# Discrete Lagrange problems with constraints valued in a Lie group

P. M. Chacón  IUFFyM and Dpto. Matemáticas, Universidad de Salamanca, Plaza de
la Merced 1-4, 37008 Salamanca, Spain<EMAIL_ADDRESS>,  A. Fernández
IUFFyM and Dpto. Matemática Aplicada, Universidad de Salamanca, Casas del
Parque 2, 37008 Salamanca, Spain<EMAIL_ADDRESS>and  P. L. García  IUFFyM-
USAL and Real Academia de Ciencias, Plaza de la Merced 1-4, 37008 Salamanca,
Spain<EMAIL_ADDRESS>
###### Abstract.

The Lagrange problem is established in the discrete field theory subject to
constraints with values in a Lie group. For the admissible sections that
satisfy a certain regularity condition, we prove that the critical sections of
such problems are the solutions of a canonically unconstrained variational
problem associated with the Lagrange problem (discrete Lagrange multiplier
rule). This variational problem has a discrete Cartan 1-form, from which a
Noether theory of symmetries and a multisymplectic form formula are
established. The whole theory is applied to the Euler-Poincaré reduction in
the discrete field theory, concluding as an illustration with the remarkable
example of the harmonic maps of the discrete plane in the Lie group $SO(n)$.

###### Key words and phrases:

Cellular complexes, Discrete Lagrange problems, Constraints valued in a Lie
group, Euler-Poincaré reduction in discrete field theory.

###### 2020 Mathematics Subject Classification:

Primary 58A20, 53A70; Secondary 57R15, 49N99, 53Z05, 58E30.

Partially supported by Consejería de Educación, Junta de Castilla y León
(Spain), grant SA090G19.

##  1\. Introduction

In [5] the Lagrange problem is posed and solved in the discrete field theory
for constraints valued in a vector space.

More specifically, with the notations and concepts of [5] (see sections 2 and
3), the starting point of the doctrine is a bundle $\pi:Y\to V_{0}$ over the
set $V_{0}$ of the vertices of an arbitrary $n$-dimensional cellular complex
$V$ with fibers $Y_{v}$, $v\in V_{0}$, differentiable manifolds of the same
dimension $m$, a Lagrangian density $\mathcal{L}:J^{1}Y\to\mathbb{R}$ on the
bundle $j^{1}\pi:J^{1}Y\to V_{n}$ of the 1-jets of $\pi:Y\to V_{0}$ over the
set $V_{n}$ of the $n$-dimensional cells (faces) of the complex $V$, and a
constraint submanifold $S=\Phi^{-1}(0)\subset J^{1}Y$ where $\Phi:J^{1}Y\to E$
is a differentiable mapping from $J^{1}Y$ to a real vector space $E$ of
dimension $l\leq m$. If $\mathcal{V}$ is a finite set of faces of $V$ and
$\mathcal{V}_{0}=\\{v\in V_{0}\,|\,v\prec\alpha\in\mathcal{V}\\}$ is the
(finite) set of the adherent vertices to the elements of $\mathcal{V}$, a
section $y\in\Gamma(\mathcal{V}_{0},Y)$ is said to be admissible when
$\textrm{img}(j^{1}y)\subset S$ ($j^{1}y$ the 1-jet extension of the section
$y$), and an infinitesimal variation $\delta y\in
T_{y}\big{(}\Gamma(\mathcal{V}_{0},Y)\big{)}$ is said to be admissible when
$j^{1}(\delta y)$ is tangent to $S$ along $\textrm{img}(j^{1}y)\subset S$
($j^{1}(\delta y)$ the 1-jet extension of the infinitesimal variation $\delta
y$). Under these conditions, the aim of the Lagrange problem is to find the
admissible sections that are critical sections of the action functional:

$\mathcal{A}_{\mathcal{V}}(\mathcal{L}):y\in\Gamma\big{(}\mathcal{V}_{0},Y)\mapsto\sum_{\alpha\in\mathcal{V}}\mathcal{L}\big{(}(j^{1}y)(\alpha)\big{)}$

respect to the admissible infinitesimal variations that vanish at the frontier
of $\mathcal{V}$.

Under a certain condition of regularity, it can be proved that these critical
sections are the solutions of the unconstrained variational problem with
Lagrangian density $\widehat{\mathcal{L}}=\mathcal{L}+\lambda\circ\Phi$ on the
fibered product $J^{1}Y\times_{V_{n}}(E^{*}\times V_{n})$, ($E^{*}$ dual of
$E$), where $\lambda$ is the tautological function $\lambda(\omega,\alpha)\in
E^{*}\times V_{n}\mapsto\omega\in E^{*}$ and $\circ$ is the bilinear product
given by the duality pairing. This problem has a discrete Cartan 1-form from
which a Noether theory of symmetries and a multisymplectic form formula can be
established.

Having established this multisymplectic formulation of discrete Lagrangian
problems with constraints valued in a vector space, it will be desirable to
generalize it to other more complex situations of interest. This is the aim of
the present paper, in which we generalize the theory of discrete Lagrangian
problems with constraint submanifold $S=\Phi^{-1}(e)\subset J^{1}Y$, where
$\Phi$ is a mapping from $J^{1}Y$ to a Lie group $G$ ($e\in G$ the identity
element of the group). We apply the doctrine thus developed to the important
case of the Euler-Poincaré reduction in principal bundles by the action of its
structural group. See [1, 2, 3, 4, 7] and references cited there. The interest
of this situation continues in a recent pre-print [6] where a discretization
of the Navier-Stokes-Fourier system is developed.

This paper is structured as follows. In Section 2 we set out the framework of
the theory following the guidelines used in [5]. The main result of this
section is Theorem 2.6 which characterizes critical admissible sections that
satisfy an adequate condition of regularity. A key concept of this formulation
is the version we give of the Cartan 1-form $\Theta(\Phi)$ associated with the
constraint map $\Phi:J^{1}Y\to G$ via the 1-form over $J^{1}Y$ with values in
the Lie algebra $\mathfrak{g}$ of $G$ resulting from the composition
$\theta\circ d\Phi$ of the tangent map $d\Phi$ with the Maurer-Cartan 1-form
of the group.

In Section 3, the above result allows us identify the critical sections of
these problems with the solutions of the Euler-Lagrange equations of an
extended unconstrained discrete variational problem adapted to the new class
of constraints. This variational problem has a discrete Cartan 1-form, from
which a Noether theory of symmetries and a multisymplectic form formula are
established.

In Section 4 the formalism thus developed is applied to the Euler-Poincaré
reduction in the discrete field theory, concluding by way of illustration with
the example of the harmonic maps of the discrete plane in the Lie group
$SO(n)$, which is discussed in Section 5.

##  2\. Discrete Lagrange problems with constraints valued in a Lie group

The data that define this type of problems are: a cellular complex, a fiber
bundle over the set of vertices, and the jet extensions, a Lagrangian density,
and a constraint map. Following the notation of [5] (sections 2 and 3), we
describe briefly these objects next.

  * •

For an $n$-dimensional abstract cellular complex $V$, $n\in\mathbb{N}$, we
denote by $V_{k}=\\{\alpha\in V|\dim\alpha=k\\}$ the set of the
$k$-dimensional cells, $k\in\mathbb{N}$. In particular, the elements of
$V_{0}$ are called vertices and the elements of $V_{n}$ are called faces.

We said that a cell $\alpha\in V_{k}$, $k\in\mathbb{N}$, is adherent to
another cell $\beta\in V_{k+l}$, $l\in\mathbb{N}$, denoted by
$\alpha\prec\beta$ , if $\alpha=\beta$ or there exists a sequence of cells
$\alpha=\gamma_{k},\gamma_{k+1},\dots,\gamma_{k+l}=\beta$, where
$\gamma_{k+i}\in V_{k+i},i=0,\dots,l$, such that each one is incident to the
following.

The notion of proximity is given by the spherical neighborhood of a vertex
$v\in V_{0}$, $S_{v}=\\{\alpha\in V_{n}|v\prec\alpha\\}$, that is, the set of
the different faces that have $v$ as adherent vertex.

Given a vertex $v\in V_{0}$ and a subset of faces $\mathcal{V}\subset V_{n}$,
we say that $v$ is interior to $\mathcal{V}$ if $S_{v}\subset\mathcal{V}$, is
exterior to $\mathcal{V}$ if $S_{v}\subset V_{n}\setminus\mathcal{V}$ or is
frontier of $\mathcal{V}$ otherwise. We denote by
$\operatorname{int}\mathcal{V}$ and $\operatorname{fr}\mathcal{V}$ the set of
interior and frontier vertices of $\mathcal{V}$, respectively.

  * •

Consider a fiber bundle $\pi:Y\to V_{0}$ over the set of vertices of an
$n$-dimensional cellular complex $V$. That is, $Y$ is a differential manifold,
and for any $v\in V_{0}$ the fiber $Y_{v}=\pi^{-1}(v)$ is a differentiable
manifold of dimension $m\in\mathbb{N}$.

  * •

The role of the bundle of 1-jets in the continuous case is given in the
discrete theory of fields by the bundle $j^{1}\pi:J^{1}Y\to V_{n}$ over the
set of faces of the cellular complex $V$. For that, given $\alpha\in V_{n}$
consider

$(J^{1}Y)_{\alpha}=(j^{1}\pi)^{-1}(\alpha)=\prod_{v\prec\alpha}Y_{v}\quad\textrm{
and }\quad J^{1}Y=\bigsqcup_{\alpha\in V_{n}}(J^{1}Y)_{\alpha}.$

For a section $y\in\Gamma(V_{0},Y)$, the 1-jet extension
$j^{1}y\in\Gamma(V_{n},J^{1}Y)$ is defined at $\alpha\in V_{n}$ by
$(j^{1}y)(\alpha)=(y(v))$, where $v\prec\alpha$ .

In this context, the 2-jet bundle extension of $\pi:Y\to V_{0}$ is the bundle
$j^{2}\pi:J^{2}Y\to V_{0}$ over the vertices of $V$ such that the fiber over a
$v\in V_{0}$ is:

$(J^{2}Y)_{v}=(j^{2}\pi)^{-1}(v)=\prod_{v^{\prime}\prec\alpha\in
S_{v}}Y_{v^{\prime}},$

where $S_{v}\subset V_{n}$ is the spherical neighborhood of $v$. That is, the
fiber over the vertex $v$ is the product of the fibers of $\pi:Y\to V_{0}$
over each vertex of each face that has $v$ as adherent vertex. If
$y\in\Gamma(V_{0},Y)$ is a section, the 2-jet extension of $y$ is
$j^{2}y\in\Gamma(V_{0},J^{2}Y)$ defined by $(j^{2}y)(v)=(y(v^{\prime}))$,
$v^{\prime}\prec\alpha\in S_{v}$ for each vertex $v\in V_{0}$.

