# Lagrangian multiforms on Lie groups and non-commuting flows

Vincent Caudrelier  School of Mathematics, University of Leeds, Leeds, UK.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Frank
Nijhoff  School of Mathematics, University of Leeds, Leeds, UK.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Duncan
Sleigh  School of Mathematics, University of Leeds, Leeds, UK.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>School
of Mathematics, University of Leeds, Leeds, UK.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Mats
Vermeeren Corresponding author Department of Mathematical Sciences,
Loughborough University,
Loughborough, UK.
<EMAIL_ADDRESS>
###### Abstract

We describe a variational framework for non-commuting flows, extending the
theories of Lagrangian multiforms and pluri-Lagrangian systems, which have
gained prominence in recent years as a variational description of integrable
systems in the sense of multidimensional consistency. In the context of non-
commuting flows, the manifold of independent variables, often called multi-
time, is a Lie group whose bracket structure corresponds to the commutation
relations between the vector fields generating the flows. Natural examples are
provided by superintegrable systems for the case of Lagrangian 1-form
structures, and integrable hierarchies on loop groups in the case of
Lagrangian 2-forms. As particular examples we discuss the Kepler problem, the
rational Calogero-Moser system, and a generalisation of the Ablowitz-Kaup-
Newell-Segur system with non-commuting flows. We view this endeavour as a
first step towards a purely variational approach to Lie group actions on
manifolds.

Keywords: Integrable hierarchies, Non-commuting flows, Lagrangian multiforms,
Symmetry algebras

MSC2020: 70G65, 37J35, 37K10

##  1 Introduction

The variational theory behind integrable systems has recently undergone a
major development through the introduction of Lagrangian multiform theory
[21], which establishes a variational framework for _multidimensional
consistency_ , a notion which generalises the idea of commuting flows of
differential equations, including the analogous property for lattice
equations. Multidimensional consistency is the key integrability phenomenon of
the coexistence and compatibility of a multitude of dynamical equations on the
same dependent variable in terms of several (often an arbitrary number)
independent variables. Thus, multidimensional consistency is a manifestation
of well-known integrability aspects, such as the existence of hierarchies of
nonlinear evolution equations, their infinite sequences of conservation laws,
associated linear problems (Lax pairs), dressing transforms and Darboux
schemes.

A conventional Lagrangian formalism only provides a single (Euler-Lagrange)
equation per component of the relevant field variables. From the perspective
of multi-dimensional consistency, it is more natural to consider a space of
independent variables of arbitrary dimension, called _multi-time_. In the
_multiform_ formalism (and in the closely related _pluri-Lagrangian_
formalism), Lagrangians are components of a differential (or difference)
$d$-form in multi-time. The associated Euler-Lagrange equations provide a
compatible system of simultaneous equations on each component of the fields.

Since the initial proposal in [21], this new variational approach has been
shown to be a quite universal structure behind integrable systems, both in the
realm of discrete and continuous equations of this class, see e.g. [23, 22,
46, 47, 7, 35, 37]. In the continuous case, the full set of variational
equations, i.e. the extended set of Euler-Lagrange equations, were worked out
in [39]. Many salient features have been elaborated in recent years, e.g. the
connection with variational symmetries and Noether’s theorem [32, 43, 36], as
well as with Lax pairs [35], Hamiltonian structures [38, 10, 43] and the
classical $r$-matrix [11].

So far the Lagrangian multiform approach has proven successful in providing a
framework for multi-time integrable systems in terms of commuting flows, which
is the traditional setting of multidimensionally consistent integrable
systems. However, the notion of integrability is not necessarily restricted to
the case of commuting flows. In fact, the pioneering paper [25] strayed beyond
the confines of flows generated by Hamiltonians in involution with regard to
the Poisson structure, and considered non-commuting flows as well,
establishing a Liouville type theorem for the case that there is a nontrivial
Lie algebra structure for the corresponding vector fields.

In [28] non-commuting flows on loop algebras were considered, generating
systems integrable through linear integral equations associated with Lax
representations of $N\times N$ matrix hierarchies. Corresponding Lagrangian
structures were established in [27]. In these extended integrable systems,
compatibility relations generate vector fields that are ‘alien’ to the
original flows in the equations. Hence any variational description would
necessarily entail a multiform structure. In the present paper we aim at
providing a solid underpinning of these ideas, which amount to a description
of dynamical systems where the set of _independent variables_ is a Lie
group111In this respect what is aimed at here is essentially different from
recent work on non-commutative integrable systems, where the _dependent
variables_ are chosen in some associative algebra.. In contrast, so far Lie
groups have mostly appeared in dynamical systems either as objects describing
the symmetries or as the phase space. What is done in the present paper is to
consider a multi-time (space of independent variables) that has the structure
of a Lie group. More ambitiously, the present set-up could be viewed as a
first step in general towards a purely variational approach to Lie group
actions on manifolds.

The outline of the paper is as follows. In section 2 we provide some
background on the Lagrangian structure of Liouville integrable systems and
sketch the multiform approach. In section 3 we set up the framework for the
Lagrangian multiform structure for systems where a Lie group is considered as
a (non-commuting) multi-time manifold. In section 4 we provide the
corresponding variational structure for those Lagrange 1-forms and provide
some compelling examples: the Kepler problem and the rational Calogero-Moser
system. These systems have the added feature that they are superintegrable, so
a full description of the group actions of their symmetries necessarily
requires non-commuting flows. In section 5 we consider the case of Lagrange
2-forms on loop groups, building on the structures of [27, 28] as well as
[11], and present, as a new example, a non-commutative multi-time version of
the AKNS hierarchy. We end with some conclusions in section 6.

##  2 Background: Lagrangian structure of Liouville integrable systems

A Hamiltonian system on a $2n$-dimensional symplectic manifold $S$ is
Liouville integrable if the Hamiltonian function
$H_{1}:S\rightarrow\mathbb{R}$ is one of $n$ functionally independent
functions $H_{i}:S\rightarrow\mathbb{R}$ ($i=1,\ldots,n$) such that
$\\{H_{i},H_{j}\\}=0$ for all $i,j$, where $\\{\cdot,\cdot\\}$ is the Poisson
bracket induced by the symplectic structure $\omega$ on $S$. The flows of the
additional Hamiltonian functions $H_{2},\ldots,H_{n}$ are symmetries of the
Hamiltonian system $(S,H_{1})$. The Liouville-Arnold theorem shows that this
setup has a rich geometric structure (see e.g. [5]). Below we will assume that
$S$ is a cotangent bundle, $S=T^{*}Q$.

It is important to note that the definition of Liouville integrability is
symmetric under relabelling of the Hamiltonian functions $H_{i}$. It makes no
difference which Hamiltonian function we consider to be physical (all others
being its symmetries). For each $i$ we could define a flow

$\Phi_{i}:\mathbb{R}\times S\rightarrow S:(t,z)\mapsto\Phi_{i}^{t}(z)\,,$

where
$\frac{\partial{\Phi_{i}^{t}(z)}}{\partial{t}}\lrcorner\,\omega=\mathrm{d}H_{i}$.
But why should we consider these flows as separate objects? We might as well
introduce a “flow” on the _multi-time_ $\mathbb{R}^{n}$,

$\Phi:\mathbb{R}^{n}\times S\rightarrow
S:(t_{1},\ldots,t_{n},z)\mapsto\Phi^{t_{1}}\circ\ldots\circ\Phi^{t_{n}}(z)\,,$
(2.1)

which captures the combined dynamics of the system and its symmetries (in the
sense of the action of the symmetry group on phase space).

The advantage of combining the physical time and the “times” of the symmetry
flows into multi-time manifests itself more clearly in the Lagrangian picture.
Assuming the Hamiltonian functions $H_{1},\ldots,H_{n}$ are non-
degenerate222In fact they need not all be non-degenerate, but rather non-
degenerate as a family. See [38, 43]., we could introduce Lagrangians
$L_{1},\ldots,L_{n}$. Then the flows $\Phi_{i}:\mathbb{R}\times
T^{*}Q\rightarrow T^{*}Q$, projected down to $Q$, produce critical curves of
the action integrals

$\int_{a}^{b}L_{i}(q,\dot{q})\,\mathrm{d}t\,.$

In the multi-time formalism we can combine the Lagrangian functions into a
1-form

$\mathcal{L}[q]=L_{1}[q]\,\mathrm{d}t_{1}+\ldots+L_{n}[q]\,\mathrm{d}t_{n}\,,$

where the square brackets denote dependence on a function
$q:\mathbb{R}^{n}\rightarrow Q$ and its derivatives. We call such a function,
from multi-time to the configuration manifold, a _field_. The main feature of
a 1-form is that it can be integrated along curves. So for every curve
$\gamma:[0,1]\rightarrow\mathbb{R}^{n}$ we can define an action functional

$A_{\gamma}:q\mapsto\int_{\gamma}\mathcal{L}[q]\,.$

We can now impose the following variational principle, which provides the
setting to recognise integrability from the Lagrangian perspective [21, 47,
38, 39].

######  Definition 2.1.

We say that a field $q:\mathbb{R}^{n}\rightarrow Q$ is _critical_ for
$\mathcal{L}$ if the corresponding action $A_{\gamma}$ is critical for every
curve $\gamma$. That is, for every $\gamma:[0,1]\rightarrow\mathbb{R}^{n}$ and
every smooth family of curves $q_{\varepsilon}$ such that $q=q_{0}$ and
$q_{\varepsilon}(\gamma(0))=q(\gamma(0))$ and
$q_{\varepsilon}(\gamma(1))=q(\gamma(1))$ there holds

$\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\Big{|}_{\varepsilon=0}\int_{\gamma}\mathcal{L}[q_{\varepsilon}]=0\,.$

If the Lagrangian depends on second or higher derivatives, we also require the
derivatives of $q$ and $q_{\varepsilon}$ to be equal at the endpoints of
$\gamma$

A system that is described by a Lagrangian 1-form via this variational
principle is known in the literature as a “pluri-Lagrangian system” [7, 6,
39]. Additionally, one often requires that $q$ is critical with respect to
variations of the curve $\gamma$ too, which is equivalent to requiring that
$\mathcal{L}[q]$ is closed. This perspective is known as “Lagrangian
multiform” theory [21, 46, 19]. The closure property $\mathrm{d}\mathcal{L}=0$
implies that the corresponding Hamiltonian functions are in involution [38,
43]. In addition to being a formalism to derive equations from a given
Lagrangian, Lagrangian multiform theory can be seen as a guiding principle to
determine integrable Lagrangians.

##### Multi-time Euler-Lagrange equations

The differential equations which characterise criticality in the sense of
Definition 2.1 are called _multi-time Euler-Lagrange equations_ (or _multiform
Euler-Lagrange equations_). They were first derived in [39] and using a
different approach in [36, 37]. Below we give a heuristic explanation of this
system of equations.

