi)g^{\mu\nu}U_{;\gamma}\Gamma_{\nu\mu}^{\gamma}+p_{\phi}^{,\mu}U_{,\mu}-\sqrt{-g}\left(X+G_{2}(\phi,X)\right)$
$\displaystyle-\sqrt{-g}g^{\rho\lambda}\left[-\frac{1}{2}g^{\sigma\epsilon}g^{\delta\zeta}g_{\epsilon\zeta,\sigma}\left(g_{\lambda\delta,\rho}+g_{\rho\delta,\lambda}-g_{\rho\lambda,\delta}\right)+\frac{1}{2}g^{\sigma\epsilon}g^{\delta\zeta}g_{\epsilon\zeta,\lambda}\left(g_{\rho\delta,\lambda}+g_{\sigma\delta,\rho}-g_{\rho\sigma,\delta}\right)+\Gamma_{\sigma\delta}^{\sigma}\Gamma_{\lambda\rho}^{\delta}-\Gamma_{\lambda\rho}^{\sigma}\Gamma_{\sigma\rho}^{\delta}\right]$

The field equations are derived again from (45) expressed using the new
coordinates. Now, the Hamilton-Cartan form $\Omega_{h}$ has the local
expression:

$\Omega_{h_{\mathfrak{V}}}={\rm d}H_{\mathfrak{V}}\wedge{\rm
d}^{4}x-\sum_{\alpha\leq\beta}{\rm d}p_{g}^{\alpha\beta,\mu}\wedge{\rm
d}g_{\alpha\beta}\wedge{\rm d}^{3}x_{\mu}-{\rm d}p_{\phi}^{,\mu}\wedge{\rm
d}\phi\wedge{\rm d}^{3}x_{\mu}-\sum_{\alpha\leq\beta}{\rm
d}L_{g}^{\alpha\beta,\mu\nu}\wedge{\rm d}V_{\alpha\beta,\mu}\wedge{\rm
d}^{3}x_{\nu}-{\rm d}L_{\phi}^{,\mu\nu}\wedge{\rm d}U_{,\mu}\wedge{\rm
d}^{3}x_{\nu}\ ,$

and the local expression of a representative of a class $\\{{\bf X}_{h}\\}$ of
semi-holonomic multivector fields in ${\cal P}$ is

${\bf X}_{h}=\bigwedge_{i=\nu}^{4}\left(\frac{\partial}{\partial
x^{\nu}}+F_{g\;\alpha\beta,\nu}\frac{\partial}{\partial
g_{\alpha\beta}}+F_{\phi\;,\nu}\frac{\partial}{\partial\phi}+G^{\alpha\beta,\mu}_{g\;\nu}\frac{\partial}{\partial
p_{g}^{\alpha\beta,\mu}}+G^{,\mu}_{\phi\;\nu}\frac{\partial}{\partial
p_{\phi}^{,\mu}}\right);.$

with $F_{g\;\alpha\beta,\nu}$, $G^{\alpha\beta,\mu}_{g\;\nu}$,
$F_{\phi\;,\nu}$, $G^{,\mu}_{\phi\;\nu}\in C^{\infty}(\mathcal{P})$.