Note that there exists a canonical projection between these jet extensions.
More precisely, for any vertex $v\in V_{0}$ and any face $\alpha\in S_{v}$,
given $(y_{v^{\prime}})\in(J^{2}Y)_{v}$, where $v^{\prime}\prec\beta\in
S_{v}$, we can collect the values over the vertices that are adherent
precisely to the face $\alpha$ (and ignore the others). So we can define
$\pi_{v\alpha}:(J^{2}Y)_{v}\to(J^{1}Y)_{\alpha}$ given by

$\pi_{v\alpha}\big{(}\underbrace{(y_{v^{\prime}})}_{v^{\prime}\prec\beta\in
S_{v}}\big{)}=\big{(}\underbrace{(y_{v^{\prime}})}_{v^{\prime}\prec\alpha}\big{)}.$

  * •

A discrete Lagrangian density $\mathcal{L}:J^{1}Y\to\mathbb{R}$ over the
bundle $j^{1}\pi:J^{1}Y\to V_{n}$ is a family of smooth functions
$\mathcal{L}_{\alpha}:(J^{1}Y)_{\alpha}\to\mathbb{R}$ where $\alpha\in V_{n}$.

  * •

Finally, a constraint with values in a Lie group $G$ is a map $\Phi:J^{1}Y\to
G$ such that, for all $\alpha\in V_{n}$,
$\Phi_{\alpha}=\Phi|_{(J^{1}Y)_{\alpha}}$ is a differential map being $e\in G$
(the identity element of the group) a regular value. Thus,
$S_{\alpha}=\Phi_{\alpha}^{-1}(e)$ is a submanifold of $(J^{1}Y)_{\alpha}$,
and we can express $S=\Phi^{-1}(e)$ as the disjoint union $S=\sqcup_{\alpha\in
V_{n}}S_{\alpha}$. We denote the tangent map of $\Phi_{\alpha}$, $\alpha\in
V_{n}$, at $j^{1}_{\alpha}y\in(J^{1}Y)_{\alpha}$ by
$(d\Phi_{\alpha})_{j^{1}_{\alpha}y}$.

Now, let $\mathcal{V}$ be a finite set of faces and $\mathcal{V}_{0}=\\{v\in
V_{0}\,|\,v\prec\alpha\in\mathcal{V}\\}$ the (finite) set of its adherent
vertices. Then, similar to those set out in [5], we may give the following
definitions.

######  Definition 2.1.

A section $y\in\Gamma(\mathcal{V}_{0},Y)$ is said to be admissible if
$\textrm{img}(j^{1}y)\subset S$, that is, $(j^{1}y)(\alpha)\in
S_{\alpha}\subset(J^{1}Y)_{\alpha}$ for all $\alpha\in\mathcal{V}$.

We shall denote this subset of section by $\Gamma_{S}(\mathcal{V}_{0},Y)$.

######  Definition 2.2.

Given an admissible section $y\in\Gamma_{S}(\mathcal{V}_{0},Y)$, an admissible
infinitesimal variation of $y$ is a tangent vector $\delta y\in
T_{y}\big{(}\Gamma(\mathcal{V}_{0},Y)\big{)}$ whose 1-jet extension
$j^{1}\delta y$ is tangent to $S\subset J^{1}Y$ along
$\textrm{img}(j^{1}y)\subset S$, that is,
$(d\Phi_{\alpha})_{(j^{1}y)(\alpha)}(j^{1}\delta y)(\alpha)=0$ for all
$\alpha\in\mathcal{V}$.

We shall denote by $T_{y}\big{(}\Gamma_{S}(\mathcal{V}_{0},Y)\big{)}$ this
subspace of infinitesimal variations, and by
$T_{y}^{c}\big{(}\Gamma_{S}(\mathcal{V}_{0},Y)\big{)}$ the subspace of those
that vanish at the frontier of $\mathcal{V}$.

######  Definition 2.3.

An admissible section $y\in\Gamma_{S}(\mathcal{V}_{0},Y)$ is critical with
fixed boundary for the discrete Lagrange problem with Lagrangian density
$\mathcal{L}:J^{1}Y\to\mathbb{R}$ and constraint submanifold
$S=\Phi^{-1}(e)\subset J^{1}Y$ if
$\big{(}d\mathcal{A}_{\mathcal{V}}(\mathcal{L})\big{)}_{y}=0$ on the subspace
$T_{y}^{c}\big{(}\Gamma_{S}(\mathcal{V}_{0},Y)\big{)}$, where
$\mathcal{A}_{\mathcal{V}}(\mathcal{L})$ is the action functional

$\mathcal{A}_{\mathcal{V}}(\mathcal{L}):y\in\Gamma(\mathcal{V}_{0},Y)\mapsto\sum_{\alpha\in\mathcal{V}}\mathcal{L}\big{(}(j^{1}y)(\alpha)\big{)}.$

As in [5], under a certain condition of regularity such critical sections can
be characterized as the solutions of an Euler-Lagrange operator by proceeding
as follows.

From an initial section $y_{0}\in\Gamma(\mathcal{V}_{0},Y)$, let
$\Gamma(\mathcal{V}_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}$
be the submanifold of $\Gamma(\mathcal{V}_{0},Y)$ defined by the sections with
fixed boundary $y_{0}(\operatorname{fr}\mathcal{V})\subset Y$ and
$\Gamma_{S}(\mathcal{V}_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}$
the subset of the admissible ones.

We define the mapping
$\Psi:\Gamma(\mathcal{V}_{0},Y)\to\textrm{Map}(\mathcal{V},G)$ by the rule:

(1)
$\Psi(y):\alpha\in\mathcal{V}\mapsto\Phi_{\alpha}\big{(}(j^{1}y)(\alpha)\big{)},\quad
y\in\Gamma(\mathcal{V}_{0},Y).$

Since $\Phi_{\alpha}:(J^{1}Y)_{\alpha}\to G$ is differentiable so is $\Psi$,
expressing its differential
$(d\Psi)_{y}:T_{y}\big{(}\Gamma(\mathcal{V}_{0},Y)\big{)}\to
T_{\Psi(y)}\textrm{Map}(\mathcal{V},G)$ at $y\in\Gamma(\mathcal{V}_{0},Y)$ as:

(2) $(d\Psi)_{y}(\delta
y):\alpha\in\mathcal{V}\mapsto(d\Phi_{\alpha})_{(j^{1}y)(\alpha)}\big{(}(j^{1}\delta
y)(\alpha)\big{)},\quad\delta y\in
T_{y}\big{(}\Gamma(\mathcal{V}_{0},Y)\big{)}.$

Otherwise, taking into account the isomorphism
$T_{\Psi(y)}\textrm{Map}(\mathcal{V},G)\approx\textrm{Map}(\mathcal{V},\mathfrak{g})$
induced by the left translations of the elements $g\in G$, $D_{g}\in
T_{g}G\mapsto L_{g^{-1}}D_{g}\in T_{e}G=\mathfrak{g}$, the map $(d\Psi)_{y}$
can also be expressed as:

(3) $(d\Psi)_{y}(\delta
y):\alpha\in\mathcal{V}\mapsto\big{(}\theta\circ(d\Phi_{\alpha})_{(j^{1}y)(\alpha)}\big{)}\big{(}(j^{1}\delta
y)(\alpha)\big{)},\quad\delta y\in
T_{y}\big{(}\Gamma(\mathcal{V}_{0},Y)\big{)},$

where $\theta$ is the Maurer-Cartan 1-form of the group $\theta:g\in
G\mapsto\theta_{g}$ ($\theta_{g}:D_{g}\in T_{g}G\mapsto L_{g^{-1}}D_{g}\in
T_{e}G=\mathfrak{g}$) and where $\circ$ is the composition of maps.

In particular, for the admissible sections $y\in\Gamma_{S}(\mathcal{V}_{0},Y)$
both expressions (2) and (3) of $(d\Psi)_{y}$ coincide because in this case
$\Phi_{\alpha}(j^{1}y)=e$, $\alpha\in\mathcal{V}$.

By restricting $\Psi$ to the submanifold
$\Gamma(V_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}\subset\Gamma(\mathcal{V}_{0},Y)$
and $(d\Psi)_{y}$ to the subspace
$T_{y}\big{(}\Gamma(\mathcal{V}_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}\big{)}\subset
T_{y}\big{(}\Gamma(\mathcal{V}_{0},Y)\big{)}$ we get:

######  Proposition 2.4.

$\Gamma_{S}(\mathcal{V}_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}=\Psi^{-1}(e),\quad
T_{y}^{c}\big{(}\Gamma_{S}(\mathcal{V}_{0},Y)\big{)}=\ker(d\Psi)_{y}.$

###### Proof.

It is enough to apply definitions 2.1 and 2.2, and formulas (1) and (2)
$\equiv$ (3) because $y$ is admissible. ∎

Now, a key point of our approach is the following condition of regularity for
admissible sections:

######  Definition 2.5.

An admissible section
$y\in\Gamma_{S}(\mathcal{V}_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}$
is said to be regular if
$(d\Psi)_{y}:T_{y}\big{(}\Gamma(\mathcal{V}_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}\big{)}\to\textrm{Map}(\mathcal{V},\mathfrak{g})$
is onto.

Under this hypothesis, the Inverse Function Theorem proves the existence in
the manifold
$\Gamma(\mathcal{V}_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}$
of an open neighborhood $\mathcal{U}(y)$ of $y$ such that
$\Gamma_{S}(\mathcal{V}_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}\cap\mathcal{U}(y)$
is a submanifold of $\mathcal{U}(y)$ whose tangent space at $y$ is
$\ker(d\Psi)_{y}=T_{y}^{c}\big{(}\Gamma_{S}(\mathcal{V}_{0},Y)\big{)}$.
Therefore, Definition 2.3 of stationarity is equivalent to stating that $y$ is
a critical point of the restriction of the action functional
$\mathcal{A}_{\mathcal{V}}(\mathcal{L})$ to the submanifold
$\Gamma_{S}(\mathcal{V}_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}\cap\mathcal{U}(y)$.
On the other hand, we have the exact sequence:

(4) $0\to T_{y}^{c}\big{(}\Gamma_{S}(\mathcal{V}_{0},Y)\big{)}\to
T_{y}\Big{(}\Gamma(\mathcal{V}_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}\Big{)}\xrightarrow{(d\Psi)_{y}}\textrm{Map}(\mathcal{V},\mathfrak{g})\to
0,$

which, finally, will allow us to characterize critical regular sections as
solutions of an Euler-Poincaré operator.

Indeed, defining the notion of Cartan 1-form associated to the constraint
morphism $\Phi:J^{1}Y\to G$ as the family of 1-forms
$\Theta_{\alpha}^{v}(\Phi)$ on $J^{1}Y$ with values in Lie algebra
$\mathfrak{g}$, $\alpha\in\mathcal{V}$, $v\in\mathcal{V}_{0}$, $v\prec\alpha$,
such that

(5)
$\theta\circ(d\Phi_{\alpha})_{j^{1}_{\alpha}y}=\sum_{v\prec\alpha}\big{(}\Theta^{v}_{\alpha}(\Phi)\big{)}_{j^{1}_{\alpha}y},\quad
j^{1}_{\alpha}y\in(J^{1}Y)_{\alpha},$

the mapping $(d\Psi)_{y}$ of (4) can be expressed as:

(6) $(d\Psi)_{y}(\delta
y):\alpha\in\mathcal{V}\mapsto\sum_{v\prec\alpha}\big{(}\Theta_{\alpha}^{v}(\Phi)\big{)}_{(j^{1}y)(\alpha)}(j^{1}\delta
y)(\alpha),\quad\delta y\in
T_{y}\Big{(}\Gamma(\mathcal{V}_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}\Big{)},$

from where we can obtain the following characterization of the critical
sections.