If we choose the curve $\gamma$ to be a straight line in the
$t_{i}$-direction, we recover a familiar action integral with the
corresponding component of $\mathcal{L}$ as the Lagrangian: $\int
L_{i}\,\mathrm{d}t_{i}$. This leads to the Euler-Lagrange equation

$\frac{\partial{L_{i}}}{\partial{q}}-\operatorname{D}_{i}\frac{\partial{L_{i}}}{\partial{q_{i}}}+\ldots=0\,,$

where $D_{i}$ denotes the total derivative with respect to $t_{i}$,
$q_{i}=\frac{\mathrm{d}q}{\mathrm{d}t_{i}}$, and the dots represent terms of
the Euler-Lagrange equation which are relevant if the Lagrangian depends on
second or higher derivatives. We call this expression a _variational
derivative_ and denote it by

$\frac{\delta_{i}{L_{i}}}{\delta{q}}=\frac{\partial{L_{i}}}{\partial{q}}-\operatorname{D}_{i}\frac{\partial{L_{i}}}{\partial{q_{i}}}+\ldots\,,$

where the first index $i$ indicates that additional derivatives, which
originate from integration by parts in the standard derivation of the Euler-
Lagrange equations, are only with respect to $t_{i}$. Because $\gamma$ was
taken in the $t_{i}$-direction, these are the only integrations by parts that
could be carried out. In particular, derivatives of $q$ with respect to other
time variables cannot be integrated away. Therefore we should consider them as
additional variables and include the corresponding Euler-Lagrange equations

$\displaystyle\frac{\delta_{i}{L_{i}}}{\delta{q_{j}}}$
$\displaystyle=\frac{\partial{L_{i}}}{\partial{q_{j}}}-\operatorname{D}_{i}\frac{\partial{L_{i}}}{\partial{q_{ji}}}+\ldots=0\,,$
$\displaystyle\frac{\delta_{i}{L_{i}}}{\delta{q_{jk}}}$
$\displaystyle=\frac{\partial{L_{i}}}{\partial{q_{jk}}}-\operatorname{D}_{i}\frac{\partial{L_{i}}}{\partial{q_{jki}}}+\ldots=0\,,$
$\displaystyle\mathmakebox[width("{}={}")][c]{\vdots}$

where $j,k,\ldots\neq i$ and subscripts of $q$ denote partial derivatives.

So far we have only considered curves $\gamma$ which are in coordinate
directions. Additional multi-time Euler-Lagrange equations are found when we
consider curves in other directions (or curves that are not straight). They
are of the form

$\frac{\delta_{i}{L_{i}}}{\delta{q_{i}}}=\frac{\delta_{j}{L_{j}}}{\delta{q_{j}}}\,,\qquad\frac{\delta_{i}{L_{i}}}{\delta{q_{ki}}}=\frac{\delta_{j}{L_{j}}}{\delta{q_{kj}}}\,,\qquad\ldots$

In summary, the variational principle of Definition 2.1 is equivalent to the
following system of multi-time Euler-Lagrange equations:

$\displaystyle\frac{\delta_{i}{L_{i}}}{\delta{q_{I}}}=0\,,\qquad\qquad
I\not\ni t_{i}\,,$
$\displaystyle\frac{\delta_{i}{L_{i}}}{\delta{q_{Ii}}}=\frac{\delta_{j}{L_{j}}}{\delta{q_{Ij}}}\,,$

where $I$ is a multi-index listing the differentiations applied to $q$,
$I\not\ni t_{i}$ means that none of them are with respect to $t_{i}$, and the
$i$ in $q_{Ii}$ denotes an additional differentiation with respect to $t_{i}$.

An even more compact expression for the multi-time Euler-Lagrange equations
can be given as

$\frac{\delta_{i}{L_{i}}}{\delta{q_{I\setminus
j}}}=\frac{\delta_{j}{L_{j}}}{\delta{q_{I\setminus i}}}\,,$

where $I\setminus i$ denotes one fewer differentiation with respect to
$t_{i}$. If $I$ does not list any differentiations with respect to $t_{i}$,
then any term containing $I\setminus i$ is taken to be zero.

##### Exterior derivative

The variational principle of Definition 2.1 gives a single Lagrangian
description of a number of commuting flows. To capture integrability in the
sense of Liouville, we need more than commutativity of the flows.
(Commutativity corresponds to constant, not necessarily vanishing, Poisson
brackets.) The key integrability feature in the multi-form approach is the
_closure relation_ : the exterior derivative $\mathrm{d}\mathcal{L}$ should
vanish when evaluated on solutions of the multi-time Euler-Lagrange equations.

Furthermore, taking variations of (the coefficients of)
$\mathrm{d}\mathcal{L}$ is equivalent to the variational principle. Hence
$\mathrm{d}\mathcal{L}$ is zero on solutions if and only if it attains a
double zero on solutions. In many examples one can write the coefficients of
$\mathrm{d}\mathcal{L}$ explicitly as a product of two expressions which
vanish on the multi-time Euler-Lagrange equations.

The closedness of the form $\mathcal{L}$ is immediately related to the
vanishing of Poisson brackets between the corresponding Hamiltonian functions
[38, 43], and to the fact that the commuting flows are variational symmetries
of each other’s Lagrangian functions [36, 33].

##### Higher forms

So far in this introduction we have only mentioned the multi-form principle
for 1-forms, which applies to systems of ODEs. In the case of hierarchies of
PDEs, a completely analogous principle applies for a higher form. For example,
in integrable hierarchies such as KdV [39] and AKNS [35, 36], the individual
equations are 2-dimensional, so the classical variational principle involves
integration over a plane. In the multi-time setting, all the equations of such
a hierarchy share the same space variable, but they each have their own time
variable. Multi-time is spanned by the full set of space and time directions.
The pluri-Lagrangian principle now requires that the integral of a 2-form is
critical regardless of which 2-dimensional surface of integration is chosen.
As before, the Lagrangian multiform principle augments this by the fact that
the action should also be critical with respect to variations of the surface
of integration. For higher-dimensional PDEs one can consider higher forms. For
example, there is a Lagrangian 3-form description of the KP hierarchy [37].

##  3 A Lie group as multi-time

Many systems have symmetries that do not all commute with each other. Of
particular interest are those Hamiltonian systems where there exist functions
$H_{1},\ldots,H_{n+\ell}:T^{*}Q\rightarrow\mathbb{R}$ such that

$\\{H_{i},H_{k}\\}=0\qquad\text{for all }i\in\\{1,\ldots,n-\ell\\}\text{ and
all }k\in\\{1,\ldots,n+\ell\\}\,,$

and the remaining Poisson brackets may be nonzero. Systems like this are
called _non-commutative integrable_ or _degenerate integrable_ and a simple
adaptation of the Liouville-Arnold theorem applies to them [25, 8]. For
$\ell=0$ we recover Liouville integrability, and it can be shown that these
conditions for $\ell>0$ also imply Liouville integrability [8], hence the term
_superintegrability_ is also used for such systems [14]. Of course, one may
also be interested in non-integrable systems that still possess some smaller
amount of symmetries with nontrivial commutation relations.

If some of the $H_{k}$ have non-constant Poisson brackets, their flows will
not commute. Hence we cannot consider the flows of all
$H_{1},\ldots,H_{n+\ell}$ together as functions of some multi-time
$\mathbb{R}^{n+\ell}$. Indeed equation (2.1) breaks down because it now
depends on the order in which we list the flows $\Phi^{t_{i}}$, which defeats
the point of putting all flows on the same footing. But all is not lost. The
infinitesimal generators of the flows of the $H_{i}$ form a Lie algebra. We
can use a copy of the (universal covering) Lie group $G$ of this Lie algebra
as multi-time. The Hamiltonian “flow” on multi-time now depends on a Lie group
element $g$ instead of a number of time coordinates,

$\Phi:G\times T^{*}Q\rightarrow T^{*}Q:(g,z_{0})\mapsto\Phi^{g}(z_{0})=g\cdot
z_{0}\,,$ (3.1)

where $\cdot$ denotes the left group action of $G$ on $T^{*}Q$ by
symplectomorphisms. Hence $\Phi$ assigns to an initial condition $z_{0}\in
T^{*}Q$ its flow under an element $g\in G$ of multi-time. In case all
symmetries commute, $g$ would be the vector of times $(t_{1},\ldots,t_{n})$.

Unlike dynamical systems where the Lie group is the phase space (such as e.g.
in rigid body dynamics, [2, 24]), in this setting it is the space of
independent variables that possesses the structure of a Lie group $G$. Here we
take phase space to be a cotangent bundle $T^{*}Q$ for which no additional
structure is assumed.

We would like to think of the Lie group $G$ not just as the multi-time, but
also as a symmetry group acting on fields, with $g\in G$ acting on a field
$z:G\rightarrow T^{*}Q$ to produce a new field $(g\bullet z):G\rightarrow
T^{*}Q$ defined by

$(g\bullet z)(h)=z(gh)\,.$

Note that we defined both actions, $\cdot$ and $\bullet$, as left actions.
Alternatively, we could have adopted a convention where both are right
actions.

We are interested in those fields $z:G\rightarrow T^{*}Q$ for which the two
actions of $G$ agree: the flow over “time” $g\in G$ maps the field to its
transformation by $g$. We call such fields “symmetry group solutions” and
define them as follows.

######  Definition 3.1.

Let $G$ act by symplectic transformations on $T^{*}Q$ and denote this action
by $\cdot$. We say that a field $z:G\rightarrow T^{*}Q$ is a _symmetry group
solution_ of this group action if for all $h\in G$ there holds
$\Phi^{g}(z(h))=(g\bullet z)(h)$ or, equivalently,

$g\cdot z(h)=z(gh),$ (3.2)

######  Example 3.2.

Let $G=(\mathbb{R},+)$ act on $\mathbb{R}^{2}$, with coordinates $(q,p)$ by
horizontal translation:

$\Phi^{g}(q,p)=g\cdot(q,p)=(q+g,p)\,.$

Note that $\Phi$ is the flow of the Hamiltonian $H(q,p)=p$, assuming the
standard symplectic structure. For any constants $q_{0}$, $p_{0}$ we have a
symmetry group solution $z_{s}:G\rightarrow\mathbb{R}^{2}$ defined as

$z_{s}(g)=(q_{0}+g,p_{0})\,.$

Indeed we have

$(g\bullet z_{s})(h)=(q_{0}+g+h,p_{0})=g\cdot z_{s}(h)=\Phi^{g}(z_{s}(h))\,.$

Any function $z:G\rightarrow\mathbb{R}^{2}$ that is not of this form will not
be a symmetry group solution. As a specific counterexample, consider

$z_{c}(g)=(q_{0},p_{0}+g)\,.$

We have $(g\bullet z_{c})(h)=(q_{0},p_{0}+g+h)$ but $\Phi^{g}(z_{c}(h))=g\cdot
z_{c}(h)=(q_{0}+g,p_{0}+h)$.

######  Example 3.3.

As an example involving a nonabelian group, consider the Lie group $SE(2)$,
parameterised by $(x,y,\theta)$, with multiplication

$(x^{\prime},y^{\prime},\theta^{\prime})(x,y,\theta)=(x^{\prime}+x\cos\theta^{\prime}-y\sin\theta^{\prime},y^{\prime}+x\sin\theta^{\prime}+y\cos\theta^{\prime},\theta+\theta^{\prime})\,.$
(3.3)

It acts by Euclidean transformations of the $(q,p)$-plane:

$\Phi^{(x,y,\theta)}\begin{pmatrix}q\\\
p\end{pmatrix}=(x,y,\theta)\cdot\begin{pmatrix}q\\\
p\end{pmatrix}=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}q\\\
p\end{pmatrix}+\begin{pmatrix}x\\\ y\end{pmatrix}\,.$ (3.4)

Note that the one-parameter flow maps $\Phi^{(x,0,0)}$, $\Phi^{(0,y,0)}$ and
$\Phi^{(0,0,\theta)}$ correspond to the Hamiltonian systems given by
$H(q,p)=p$, $H(q,p)=-q$ and $H(q,p)=-\frac{1}{2}(p^{2}+q^{2})$, respectively.