From (45) we obtain

$\displaystyle\frac{\partial H_{\mathfrak{V}}}{\partial g_{\alpha\beta}}=$
$\displaystyle-G^{\alpha\beta,\mu}_{g\;\mu}+G^{\rho\lambda,\gamma}_{g\;\nu}\frac{\partial
V_{\delta\zeta,\epsilon}}{\partial p_{g}^{\rho\lambda,\gamma}}\frac{\partial
L^{\delta\zeta,\epsilon\nu}}{\partial
g_{\alpha\beta}}+F_{g\;\rho\lambda,\nu}\left(\frac{\partial
V_{\delta\zeta,\epsilon}}{\partial g_{\rho\lambda}}\frac{\partial
L^{\delta\zeta,\epsilon\nu}}{\partial g_{\alpha\beta}}-\frac{\partial
V_{\delta\zeta,\epsilon}}{\partial g_{\alpha\beta}}\frac{\partial
L^{\delta\zeta,\epsilon\nu}}{\partial g_{\rho\lambda}}\right)$
$\displaystyle+G_{\phi\;\rho}^{,\rho}\left(\frac{\partial U_{,\nu}}{\partial
p_{\phi}^{,\mu}}\frac{\partial L_{\phi}^{,\mu\nu}}{\partial
g_{\alpha\beta}}-\frac{\partial U_{,\nu}}{\partial
g_{\alpha\beta}}\frac{\partial L_{\phi}^{,\mu\nu}}{\partial
p_{\phi}^{,\mu}}\right)+F_{\phi\;,\mu}\left(\frac{\partial
U_{,\nu}}{\partial\phi}\frac{\partial L_{\phi}^{,\mu\nu}}{\partial
g_{\alpha\beta}}-\frac{\partial U_{,\nu}}{\partial
g_{\alpha\beta}}\frac{\partial L_{\phi}^{,\mu\nu}}{\partial\phi}\right),$ (50)
$\displaystyle\frac{\partial H_{\mathfrak{V}}}{\partial p^{\alpha\beta,\mu}}=$
$\displaystyle F_{g\;\alpha\beta,\mu}-F_{g\;\rho\lambda,\nu}\frac{\partial
V_{ab,c}}{\partial p_{g}^{\alpha\beta,\mu}}\frac{\partial
L_{g}^{ab,c\nu}}{\partial g_{\rho\lambda}},$ (51) $\displaystyle\frac{\partial
H_{\mathfrak{V}}}{\partial\phi}=$ $\displaystyle-
G_{\phi\;\nu}^{,\rho}\left(\delta_{\rho}^{\nu}+\frac{\partial
U_{,\mu}}{\partial\phi}\frac{\partial L_{\phi}^{,\mu\nu}}{\partial
p_{\phi}^{,\rho}}-\frac{\partial U_{,\mu}}{\partial
p_{\phi}^{,\rho}}\frac{\partial L_{\phi}^{,\mu\nu}}{\partial\phi}\right)$
$\displaystyle+F_{g\;\alpha\beta,\nu}\left(\frac{\partial
V_{\rho\lambda,\mu}}{\partial\phi}\frac{\partial
L_{g}^{\alpha\beta,\mu\nu}}{\partial g_{\rho\lambda}}+\frac{\partial
U_{,\mu}}{\partial g_{\alpha\beta}}\frac{\partial
L_{\phi}^{,\mu\nu}}{\partial\phi}-\frac{\partial
U_{,\mu}}{\partial\phi}\frac{\partial L_{\phi}^{,\mu\nu}}{\partial
g_{\alpha\beta}}\right),$ (52) $\displaystyle\frac{\partial
H_{\mathfrak{V}}}{\partial p_{\phi}^{,\mu}}=$ $\displaystyle
F_{\phi\;,\mu}+F_{g\;\alpha\beta,\nu}\left(\frac{\partial U_{,\rho}}{\partial
g_{\alpha\beta}}\frac{\partial L_{\phi}^{,\rho\nu}}{\partial
p_{\phi}^{,\mu}}-\frac{\partial U_{,\rho}}{\partial
p_{\phi}^{,\mu}}\frac{\partial L_{\phi}^{,\rho\nu}}{\partial
g_{\alpha\beta}}-\frac{\partial V_{\delta\zeta,\epsilon}}{\partial
p_{\phi}^{,\mu}}\frac{\partial L_{g}^{\delta\zeta,\epsilon\nu}}{\partial
g_{\alpha\beta}}\right)$ $\displaystyle+F_{\phi\;,\nu}\left(\frac{\partial
U_{,\rho}}{\partial\phi}\frac{\partial L_{\phi}^{,\rho\nu}}{\partial
p_{\phi}^{,\mu}}-\frac{\partial U_{,\rho}}{\partial
p_{\phi}^{\mu}}\frac{\partial L_{\phi}^{,\rho\nu}}{\partial\phi}\right).$ (53)

Expresions (5.1) through (5.1) would be the classical Hamilton-De Donder-Weil
equations for a first order field theory except by the fact that they contain
extra-terms because the cubic Horndeski Lagrangian is a second order theory
with respect to the metric and the scalar field, and neither $\displaystyle
L_{g}^{\alpha\beta,\mu\nu}=\frac{1}{n(\mu\nu)}\frac{\partial L}{\partial
g_{\alpha\beta,\mu\nu}}$ nor
$L_{\phi}^{,\mu\nu}=\frac{1}{n(\mu\nu)}\frac{\partial
L}{\partial\phi_{;\mu\nu}}$ vanish.