######  Theorem 2.6.

A regular admissible section
$y\in\Gamma_{S}(\mathcal{V}_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}$
is critical with fixed boundary for the Lagrange problem with Lagrangian
density $\mathcal{L}:J^{1}Y\to\mathbb{R}$ and constraint submanifold
$S=\Phi^{-1}(e)\subset J^{1}Y$ if and only if there exists a mapping
$\lambda\in\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$ such that for any
$v\in\operatorname{int}\mathcal{V}$:

(7) $\Big{(}\mathcal{E}_{v}(\mathcal{L})+\sum_{\alpha\in
S_{v}}\pi^{*}_{v\alpha}\big{(}\lambda(\alpha)\circ\Theta_{\alpha}^{v}(\Phi)\big{)}\Big{)}_{(j^{2}y)(v)}=0,$

where $\mathcal{E}_{v}(\mathcal{L})$ is the Euler-Lagrange 1-form of
$\mathcal{L}$ at $v$ as unconstrained variational problem, $S_{v}$ is the
spherical neighborhood of $v$, $\circ$ is the bilinear product of the duality
between $\mathfrak{g}$ and $\mathfrak{g}^{*}$, and $\pi_{v\alpha}$ is the
canonical projection from $(J^{2}Y)_{v}$ to $(J^{1}Y)_{\alpha}$. This mapping
$\lambda$ is unique and we shall call the multiplier associated to the
critical regular section $y$.

###### Proof.

It is enough to repeat mutatis mutandis the proof of the Theorem 4.6 of
Section 4 in [5]. Indeed, from the exact dual sequence of (4):

$0\to\textrm{Map}(\mathcal{V},\mathfrak{g})^{*}\xrightarrow{(d\Psi)_{y}^{*}}T_{y}\Big{(}\Gamma(\mathcal{V}_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}\Big{)}^{*}\to
T_{y}^{c}\big{(}\Gamma_{S}(\mathcal{V}_{0},Y)\big{)}^{*}\to 0,$

follows the identification of $\textrm{img}(d\Psi)_{y}^{*}$ with the elements
of
$T_{y}\Big{(}\Gamma(\mathcal{V}_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}\Big{)}^{*}$
incidents with $T_{y}^{c}\big{(}\Gamma_{S}(\mathcal{V}_{0},Y)\big{)}$.

So, if the section $y$ is critical (Definition 2.3), then there exists a map
$\lambda\in\textrm{Map}(\mathcal{V},\mathfrak{g})^{*}$
$=\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$, that is unique by the
injectivity of $(d\Psi)_{y}^{*}$, such that:

(8) $-\big{(}d\mathcal{A}_{\mathcal{V}}(\mathcal{L})\big{)}_{y}(\delta
y)=\big{(}(d\Psi)_{y}^{*}\lambda\big{)}(\delta y),\quad\delta y\in
T_{y}\Big{(}\Gamma(\mathcal{V}_{0},Y)\big{(}y_{0}(\operatorname{fr}\mathcal{V})\big{)}\Big{)}.$

By the variation formula (10) of [5], Section 3:

$\big{(}d\mathcal{A}_{\mathcal{V}}(\mathcal{L})\big{)}_{y}(\delta
y)=\sum_{v\in\operatorname{int}\mathcal{V}}\big{(}\mathcal{E}_{v}(\mathcal{L})\big{)}_{(j^{2}y)(v)}(j^{2}\delta
y)(v),$

since $(\delta y)_{v}=0$ at the vertices $v\in\operatorname{fr}\mathcal{V}$.

On the other hand, by (6) we have:

$\displaystyle\big{(}(d\Psi)_{y}^{*}\lambda\big{)}(\delta y)=$
$\displaystyle\lambda\circ(d\Psi)_{y}(\delta
y)=\sum_{\alpha\in\mathcal{V}}\lambda(\alpha)\circ\bigg{(}\sum_{v\prec\alpha}\big{(}\Theta_{\alpha}^{v}(\Phi)\big{)}_{(j^{1}y)(\alpha)}(j^{1}\delta
y)(\alpha)\bigg{)}$ $\displaystyle=$
$\displaystyle\sum_{v\in\operatorname{int}\mathcal{V}}\sum_{\alpha\in
S_{v}}\lambda(\alpha)\circ\big{(}\Theta_{\alpha}^{v}(\Phi)\big{)}_{(j^{1}y)(\alpha)}(j^{1}\delta
y)(\alpha),$

since by swapping the sums, as $(\delta y)_{v}=0$ at the vertices
$v\in\operatorname{fr}\mathcal{V}$, we are left with only the interior
vertices and the faces of the corresponding spherical neighborhood. By
replacing it now in equality (8), and using the canonical projection
$\pi_{v\alpha}:(J^{2}Y)_{v}\to(J^{1}Y)_{\alpha}$, we finally obtain:

$\sum_{v\in\operatorname{int}\mathcal{V}}\Big{(}\mathcal{E}_{v}(\mathcal{L})+\sum_{\alpha\in
S_{v}}\pi^{*}_{v\alpha}\big{(}\lambda(\alpha)\circ\Theta_{\alpha}^{v}(\Phi)\big{)}\Big{)}_{(j^{2}y)(v)}(j^{2}\delta
y)(v)=0.$

From here the result follows because $(\delta y)_{v}$ is an arbitrary
variation at the vertices $v\in\operatorname{int}\mathcal{V}$.

∎

##  3\. Lagrange multiplier rule. Noether’s theorem. Multisymplectic form
formula

The aim of section is to formulate the Lagrange problems that we have seen in
the previous section as free variational problems via a Lagrange multipliers
rule. In this sense the equations (7) that characterize the critical sections
of the Lagrange problem will be interpreted as the Euler-Lagrange equations of
an unconstrained variational problem extended to the multipliers
$\lambda\in\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$ as follows.

Let $\alpha\in V_{n}$ and let us denote by $\mathcal{L}_{\alpha}$ and
$\Phi_{\alpha}$ the restriction of $\mathcal{L}$ and $\Phi$ to the fibers
$(J^{1}Y)_{\alpha}$ as before. Then, for every
$\lambda_{\alpha}\in\mathfrak{g}^{*}$ we can define the 1-form

$d\mathcal{L}_{\alpha}+\lambda_{\alpha}\circ(\theta\circ d\Phi_{\alpha})$

on $(J^{1}Y)_{\alpha}$; where the first $\circ$ from the left denotes the
duality pairing between $\mathfrak{g}$ and $\mathfrak{g}^{*}$. That is, for
every $\lambda\in\textrm{Map}(V_{n},\mathfrak{g}^{*})$ we have the 1-form

(9) $d\mathcal{L}+\lambda\circ(\theta\circ d\Phi),$

on $J^{1}Y\times\mathfrak{g}^{*}\equiv J^{1}Y\times_{V_{n}}\mathfrak{G}^{*}$,
where $\mathfrak{G}^{*}=\mathfrak{g}^{*}\times V_{n}\to V_{n}$ is the trivial
bundle on $V_{n}$ with fiber $\mathfrak{g}^{*}$. We will use
$\lambda_{\alpha}$ or $\lambda(\alpha)$ to denote the same element of
$\mathfrak{g}^{*}$.

This 1-form defines an unconstrained variational problem on the manifold
$\Gamma(\mathcal{V}_{0}\times
Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$ with action 1-form given
by:

(10)
$\big{(}\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)\big{)}_{(y,\lambda)}(\delta
y,\delta\lambda)=\sum_{\alpha\in\mathcal{V}}(d\mathcal{L}_{\alpha})_{(j^{1}y)(\alpha)}(j^{1}\delta
y)(\alpha)+\lambda_{\alpha}\circ(\theta\circ
d\Phi_{\alpha})_{(j^{1}y)(\alpha)}(j^{1}\delta y)(\alpha),$

$(y,\lambda)\in\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$,
$(\delta y,\delta\lambda)\in
T_{y}\big{(}\Gamma(\mathcal{V}_{0},Y)\big{)}\oplus
T_{\lambda}\big{(}\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})\big{)}$, from
where the following notion of stationarity is given.

######  Definition 3.1.

We say that a section-mapping
$(y,\lambda)\in\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$
is critical for the action 1-form $\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)$
if the section $y\in\Gamma(\mathcal{V}_{0},Y)$ is admissible and
$\big{(}\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)\big{)}_{(y,\lambda)}$
vanishes on the subspace
$T_{y}^{c}\big{(}\Gamma(\mathcal{V}_{0},Y)\big{)}\oplus
T_{\lambda}\big{(}\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})\big{)}$.

From (10) we get that the subspace
$T_{\lambda}\big{(}\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})\big{)}$ is
incident with the 1-form $\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)$, i.e.:

(11)
$\big{(}\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)\big{)}_{(y,\lambda)}(0,\delta\lambda)=0\quad\textrm{
for all }\quad\delta\lambda\in
T_{\lambda}\big{(}\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})\big{)}.$

######  Remark 3.2.

The unconstrained variational problem defined by the 1-form (9) differs
substantially from that introduced in Section 5 of [5] for Lagrange problems
with constraints $\Phi:J^{1}Y\to E$ valued in vector spaces. As noted in the
Introduction, these problems are defined by an unconstrained Lagrangian
density $\widehat{\mathcal{L}}=\mathcal{L}+\lambda\circ\Phi$ ($\circ$ the
bilinear product of the duality between $E$ and $E^{*}$), which is not
generalizable to constraints $\Phi:J^{1}Y\to G$ valued in Lie groups. However,
differentiating $\widehat{\mathcal{L}}$ we obtain:

$d\widehat{\mathcal{L}}=d\mathcal{L}+\lambda\circ
d\Phi+d\lambda\barwedge\Phi,$

where $\barwedge$ denotes the wedge product of valued forms with respect to
the bilinear form $\circ$ induced by the duality pairing. The first two terms
of this formula are generalizable via the expression (9) in which the Maurer-
Cartan 1-form of $G$ is inserted. In this sense, it is important to observe
that by dispensing with the term $d\lambda\barwedge\Phi$ in this formulation
(which is also not generalizable), condition (11) no longer allows to obtain
the admissibility of the critical section-mappings as part of the Euler-
Lagrange equations of the extended problem. This condition must be imposed
separately, as contemplated in Definition 3.1 that we have given of
stationarity.