Any function $z:SE(2)\to\mathbb{R}^{2}$ of the form

$z(x,y,\theta)=(x,y,\theta)\cdot\begin{pmatrix}q_{0}\\\ p_{0}\end{pmatrix},$

with constant $q_{0}$ and $p_{0}$, is a symmetry group solution. Indeed:

$\displaystyle\big{(}(x^{\prime},y^{\prime},\theta^{\prime})\bullet
z\big{)}(x,y,\theta)$
$\displaystyle=z\big{(}(x^{\prime},y^{\prime},\theta^{\prime})(x,y,\theta)\big{)}$
$\displaystyle=\big{(}(x^{\prime},y^{\prime},\theta^{\prime})(x,y,\theta)\big{)}\cdot\begin{pmatrix}q_{0}\\\
p_{0}\end{pmatrix}$
$\displaystyle=(x^{\prime},y^{\prime},\theta^{\prime})\cdot\left((x,y,\theta)\cdot\begin{pmatrix}q_{0}\\\
p_{0}\end{pmatrix}\right)$
$\displaystyle=(x^{\prime},y^{\prime},\theta^{\prime})\cdot z(x,y,\theta).$

The following proposition shows that all symmetry group solutions are of this
form.

######  Proposition 3.4.

The field $z:G\rightarrow T^{*}Q$ is a symmetry group solution if and only if

$g\cdot z(e)=z(g)\,,$ (3.5)

where $e$ is the unit element of the Lie group $G$.

###### Proof.

Equation (3.5) follows form Equation (3.2) by choosing $h=e$.

Considering both sides of Equation (3.5) as a function of $g$ and acting with
$h\bullet$ we find

$hg\cdot z(e)=z(hg)\,.$

On the other hand, considering both sides of Equation (3.5) as an element of
$T^{*}Q$ and acting with $h\cdot$ we find

$hg\cdot z(e)=h\cdot z(g)\,.$

Hence $h\cdot z(g)=z(hg)$, which is Equation (3.2) with $g$ and $h$
interchanged. ∎

Let $\mathfrak{g}$ be the Lie algebra of $G$. An abstract Lie algebra element
$\xi\in\mathfrak{g}$ has two differential geometric interpretations. First,
there is its representation as a left-invariant vector field
$\partial_{\xi}\in\mathfrak{X}(G)$. It acts on functions $z$ on $G$ as

$\partial_{\xi}z=\frac{\mathrm{d}}{\mathrm{d}s}\Big{|}_{s=0}\left(\exp(s\xi)\bullet
z\right)\,,$ (3.6)

i.e.

$(\partial_{\xi}z)(g)=\frac{\mathrm{d}}{\mathrm{d}s}\Big{|}_{s=0}z\big{(}\exp(s\xi)g\big{)}\,.$

In the abelian case, where $G=\mathbb{R}^{n}$ with coordinates
$t_{1},\ldots,t_{n}$, we can identify
$\partial_{\xi_{i}}=\frac{\partial}{\partial t_{i}}$. Second, there is the
infinitesimal generator $V_{\xi}\in\mathfrak{X}(T^{*}Q)$ of its action on
phase space. The vector field $V_{\xi}$ is defined by, for $z_{0}\in T^{*}Q$,

$V_{\xi}(z_{0})=\frac{\mathrm{d}}{\mathrm{d}s}\Big{|}_{s=0}\exp(s\xi)\cdot
z_{0}\,.$ (3.7)

If the action of $G$ on $T^{*}Q$ is locally effective, then the Lie algebra of
vector fields $V_{\xi}$ is isomorphic to $\mathfrak{g}$ [30, Theorem 2.62].

A symmetry group solution is characterised infinitesimally as follows:

######  Proposition 3.5.

The field $z:G\rightarrow T^{*}Q$ is a symmetry group solution if and only if,
for all $g\in G$,

$(\partial_{\xi}z)(g)=V_{\xi}(z(g))\,.$ (3.8)

###### Proof.

Equation (3.8) is the infinitesimal form of

$z\big{(}\exp(\xi)g\big{)}=\exp(\xi)\cdot z(g)\,,$

which is equivalent to Equation (3.2). ∎

The motivation behind these definitions and propositions is to extend to the
non-abelian case the familiar situation where one associates a time derivative
$\frac{\partial}{\partial t_{i}}$ to each Hamiltonian vector field $V_{H_{i}}$
on the phase space $T^{*}Q$, for a family of functions $H_{i}$ on $T^{*}Q$
which are in involution, $\\{H_{i},H_{j}\\}=0$. In that abelian case, for a
basis $\xi_{1},\ldots,\xi_{n}$ of the abelian Lie algebra $\mathfrak{g}$,
(3.8) reduces to

$\frac{\partial}{\partial
t_{i}}z(t_{1},\dots,t_{n})=V_{H_{i}}(z(t_{1},\dots,t_{n}))$

and we consider the collection of these $n$ differential equations on the
function $z$, giving rise to $n$ commuting time flows on the phase space
$T^{*}Q$.

###  3.1 Jet bundles over a Lie group

As sketched in Section 2, the variational principle on multi-time involves a
differential form depending on configuration variables and their derivatives,
i.e. depending on elements of a jet bundle. See for example [34] or [31,
Section 2.3] for a detailed treatment of jet bundles. Here, we will introduce
jet bundles over Lie groups, in a slightly unusual way which allows us to
understand how the Lie algebra structure affects prolongations of functions.
This will help us derive multi-time Euler-Lagrange equations after we have
formulated the variational principle. In the present section we discuss the
jet bundle of a real function on a Lie group. It is easy to extend this to
vector-valued functions (or functions into a single coordinate patch of
configuration space $Q$), but for ease of presentation we restrict the
discussion to real functions.

Consider a Lie group $G$ and its Lie algebra $\mathfrak{g}$, generated by
$\xi_{1},\ldots,\xi_{N}$, with structure relations

$[\xi_{i},\xi_{j}]=\sum_{k}\mathrm{C}_{ij}^{k}\xi_{k}\,.$ (3.9)

We consider fields $q:G\to\mathbb{R}$ as sections of the trivial bundle
$G\times\mathbb{R}\rightarrow G$ with coordinates $(g,q)$. The first jet
bundle is $\mathcal{J}_{1}\rightarrow G$ with
$\mathcal{J}_{1}=G\times\mathbb{R}\times\mathbb{R}^{N}$ and has coordinates
$(g,q,q_{1},\ldots,q_{n})$. The prolongation of a smooth function $q$ to the
first jet bundle is

$\operatorname{pr}_{1}q:G\to\mathcal{J}_{1}:g\mapsto(g,q(g),\partial_{\xi_{1}}q(g),\ldots,\partial_{\xi_{N}}q(g))\,.$

Starting from the second jet bundle we need to take into account the possibly
non-commuting derivatives. The derivatives $q_{ij}$ and $q_{ji}$ ($i\neq j$)
are not necessarily the same, but neither are they independent, because
$\mathfrak{g}$ comes with commutation relations. There are two jet bundles we
could consider. Below we only present their definition for the second order
jet bundle. Higher jet bundles can be constructed in an analogous way.

The first definition we present is of a jet bundle which ignores any relation
between $q_{ij}$ and $q_{ji}$ ($i\neq j$), hence it is “free” in a similar
sense as in “free algebra”.

######  Definition 3.6.

The _free jet bundle_ $\mathcal{J}_{2}\rightarrow G$ has coordinates
$(g,q,q_{i},q_{ij})$, where both $i\leq j$ and $i>j$ are allowed, i.e.

$(g,q,q_{i},q_{ij})=(g,q,q_{i},q_{ii},q_{ij},q_{ji})|_{i<j}\,.$

The fibre is thought of as the vector space spanned by $(q,q_{i},q_{ij})$. The
prolongation of $q$ to the second jet bundle is

$\operatorname{pr}_{2}q:G\to\mathcal{J}_{2}:g\mapsto(g,q(g),\partial_{\xi_{i}}q(g),\partial_{\xi_{j}}\partial_{\xi_{i}}q(g))\,.$

Taking into account the Lie algebra structure, we can reduce the free jet
bundle as follows. Guided by the commutation relation

$\partial_{\xi_{i}}\partial_{\xi_{j}}=\partial_{\xi_{j}}\partial_{\xi_{i}}+\sum_{k}\mathrm{C}_{ij}^{k}\partial_{\xi_{k}},$

we define the equivalence relation

$q_{ji}\sim q_{ij}+\sum_{k}\mathrm{C}_{ij}^{k}q_{k},$

and quotient the fibres of the free jet bundle by $\sim$. Since these
relations reflect the Lie group structure, they will become identities for
prolongations of fields. In particular, the variational principle involves
prolonged fields rather than abstract bundle variables, so it will be
independent of the choice of representative.

######  Definition 3.7.

The _quotiented jet bundle_ is $\widetilde{\mathcal{J}}_{2}\rightarrow G$,
with
$\widetilde{\mathcal{J}}_{2}=G\times\mathbb{R}\times\mathbb{R}^{N}\times\mathbb{R}^{\frac{N(N+1)}{2}}$,
and has coordinates $(g,q,q_{i},q_{ij})|_{i\leq j}$. The prolongation of $q$
to the second jet bundle is

$\operatorname{pr}_{2}q:G\to\mathcal{J}_{2}:g\mapsto(g,q(g),\partial_{\xi_{i}}q(g),\partial_{\xi_{j}}\partial_{\xi_{i}}q(g))|_{i\leq
j}\,.$

For a function $f:\mathcal{J}_{2}\rightarrow\mathbb{R}$ of the free jet
bundle, we denote by $\widetilde{f}$ its projection to the quotiented jet
bundle:

$\widetilde{f}(g,q,q_{i},q_{ij})|_{i\leq
j}=f\Big{(}g,q,q_{i},q_{ii},q_{ij},q_{ij}+\sum_{k}\mathrm{C}_{ij}^{k}q_{k}\Big{)}\Big{|}_{i<j}\,.$

To easily denote elements of higher free jet bundles we use index-strings:

######  Definition 3.8.

An _index-string_ is a finite sequence $\mathfrak{I}=s_{1},\ldots,s_{k}$ where
$k\in\mathbb{N}$ and $s_{i}\in\\{1,\ldots,N\\}$, where $N$ is the dimension of
the Lie group $G$. An index-string defines a derivative of the field,
$q_{\mathfrak{I}}=\partial_{\xi_{s_{k}}}\ldots\partial_{\xi_{s_{2}}}\partial_{\xi_{s_{1}}}q$,
which is a coordinate of the prolongation of $q$ to the $k$-th free jet
bundle.

To easily denote elements of higher quotiented jet bundles we use multi-
indices:

######  Definition 3.9.

A _multi-index_ is an element of $\mathbb{N}^{N}$. A multi-index
$I=(i_{1},\ldots,i_{N})\in\mathbb{N}^{N}$ defines a derivative of the field,
$q_{I}=\partial_{\xi_{N}}^{i_{N}}\ldots\partial_{\xi_{1}}^{i_{1}}q$, which is
a coordinate of the prolongation of $q$ to the $k$-th quotiented jet bundle.