###  5.2 Hamiltonian formalism for the general case

On this subsection we consider the full cubic Horndeski Lagrangian

$L_{\mathfrak{V}}=\frac{1}{16\pi
G}\sqrt{|g|}\left[R+X+G_{2}(\phi,X)+G_{3}(\phi,X)\square\phi\right].$ (54)

We will assume that $L$ is almost-regular. The multimomenta for the general
cubic case are

$\displaystyle p_{g}^{\alpha\beta,\mu}=$
$\displaystyle-\frac{1}{2}\sqrt{-g}\Big{\\{}G_{3}(\phi,X)\phi_{;\gamma}\left(g^{\gamma\beta}g^{\alpha\mu}+g^{\gamma\alpha}g^{\beta\mu}-g^{\gamma\mu}g^{\alpha\beta}\right)$
$\displaystyle-
g_{\rho\lambda,\sigma}\Big{[}-3g^{\rho\sigma}g^{\mu(\alpha}g^{\beta)\lambda}+2g^{\rho\lambda}g^{\mu(\alpha}g^{\beta)\sigma}+2g^{\alpha\beta}g^{\rho\sigma}g^{\mu\lambda}$
$\displaystyle+3g^{\mu\sigma}g^{\lambda(\alpha}g^{\beta)\rho}-2g^{\rho\mu}g^{\lambda(\alpha}g^{\beta)\sigma}-g^{\alpha\beta}g^{\rho\lambda}g^{\mu\sigma}$
$\displaystyle-\frac{1}{2}g^{\rho\lambda}g^{\mu\sigma}g^{\alpha\beta}+g^{\mu(\rho}g^{\lambda)\sigma}g^{\alpha\beta}+g^{\mu\sigma}g^{\alpha(\rho}g^{\lambda)\beta}\Big{]}\Big{\\}}\,,$
(55)

$\displaystyle p_{\phi}^{,\mu}=$
$\displaystyle-\sqrt{-g}g^{\mu\nu}\Bigg{\\{}\phi_{;\nu}\left[1+\frac{\partial
G_{2}(\phi,X)}{\partial X}+\frac{\partial
G_{3}(\phi,X)}{\partial\phi}+\square\phi\frac{\partial G_{3}(\phi,X)}{\partial
X}\right]$
$\displaystyle+\left(\phi_{;\alpha\nu}+\phi_{;\gamma}\Gamma^{\gamma}_{\alpha\nu}\right)g^{\alpha\beta}\phi_{;\beta}\frac{\partial
G_{3}(\phi,X)}{\partial X}\Bigg{\\}}=L_{\phi_{h}}^{\mu}.$ (56)

The multimomentum (5.2) is not a constraint for the general case, in which
$G_{3}\neq 0$. In contrast, (5.2) is indeed a constraint and we must demand
tangency of the multivector field to the submanifold defined by this
constraint.