Following the route of Section 5 of [5] with the obvious changes we will have:

######  Proposition 3.3 (Variational formula).

For any section-mapping $(y,\lambda)\in\Gamma(\mathcal{V}_{0},Y)\times$
$\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$ and for any tangent vector
$(\delta y,\delta\lambda)\in
T_{(y,\lambda)}\big{(}\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})\big{)}$,
we have now:

(12)
$\displaystyle\big{(}\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)\big{)}_{(y,\lambda)}(\delta
y,\delta\lambda)=$
$\displaystyle\sum_{v\in\operatorname{int}\mathcal{V}}\big{(}\mathcal{E}_{v}(\mathcal{L})+\sum_{\alpha\in
S_{v}}\lambda(\alpha)\circ\pi^{*}_{v\alpha}\Theta_{\alpha}^{v}(\Phi)\big{)}_{(j^{2}y)(v)}(j^{2}\delta
y)(v)$
$\displaystyle+\sum_{(v\in\operatorname{fr}\mathcal{V})\prec\alpha\in\mathcal{V}}\big{(}\Theta_{\alpha}^{v}(\mathcal{L})+\lambda(\alpha)\circ\Theta_{\alpha}^{v}(\Phi)\big{)}_{(j^{1}y)(\alpha)}(j^{1}\delta
y)(\alpha).$

######  Corollary 3.4 (Euler-Lagrange equations).

A section-mapping $(y,\lambda)\in\Gamma(\mathcal{V}_{0},Y)\times$
$\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$ is critical for the action 1-form
$\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)$ if and only if the section
$y\in\Gamma(\mathcal{V}_{0},Y)$ is admissible and for any
$v\in\operatorname{int}\mathcal{V}$:

(13) $\big{(}\mathcal{E}_{v}(\mathcal{L})+\sum_{\alpha\in
S_{v}}\lambda(\alpha)\circ\pi^{*}_{v\alpha}\Theta_{\alpha}^{v}(\Phi)\big{)}_{(j^{2}y)(v)}=0.$

######  Theorem 3.5 (Lagrange Multiplier Rule).

A section-mapping $(y,\lambda)\in\Gamma(\mathcal{V}_{0},Y)\times$
$\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$, being $y$ an admissible regular
section, is critical for the action 1-form
$\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)$ if and only if its component
$y\in\Gamma(\mathcal{V}_{0},Y)$ is an admissible regular critical section for
the Lagrange problem with Lagrangian $\mathcal{L}:J^{1}Y\to\mathbb{R}$ and
constraint submanifold $S=\Phi^{-1}(e)\subset J^{1}Y$, being its component
$\lambda$ the multiplier associated to $y$.

On the other hand, the expression of the boundary term of the variation
formula (12) suggests to take as Cartan 1-form associated to the action 1-form
$\mathbb{A}_{\mathcal{V}}(\mathcal{L},\phi)$ the family of 1-forms:

$\Theta_{\alpha}^{v}(\mathcal{L})+\lambda_{\alpha}\circ\Theta_{\alpha}^{v}(\Phi),\quad
v\in\mathcal{V}_{0},\quad\alpha\in\mathcal{V}_{n},\quad v\prec\alpha.$

This concept will allow us to formulate a Noether’s theory of symmetries for
the Lagrange problems with constraints valued in Lie groups, as well as to
establish a corresponding formula of the multisymplectic form.

######  Definition 3.6.

An infinitesimal symmetry of a Lagrange problem with Lagrangian density
$\mathcal{L}:J^{1}Y\to\mathbb{R}$ and constraint submanifold $S=\Phi^{-1}(e)$,
with $\Phi:J^{1}Y\to G$, is a vector field $D\in\mathfrak{X}(Y)$ such that
$(j^{1}D)\mathcal{L}=0$ and $(j^{1}D)(\Phi)=(d\Phi)(j^{1}D)=0$ where:

$(d\Phi)(j^{1}D):j^{1}_{\alpha}y\in(J^{1}Y)_{\alpha}\mapsto(d\Phi)_{j^{1}_{\alpha}y}(j^{1}D)_{j^{1}_{\alpha}y}\in
T_{\Phi(j^{1}_{\alpha}y)}G.$

######  Theorem 3.7 (Noether).

If $y\in\Gamma_{S}(\mathcal{V}_{0},Y)$ is a regular critical section of the
Lagrange problem given by $(\mathcal{L},\Phi)$ and $D\in\mathfrak{X}(Y)$ is an
infinitesimal symmetry, then:

$\sum_{(v\prec\operatorname{fr}\mathcal{V})\prec\alpha}\left[\big{(}\Theta^{v}_{\alpha}(\mathcal{L})+\lambda(\alpha)\circ\Theta^{v}_{\alpha}(\Phi)\big{)}(j^{1}D)_{\alpha}\right](j^{1}y)(\alpha)=0,$

where $\lambda\in\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$ is the multiplier
associated to the regular critical section.

###### Proof.

If we call $(\delta y)(v)=(D_{v})_{y(v)}$, $v\in\mathcal{V}_{0}$, then
$(j^{1}\delta y)(\alpha)=\big{(}(j^{1}D)_{\alpha}\big{)}_{(j^{1}y)(\alpha)}$,
$\alpha\in\mathcal{V}$, and we obtain
$\big{(}\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)\big{)}_{(y,\lambda)}(\delta
y,0)=0$ by (10) and for being $D$ an infinitesimal symmetry of the Lagrange
problem. On the other hand, if $y\in\Gamma_{S}(\mathcal{V}_{0},Y)$ is a
regular critical section of the Lagrange problem with associated multiplier
$\lambda\in\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$, then for any
$v\in\operatorname{int}\mathcal{V}$ Theorem 2.6 provides:

$\left(\mathcal{E}_{v}(\mathcal{L})+\sum_{\alpha\in
S_{v}}\lambda(\alpha)\circ\pi^{*}_{v\alpha}\Theta^{v}_{\alpha}(\Phi)\right)_{(j^{2}y)(v)}=0.$

Now the result follows if we substitute these both zero terms in the variation
formula (12). ∎

As for the establishment of a multisymplectic form formula for this kind of
Lagrange problems we will proceed as follows.

The variation formula (12) can be expressed on the manifold
$\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$
as:

(14)
$\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)=\sum_{v\in\operatorname{int}\mathcal{V}}\big{(}\mathcal{E}_{v}(\mathcal{L})+\sum_{\alpha\in
S_{v}}\lambda_{\alpha}\circ\Theta_{\alpha}^{v}(\Phi)\big{)}+\sum_{(v\in\operatorname{fr}\mathcal{V})\prec\alpha\in\mathcal{V}}\big{(}\Theta_{\alpha}^{v}(\mathcal{L})+\lambda_{\alpha}\circ\Theta_{\alpha}^{v}(\Phi)\big{)},$

where the terms of this sum are considered 1-forms on
$\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$
through the canonical projections from
$\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$ to
$(J^{1}Y\times_{\mathcal{V}}\mathfrak{G}^{*})_{\alpha}$ and
$(J^{2}Y\times_{\mathcal{V}_{0}}j^{1}\mathfrak{G}^{*})_{v}$ respectively,
$(y,\lambda)\mapsto\big{(}(j^{1}y)(\alpha),e^{*}(\alpha)=(\alpha,\lambda(\alpha))\big{)}$
and $(y,\lambda)\mapsto\big{(}(j^{2}y)(v),j^{1}e^{*}(v)\big{)}$, being
$j^{1}\mathfrak{G}^{*}\to\mathcal{V}_{0}$ the 1-jet bundle of
$\mathfrak{G}^{*}\to\mathcal{V}$ defined by
$(j^{1}\mathfrak{G}^{*})_{v}=\prod_{\alpha\in
S_{v}}\mathfrak{G}^{*}_{\alpha}$, $v\in\mathcal{V}_{0}$, and where
$j^{1}e^{*}:v\in\mathcal{V}_{0}\mapsto\big{(}e^{*}(\alpha)\big{)}$, $\alpha\in
S_{v}$, denotes the 1-jet extension of
$e^{*}\in\Gamma(\mathcal{V},\mathfrak{G}^{*})$.

If $\mathcal{E}(\mathcal{L},\Phi)$ is the 1-form on
$\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$:

(15)
$\mathcal{E}(\mathcal{L},\Phi)=\sum_{v\in\operatorname{int}\mathcal{V}}\big{(}\mathcal{E}_{v}(\mathcal{L})+\sum_{\alpha\in
S_{v}}(\lambda_{\alpha}\circ\Theta_{\alpha}^{v})(\Phi)\big{)}$

then the critical section-mappings
$(y,\lambda)\in\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$
are characterized through Corollary 3.4 by the conditions:

$y\quad\textrm{ admisible and
}\quad\big{(}\mathcal{E}(\mathcal{L},\Phi)\big{)}_{(y,\lambda)}=0.$

Local expression. Let $\\{y^{i}_{v}|1\leq i\leq m\\}$ be a local coordinate
system in a neighborhood of each $y_{v}\in Y_{v}$ and $\\{y^{i}_{v}|1\leq
i\leq m,v\prec\alpha\\}$, the local coordinate system induced in
$(J^{1}Y)_{\alpha}$. On the other hand, let $\\{g^{I}|1\leq I\leq l\\}$,
$l=\dim G$, be a local system of coordinates of $G$ in a neighborhood of $e\in
G$ and $\big{\\{}\left(\frac{\partial}{\partial g^{I}}\right)_{e}|1\leq I\leq
l\big{\\}}$ and $\big{\\{}(dg^{I})_{e}|1\leq I\leq l\big{\\}}$ the induced
basis into the Lie algebra $\mathfrak{g}$ and its dual $\mathfrak{g}^{*}$
respectively. If $L_{g}:G\to G$ is the left translation by $g\in G$,
$L_{g}(\overline{g})=g\cdot\overline{g}$, $L_{g}^{I}:G\to\mathbb{R}$ is the
function $L_{g}^{I}=g^{I}\circ L_{g}$ and
$\Phi_{\alpha}^{I}=g^{I}\circ\Phi_{\alpha}$, $1\leq I\leq l$, then for any
$\alpha\in\mathcal{V}_{n}$, $\theta\circ d\Phi_{\alpha}$ is the 1-form over
$(J^{1}Y)_{\alpha}$ with values in the Lie algebra $\mathfrak{g}$ given by:

(16) $(\theta\circ
d\Phi_{\alpha})_{j^{1}_{\alpha}y}=\sum_{\begin{subarray}{c}v\prec\alpha\\\
1\leq I\leq l\\\ 1\leq i\leq
m\end{subarray}}\left(\sum_{J=1}^{l}\bigg{(}\frac{\partial
L^{I}_{g^{-1}}}{\partial
g^{J}}\bigg{)}_{g}\bigg{(}\frac{\partial\Phi^{J}_{\alpha}}{\partial
y^{i}_{v}}\bigg{)}_{j^{1}_{\alpha}y}\right)(dy^{i}_{v})_{j^{1}_{\alpha}y}\otimes\Big{(}\frac{\partial}{\partial
g^{I}}\Big{)}_{e}$

for $j^{1}_{\alpha}y\in(J^{1}Y)_{\alpha}$ and
$g=\Phi_{\alpha}(j^{1}_{\alpha}y)\in G$.