When no confusion is possible, we will also use a string notation for multi-
indices, for example both “$12$” and “$21$” represent the multi-index
$(1,1,0,\ldots,0)$. We use the notation $\emptyset$ for the empty index-string
and for the corresponding multi-index $(0,\ldots,0)$.

For a function $f:\mathcal{J}_{k}\rightarrow\mathbb{R}$ of a free jet bundle
we denote by $\operatorname{D}_{i}$ the total derivative

$\operatorname{D}_{i}f=\sum_{\mathfrak{I}}\frac{\partial{f}}{\partial{q_{\mathfrak{I}}}}\partial_{\xi_{i}}q_{\mathfrak{I}}+\partial_{\xi_{i}}f\,,$

where the sum is over all index-strings $\mathfrak{I}$ and the final term is
the analogue of $\frac{\partial{f}}{\partial{t_{i}}}$ in the case of commuting
flows, which vanishes in case $f$ does not depend explicitly on $g\in G$. For
a function $\widetilde{f}:\widetilde{\mathcal{J}}_{k}\rightarrow\mathbb{R}$ of
a quotiented jet bundle we denote by $\widetilde{\operatorname{D}}_{i}$ the
total derivative

$\widetilde{\operatorname{D}}_{i}\widetilde{f}=\sum_{I}\frac{\partial{\widetilde{f}}}{\partial{q_{I}}}\widetilde{\partial_{\xi_{i}}q_{I}}+\partial_{\xi_{i}}\widetilde{f}\,,$

where the sum is over all multi-indices $I$. Recall that $\widetilde{\ }$
denotes projection onto the quotiented jet bundle, so
$\widetilde{\partial_{\xi_{i}}q_{I}}$ means that we evaluate
$\partial_{\xi_{i}}q_{I}$ while taking into account the commutation relations
to write it as a linear combination of well-ordered derivatives. For example,
if $i=1$ and $\widetilde{f}=q_{2}$ the only nonzero term in the sum is the one
with $I=(0,1,0,\ldots,0)$, i.e. with $q_{I}=q_{2}$, yielding

$\widetilde{\operatorname{D}}_{1}q_{2}=\widetilde{\partial_{\xi_{1}}q_{2}}=\widetilde{q_{21}}=q_{12}+\sum_{k}\mathrm{C}_{12}^{k}q_{k}\,.$

Note that for any function $f:G\times\mathcal{J}^{2}\rightarrow\mathbb{R}$
there holds
$\widetilde{\operatorname{D}}_{i}\widetilde{f}=\widetilde{\operatorname{D}_{i}f}$.

######  Proposition 3.10.

For functions $f:\mathcal{J}_{1}\rightarrow\mathbb{R}$ of the first free jet
bundle there holds

$\displaystyle\frac{\partial{}}{\partial{q}}\operatorname{D}_{i}f=\operatorname{D}_{i}\frac{\partial{f}}{\partial{q}}\,,$
(3.10a)
$\displaystyle\frac{\partial{}}{\partial{q_{j}}}\operatorname{D}_{i}f=\operatorname{D}_{i}\frac{\partial{f}}{\partial{q_{j}}}+\delta_{i}^{j}\frac{\partial{f}}{\partial{q}}\,,$
(3.10b)
$\displaystyle\frac{\partial{}}{\partial{q_{ji}}}\operatorname{D}_{i}f=\frac{\partial{f}}{\partial{q_{j}}}\,,$
(3.10c)
$\displaystyle\frac{\partial{}}{\partial{q_{ij}}}\operatorname{D}_{i}f=\delta_{i}^{j}\frac{\partial{}}{\partial{q_{j}}}\,,$
(3.10d)
$\displaystyle\frac{\partial{}}{\partial{q_{jk}}}\operatorname{D}_{i}f=0\qquad\text{if
}j,k\neq i\,,$ (3.10e)

where $\delta_{i}^{j}$ is the Kronecker delta.

###### Proof.

We have

$\operatorname{D}_{i}=q_{i}\frac{\partial{}}{\partial{q}}+\sum_{k}q_{ki}\frac{\partial{}}{\partial{q_{k}}}+\partial_{\xi_{i}}\,.$

Since none of the coefficients contain an undifferentiated $q$, it follows
that

$\frac{\partial{}}{\partial{q}}\operatorname{D}_{i}=\operatorname{D}_{i}\frac{\partial{}}{\partial{q}}\,.$

Furthermore, we find

$\frac{\partial{}}{\partial{q_{j}}}\operatorname{D}_{i}=\operatorname{D}_{i}\frac{\partial{}}{\partial{q_{j}}}+\delta_{i}^{j}\frac{\partial{}}{\partial{q}}$

and

$\frac{\partial{}}{\partial{q_{ji}}}\operatorname{D}_{i}=\operatorname{D}_{i}\frac{\partial{}}{\partial{q_{ji}}}+\frac{\partial{}}{\partial{q_{j}}}=\frac{\partial{}}{\partial{q_{j}}}\,,$

where the last equality holds because we are only considering function of the
first jet bundle. Similarly, if $j\neq i$ and $k\neq i$ we find
$\frac{\partial{}}{\partial{q_{ij}}}\operatorname{D}_{i}=0$ and
$\frac{\partial{}}{\partial{q_{jk}}}\operatorname{D}_{i}=0$. ∎

For the sake of completeness, we also state the corresponding result in the
quotiented jet bundle. However, in the calculations to follow we will always
commute total and partial derivatives using Proposition 3.10, before
projecting to the quotiented bundle.

######  Proposition 3.11.

For functions $f:\widetilde{\mathcal{J}}_{1}\rightarrow\mathbb{R}$ of the
first quotiented jet bundle there holds

$\displaystyle\frac{\partial{}}{\partial{q}}\widetilde{\operatorname{D}}_{i}f=\widetilde{\operatorname{D}}_{i}\frac{\partial{f}}{\partial{q}}\,,$
(3.11a)
$\displaystyle\frac{\partial{}}{\partial{q_{j}}}\widetilde{\operatorname{D}}_{i}f=\widetilde{\operatorname{D}}_{i}\frac{\partial{f}}{\partial{q_{j}}}+\delta_{i}^{j}\frac{\partial{f}}{\partial{q}}+\sum_{k>i}\mathrm{C}_{ik}^{j}\frac{\partial{f}}{\partial{q_{k}}}\,,$
(3.11b)
$\displaystyle\frac{\partial{}}{\partial{q_{ij}}}\widetilde{\operatorname{D}}_{i}f=\frac{\partial{}}{\partial{q_{ji}}}\widetilde{\operatorname{D}}_{i}f=\frac{\partial{}}{\partial{q_{j}}}\,,$
(3.11c)
$\displaystyle\frac{\partial{}}{\partial{q_{jk}}}\widetilde{\operatorname{D}}_{i}f=0\qquad\text{if
}j,k\neq i\,,$ (3.11d)

where $\delta_{i}^{j}$ is the Kronecker delta.

###### Proof.

The proof is analogous to the proof of Proposition 3.10, starting from the
expansion

$\displaystyle\widetilde{\operatorname{D}}_{i}$
$\displaystyle=q_{i}\frac{\partial{}}{\partial{q}}+\sum_{k}\widetilde{q_{ki}}\frac{\partial{}}{\partial{q_{k}}}+\partial_{\xi_{i}}$
$\displaystyle=q_{i}\frac{\partial{}}{\partial{q}}+\sum_{k\leq
i}q_{ki}\frac{\partial{}}{\partial{q_{k}}}+\sum_{k>i}\left(q_{ik}+\sum_{\ell}\mathrm{C}_{ik}^{\ell}q_{\ell}\right)\frac{\partial{}}{\partial{q_{k}}}+\partial_{\xi_{i}}\,.\qed$

###  3.2 Variational principle for functions on a Lie group

We are now in a position to formulate the variational principle on a Lie group
and derive the corresponding multi-time Euler-Lagrange equations. In this
subsection we will state the definition for general $d$-forms and derive some
results which will be helpful to carry out the calculus of variations. In
Sections 4 and 5 we will specialise the discussion to $d=1$ and $d=2$.

######  Definition 3.12.

Consider a $d$-form $\mathcal{L}[q]$ on a Lie group $G$. We say that
$q:G\rightarrow Q$ is _critical_ if for every $d$-dimensional submanifold
$\gamma\subset G$, we have

$\frac{\partial{}}{\partial{\varepsilon}}\Big{|}_{\varepsilon=0}\int_{\gamma}\mathcal{L}[q+\varepsilon\eta]=0\,,$
(3.12)

for any variation $\eta$ that vanishes (along with all its derivatives) at the
boundary of $\gamma$.

The following characterisation of critical fields makes use of the vertical
exterior derivative $\delta$ in the variational bicomplex (see for example
[3], [39, Appendix A], or [10]). This $\delta$ can be thought of as taking an
infinitesimal variation (Gateaux derivative) in a direction yet to be
specified. For example, $\delta q$ is an operator which maps a vector field to
the variation of $q$ in the direction of this vector field. In the variational
bicomplex, $\delta$ anti-commutes with $\mathrm{d}$.

######  Lemma 3.13.

The following are equivalent:

  1. $(i)$

$q:G\rightarrow Q$ is critical,

  2. $(ii)$

All infinitesimal variations of the exterior derivative of $\mathcal{L}$
vanish, i.e.

$\delta\mathrm{d}\mathcal{L}[q]=0\,.$

###### Proof.

$(i)\Rightarrow(ii)$ Let $q$ be critical and consider a $(d+1)$-dimensional
oriented submanifold $D$ of $G$ with boundary $\partial D$. Then by Stokes
theorem and the variational principle (3.12) with $\gamma=\partial D$ there
holds

$\int_{D}\delta\mathrm{d}\mathcal{L}=-\int_{D}\mathrm{d}\delta\mathcal{L}=-\int_{\gamma}\delta\mathcal{L}=0\,.$
(3.13)

Since $D$ is arbitrary, this implies that $\delta\mathrm{d}\mathcal{L}=0$.

$(ii)\Rightarrow(i)$ Reading Equation (3.13) from right to left, we see that
if $\delta\mathrm{d}\mathcal{L}=0$ then the variational principle (3.12) is
satisfied for all closed $d$-dimensional submanifolds $\gamma=\partial D$.
Below we argue that this implies that the variational principle holds on all
$d$-dimensional submanifolds.

In the variational principle it is sufficient to consider variations supported
in a small neighbourhood, because using a partition of unity we can write any
variation as a sum of variations with smaller supports. Hence we can assume
that the manifold $\gamma$ in Equation (3.12) is bounded, so we can extend it
to a closed $d$-dimensional submanifold $\bar{\gamma}$, such that
$\gamma\subset\bar{\gamma}$. Since we already established that the variational
principle holds on closed submanifolds, it now follows that is also holds on
$\gamma$. ∎

In the following sections, we will use Lemma 3.13 to derive the multi-time
Euler-Lagrange equations in the case of 1-forms and 2-forms. Before doing so,
we explore some properties of the vertical exterior derivative $\delta$.