$\mathop{\rm
L}\nolimits(X_{h\;\tau})\left(p_{\phi}^{,\mu}-L_{\phi_{h}}^{\mu}\right)|_{\mathcal{W}_{c}}=0,$

which yields

$\displaystyle 0=$ $\displaystyle
G_{g\;\tau}^{\alpha\beta,\mu}-\frac{1}{2}\sqrt{-g}\Bigg{\\{}g^{\rho\lambda}F_{g\;\;\rho\lambda,\tau}N^{\alpha\beta\mu}-G_{3}(\phi,X)F_{g\;\rho\lambda,\tau}\phi_{;\gamma}\frac{\partial}{\partial
g_{\rho\lambda}}\left(g^{\gamma\beta}g^{\alpha\mu}+g^{\gamma\alpha}g^{\beta\mu}-g^{\gamma\mu}g^{\alpha\beta}\right)$
$\displaystyle-F_{\phi,\tau}\frac{\partial
G_{3}}{\partial\phi}\phi_{;\gamma}\left(g^{\gamma\beta}g^{\alpha\mu}+g^{\gamma\alpha}g^{\beta\mu}-g^{\gamma\mu}g^{\alpha\beta}\right)-g_{\rho\lambda,\sigma}F_{g\;\delta\zeta,\tau}\frac{\partial
M^{\rho\lambda\sigma\alpha\beta\mu}}{\partial
g_{\delta\zeta}}+F_{g\;\delta\zeta,\eta\tau}\delta_{\rho\lambda}^{\delta\zeta}\delta_{\sigma}^{\eta}M^{\rho\lambda\sigma\alpha\beta\mu}$
$\displaystyle-
F_{\phi\;\rho,\tau}\delta_{\gamma}^{\;\rho}\left(g^{\gamma\beta}g^{\alpha\mu}+g^{\gamma\alpha}g^{\beta\mu}-g^{\gamma\mu}g^{\alpha\beta}\right)\Bigg{\\}},$
(57)

where

$\displaystyle M^{\rho\lambda\sigma\alpha\beta\mu}=$
$\displaystyle-3g^{\rho\sigma}g^{\mu(\alpha}g^{\beta)\lambda}+2g^{\rho\lambda}g^{\mu(\alpha}g^{\beta)\sigma}+2g^{\alpha\beta}g^{\rho\sigma}g^{\mu\lambda}$
$\displaystyle+3g^{\mu\sigma}g^{\lambda(\alpha}g^{\beta)\rho}-2g^{\rho\mu}g^{\lambda(\alpha}g^{\beta)\sigma}-g^{\alpha\beta}g^{\rho\lambda}g^{\mu\sigma}$
$\displaystyle-\frac{1}{2}g^{\rho\lambda}g^{\mu\sigma}g^{\alpha\beta}+g^{\mu(\rho}g^{\lambda)\sigma}g^{\alpha\beta}+g^{\mu\sigma}g^{\alpha(\rho}g^{\lambda)\beta}$

and

$N^{\alpha\beta\mu}=G_{3}(\phi,X)\phi_{;\gamma}\left(g^{\gamma\beta}g^{\alpha\mu}+g^{\gamma\alpha}g^{\beta\mu}-g^{\gamma\mu}g^{\alpha\beta}\right)-g_{\rho\lambda,\sigma}M^{\rho\lambda\sigma\alpha\beta\mu}.$

The second-order multimomenta $p_{g}^{\alpha\beta,\mu\nu}$ and
$p_{\phi}^{,\mu\nu}$ are completely determined by the constrains defining
$\mathcal{P}$ 1. It is not possible, in general, to isolate the velocities in
terms of the momenta unless $G_{2}(\phi,X)$ and $G_{3}(\phi,X)$ are explicitly
specified. Moreover, the momenta depend on the acceleration of the scalar
field, i.e. $\mathcal{P}$ does not project on $J^{1}\pi$ as expected 2. It is
impossible, in general, to explicitly isolate the velocities purely in terms
of the momenta, so we will use the mix coordinates
$(x^{\mu},g_{\alpha\beta},\phi,g_{\alpha\beta,\mu},\phi_{;\mu},p_{g}^{\alpha\beta,\mu},p_{\phi}^{,\mu},p_{g}^{\alpha\beta,\mu\nu},p_{\phi}^{,\mu\nu})$
for the Hamiltonian formulation, which contains a positions, momenta and
velocities.