From here it follows that:

(17)
$\mathcal{E}(\mathcal{L},\Phi)=\sum_{v\in\operatorname{int}\mathcal{V}}\sum_{i=1}^{m}\mathcal{E}^{i}_{v}(\mathcal{L},\Phi)dy^{i}_{v},$

where, for $\lambda(\alpha)=\sum_{1\leq I\leq
l}\lambda(\alpha)^{I}(dg^{I})_{e}$:

$\mathcal{E}^{i}_{v}(\mathcal{L},\Phi)=\sum_{\alpha\in
S_{v}}\frac{\partial\mathcal{L}_{\alpha}}{\partial
y^{i}_{v}}+\sum_{I,J=1}^{l}\lambda(\alpha)^{I}\bigg{(}\frac{\partial
L^{I}_{g^{-1}}}{\partial
g^{J}}\bigg{)}\bigg{(}\frac{\partial\Phi^{J}_{\alpha}}{\partial
y^{i}_{v}}\bigg{)}.$

######  Definition 3.8.

A Jacobi field along a critical section-mapping
$(y,\lambda)\in\Gamma(\mathcal{V}_{0},Y)\times$
$\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$ of the action 1-form
$\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)$ is a tangent vector $(\delta
y,\delta\lambda)\in
T_{(y,\lambda)}\big{(}\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})\big{)}$
that verifies: $\delta y$ is admissible and $(\delta y,\delta\lambda)$ has an
extension
$D\in\mathfrak{X}\big{(}\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})\big{)}$
such that:

(18) $\big{(}L_{D}\mathcal{E}(\mathcal{L},\Phi)\big{)}_{(y,\lambda)}=0.$

The definition is correct since from (17) we have:

$L_{D}\mathcal{E}(\mathcal{L},\Phi)=\sum_{v\in\operatorname{int}\mathcal{V}}(D\mathcal{E}^{i}_{v})dy^{i}_{v}+\mathcal{E}^{i}_{v}(\mathcal{L},\Phi)L_{D}dy^{i}_{v},$

and therefore, evaluating this expression at $(y,\lambda)$, the second term
vanishes because $(y,\lambda)$ is critical and the first one depends only on
$(\delta y,\delta\lambda)$.

Continuing with this approach, a key point for obtaining a multisymplectic
form formula for this new class of Lagrange problems is the following
property.

######  Proposition 3.9.

If
$(y,\lambda)\in\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$
is a section-mapping with $y$ admissible and $\big{(}(\delta
y)^{i},(\delta\lambda)^{i}\big{)}\in
T_{(y,\lambda)}\big{(}\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})\big{)}$
are two tangent vectors at $(y,\lambda)$ with $(\delta y)^{i}$ admissible,
$i=1,2$, we have:

$\big{(}d\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)\Big{)}_{(y,\lambda)}\big{(}\big{(}(\delta
y)^{1},(\delta\lambda)^{1}\big{)},\big{(}(\delta
y)^{2},(\delta\lambda)^{2}\big{)}\big{)}=0.$

###### Proof.

From (10):

$\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)=\sum_{\alpha\in\mathcal{V}}d\mathcal{L}_{\alpha}+\lambda_{\alpha}\circ(\theta\circ
d\Phi_{\alpha})$

where the terms of this sum are considered 1-forms over
$\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$
via the canonical projection
$(y,\lambda)\mapsto\big{(}(j^{1}y)(\alpha),e^{*}(\alpha)=(\alpha,\lambda(\alpha))\big{)}$
from
$\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$ to
$(J^{1}Y\times_{\mathcal{V}}\mathfrak{G}^{*})_{\alpha}$.

From here we get:

$d\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)=\sum_{\alpha\in\mathcal{V}}\underbrace{d^{2}\mathcal{L}_{\alpha}}_{=0}+d\lambda_{\alpha}\barwedge(\theta\circ
d\Phi_{\alpha})+\lambda_{\alpha}\circ d(\theta\circ d\Phi_{\alpha}),$

where, as before, $\barwedge$ is the wedge product of valued forms in
$\mathfrak{g}^{*}$ with forms valued in $\mathfrak{g}$, applying the duality
pairing. This gives:

$\displaystyle\big{(}d\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)\big{)}_{(y,\lambda)}$
$\displaystyle\big{(}\big{(}(\delta
y)^{1},(\delta\lambda)^{1}\big{)},\big{(}(\delta
y)^{2},(\delta\lambda)^{2}\big{)}\big{)}=$
$\displaystyle\sum_{\alpha\in\mathcal{V}}\big{(}d\lambda_{\alpha}\barwedge(\theta\circ
d\Phi_{\alpha})\big{)}_{(y,\lambda)}\big{(}\big{(}(\delta
y)^{1},(\delta\lambda)^{1}\big{)},\big{(}(\delta
y)^{2},(\delta\lambda)^{2}\big{)}\big{)}$
$\displaystyle+\big{(}\lambda_{\alpha}\circ d(\theta\circ
d\Phi_{\alpha})\big{)}_{(y,\lambda)}\big{(}\big{(}(\delta
y)^{1},(\delta\lambda)^{1}\big{)},\big{(}(\delta
y)^{2},(\delta\lambda)^{2}\big{)}\big{)}.$

The first term vanishes because $(\delta y)^{i}$, $i=1,2$ are admissible.
Indeed:

$\displaystyle\sum_{\alpha\in\mathcal{V}}$
$\displaystyle\big{(}d\lambda_{\alpha}\barwedge(\theta\circ
d\Phi_{\alpha})\big{)}_{(y,\lambda)}\big{(}\big{(}(\delta
y)^{1},(\delta\lambda)^{1}\big{)},\big{(}(\delta
y)^{2},(\delta\lambda)^{2}\big{)}\big{)}=$
$\displaystyle\sum_{\alpha\in\mathcal{V}}$
$\displaystyle\Big{(}(d\lambda_{\alpha})_{(y,\lambda)}\big{(}(\delta
y)^{1},(\delta\lambda)^{1}\big{)}\cdot(\theta\circ
d\Phi_{\alpha}))_{(y,\lambda)}\big{(}(\delta
y)^{2},(\delta\lambda)^{2}\big{)}$
$\displaystyle-(d\lambda_{\alpha})_{(y,\lambda)}\big{(}(\delta
y)^{2},(\delta\lambda)^{2}\big{)}\cdot(\theta\circ
d\Phi_{\alpha}))_{(y,\lambda)}\big{(}(\delta
y)^{1},(\delta\lambda)^{1}\big{)}\Big{)}=0$

since we know from the admissibility of $(\delta y)^{i}$, by Definition 2.2,
that

$(d\Phi_{\alpha})_{(j^{1}y)(\alpha)}(j^{1}\delta y^{i})(\alpha)\big{)}=0,$

for $i=1,2$, and so:

$(\theta\circ d\Phi_{\alpha}))_{(y,\lambda)}\big{(}(\delta
y)^{i},(\delta\lambda)^{i}\big{)}=\theta_{\Phi_{\alpha}((j^{1}y)(\alpha))}\big{(}(d\Phi_{\alpha})_{(j^{1}y)(\alpha)}(j^{1}\delta
y^{i})(\alpha)\big{)}=0\quad i=1,2.$

Regarding the second term, from the local expression (16) we obtain:

$\displaystyle d(\theta\circ d\Phi_{\alpha})=$
$\displaystyle\left(d\left(\sum_{v\prec\alpha,I,i}\left(\sum_{J}\bigg{(}\frac{\partial
L^{I}_{g^{-1}}}{\partial
g^{J}}\bigg{)}\bigg{(}\frac{\partial\Phi^{J}_{\alpha}}{\partial
y^{i}_{v}}\bigg{)}\right)\right)\wedge
dy^{i}_{v}\right)\otimes\Big{(}\frac{\partial}{\partial g^{I}}\Big{)}_{e}$
$\displaystyle=$ $\displaystyle\sum_{I,J}\left(d\bigg{(}\frac{\partial
L^{I}_{g^{-1}}}{\partial g^{J}}\bigg{)}\wedge
d\Phi^{J}_{\alpha}+\bigg{(}\frac{\partial L^{I}_{g^{-1}}}{\partial
g^{J}}\bigg{)}\underbrace{d^{2}\Phi^{J}_{\alpha}}_{=0}\right)\otimes\Big{(}\frac{\partial}{\partial
g^{I}}\Big{)}_{e}$

from which, if $\lambda(\alpha)=\sum_{I}\lambda(\alpha)^{I}(dg^{I})_{e}$ as
before, we get:

$\displaystyle\sum_{\alpha\in\mathcal{V}}$
$\displaystyle\big{(}\lambda_{\alpha}\circ d(\theta\circ
d\Phi_{\alpha})\big{)}_{(y,\lambda)}\big{(}\big{(}(\delta
y)^{1},(\delta\lambda)^{1}\big{)},\big{(}(\delta
y)^{2},(\delta\lambda)^{2}\big{)}\big{)}=$
$\displaystyle\sum_{\alpha\in\mathcal{V},I,J}$
$\displaystyle\lambda(\alpha)^{I}\left(d\bigg{(}\frac{\partial
L^{I}_{g^{-1}}}{\partial g^{J}}\bigg{)}\wedge
d\Phi^{J}_{\alpha}\right)_{(j^{1}y)(\alpha)}\big{(}j^{1}(\delta
y)^{1}(\alpha),j^{1}(\delta y)^{2}(\alpha)\big{)}=0$

because of, again, the admissibility of $(\delta y)^{i}$, $i=1,2$. ∎

######  Remark 3.10.

As noted in Remark 3.2, the action 1-form
$\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)$ generalizes to discrete Lagrange
problems valued in a Lie group the differential,
$d\mathbb{A}_{\mathcal{V}}(\widehat{\mathcal{L}})$, of the action function
$\mathbb{A}_{\mathcal{V}}(\widehat{\mathcal{L}})$ of the extended Lagrangian
$\widehat{\mathcal{L}}=\mathcal{L}+\lambda\circ\Phi$ of discrete Lagrange
problems valued in a vector space. In this case, Proposition 3.9 is trivial
because:

$d\big{(}d\mathbb{A}_{\mathcal{V}}(\widehat{\mathcal{L}})\big{)}=d^{2}\mathbb{A}_{\mathcal{V}}(\widehat{\mathcal{L}})=0.$

At this point we finally have:

######  Theorem 3.11 (Multisymplectic form formula).

If
$(y,\lambda)\in\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})$
is a critical section-mapping of the action 1-form
$\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)$ and $\big{(}(\delta
y)^{i},(\delta\lambda)^{i}\big{)}\in
T_{(y,\lambda)}\big{(}\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})\big{)}$,
$i=1,2$, are two Jacobi fields along $(y,\lambda)$, then we have:

$\sum_{(v\prec\operatorname{fr}\mathcal{V})\prec\alpha}\Big{(}d\big{(}\Theta^{v}_{\alpha}(\mathcal{L})+\lambda_{\alpha}\circ\Theta^{v}_{\alpha}(\Phi)\big{)}\Big{)}_{(y,\lambda)}\big{(}\big{(}(\delta
y)^{1},(\delta\lambda)^{1}\big{)},\big{(}(\delta
y)^{2},(\delta\lambda)^{2}\big{)}\big{)}=0.$

###### Proof.

Taking the differential of (14) and having in mind (15), we get:

$d\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)=d\mathcal{E}(\mathcal{L},\Phi)+\sum_{(v\in\operatorname{fr}\mathcal{V})\prec\alpha}d\big{(}\Theta^{v}_{\alpha}(\mathcal{L})+\lambda_{\alpha}\circ\Theta^{v}_{\alpha}(\Phi)\big{)},$

from which, if
$D^{i}\in\mathfrak{X}\big{(}\Gamma(\mathcal{V}_{0},Y)\times\textrm{Map}(\mathcal{V},\mathfrak{g}^{*})\big{)}$,
$i=1,2$, are extensions of $\big{(}(\delta y)^{i},(\delta\lambda)^{i}\big{)}$
verifying the Jacobi equation (18), then by Proposition 3.9 we have:

$\displaystyle 0=$
$\displaystyle\Big{(}\big{(}d\mathbb{A}_{\mathcal{V}}(\mathcal{L},\Phi)\big{)}(D^{1},D^{2})\Big{)}(y,\lambda)=\Big{(}\big{(}d\mathcal{E}(\mathcal{L},\Phi)\big{)}(D^{1},D^{2})\Big{)}(y,\lambda)$
$\displaystyle+\sum_{(v\in\operatorname{fr}\mathcal{V})\prec\alpha}\Big{(}d\big{(}\Theta^{v}_{\alpha}(\mathcal{L})+\lambda_{\alpha}\circ\Theta^{v}_{\alpha}(\Phi)\big{)}\Big{)}_{(y,\lambda)}\big{(}\big{(}(\delta
y)^{1},(\delta\lambda)^{1}\big{)},\big{(}(\delta
y)^{2},(\delta\lambda)^{2}\big{)}\big{)}.$

Now, applying the Cartan formula to the first term of the previous sum, and
taking into account the Jacobi equation (18) and Euler-Lagrange equations
$\big{(}\mathcal{E}(\mathcal{L},\Phi)\big{)}_{(y,\lambda)}=0$, we obtain:

$\displaystyle\Big{(}\big{(}d\mathcal{E}$
$\displaystyle(\mathcal{L},\Phi)\big{)}(D^{1},D^{2})\Big{)}(y,\lambda)$
$\displaystyle=$
$\displaystyle\Big{(}D^{1}\big{(}\mathcal{E}(\mathcal{L},\Phi)(D^{2})\big{)}\Big{)}(y,\lambda)-\Big{(}D^{2}\big{(}\mathcal{E}(\mathcal{L},\Phi)(D^{1})\big{)}\Big{)}(y,\lambda)-\big{(}\mathcal{E}(\mathcal{L},\Phi)([D^{1},D^{2}])\big{)}(y,\lambda)$
$\displaystyle=$
$\displaystyle\big{(}L_{D^{1}}\mathcal{E}(\mathcal{L},\Phi)\big{)}_{(y,\lambda)}(D^{2}_{(y,\lambda)})-\big{(}L_{D^{2}}\mathcal{E}(\mathcal{L},\Phi)\big{)}_{(y,\lambda)}(D^{1}_{(y,\lambda)})+\big{(}\mathcal{E}(\mathcal{L},\Phi)([D^{1},D^{2}])\big{)}(y,\lambda)$
$\displaystyle=$ $\displaystyle 0.$

So, the result is concluded. ∎

##  4\. Application to Euler-Poincaré reduction in discrete field theory

In its simplest original version, this discrete reduction problem arises when
trying to solve an unconstrained discrete problem for a principal bundle over
the standard simplicial complex of $\mathbb{R}^{2}$ with Lagrangian density
invariant by the action of the structural group of the bundle (see for example
[7] and the references cited therein).

More precisely, following [7], let $V$ be the simplicial complex with vertices
$V_{0}=\big{\\{}(i,j)\in\mathbb{Z}\times\mathbb{Z}\big{\\}}$, oriented edges
$V_{1}=\big{\\{}[(i,j),(i+1,j)],[(i,j),(i,j+1)],\,i,j\in\mathbb{Z}\big{\\}}$,
and oriented faces
$V_{2}=\big{\\{}\Delta_{ij}=[(i,j),(i+1,j),(i,j+1)],\,i,j\in\mathbb{Z}\big{\\}}$.
In this case, $V_{2}$ is identified with $V_{0}$ by the bijection
$\Delta_{ij}\mapsto(i,j)$ which we will assume in what follows. Given a Lie
group $G$, let $P=G\times V_{0}\to V_{0}$ be the (left) trivial principal
bundle, and we consider a Lagrangian density $\mathcal{L}:(J^{1}P=G\times
G\times G\times V_{0}\to V_{0})\to\mathbb{R}$ invariant by the diagonal action
of $G$ on
$J^{1}P:g\big{(}(g_{ij},g_{i+1j},g_{ij+1}),(i,j)\big{)}=\big{(}(gg_{ij},gg_{i+1j},gg_{ij+1}),(i,j)\big{)}$.
Identifying $J^{1}P/G$ with the fiber bundle $Y=G\times G\times V_{0}\to
V_{0}$ by the rule
$G\big{(}(g_{ij},g_{i+1j},g_{ij+1}),(i,j)\big{)}=\big{(}(g_{ij}^{-1}g_{i+1j},g_{ij}^{-1}g_{ij+1}),(i,j)\big{)}$,
the Lagrangian density $\mathcal{L}$ is projected onto a Lagrangian density
$\mathit{l}:Y=J^{1}P/G\to\mathbb{R}$ by the projection (reduction map)

$\displaystyle\pi:J^{1}P\hskip 10.00002pt\longrightarrow$ $\displaystyle\hskip
10.00002ptY=J^{1}P/G$
$\displaystyle\big{(}(g_{ij},g_{i+1j},g_{ij+1}),(i,j)\big{)}\mapsto$
$\displaystyle\big{(}(u_{ij}=g^{-1}_{ij}g_{i+1j},v_{ij}=g^{-1}_{ij}g_{ij+1}),(i,j)\big{)}$

On the other hand, the sections $g:(i,j)\in
V_{0}\mapsto\big{(}g_{ij},(i,j)\big{)}$ of $P$ are projected by $\pi\circ
j^{1}$ in the sections $y:(i,j)\in
V_{0}\mapsto\big{(}(u_{ij}=g^{-1}_{ij}g_{i+1j},v_{ij}=g^{-1}_{ij}g_{ij+1}),(i,j)\big{)}$
of $Y$ satisfying the constraint $\Phi(j^{1}y)=e$ where $\Phi:J^{1}Y\to G$ on
$(J^{1}Y)_{(ij)}$ is given by the formula

(19)
$\Phi_{ij}\big{(}(u_{ij},v_{ij},u_{i+1j},v_{i+1j},u_{ij+1},v_{ij+1}),(i,j)\big{)}=u_{ij}v_{i+1j}u_{ij+1}^{-1}v_{ij}^{-1}.$

Similarly, the infinitesimal variations $\delta g:(i,j)\in
V_{0}\mapsto\big{(}\delta g_{ij},(i,j)\big{)}$ of each section
$g\in\Gamma(V_{0},P)$ are projected by $\pi\circ j^{1}$ in the infinitesimal
variations $\delta y:(i,j)\in V_{0}\mapsto\big{(}(\delta u_{ij},\delta
v_{ij}),(i,j)\big{)}$ of the section $y\in\Gamma(V_{0},Y)$ projection of $g$,
where:

(20) $\begin{split}\delta
u_{ij}=&(L_{u_{ij}})_{*}(\theta_{i+1j})-(R_{u_{ij}})_{*}(\theta_{ij})\in
T_{u_{ij}}G,\\\ \delta
v_{ij}=&(L_{v_{ij}})_{*}(\theta_{ij+1})-(R_{v_{ij}})_{*}(\theta_{ij})\in
T_{v_{ij}}G,\end{split}$

being $\theta_{ij}=(L_{g_{ij}^{-1}})_{*}(\delta g_{ij})\in\mathfrak{g}$.

Thus we have a constrained variational problem on $Y=J^{1}P/G$ (reduced
problem) whose action functional with respect to a finite set of faces
$\mathcal{V}\subset V_{2}$ ($\mathcal{V}\equiv\mathcal{U}\subset V_{0}$ by the
bijection $\Delta_{ij}\mapsto(i,j)$) is:

$\mathcal{A}_{\mathcal{U}}(\mathit{l}):y\in\Gamma(\mathcal{U}_{0},Y)\mapsto\sum_{(i,j)\in\mathcal{U}}\mathit{l}\big{(}(u_{ij},v_{ij}),(i,j)\big{)},$

where $\mathcal{U}_{0}$ is the (finite) set of adherent vertices to
$\mathcal{V}$.

Under these conditions, we will say that a section
$y\in\Gamma(\mathcal{U}_{0},Y)$ is critical for the reduced problem if
$(d\mathcal{A}_{\mathcal{U}}(\mathit{l}))_{y}$ is zero over the infinitesimal
variations (20) such that $\theta_{ij}$ vanishes at the frontier of
$\mathcal{U}$.

######  Remark 4.1.

By similarity with the continuous case [3], the sections $y\in\Gamma(V_{0},Y)$
of the reduced bundle $Y=J^{1}P/G$ are identified with the connections of the
principal bundle $P$, the constraint morphism (19) defines the curvature of
the connection, and the reduced infinitesimal variations (20) are interpreted
as the action on the connections of the infinitesimal gauge transformations of
$P$, $\Gamma(V_{0},\mathfrak{g}\times V_{0}\to V_{0})$.

Under this approach, the main result of [7] is as follows.

######  Theorem 4.2 (Reduction).

Let $\mathcal{L}$ be a $G$-invariant Lagrangian density on $J^{1}P$ and
consider the reduced Lagrangian density $\mathit{l}$ on $Y=J^{1}P/G$. Consider
a section $g\in\Gamma(\mathcal{U}_{0},P)$ and let
$y\in\Gamma(\mathcal{U}_{0},Y)$ the induced reduced section. Then the
following conditions are equivalent:

  1. a)

$g$ is a solution of the discrete Euler-Lagrange equations for $\mathcal{L}$
in the interior points of $\mathcal{U}$.

  2. b)

$g$ is critical for arbitrary variations $\delta g$ that vanishes in the
frontier of $\mathcal{U}$.

  3. c)

the reduced section $y$ is a solution of the discrete Euler-Poincaré equations
in the interior points of $\mathcal{U}$:

(21)
$R^{*}_{u_{ij}}\big{(}d\mathit{l}(\cdot,v_{ij})\big{)}_{e}-L^{*}_{u_{i-1j}}\big{(}d\mathit{l}(\cdot,v_{i-1j})\big{)}_{e}+R^{*}_{v_{ij}}\big{(}d\mathit{l}(u_{ij},\cdot)\big{)}_{e}-L^{*}_{v_{ij-1}}\big{(}d\mathit{l}(u_{ij-1},\cdot)\big{)}_{e}=0.$

  4. d)

the reduced section $y$ is critical for infinitesimal variations $\delta y$
(20) where $\theta_{ij}$ are arbitrary variations that vanishes in the
frontier of $\mathcal{U}$.

From Theorem 4.2, a solution $g\in\Gamma(\mathcal{U}_{0},P)$ of the unreduced
Euler-Lagrange equations gives rise to a reduced section
$y\in\Gamma(\mathcal{U}_{0},Y)$ verifying the constraint $\Phi(j^{1}y)=e$.

Reciprocally [7]:

######  Theorem 4.3 (Reconstruction).

Let $y\in\Gamma(\mathcal{U}_{0},Y)$ be a solution of the discrete Euler-
Poincaré equations (21) verifying the constraint $\Phi(j^{1}y)=e$. Then there
exists a solution $g\in\Gamma(\mathcal{U}_{0},P)$ of the unreduced Euler-
Lagrange equations that projects over $y$. In this case, $g$ is uniquely
determined up to right translation by an element of $G$.