For any function $P$ of the free jet bundle, there holds

$\delta P=\sum_{\mathfrak{I}}A^{\mathfrak{I}}\delta q_{\mathfrak{I}}\,,$
(3.14)

where

$A^{\mathfrak{I}}=\frac{\partial{P}}{\partial{q_{\mathfrak{I}}}}$

for every index-string $\mathfrak{I}$. Because of the redundant nature of the
set of index-strings, the $\delta q_{\mathfrak{I}}$ in this sum are not
independent when we take into account the commutation relations. An expansion
into $\delta q_{I}$, which are independent in the quotiented jet bundle, is
obtained in the following Lemma.

######  Lemma 3.14.

For any function $P:\mathcal{J}^{2}\rightarrow\mathbb{R}$ of the free second
jet bundle. The vertical exterior derivative of its projection onto the
quotiented bundle reads

$\delta\widetilde{P}=\sum_{I}B^{I}\delta q_{I}\,,$

where

$\begin{split}&B^{kk}=\widetilde{A^{kk}}\,,\\\
&B^{k\ell}=\widetilde{A^{k\ell}}+\widetilde{A^{\ell k}}\quad\text{where
}k<\ell\,,\\\
&B^{k}=\widetilde{A^{k}}+\sum_{m<n}\mathrm{C}_{mn}^{k}\widetilde{A^{nm}}\,,\\\
&B^{\emptyset}=\widetilde{A^{\emptyset}}\,,\end{split}$ (3.15)

$A^{J}=\frac{\partial{P}}{\partial{v_{J}}}$, and $\widetilde{\ }$ denotes the
projection onto the quotiented jet. In particular, we define
$\widetilde{\delta q_{\mathfrak{I}}}=\delta\widetilde{q_{\mathfrak{I}}}$,
hence $\widetilde{\delta P}=\delta\widetilde{P}$.

###### Proof.

We start from the expansion (3.14) in the free jet bundle. From the Lie
bracket relations (Equation (3.9)) it follows that

$\delta q_{ji}=\delta q_{ij}+\sum_{k}\mathrm{C}_{ij}^{k}\delta q_{k}\,.$

Using this relation we find the $B^{I}$ in terms of the $A^{\mathfrak{I}}$ as
in Equation (3.15). ∎

##  4 Lagrangian 1-forms on Lie groups

Let $\mathcal{L}[q]$ be a 1-form on $G$, depending on the first jet of a field
$q$. It is determined by its pairings $L_{i}[q]$ with the generators
$\partial_{\xi_{i}}\in\mathfrak{X}(G)$ of the Lie algebra,

$\partial_{\xi_{i}}\lrcorner\,\mathcal{L}[q]=L_{i}[q]\,.$

Once again we pose the variational principle that the action along every curve
must have a critical value with respect to variations of $q:G\rightarrow Q$.
The same multi-time Euler-Lagrange equations as in the commutative case apply:

######  Theorem 4.1.

If $\mathcal{L}$ only depends on the first jet of $q$, the variational
principle of Definition 3.12 is equivalent to the following set of multi-time
Euler-Lagrange equations

$\displaystyle\frac{\partial{L_{j}}}{\partial{q_{j}}}=\frac{\partial{L_{i}}}{\partial{q_{i}}}\,,$
(4.1) $\displaystyle\frac{\partial{L_{j}}}{\partial{q_{k}}}=0\qquad\text{ if
}k\neq j\,,$ (4.2) $\displaystyle\frac{\delta_{j}{L_{j}}}{\delta{q}}=0\,.$
(4.3)

###### Proof.

Let
$P_{ij}[q]=\partial_{\xi_{j}}\lrcorner\,\partial_{\xi_{i}}\lrcorner\,\mathrm{d}\mathcal{L}[q]$,
with $i<j$. Then

$\displaystyle P_{ij}$
$\displaystyle=\operatorname{D}_{i}(\partial_{\xi_{j}}\lrcorner\,\mathcal{L})-\operatorname{D}_{j}(\partial_{\xi_{i}}\lrcorner\,\mathcal{L})-\partial_{[\xi_{i},\xi_{j}]}\lrcorner\,\mathcal{L}$
$\displaystyle=\operatorname{D}_{i}L_{j}-\operatorname{D}_{j}L_{i}-\sum_{k}\mathrm{C}_{ij}^{k}L_{k}\,.$

We consider $P_{ij}$ as a function of the free jet bundle, i.e. we allow it to
contain derivatives that are not well-ordered. We have

$\delta P_{ij}=\sum_{\mathfrak{I}}A^{\mathfrak{I}}\delta q_{\mathfrak{I}}\,,$

where the sum is over all index-strings and

$A^{\mathfrak{I}}=\frac{\partial{P_{ij}}}{\partial{q_{\mathfrak{I}}}}\,.$

Since we are on the free jet bundle, we are ignoring the commutation
relations, hence the $\delta q_{\mathfrak{I}}$ are not independent. To remedy
this we use Lemma 3.14 and find

$\delta P_{ij}=\sum_{I}B^{I}\delta q_{I}\,,$

where the sum is over all multi-indices and the $B^{I}$ are given by Equation
(3.15). In particular, using Proposition 3.10 we find

$\displaystyle
B^{ii}=\widetilde{A^{ii}}=\widetilde{\frac{\partial{P_{ij}}}{\partial{q_{ii}}}}=\frac{\partial{L_{j}}}{\partial{q_{i}}}\,,$
$\displaystyle
B^{ij}=\widetilde{A^{ij}}+\widetilde{A^{ji}}=\widetilde{\frac{\partial{P_{ij}}}{\partial{q_{ij}}}}+\widetilde{\frac{\partial{P_{ij}}}{\partial{q_{ji}}}}=\frac{\partial{L_{i}}}{\partial{q_{i}}}-\frac{\partial{L_{j}}}{\partial{q_{j}}}\,,$
$\displaystyle
B^{ik}=\widetilde{A^{ik}}+\widetilde{A^{ki}}=\widetilde{\frac{\partial{P_{ij}}}{\partial{q_{ik}}}}+\widetilde{\frac{\partial{P_{ij}}}{\partial{q_{ki}}}}=-\frac{\partial{L_{j}}}{\partial{q_{k}}}\,.$

Hence $\delta P_{ij}=0$ implies Equations (4.1) and (4.2). Furthermore, using
Equations (4.1)–(4.2),

$\displaystyle
B^{i}=\widetilde{A^{i}}+\sum_{m<n}\mathrm{C}_{mn}^{i}\widetilde{A^{nm}}$
$\displaystyle=\widetilde{\frac{\partial{P_{ij}}}{\partial{q_{i}}}}+\mathrm{C}_{ji}^{i}\widetilde{\frac{\partial{P_{ij}}}{\partial{q_{ij}}}}$
$\displaystyle=\frac{\partial{L_{j}}}{\partial{q}}-\operatorname{D}_{j}\frac{\partial{L_{i}}}{\partial{q_{i}}}+\mathrm{C}_{ij}^{i}\frac{\partial{L_{i}}}{\partial{q_{i}}}+\mathrm{C}_{ji}^{i}\frac{\partial{L_{i}}}{\partial{q_{i}}}$
$\displaystyle=\frac{\delta_{j}{L_{j}}}{\delta{q}}\,,$

hence $\delta P_{ij}=0$ also implies Equation (4.3).

We have shown, using Lemma 3.13, that the variational principle implies
Equations (4.1)–(4.3). The converse can be proved by retracing our steps and
observing that the equations $B^{k}=0$ ($k\neq i,j$) and $B^{\emptyset}=0$ are
consequences of Equations (4.1)–(4.3) as well. ∎

An alternative proof can be given using the stepped curve approach of [39],
which for 1-forms easily generalises to the Lie group setting. The stepped
curve approach also applies to forms depending on higher jets, and again leads
to the same multi-time Euler-Lagrange equations as in the commutative case.

###  4.1 Building a 1-form from symmetries of a given Lagrangian

In this subsection we start from a mechanical Lagrangian (of Newtonian type)
and build a Lagrangian 1-form describing its variational symmetry group. As is
common in this context, we use $\dot{q}$ as shorthand for the time-derivative
of $q$, which will be identified $\partial_{\xi_{1}}q$ in the multi-time
setting. Suppose we are given a mechanical system on $TQ$ with Lagrangian

$L_{1}(q,\dot{q})=\frac{1}{2}|\dot{q}|^{2}-U(q)$ (4.4)

and a group $G_{0}$ of variational symmetries of $L_{1}$. We allow elements of
$G_{0}$ to be generalised symmetries, meaning that they do not necessarily act
as point transformations on $Q$ and have infinitesimal generators that
potentially depend on derivatives of the curve. We do require the symmetries
in $G_{0}$ to act on the space of curves from $\mathbb{R}$ to $Q$. This rules
out some generalised symmetries which are defined infinitesimally as a
generalised vector field but cannot be integrated to a symmetry transformation
[31, Chapter 5]. By definition, $g$ is a variational symmetry if for any
$q:\mathbb{R}\rightarrow Q$:

$L_{1}\left((g\cdot q)(t),\frac{\mathrm{d}}{\mathrm{d}t}(g\cdot
q)(t)\right)=L_{1}(q(t),\dot{q}(t))+\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{F}_{g}(q(t),\dot{q}(t))$

for some function $\mathcal{F}_{g}:TQ\rightarrow\mathbb{R}$.

We assume that the infinitesimal generators of all $g\in G_{0}$ are
prolongations of vector fields of the form $W=w(q,\dot{q})\partial_{q}$:

$\operatorname{pr}(W)=w(q,\dot{q})\partial_{q}+\left(\frac{\partial{w}}{\partial{q}}\dot{q}+\frac{\partial{w}}{\partial{\dot{q}}}\ddot{q}\right)\partial_{\dot{q}}\,.$

The infinitesimal characterisation of a variational symmetry reads

$\operatorname{pr}(W)L_{1}(q,\dot{q})=\frac{\mathrm{d}}{\mathrm{d}t}F_{W}(q,\dot{q})\,,$
(4.5)

where the function $F_{W}$ is called the _flux_ of the variational symmetry.

In this setting, a natural choice of multi-time is $G=\mathbb{R}\times G_{0}$.
The additional $\mathbb{R}$ represents translations in time $t$. Picking some
reference time $(t_{0},e)\in G$ and initial conditions $(q_{0},v_{0})\in TQ$
such that $q(t_{0},e)=q_{0}$, $\dot{q}(t_{0},e)=v_{0}$, the Euler-Lagrange
equation of the Lagrangian (4.4) defines $q(t,e)$ for all $t\in\mathbb{R}$.
This solution can be extended to a symmetry group solution on $G$ by

$q(t,h)=\Phi^{h}(q(t,e))=h\cdot q(t,e)$ (4.6)

for $h\in G_{0}$ (see Proposition 3.4). When writing (4.6), we took advantage
of the fact that the $t$-flow and $\Phi^{h}$ commute (because $G_{0}$ is a
symmetry group of $L_{1}$), hence the order in which the two flows are applied
does not matter.

The infinitesimal characterisation (see Proposition 3.5) of a symmetry group
solution is that for all $g\in G$ and $\xi\in\mathfrak{g}$:

$\displaystyle\ddot{q}(g)$ $\displaystyle=-U^{\prime}(q(g))\,,$
$\displaystyle\partial_{\xi}q(g)$
$\displaystyle=W_{\xi}q(g)=w_{\xi}(q(g),\dot{q}(g))\,,$

where $W_{\xi}=w_{\xi}\partial_{q}$ is the infinitesimal generator of the
group action and the differential operator $\partial_{\xi}$ is defined in
Equation (3.6).