In terms of these coordinates, the Hamiltonian function is

$\displaystyle H_{\mathfrak{V}}=$
$\displaystyle\sum_{\alpha\leq\beta}p_{g}^{\alpha\beta,\mu}g_{\alpha\beta,\mu}+\sqrt{-g}G_{3}(\phi,X)g^{\mu\nu}\phi_{;\gamma}\Gamma_{\nu\mu}^{\gamma}+p_{\phi}^{,\mu}\phi_{;\mu}-\sqrt{-g}\left(X+G_{2}(\phi,X)\right)$
$\displaystyle-\sqrt{-g}g^{ab}\left[-\frac{1}{2}g^{ce}g^{df}g_{ef,c}\left(g_{bd,a}+g_{ad,b}-g_{ab,d}\right)+\frac{1}{2}g^{ce}g^{df}g_{ef,b}\left(g_{ad,c}+g_{cd,a}-g_{ac,d}\right)+\Gamma_{cd}^{c}\Gamma_{ba}^{d}-\Gamma_{ba}^{c}\Gamma_{ca}^{d}\right]$
(58)

The Hamilton-Cartan form $\Omega_{h}$ has the local expression:

$\Omega_{h_{\mathfrak{V}}}={\rm d}H_{\mathfrak{V}}\wedge{\rm
d}^{4}x-\sum_{\alpha\leq\beta}{\rm d}p_{g}^{\alpha\beta,\mu}\wedge{\rm
d}g_{\alpha\beta}\wedge{\rm d}^{3}x_{\mu}-{\rm d}p_{\phi}^{,\mu}\wedge{\rm
d}\phi\wedge{\rm d}^{3}x_{\mu}-\sum_{\alpha\leq\beta}{\rm
d}L_{g}^{\alpha\beta,\mu\nu}\wedge{\rm d}g_{\alpha\beta,\mu}\wedge{\rm
d}^{3}x_{\nu}-{\rm d}L_{\phi}^{,\mu\nu}\wedge{\rm d}\phi_{;\mu}\wedge{\rm
d}^{3}x_{\nu}\ ,$

and the local expression of a representative of a class $\\{{\bf X}_{h}\\}$ of
semi-holonomic multivector fields in ${\cal P}$ is

${\bf X}_{h}=\bigwedge_{i=\nu}^{4}\left(\frac{\partial}{\partial
x^{\nu}}+F_{g\;\alpha\beta,\nu}\frac{\partial}{\partial
g_{\alpha\beta}}+F_{\phi\;,\nu}\frac{\partial}{\partial\phi}+F_{g\;\alpha\beta,\mu,\nu}\frac{\partial}{\partial
g_{\alpha\beta,\mu}}+F_{\phi\;\mu,\nu}\frac{\partial}{\partial\phi_{;\mu}}+G^{\alpha\beta,\mu}_{g\;\nu}\frac{\partial}{\partial
p_{g}^{\alpha\beta,\mu}}+G^{,\mu}_{\phi\;\nu}\frac{\partial}{\partial
p_{\phi}^{,\mu}}\right),$

with $F_{g\;\alpha\beta,\nu}$,
$G^{\alpha\beta,\mu}_{g\;\nu}$,$F_{g\;\alpha\beta,\mu,\nu}$, $F_{\phi\;,\nu}$,
$F_{\phi\;,\mu,\nu}$,$G^{,\mu}_{\phi\;\nu}$ $\in C^{\infty}(\mathcal{P})$.