The Euler-Poincaré reduction just summarized provides an interesting example
of a discrete Lagrangian problem valued in a Lie group taking as problem data
the reduced Lagrangian density $\mathit{l}:Y=J^{1}P/G\to\mathbb{R}$ and the
constraint morphism $\Phi:J^{1}Y\to G$ given by formula (19).

In particular, the solutions of this problem will be solutions of the discrete
Euler-Poincaré equations (21), which has the nice consequence of being able to
apply to the discrete Euler-Poincaré reduction the formalism that we have
developed in this paper for Lagrange problems.

The procedure we are going to follow is to eliminate the Lagrange multiplier
from equations (13) and observe that the result of such elimination satisfies
the Euler-Poincaré equations (21).

Indeed. Equations (13) are in this case:

(22)
$(d\mathit{l})_{y_{ij}}+\lambda_{\Delta_{ij}}\circ\big{(}\Theta_{\Delta_{ij}}^{ij}(\Phi)\big{)}_{j^{1}y(\Delta_{ij})}+\lambda_{\Delta_{i-1j}}\circ\big{(}\Theta_{\Delta_{i-1j}}^{ij}(\Phi)\big{)}_{j^{1}y(\Delta_{ij})}\\\
+\lambda_{\Delta_{ij-1}}\circ\big{(}\Theta_{\Delta_{ij-1}}^{ij}(\Phi)\big{)}_{j^{1}y(\Delta_{ij})}=0$

where, because $y$ is admissible, the 1-forms
$\Theta_{\Delta_{ij}}^{ij}(\Phi)$ are calculated along $j^{1}y$ by the
formula:

(23)
$\big{(}d\Phi_{\Delta_{ij}}\big{)}_{j^{1}y(\Delta_{ij})}=\big{(}\Theta_{\Delta_{ij}}^{ij}(\Phi)\big{)}_{j^{1}y(\Delta_{ij})}+\big{(}\Theta_{\Delta_{ij}}^{i+1j}(\Phi)\big{)}_{j^{1}y(\Delta_{ij})}+\big{(}\Theta_{\Delta_{ij}}^{ij+1}(\Phi)\big{)}_{j^{1}y(\Delta_{ij})}.$

Applying (22) to $\delta y_{ij}=(\delta u_{ij},\delta v_{ij})$ these equations
are equivalent to the system:

(24) $\left\\{\begin{aligned} d\mathit{l}(\cdot,v_{ij})(\delta
u_{ij})+&\lambda_{\Delta_{ij}}\circ\big{(}\Theta_{\Delta_{ij}}^{ij}(\Phi)\big{)}_{j^{1}y(\Delta_{ij})}(\delta
u_{ij})+\lambda_{\Delta_{i-1j}}\circ\big{(}\Theta_{\Delta_{i-1j}}^{ij}(\Phi)\big{)}_{j^{1}y(\Delta_{ij})}(\delta
u_{ij})\\\
+&\lambda_{\Delta_{ij-1}}\circ\big{(}\Theta_{\Delta_{ij-1}}^{ij}(\Phi)\big{)}_{j^{1}y(\Delta_{ij})}(\delta
u_{ij})=0\\\ d\mathit{l}(u_{ij},\cdot)(\delta
v_{ij})+&\lambda_{\Delta_{ij}}\circ\big{(}\Theta_{\Delta_{ij}}^{ij}(\Phi)\big{)}_{j^{1}y(\Delta_{ij})}(\delta
v_{ij})+\lambda_{\Delta_{i-1j}}\circ\big{(}\Theta_{\Delta_{i-1j}}^{ij}(\Phi)\big{)}_{j^{1}y(\Delta_{ij})}(\delta
v_{ij})\\\
+&\lambda_{\Delta_{ij-1}}\circ\big{(}\Theta_{\Delta_{ij-1}}^{ij}(\Phi)\big{)}_{j^{1}y(\Delta_{ij})}(\delta
v_{ij})=0\end{aligned}\right.$

On the other hand, by the expression (19) of the constraint morphism, the only
nonzero components of $(d\Phi_{\Delta_{ij}})_{j^{1}y(\Delta_{ij})}$ are those
resulting from applying this 1-form to $\delta u_{ij}$, $\delta v_{ij}$,
$\delta u_{ij+1}$, $\delta v_{i+1j}$ which, taking into account the constraint
condition of $y_{ij}=(u_{ij},v_{ij})$, are computed as follows:

$\displaystyle(d\Phi_{\Delta_{ij}})_{j^{1}y(\Delta_{ij})}$
$\displaystyle(\delta u_{ij})=\delta
u_{ij}v_{i+1j}u^{-1}_{ij+1}v^{-1}_{ij}=\delta
u_{ij}u^{-1}_{ij}=(R_{u^{-1}_{ij}})_{*}(\delta u_{ij}),$
$\displaystyle(d\Phi_{\Delta_{ij}})_{j^{1}y(\Delta_{ij})}$
$\displaystyle(\delta v_{ij})=u_{ij}v_{i+1j}u^{-1}_{ij+1}(-v^{-1}_{ij}\delta
v_{ij}v^{-1}_{ij})=-(R_{v^{-1}_{ij}})_{*}(\delta v_{ij}),$
$\displaystyle(d\Phi_{\Delta_{ij}})_{j^{1}y(\Delta_{ij})}$
$\displaystyle(\delta u_{ij+1})=u_{ij}v_{i+1j}(-u^{-1}_{ij+1}\delta
u_{ij+1}u^{-1}_{ij+1})v^{-1}_{ij}=-Ad_{v_{ij}}(\delta u_{ij+1}u^{-1}_{ij+1}),$
$\displaystyle(d\Phi_{\Delta_{ij}})_{j^{1}y(\Delta_{ij})}$
$\displaystyle(\delta v_{i+1j})=u_{ij}\delta
v_{i+1j}u^{-1}_{ij+1}v^{-1}_{ij}=u_{ij}\delta
v_{i+1j}v^{-1}_{i+1j}u^{-1}_{ij}=Ad_{u_{ij}}(\delta v_{i+1j}v^{-1}_{i+1j}).$

From here, formula (23) allows to calculate all the terms of the system of
equations (24) obtaining:

$\left\\{\begin{aligned} d\mathit{l}(\cdot,v_{ij})(\delta
u_{ij})+\lambda_{\Delta_{ij}}\circ(R_{u^{-1}_{ij}})_{*}(\delta
u_{ij})-\lambda_{\Delta_{ij-1}}\circ Ad_{v_{ij-1}}(R_{u^{-1}_{ij}})_{*}(\delta
u_{ij})&=0\\\ d\mathit{l}(u_{ij},\cdot)(\delta
v_{ij})-\lambda_{\Delta_{ij}}\circ(R_{v^{-1}_{ij}})_{*}(\delta
v_{ij})+\lambda_{\Delta_{i-1j}}\circ Ad_{u_{i-1j}}(R_{v^{-1}_{ij}})_{*}(\delta
v_{ij})&=0\end{aligned}\right.$

By transposition and translating to the identity element $e\in G$ by
$R_{u_{ij}}$ and $R_{v_{ij}}$ respectively we obtain the system of equations:

(25) $\left\\{\begin{aligned}
(R_{u_{ij}})^{*}\big{(}d\mathit{l}(\cdot,v_{ij})\big{)}+\lambda_{\Delta_{ij}}-Ad^{*}_{v_{ij-1}}\lambda_{\Delta_{ij-1}}&=0\\\
(R_{v_{ij}})^{*}\big{(}d\mathit{l}(u_{ij},\cdot)\big{)}-\lambda_{\Delta_{ij}}+Ad^{*}_{u_{i-1j}}\lambda_{\Delta_{i-1j}}&=0\end{aligned}\right.$

Under these conditions we have the following:

######  Theorem 4.4.

If
$(y,\lambda)\in\Gamma(\mathcal{U}_{0},Y)\times\textrm{Map}(\mathcal{U},\mathfrak{g}^{*})$
is a section-mapping, with $y$ admissible, solution of equations (25) of the
Lagrange problem with Lagrangian density $\mathit{l}:Y=J^{1}P/G\to\mathbb{R}$
and constraint morphism $\Phi:J^{1}Y\to G$ given by formula (19), then its
component $y$ is a solution of the Euler-Poincaré equations (21).

###### Proof.

Passing the multipliers to the second member in (25) we have:

(26) $\left\\{\begin{aligned}
(R_{u_{ij}})^{*}\big{(}d\mathit{l}(\cdot,v_{ij})\big{)}=&-\lambda_{\Delta_{ij}}+Ad^{*}_{v_{ij-1}}\lambda_{\Delta_{ij-1}}\\\
(R_{v_{ij}})^{*}\big{(}d\mathit{l}(u_{ij},\cdot)\big{)}=&\lambda_{\Delta_{ij}}-Ad^{*}_{u_{i-1j}}\lambda_{\Delta_{i-1j}}\end{aligned}\right.$

Subtracting
$Ad^{*}_{u_{i-1j}}(R_{u_{i-1j}})^{*}\big{(}d\mathit{l}(\cdot,v_{i-1j})\big{)}$
in each member of the first equation and
$Ad^{*}_{v_{ij-1}}(R_{v_{ij-1}})^{*}\big{(}d\mathit{l}(u_{ij-1},\cdot)\big{)}$
in the second one, and then adding both expressions then the left-hand side of
the new equality will be:

$\displaystyle(R_{u_{ij}})^{*}\big{(}d\mathit{l}(\cdot,v_{ij})\big{)}-$
$\displaystyle
Ad^{*}_{u_{i-1j}}(R_{u_{i-1j}})^{*}\big{(}d\mathit{l}(\cdot,v_{i-1j})\big{)}+(R_{v_{ij}})^{*}\big{(}d\mathit{l}(u_{ij},\cdot)\big{)}$
$\displaystyle-$ $\displaystyle
Ad^{*}_{v_{ij-1}}(R_{v_{ij-1}})^{*}\big{(}d\mathit{l}(u_{ij-1},\cdot)\big{)}$

which, taking into account the identities
$Ad^{*}_{u_{i-1j}}R^{*}_{u_{i-1j}}=L^{*}_{u_{i-1j}}$ and
$Ad^{*}_{v_{ij-1}}R^{*}_{v_{ij-1}}=L^{*}_{v_{ij-1}}$, is precisely the first
member of the Euler-Poincaré equations (21).