Let $\xi_{1}=\partial_{t},\xi_{2},\ldots,\xi_{N}$ be a basis of the Lie
algebra $\mathfrak{g}$ of $G$ and $w_{\xi_{i}}(q,\dot{q})\partial_{q}$ the
corresponding generalised vector fields, which are assumed to be variational
symmetries of (4.4) with fluxes $F_{i}(q,\dot{q})$. Following [32] we consider
for $i\geq 2$

$\displaystyle L_{i}(q,q_{1},q_{i})$
$\displaystyle=\frac{\partial{L_{1}}}{\partial{q_{1}}}(q_{i}-w_{\xi_{i}}(q,q_{1}))+F_{i}(q,q_{1})$
$\displaystyle=q_{1}q_{i}-q_{1}w_{\xi_{i}}(q,q_{1})+F_{i}(q,q_{1})\,,$ (4.7)

where $q_{i}=\partial_{\xi_{i}}q$. The key point here is that we do not assume
that the variational symmetries commute. This is a departure from the setting
of [32] and other previous works on Lagrangian multiforms. We allow a general
Lie algebra structure:

$[\xi_{i},\xi_{j}]=\sum_{k}\mathrm{C}_{ij}^{k}\xi_{k}\,.$ (4.8)

Given $q:G\rightarrow Q$, we define the _Lagrangian 1-form $\mathcal{L}[q]$_
on $G$ by

$\xi_{i}\,\lrcorner\,\mathcal{L}[q]=L_{i}(q,q_{1},q_{i})\,,$ (4.9)

where $L_{i}$ is defined by Equation (4.7) for $i\geq 2$ and by Equation (4.4)
for $i=1$. Note that if all $\xi_{i}$ commute, then $G\cong\mathbb{R}^{N}$ and
we can choose coordinates $t_{i}$ such that
$\mathcal{L}=\sum_{i=1}^{N}L_{i}\,\mathrm{d}t_{i}$, which is the familiar
expression for a Lagrangian $1$-form, as it appears in the literature in the
context of commuting symmetries. The definition of $\mathcal{L}$ is
independent of the basis $\xi_{1},\ldots,\xi_{n}$, as the following
proposition shows.

######  Proposition 4.2.

For every $\xi=\sum_{k}\alpha_{k}\xi_{k}\in\mathfrak{g}$ and every
$q:G\rightarrow Q$ there holds

$\xi\,\lrcorner\,\mathcal{L}[q]=q_{1}\partial_{\xi}q-q_{1}w_{\xi}(q,q_{1})+\sum_{k}\alpha_{k}F_{k}(q,q_{1})\,,$

where $w_{\xi}=\sum_{k}\alpha_{k}w_{\xi_{k}}$ is the characteristic of the
generalised vector field on $Q$ induced by $\xi$ and $F_{k}$ are the fluxes of
the variational symmetries $\xi_{k}$, with $F_{1}=\mathcal{L}_{1}$ and
$w_{\xi_{1}}=q_{1}$.

###### Proof.

We have

$\displaystyle\xi\,\lrcorner\,\mathcal{L}=\sum_{k}\alpha_{k}\xi_{k}\,\lrcorner\,\mathcal{L}$
$\displaystyle=\sum_{k}\alpha_{k}L_{k}(q,q_{1},q_{k})$
$\displaystyle=\sum_{k}\alpha_{k}\left(q_{1}q_{k}-q_{1}w_{\xi_{k}}(q,q_{1})+F_{k}(q,q_{1})\right)$
$\displaystyle=q_{1}\partial_{\xi}q-\sum_{k}\alpha_{k}q_{1}w_{\xi_{k}}(q,q_{1})+\sum_{k}\alpha_{k}F_{k}(q,q_{1})\,.\qed$

######  Theorem 4.3.

A field $q:G\rightarrow Q$ is a symmetry group solution of the group action of
$G$ if and only if it is critical in the sense of Definition 3.12 for the
Lagrangian 1-form $\mathcal{L}$.

The proof is essentially the same as that of [32, Prop. 7.1]. First we prove
some Lemmas.

######  Lemma 4.4 ([32, Lemma 3.1]).

Let $F$ be the flux of a variational symmetry $W=w\partial_{q}$ of the
Lagrangian (4.4). There holds

$\frac{\partial{F}}{\partial{\dot{q}^{\alpha}}}=\frac{\partial{w^{\beta}}}{\partial{\dot{q}^{\alpha}}}\dot{q}^{\beta}\,,$

where the Greek upper indices denote vector components and summation over
repeated indices is assumed.

###### Proof.

Using the chain rule we find from Equation (4.5)

$-\frac{\partial{U}}{\partial{q^{\beta}}}w^{\beta}+\dot{q}^{\beta}\left(\frac{\partial{w^{\beta}}}{\partial{q^{\alpha}}}\dot{q}^{\alpha}+\frac{\partial{w^{\beta}}}{\partial{\dot{q}^{\alpha}}}\ddot{q}^{\alpha}\right)=\frac{\partial{F}}{\partial{q^{\alpha}}}\dot{q}^{\alpha}+\frac{\partial{F}}{\partial{\dot{q}^{\alpha}}}\ddot{q}^{\alpha}\,.$

Since this holds for any curve $q$, the coefficients of $\ddot{q}$ must match,
hence
$\dot{q}^{\beta}\frac{\partial{w^{\beta}}}{\partial{\dot{q}^{\alpha}}}=\frac{\partial{F}}{\partial{\dot{q}^{\alpha}}}$.
∎

######  Lemma 4.5.

On solutions of the Euler-Lagrange equation $\ddot{q}+U^{\prime}(q)=0$ there
holds

$\frac{\partial{F}}{\partial{q^{\alpha}}}=\frac{\mathrm{d}w^{\alpha}}{\mathrm{d}t}+\dot{q}^{\beta}\frac{\partial{w^{\beta}}}{\partial{q^{\alpha}}}\,,$

where summation over repeated indices is assumed.

###### Proof.

We have

$\displaystyle\frac{\partial{F}}{\partial{q^{\alpha}}}$
$\displaystyle=\frac{\partial{}}{\partial{\dot{q}^{\alpha}}}\frac{\mathrm{d}F}{\mathrm{d}t}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial{F}}{\partial{\dot{q}^{\alpha}}}$
$\displaystyle=\frac{\partial{}}{\partial{\dot{q}^{\alpha}}}\operatorname{pr}(W)L_{1}-\frac{\mathrm{d}}{\mathrm{d}t}\left(\dot{q}^{\beta}\frac{\partial{w^{\beta}}}{\partial{\dot{q}^{\alpha}}}\right)$
$\displaystyle=\frac{\partial{}}{\partial{\dot{q}^{\alpha}}}\left(\dot{q}^{\beta}\frac{\mathrm{d}w^{\beta}}{\mathrm{d}t}-\frac{\partial{U}}{\partial{q^{\beta}}}w^{\beta}\right)-\ddot{q}^{\beta}\frac{\partial{w^{\beta}}}{\partial{\dot{q}^{\alpha}}}-\dot{q}^{\beta}\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial{w^{\beta}}}{\partial{\dot{q}^{\alpha}}}$
$\displaystyle=\frac{\mathrm{d}w^{\alpha}}{\mathrm{d}t}+\dot{q}^{\beta}\frac{\partial{}}{\partial{\dot{q}^{\alpha}}}\frac{\mathrm{d}w^{\beta}}{\mathrm{d}t}-\frac{\partial{U}}{\partial{q^{\beta}}}\frac{\partial{w^{\beta}}}{\partial{\dot{q}^{\alpha}}}-\ddot{q}^{\beta}\frac{\partial{w^{\beta}}}{\partial{\dot{q}^{\alpha}}}-\dot{q}^{\beta}\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial{w^{\beta}}}{\partial{\dot{q}^{\alpha}}}$
$\displaystyle=\frac{\mathrm{d}w^{\alpha}}{\mathrm{d}t}+\dot{q}^{\beta}\frac{\partial{w^{\beta}}}{\partial{q^{\alpha}}}-\left(\ddot{q}^{\beta}+\frac{\partial{U}}{\partial{q^{\beta}}}\right)\frac{\partial{w^{\beta}}}{\partial{\dot{q}^{\alpha}}}\,.\qed$

###### Proof of Theorem 4.3.

Critical fields $q:G\rightarrow Q$ are characterised by the multi-time Euler-
Lagrange equations (4.1)–(4.3). In particular, Equation (4.3) with $j=1$
yields

$q_{11}+U^{\prime}(q)=0$

and Equation (4.2) with $k=1$ yields

$q_{j}^{\alpha}-w_{\xi_{j}}^{\alpha}-q_{1}^{\beta}\frac{\partial{w_{\xi_{j}}^{\alpha}}}{\partial{q_{1}^{\beta}}}+\frac{\partial{F}}{\partial{q_{1}^{\alpha}}}=0\,,$

which by Lemma 4.4 is equivalent to

$q_{j}-w_{\xi_{j}}=0\,.$

Equation (4.1) is equivalent to the trivial equation $\dot{q}=\dot{q}$.
Equation (4.3) with $j\neq 1$ is a differential consequence of the previous
equations, as the following computation shows. We have

$\displaystyle\eqref{EL1-line}$ $\displaystyle\Leftrightarrow-
q_{1}^{\beta}\frac{\partial{w_{\xi_{j}}^{\beta}}}{\partial{q^{\alpha}}}+\frac{\partial{F_{j}}}{\partial{q^{\alpha}}}-q_{1j}^{\alpha}=0\xLeftrightarrow{\text{Lemma
\ref{lemma-Fq}}}\partial_{1}w_{\xi_{j}}^{\alpha}-q_{1j}^{\alpha}=0\,,$

which is equivalent to $\partial_{1}(w_{\xi_{j}}^{\alpha}-q_{j}^{\alpha})$,
hence it is a consequence of Equation (4.2) with $k=1$. ∎

###  4.2 Building a 1-form from Hamiltonians

Suppose we have a Hamiltonian of Newtonian type,
$H(q,p)=\frac{1}{2}|p|^{2}+U(q)$, with a number of symmetries defined by
Hamiltonians $H_{2},\ldots,H_{N}$. Let $X_{H_{1}},\ldots,X_{H_{N}}$ be the
corresponding Hamiltonian vector fields on $T^{*}Q$.

######  Proposition 4.6.

If there exist constants $\mathrm{C}_{ij}^{k}$ such that

$\\{H_{i},H_{j}\\}=\sum_{k}\mathrm{C}_{ij}^{k}H_{k}\,,$ (4.10)

then the vector space spanned by the Hamiltonian vector fields
$X_{H_{1}},\ldots,X_{H_{N}}$ is a Lie algebra with Lie bracket given by the
commutator.

###### Proof.

We have

$\displaystyle[X_{H_{i}},X_{H_{j}}]f$
$\displaystyle=\\{H_{i},\\{H_{j},f\\}\\}-\\{H_{j},\\{H_{i},f\\}\\}$
$\displaystyle=\\{\\{H_{i},H_{j}\\},f\\}$
$\displaystyle=\sum_{k}\mathrm{C}_{ij}^{k}\\{H_{k},f\\}\,.$ (4.11)

Hence $[X_{H_{i}},X_{H_{j}}]=\sum_{k}\mathrm{C}_{ij}^{k}X_{H_{k}}$. ∎

In this case, we can take as our multi-time the universal covering Lie group
$G$ of the Lie algebra $\mathfrak{g}$. In other words, we take the group of
symmetries generated by $H_{1},\ldots,H_{N}$ as multi-time.