From (45) we get

$\displaystyle\frac{\partial H_{\mathfrak{V}}}{\partial g_{\alpha\beta}}=$
$\displaystyle-G^{\alpha\beta,\mu}_{g\;\mu}+F_{g\;ab,\mu,\nu}\frac{\partial
L_{g}^{ab,\mu\nu}}{\partial g_{\alpha\beta}}+F_{\phi\;\mu,\nu}\frac{\partial
L_{\phi}^{,\mu\nu}}{\partial g_{\alpha\beta}}$ (59)
$\displaystyle\frac{\partial H_{\mathfrak{V}}}{\partial g_{\alpha\beta,\mu}}=$
$\displaystyle F_{g\;ab,\nu}\frac{\partial
L_{g}^{\alpha\beta,\mu\nu}}{\partial g_{ab}}$ (60)
$\displaystyle\frac{\partial H_{\mathfrak{V}}}{\partial
p_{g}^{\alpha\beta,\mu}}=$ $\displaystyle F_{g\;\alpha\beta,\nu}$ (61)
$\displaystyle\frac{\partial H_{\mathfrak{V}}}{\partial\phi}=$ $\displaystyle-
G_{\phi\;\mu}^{,\mu}+F_{\phi\;\mu\nu}\frac{\partial
L_{\phi}^{,\mu\nu}}{\partial\phi}$ (62) $\displaystyle\frac{\partial
H_{\mathfrak{V}}}{\partial\phi_{;\mu}}=$ $\displaystyle
F_{\phi\;,\nu}\frac{\partial
L_{\phi}^{,\mu\nu}}{\partial\phi}-g^{\rho\sigma}\phi_{;\sigma}F_{\phi\;\rho,\nu}\frac{\partial
L_{\phi}^{,\mu\nu}}{\partial X}+F_{g\;ab,\nu}\frac{\partial
L_{\phi}^{,\mu\nu}}{\partial g_{ab}}$ (63) $\displaystyle\frac{\partial
H_{\mathfrak{V}}}{\partial p_{\phi}^{,\mu}}=$ $\displaystyle F_{\phi\;,\mu}.$
(64)

These are the covariant Hamilton equations for the cubic Horndeski’s theory.
Depending on the values of $G_{2}(\phi,X)$ and $G_{3}(\phi,X)$, these
equations may not be compatible and the constraint algorithm should be
continued. The dynamics of the Hamiltonian theory are determined by the
Hamilton equations 59 to (64) and the tangency condition of the multivector
field to the constraint (5.2).

##  6 Conclusion

In this work, we presented a multisymplectic covariant description of the
cubic Horndeski theory using the unified Lagrangian-Hamiltonian formalism. The
constraint algorithm was employed to determine a submanifold of the higher-
order jet-multimomentum bundle $\mathcal{W}_{r}$ and the corresponding
constraints that provide the main features of the theory.

The constraints (11), (11), (15) and (16) appear as a consequence of this
formalism and define the Legendre map which further allows to pose a covariant
Hamiltonian formulation and the corresponding Hamilton-de Donder-Weyl-like
equations of the theory. Although more constraints appear, they have no
physical relevance and are a mere consequence of the projectability of the
theory, as we would expect a second order Lagrangian to produce fourth-order
equations of motion, but it does produce second-order equations of motion.

We showed that the Poincaré-Cartan form of the theory form does not
necessarily project onto $J^{1}\pi$, unless $\frac{\partial
G_{3}(\phi,X)}{\partial X}=0$. This makes it impossible, in general, to obtain
a covariant Hamiltonian formulation with first order equations of motion.
Moreover, this is a counterexample that proves that the projectability of the
equations does not implies the projectability of the geometric structures.
Hence, the reciprocal of proposition 1 in [42] does not hold.

With extra assumptions on the Lagrangian, we provide the expression of the
velocities in function of the momenta, providing a covariant formulation of
the Hamiltonian formalism which involves only multimomenta. We also present
situation where this is not possible.

For a general cubic Horndeski’s theory, we provide the covariant Hamiltonian
formalism and we present the field equations. In general, they involve
velocities of the metric and the scalar field, as well as the accelerations of
the scalar field.

It is yet to be seen the map between our covariant formulation and the
instantaneous, or ADM-like, Hamiltonian formulation of these theories already
presented in [27]. Recently, a proof of the equivalence of the symplectic
forms derived from the canonical and the covariant phase space formalisms was
presented in [23]. It is yet to be proven if there’s a similar equivalence
between the symplectic forms derived from the instantaneous and the
multisymplectic forms in the multisymplectic formalism. If they are
equivalent, it will be needed to determine how to map all these forms. This
will be explored in a future work.

## Acknowledgments

The authors acknowledge financial support from the Ministerio de Ciencia,
Innovación y Universidades (Spain), projects PGC2018-098265-B-C33 and
D2021-125515NB-21.

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