As for the right-hand side of this new equation, applying again (26) we get:

$\displaystyle\Big{(}$
$\displaystyle-\lambda_{\Delta_{ij}}+Ad^{*}_{v_{ij-1}}\lambda_{\Delta_{ij-1}}-Ad^{*}_{u_{i-1j}}\big{(}-\lambda_{\Delta_{i-1j}}+Ad^{*}_{v_{i-1j-1}}\lambda_{\Delta_{i-1j-1}}\big{)}\Big{)}$
$\displaystyle+\Big{(}\lambda_{\Delta_{ij}}-Ad^{*}_{u_{i-1j}}\lambda_{\Delta_{i-1j}}-Ad^{*}_{v_{ij-1}}\big{(}\lambda_{\Delta_{ij-1}}-Ad^{*}_{u_{i-1j-1}}\lambda_{\Delta_{i-1j-1}}\big{)}\Big{)}$
$\displaystyle=$ $\displaystyle(Ad^{*}_{v_{ij-1}}\circ
Ad^{*}_{u_{i-1j-1}}-Ad^{*}_{u_{i-1j}}\circ
Ad^{*}_{v_{i-1j-1}})\lambda_{\Delta_{i-1j-1}}$

However, since the section $y$ is admissible, it satisfies the constraint

$u_{i-1j-1}v_{ij-1}=v_{i-1j-1}u_{i-1j},$

and from here we obtain:

$\displaystyle Ad^{*}_{v_{ij-1}}\circ Ad^{*}_{u_{i-1j-1}}=$
$\displaystyle(Ad_{u_{i-1j-1}}\circ
Ad_{v_{ij-1}})^{*}=Ad^{*}_{u_{i-1j-1}v_{ij-1}}=Ad^{*}_{v_{i-1j-1}u_{i-1j}}$
$\displaystyle=$ $\displaystyle Ad^{*}_{u_{i-1j}}\circ Ad^{*}_{v_{i-1j-1}}.$

Thus, this right-hand side vanishes, and the result can be concluded. ∎

Reciprocally, in an analogous way to the continuous case [3], the following
result is verified.

######  Theorem 4.5.

If $y\in\Gamma(\mathcal{U}_{0},Y)$ is an admissible section solution of the
Euler-Poincaré equations (21), then locally there exists a non-unique mapping
$\lambda\in\textrm{Map}(\mathcal{U},\mathfrak{g}^{*})$ such that $(y,\lambda)$
is a solution of equations (25) of the Lagrange problem.

###### Proof.

For each vertex $(i,j)\in\operatorname{int}\mathcal{U}$, choosing arbitrarily
$\lambda_{\Delta_{ij}}$, you can solve uniquely $\lambda_{\Delta_{i-1j}}$ and
$\lambda_{\Delta_{ij-1}}$ in the system of equations (25) over the spherical
neighborhood $S_{(i,j)}=\\{\Delta_{ij},\Delta_{i-1j},\Delta_{ij-1}\\}$ of
$(i,j)$, due to the isomorphisms $Ad^{*}_{u_{i-1j}}$ and $Ad^{*}_{v_{ij-1}}$
respectively. ∎

##  5\. Example: discrete harmonic mappings from the discrete plane into the
Lie group $SO(n)$

For this example, $P=SO(n)\times V_{0}\to V_{0}$ and the reduced Lagrangian
density $\mathit{l}:(Y=J^{1}P/SO(n)=SO(n)\times SO(n)\times V_{0}\to
V_{0})\to\mathbb{R}$ is given by the formula

(27)
$\mathit{l}\big{(}(u_{ij},v_{ij}),(i,j)\big{)}=\operatorname{tr}u_{ij}+\operatorname{tr}v_{ij}$

where $SO(n)$ is considered immersed in the standard way in the space of the
square matrices $M_{n}(\mathbb{R})$ [7].

If $(a_{kl})$ and $(b_{kl})$ are two copies of the usual local coordinates of
$M_{n}(\mathbb{R})$ coordinating, respectively, the components $u_{ij}\in
SO(n)\subset M_{n}(\mathbb{R})$ and $v_{ij}\in SO(n)\subset
M_{n}(\mathbb{R})$, then the Lagrangian density (27) is expressed as:

(28)
$\mathit{l}=\sum_{i^{\prime}=1}^{n}a_{i^{\prime}i^{\prime}}(u_{ij})+b_{i^{\prime}i^{\prime}}(v_{ij}).$

On the other hand, the Lie algebra of $SO(n)$ has as basis the tangent vectors
$E_{kl}=\big{(}\frac{\partial}{\partial a_{kl}}-\frac{\partial}{\partial
a_{lk}}\big{)}_{e}$, $k<l$, for the coordinates $(a_{kl})$ and
$E_{kl}=\big{(}\frac{\partial}{\partial b_{kl}}-\frac{\partial}{\partial
b_{lk}}\big{)}_{e}$, $k<l$, for the coordinates $(b_{kl})$.

Under these conditions, the four terms of the Euler-Poincaré equations (21)
can be calculated as follows.

Given $u_{ij}\in SO(n)$, the equations of the translation $R_{u_{ij}}:a\in
SO(n)\mapsto au_{ij}$ in the coordinates $(a_{kl})$ are: $a_{st}\circ
R_{u_{ij}}=\sum_{v}a_{vt}(u_{ij})a_{sv}$ from where:

$\displaystyle\big{(}(R_{u_{ij}})_{*,e}E_{kl}\big{)}a_{st}=$ $\displaystyle
E_{kl}(a_{st}\circ R_{u_{ij}})=\big{(}\frac{\partial}{\partial
a_{kl}}-\frac{\partial}{\partial
a_{lk}}\big{)}_{e}\big{(}\sum_{v}a_{vt}(u_{ij})a_{sv}\big{)}$ $\displaystyle=$
$\displaystyle\delta_{kj}a_{lt}(u_{ij})-\delta_{ls}a_{kt}(u_{ij}).$

From this it follows:

(29)
$\big{(}(R_{u_{ij}})_{*,e}E_{kl}\big{)}=\sum_{j^{\prime}=1}^{n}a_{lj^{\prime}}(u_{ij})\frac{\partial}{\partial
a_{kj^{\prime}}}-a_{kj^{\prime}}(u_{ij})\frac{\partial}{\partial
a_{lj^{\prime}}}.$

Then, taking into account (28) and (29) we have:

$\displaystyle\big{(}R^{*}_{u_{ij}}d\mathit{l}(\cdot,v_{ij})\big{)}_{e}E_{kl}=$
$\displaystyle\big{(}d\mathit{l}(\cdot,v_{ij})\big{)}_{u_{ij}}\big{(}(R_{u_{ij}})_{*,e}E_{kl}\big{)}=\big{(}(R_{u_{ij}})_{*,e}E_{kl}\big{)}\mathit{l}(\cdot,v_{ij})$
$\displaystyle=$
$\displaystyle\Big{(}\sum_{j^{\prime}=1}^{n}a_{lj^{\prime}}(u_{ij})\frac{\partial}{\partial
a_{kj^{\prime}}}-a_{kj^{\prime}}(u_{ij})\frac{\partial}{\partial
a_{lj^{\prime}}}\Big{)}\big{(}\sum_{i^{\prime}=1}^{n}a_{i^{\prime}i^{\prime}}+b_{i^{\prime}i^{\prime}}\big{)}$
$\displaystyle=$ $\displaystyle a_{lk}(u_{ij})-a_{kl}(u_{ij}).$

Proceeding analogously with the other terms of equations (21) we obtain:

$\displaystyle\big{(}$ $\displaystyle
L^{*}_{u_{i-1j}}d\mathit{l}(\cdot,v_{i-1j})\big{)}_{e}E_{kl}=a_{lk}(u_{i-1j})-a_{kl}(u_{i-1j}),$
$\displaystyle\big{(}$ $\displaystyle
R^{*}_{v_{ij}}d\mathit{l}(u_{ij},\cdot)\big{)}_{e}E_{kl}=b_{lk}(v_{ij})-b_{kl}(v_{ij}),$
$\displaystyle\big{(}$ $\displaystyle
L^{*}_{v_{ij-1}}d\mathit{l}(u_{ij-1},\cdot)\big{)}_{e}E_{kl}=b_{lk}(v_{ij-1})-b_{kl}(v_{ij-1}).$

Then, equations (21) applied to the elements $E_{kl}$, $k<l$ of the Lie
algebra of $SO(n)$ are:

$\displaystyle\big{(}a_{lk}(u_{ij})-a_{kl}(u_{ij})\big{)}-$
$\displaystyle\big{(}a_{lk}(u_{i-1j})-a_{kl}(u_{i-1j})\big{)}+\big{(}b_{lk}(v_{ij})-b_{kl}(v_{ij})\big{)}$
$\displaystyle-$
$\displaystyle\big{(}b_{lk}(v_{ij-1})-b_{kl}(v_{ij-1})\big{)}=0$

or equivalently, with the notation of $a^{t}$ for the transposition of $a$,

$a_{kl}(u^{t}_{ij}-u_{ij}+u_{i-1j}-u^{t}_{i-1j})+b_{kl}(v^{t}_{ij}-v_{ij}+v_{ij-1}-v^{t}_{ij-1})=0.$

In other words, equations (23) are equivalent to the fact that, for all $k<l$,
the $(k,l)$ element of the matrix $u^{t}_{ij}-u_{ij}+u_{i-1j}-u^{t}_{i-1j}$
coincides with the $(k,l)$ element of the matrix
$-(v^{t}_{ij}-v_{ij}+v_{ij-1}-v^{t}_{ij-1})$. This, by virtue of the skew-
symmetry of both matrices, is equivalent to this equality being verified for
every $(k,l)$ element. Thus, the Euler-Poincaré equations (21) of our example
are:

$u_{ij}+v_{ij}-u_{i-1j}-v_{ij-1}=u^{t}_{ij}+v^{t}_{ij}-u^{t}_{i-1j}-v^{t}_{ij-1},$

which were obtained in [7] by another procedure.

Applying again the above calculations to equations (25) of the corresponding
problem of Lagrange, they can be expressed as follows:

$\left\\{\begin{aligned}
-\lambda_{\Delta_{ij}}+Ad^{*}_{v_{ij-1}}\lambda_{\Delta_{ij-1}}&=\sum_{k<l}\big{(}a_{lk}(u_{ij})-a_{kl}(u_{ij})\big{)}E^{*}_{kl}\\\
\lambda_{\Delta_{ij}}-Ad^{*}_{u_{i-1j}}\lambda_{\Delta_{i-1j}}&=\sum_{k<l}\big{(}b_{lk}(v_{ij})-b_{kl}(v_{ij})\big{)}E^{*}_{kl}\end{aligned}\right.$

## References

  * [1] A.C. Casimiro, C. Rodrigo, Variational integrators for reduced field equations, Stat. Optim. Inf. Comput. 6 (2018), 86–115.
  * [2] M. Castrillón López, Field theories: reduction, constraints and variational integrators, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 106 (2012), 67–74.
  * [3] M. Castrillón López, P.L. García, C. Rodrigo, Euler-Poincaré reduction in principal fibre bundles and the problem of Lagrange, Differential Geom. Appl. 25 (2007), 585–593.
  * [4] M. Castrillón López, P.L. García, C. Rodrigo, Euler-Poincaré reduction in principal bundles by a subgroup of the structure group, J. Geom. Phys. 74 (2013), 352–369.
  * [5] P.M. Chacón, A. Fernández, P.L. García, C. Rodrigo, The problem of Lagrange in discrete field theory, J. Geom. Phys. 146 (2019), 103511, 18 pp.
  * [6] B. Couéraud, F. Gay-Balmaz, Variational discretization of the Navier-Stokes-Fourier system, 2002, arXiv: 2202.03502.
  * [7] J. Vankerschaver, Euler-Poincaré reduction for discrete field theories, J. Math. Phys. 48 (2007), 032902, 17 pp.