The role played in the commuting case by the time derivatives
$\partial_{t_{i}}$ is now played by a basis of left-invariant vector fields
$\partial_{\xi_{i}}$ on $G$. They satisfy the same Lie-algebraic relations as
the $X_{H_{i}}$, but we choose not to identify them. This is to emphasise the
conceptual difference between the vector fields $\partial_{\xi_{i}}$ on multi-
time $G$ and the vector fields $X_{H_{i}}$ on the phase space $T^{*}Q$. The
Hamiltonian vector fields $X_{H_{i}}$ play the role of the symmetry generators
$V_{\xi_{i}}$ in Proposition 3.5.

If the Poisson relations fail to be linear, i.e. if Equation (4.10) does not
hold for any constants $\mathrm{C}_{ij}^{k}$, then we need a different
approach to find a Lie algebra and Lie group on which to formulate the
variational principle. This case will be handled in Section 4.2.1. First we
will discuss the construction of a suitable 1-form on $G$ in case Equation
(4.10) does hold.

In the context of commuting flows, the relation between Lagrangian 1-forms and
Hamiltonian structures is well understood [38, 43], at least if they are of
Newtonian form. We use the same construction here and define the 1-form
$\mathcal{L}[q]$ by

$\partial_{\xi_{i}}\lrcorner\,\mathcal{L}[q]=L_{i}[q]:=q_{1}q_{i}-H_{i}(q,q_{1})\,,$
(4.12)

where we identify $p=q_{1}$. Note that $L_{1}=\frac{1}{2}q_{1}-U(q)$.

######  Proposition 4.7.

The multi-time Euler-Lagrange equations for $\mathcal{L}$, as defined in
Equation (4.12), are equivalent to the set of canonical Hamiltonian equations
for $H_{1},\ldots,H_{N}$, under the identification $p=q_{1}.$

###### Proof.

The multi-time Euler-Lagrange equations of type (4.1) are trivially satisfied.
The multi-time Euler-Lagrange equations of type (4.2) yield

$q_{i}=\frac{\partial{H_{i}(q,q_{1})}}{\partial{q_{1}}}$

for $i>0$ (and nothing for $L_{1}$). Finally, the multi-time Euler-Lagrange
equations of type (4.3) yield

$q_{1i}=-\frac{\partial{H_{i}(q,q_{1})}}{\partial{q}}$

for $i>0$ and

$q_{11}=-\frac{\partial{V(q)}}{\partial{q}}\,.\qed$

######  Proposition 4.8.

The exterior derivative $\mathrm{d}\mathcal{L}$, where $\mathcal{L}$ is as in
Equation (4.12), attains a double zero on solutions to the multi-time Euler-
Lagrange equations.

###### Proof.

First, observe that

$\displaystyle\partial_{[\xi_{i},\xi_{j}]}\lrcorner\,\mathcal{L}$
$\displaystyle=\sum_{k}\mathrm{C}_{ij}^{k}L_{k}$
$\displaystyle=q_{1}\sum_{k}\mathrm{C}_{ij}^{k}q_{k}-\sum_{k}\mathrm{C}_{ij}^{k}H_{k}(q,q_{1})$
$\displaystyle=q_{1}[\partial_{\xi_{i}},\partial_{\xi_{j}}]q-\\{H_{i},H_{j}\\}(q,q_{1})\,.$

Hence, by linearity, for all $\xi,\nu\in\mathfrak{g}$ there holds

$\partial_{[\xi,\nu]}\lrcorner\,\mathcal{L}=q_{1}[\partial_{\xi},\partial_{\nu}]q-\\{H_{\xi},H_{\nu}\\}(q,q_{1})\,.$

Now, identifying $p=q_{1}$, we find

$\displaystyle\partial_{\nu}\lrcorner\,\partial_{\xi}\lrcorner\,\mathrm{d}\mathcal{L}$
$\displaystyle=\partial_{\xi}L_{\nu}-\partial_{\nu}L_{\xi}-\partial_{[\xi,\nu]}\lrcorner\,\mathcal{L}$
$\displaystyle=\partial_{\xi}p\partial_{\nu}q-\partial_{\nu}p\partial_{\xi}q-\partial_{\xi}H_{\nu}+\partial_{\nu}H_{\xi}+\\{H_{\xi},H_{\nu}\\}$
$\displaystyle=\partial_{\xi}p\partial_{\nu}q-\partial_{\nu}p\partial_{\xi}q-\frac{\partial{H_{\nu}}}{\partial{q}}\partial_{\xi}q-\frac{\partial{H_{\nu}}}{\partial{p}}\partial_{\xi}p+\frac{\partial{H_{\xi}}}{\partial{q}}\partial_{\nu}q-\frac{\partial{H_{\xi}}}{\partial{p}}\partial_{\nu}p+\\{H_{\xi},H_{\nu}\\}$
$\displaystyle=\left(\partial_{\xi}p+\frac{\partial{H_{\xi}}}{\partial{q}}\right)\left(\partial_{\nu}q-\frac{\partial{H_{\nu}}}{\partial{p}}\right)-\left(\partial_{\nu}p+\frac{\partial{H_{\nu}}}{\partial{q}}\right)\left(\partial_{\xi}q-\frac{\partial{H_{\xi}}}{\partial{p}}\right)\,,$

where the last equality makes use of the canonical form of the Poisson
bracket,
$\\{f,g\\}=\frac{\partial{f}}{\partial{p}}\frac{\partial{g}}{\partial{q}}-\frac{\partial{g}}{\partial{p}}\frac{\partial{f}}{\partial{q}}.$
∎

####  4.2.1 Nonlinear Poisson relations

If the Poisson relations are not linear, then (some of) the coefficients of
$\\{H_{k},f\\}$ in Equation (4.11) will depend on the $H_{i}$ instead of being
constant. This would mean that the commutators between the vector fields
depend on the values of the Hamiltonians, which means that the vector fields
themselves do not constitute a Lie algebra. However, even in this case, we can
find a Lie algebra underlying the system.

Consider the commutative algebra $\mathcal{F}$ generated by the functions
$H_{1},\ldots,H_{N}:T^{*}Q\rightarrow\mathbb{R}$, i.e. the algebra of
functions of $H_{1},\ldots,H_{N}$. If this algebra is closed under the
canonical Poisson bracket, then it is an example of a _function group_ in the
sense of Lie [20].333For all examples in this work it would be sufficient to
consider the subalgebra of this function group consisting of polynomials in
$H_{1},\ldots,H_{N}$. Such function groups have been considered in geometric
mechanics [44, 18] and control theory [41]. The canonical Poisson bracket on
$T^{*}Q$ turns $\mathcal{F}$ into a Lie algebra

$\mathfrak{h}=(\mathcal{F},\\{\cdot,\cdot\\})\,.$

Even though structure constants in the sense of Equation (4.10) do not exist
in this case, one can define structure constants on $\mathfrak{h}$ by choosing
a basis $\\{f_{\alpha}\\}$ of $\mathfrak{h}$. (Every vector space has a basis
if the axiom of choice is assumed.) We then take
$\mathrm{C}_{\alpha\beta}^{\gamma}$ such that

$\\{f_{\alpha},f_{\beta}\\}=\sum_{\gamma}\mathrm{C}_{\alpha\beta}^{\gamma}f_{\gamma}\,.$

Even though the index set that $\gamma$ belongs to is infinite, the above sum
will only contain a finite number of nonzero terms, because every vector can
be written as a linear combination of finitely many basis elements.

Using the correspondence between functions $f\in\mathcal{F}$ and their
canonical Hamiltonian vector fields $X_{f}$, we can see that $\mathfrak{h}$,
modulo additive constants, is isomorphic to the Lie algebra

$(\\{X_{f}\mid f\in\mathcal{F}\\},[\cdot,\cdot])\,,$

where $[\cdot,\cdot]$ denotes the commutator of vector fields on $T^{*}Q$.
Note that the finite set of vector fields $\\{X_{H_{1}},\ldots,X_{H_{N}}\\}$
does not necessarily span a Lie algebra. On the other hand, the infinite-
dimensional $\mathfrak{h}$ is always a Lie algebra, regardless of what the
Poisson relations look like.

We would like to define a 1-form $\mathcal{L}$ and impose the variational
principle on a Lie group that has $\mathfrak{h}$ as its Lie algebra. However,
since Lie’s (converse) third theorem does not hold in an infinite-dimensional
setting [42], the existence of such a Lie group cannot be guaranteed. At first
sight this seems to be an insurmountable obstruction, since we need a Lie
group (or at least a manifold) to formulate the variational principle of
Definition 3.12. However, the characterisation $\delta\mathrm{d}\mathcal{L}=0$
is formulated within a single tangent space of multi-time, so it can be
considered without referring to the manifold structure. In particular, we can
impose it on the Lie algebra $\mathfrak{h}$ with no requirement for a Lie
group structure. Lemma 3.13 shows the characterisation
$\delta\mathrm{d}\mathcal{L}=0$ to be equivalent to the variational principle
whenever a Lie group exists, and the formal calculations will be the same
whether or not a Lie group associated to $\mathfrak{h}$ exists. Below we
provide some details of this construction.

As multi-time we consider a subgroup of the group of symplectic
diffeomorphisms on Q, given by

$G_{\mathcal{F}}=\\{\exp(X_{f_{1}})\circ\ldots\circ\exp(X_{f_{m}})\mid
m\in\mathbb{N},f_{1},\ldots,f_{m}\in\mathcal{F}\\}\,,$ (4.13)

where $\exp$ denotes the flow of the vector field over one unit time. This is
the group of all transformations which can be obtained as concatenation of
finitely many flows of Hamiltonian vector fields with Hamiltonian function in
$\mathcal{F}$. We do not claim that $G_{\mathcal{F}}$ has a manifold
structure. For each pair $(g,f)\in G_{\mathcal{F}}\times\mathcal{F}$ we
consider the smooth one-parameter subgroup $\\{\exp(tX_{f})\circ g\mid
t\in\mathbb{R}\\}$. This allows us to define the derivative of a field
$q:G_{\mathcal{F}}\rightarrow Q$ with respect to $f\in\mathcal{F}$:

$(\partial_{f}q)(g)=\frac{\mathrm{d}}{\mathrm{d}t}q(\exp(tX_{f})\circ
g)\big{|}_{t=0}\,,$ (4.14)

where we assume that $q$ is smooth in the sense that all such derivatives
exist. We check that this definition leads to the usual relation between
commutators and Poisson brackets:

######  Proposition 4.9.

There holds
$[\partial_{f_{1}},\partial_{f_{2}}]=\partial_{\\{f_{1},f_{2}\\}}$.

###### Proof.

For $i=1,2$ we denote by $\Phi_{i}^{t}=\exp(tX_{f_{i}})$ the flow of
$X_{f_{i}}$ over time $t$. For any $q:G_{\mathcal{F}}\rightarrow Q$ and $g\in
G_{\mathcal{F}}$ we have

$\displaystyle[\partial_{f_{1}},\partial_{f_{2}}]q(g)$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}s}\frac{\mathrm{d}}{\mathrm{d}t}\left(q\left(\Phi_{2}^{s}\circ\Phi_{1}^{t}\circ
g\right)-q\left(\Phi_{1}^{t}\circ\Phi_{2}^{s}\circ
g\right)\right)\Big{|}_{s=t=0}$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}s}\frac{\mathrm{d}}{\mathrm{d}t}\left(q\left(\Phi_{2}^{s}\circ\Phi_{1}^{t}\circ\Phi_{2}^{-s}\circ\Phi_{1}^{-t}\circ
y\right)-q(y)\right)\Big{|}_{s=t=0}\,,$

where $y=\Phi_{1}^{t}\circ\Phi_{2}^{s}\circ g$. Denote by
$\Phi_{\\{f_{1},f_{2}\\}}$ the flow of the commutator of $X_{f_{1}}$ and
$X_{f_{2}}$. We have that

$\Phi_{2}^{s}\circ\Phi_{1}^{t}\circ\Phi_{2}^{-s}\circ\Phi_{1}^{-t}=\Phi_{\\{f_{1},f_{2}\\}}^{st}+O(s^{2}+t^{2})\,.$

Hence we find

$\displaystyle[\partial_{f_{1}},\partial_{f_{2}}]q(g)$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}s}\frac{\mathrm{d}}{\mathrm{d}t}\left(q\left(\Phi_{\\{f_{1},f_{2}\\}}^{st}\circ
y\right)-q(y)\right)\Big{|}_{s=t=0}$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}s}\frac{\mathrm{d}}{\mathrm{d}t}\left(st\partial_{\\{f_{1},f_{2}\\}}q(y)+O(s^{2}t^{2})\right)\Big{|}_{s=t=0}=\partial_{\\{f_{1},f_{2}\\}}q(g)\,.\qed$

Given a field $q:G_{\mathcal{F}}\rightarrow Q$, we define a 1-tensor
$\mathcal{L}[q]$ by the condition that for all $f\in\mathcal{F}$ there holds

$f\lrcorner\,\mathcal{L}[q]=q_{1}\cdot\partial_{f}q-f(q,q_{1})\,,$

where $q_{1}=\partial_{H_{1}}q$. We call $\mathcal{L}[q]$ a 1-tensor, not a
1-form, because we do not assume a manifold structure on $G_{\mathcal{F}}$.
Nevertheless, we can consider a formal exterior derivative
$\mathrm{d}\mathcal{L}[q]$ defined by

$f_{2}\lrcorner\,f_{1}\lrcorner\,\mathrm{d}\mathcal{L}[q]=\partial_{f_{1}}(f_{2}\lrcorner\,\mathcal{L}[q])-\partial_{f_{2}}(f_{1}\lrcorner\,\mathcal{L}[q])-\\{f_{1},f_{2}\\}\lrcorner\,\mathcal{L}[q]\,.$

In this context we use $\delta\mathrm{d}\mathcal{L}[q]=0$ as definition of
critical fields, instead of the variational principle of Definition 3.12. The
operator $\delta$ is a Gateaux derivative (in a direction to be specified) of
tensors. It can be calculated coefficient-wise:

$f_{2}\lrcorner\,f_{1}\lrcorner\,(\delta\mathrm{d}\mathcal{L}[q])=\delta(f_{2}\lrcorner\,f_{1}\lrcorner\,\mathrm{d}\mathcal{L}[q])\,.$

The calculation of $\mathrm{d}\mathcal{L}$ formally takes the same form as
before in the proof of Proposition 4.8. Using Proposition 4.9, we find

$f_{2}\lrcorner\,f_{1}\lrcorner\,\mathrm{d}\mathcal{L}[q]=\left(\partial_{f_{1}}q_{1}+\frac{\partial{f_{1}}}{\partial{q}}\right)\left(\partial_{f_{2}}q-\frac{\partial{f_{2}}}{\partial{q_{1}}}\right)-\left(\partial_{f_{2}}q_{1}+\frac{\partial{f_{2}}}{\partial{q}}\right)\left(\partial_{f_{1}}q-\frac{\partial{f_{1}}}{\partial{q_{1}}}\right)\,.$

In particular, $f_{2}\lrcorner\,f_{1}\lrcorner\,\mathrm{d}\mathcal{L}$ has a
double zero on solutions to the canonical Hamiltonian equations for $f_{1}$
and $f_{2}$, hence $\delta\mathrm{d}\mathcal{L}=0$ is equivalent to the system
of all Hamiltonian equations for Hamiltonian functions in $\mathcal{F}$.

Since the multi-time Euler-Lagrange equations (4.1)–(4.3) are equivalent to
the condition $\delta\mathrm{d}\mathcal{L}=0$, they will also be equivalent to
the system of Hamiltonian equations. Indeed, we find

$\displaystyle
0=\frac{\partial{}}{\partial{q_{1}}}(f\lrcorner\,\mathcal{L})=\partial_{f}q-\frac{\partial{f}}{\partial{q_{1}}}\qquad(f\neq
H_{1})\,,$ $\displaystyle
0=\frac{\partial{}}{\partial{q}}(f\lrcorner\,\mathcal{L})-\partial_{f}\frac{\partial{}}{\partial{(\partial_{f}q)}}(f\lrcorner\,\mathcal{L})=-\partial_{f}q_{1}-\frac{\partial{f}}{\partial{q}}\,,$

while

$\frac{\partial{}}{\partial{(\partial_{f_{1}}q)}}(f_{1}\lrcorner\,\mathcal{L})=\frac{\partial{}}{\partial{(\partial_{f_{2}}q)}}(f_{2}\lrcorner\,\mathcal{L})$

is trivially satisfied.

The above construction can be thought of as a generalisation of the procedure
described in [13] to obtain an infinite-dimensional Lie algebra for the Kepler
problem. As we will see below, this particular problem actually admits a
finite dimensional Lie group, because the Poisson relations can be linearised
by rescaling the Runge-Lenz vector.

###  4.3 Exterior derivative and Poisson bracket

We know from Lemma 3.13 that every critical field $q$ satisfies
$\delta\mathrm{d}\mathcal{L}=0$. In the case of commuting flows, it has been
shown that $\mathrm{d}\mathcal{L}=0$ implies that the corresponding
Hamiltonian functions are in involution [38, 43]. In this section we
generalise this property to the Lie group setting describing non-commuting
flows.

For basis elements $\xi_{i},\xi_{j}\in\mathfrak{g}$ (or
$\xi_{i},\xi_{j}\in\mathfrak{h}$) we have

$\displaystyle\partial_{\xi_{j}}\lrcorner\,\partial_{\xi_{i}}\lrcorner\,\mathrm{d}\mathcal{L}[q]$
$\displaystyle=\partial_{\xi_{i}}L_{j}(q,q_{1},q_{j})-\partial_{\xi_{j}}L_{i}(q,q_{1},q_{i})-\partial_{[\xi_{i},\xi_{j}]}\lrcorner\,\mathcal{L}[q]$
$\displaystyle=\partial_{\xi_{i}}L_{j}(q,q_{1},q_{j})-\partial_{\xi_{j}}L_{i}(q,q_{1},q_{i})-\sum_{k}\mathrm{C}_{ij}^{k}L_{k}(q,q_{1},q_{k})\,.$

A basis-independent form of this expression is obtained in terms of
$L_{\xi}=\partial_{\xi}\lrcorner\,\mathcal{L}[q]$ :

$\partial_{\nu}\lrcorner\,\partial_{\xi}\lrcorner\,\mathrm{d}\mathcal{L}[q]=\partial_{\xi}L_{\nu}(q,q_{1},\partial_{\nu}q)-\partial_{\nu}L_{\xi}(q,q_{1},\partial_{\xi}q)-L_{[\xi,\nu]}(q,q_{1},\partial_{[\xi,\nu]}q)\,.$

Hence if $\mathrm{d}\mathcal{L}=0$ (as is typically the case on solutions)
then

$L_{[\xi,\nu]}(q,q_{1},\partial_{[\xi,\nu]}q)=\partial_{\xi}L_{\nu}(q,q_{1},\partial_{\nu}q)-\partial_{\nu}L_{\xi}(q,q_{1},\partial_{\xi}q)\,.$
(4.15)

As we will argue below, Equation (4.15) is the Lagrangian form of the
fundamental relation between the Poisson bracket of two Hamiltonian functions
and the commutator of the corresponding vector fields.

Regardless of how a Lagrangian 1-form $\mathcal{L}$ on a Lie group was
constructed, we can find Hamiltonian functions corresponding to $\mathcal{L}$
if $L_{1}$ is of a suitable form. To do this, we generalise the construction
from [38, 43] to our setting. Assume that $\mathcal{L}$ is of the form

$\begin{split}&\partial_{\xi_{1}}\lrcorner\,\mathcal{L}=L_{1}(q,q_{1})=\frac{1}{2}q_{1}^{2}-V\,,\\\
&\partial_{\xi_{i}}\lrcorner\,\mathcal{L}=L_{i}(q,q_{1},q_{i})\,,\end{split}$
(4.16)

and produces multi-time Euler-Lagrange equations of the form

$\displaystyle q_{11}=f(q,q_{1})\,,$ $\displaystyle
q_{i}=w_{\xi_{i}}(q,q_{1})\,.$

The Hamiltonian function associated to $L_{\xi}$, for
$\xi=\sum_{k}\alpha_{k}\xi_{k}$, is

$H_{\xi}(q,p)=pw_{\xi}(q,p)-L_{\xi}(q,p,w_{\xi}(q,p))\,,$

where $w_{\xi}=\sum_{k}\alpha_{k}w_{k}$. Note that on solutions, we have
$w_{\xi}(q,p)=\partial_{\xi}q$. If the Lagrangian 1-form is constructed from
variational symmetries as in Section 4.1, the Hamiltonian can also be written
as

$H_{\xi}(q,p)=pw_{\xi}(q,p)-\sum_{k}\alpha_{k}F_{k}(q,p)\,.$

######  Proposition 4.10.

The canonical Hamilton equations for $H_{\xi}$ are equivalent to the Euler-
Lagrange equations, i.e. to the lifted action of $w_{\xi}\partial_{q}$ on
$(q,q_{1})$, under the identification $p=q_{1}$.

Note that this can be thought of as an inverse statement to Proposition 4.7.
Accordingly, the proof will be quite similar, but we find it instructive to
include it.

###### Proof of Proposition 4.10.

If $\xi=\xi_{1}$ this is nothing but the Legendre transform. Below we prove
the result for $\xi$ an arbitrary linear combination of
$\xi_{2},\ldots,\xi_{N}$. Then the general claim follows from linearity. We
have

$\displaystyle\frac{\partial{H_{\xi}}}{\partial{p}}$
$\displaystyle=w_{\xi}+p\frac{\partial{w_{\xi}}}{\partial{p}}-\frac{\partial{L_{\xi}}}{\partial{q_{1}}}-\frac{\partial{L_{\xi}}}{\partial{q_{\xi}}}\frac{\partial{w_{\xi}}}{\partial{p}}$
$\displaystyle=w_{\xi}+\left(p-\frac{\partial{L_{\xi}}}{\partial{q_{\xi}}}\right)\frac{\partial{w_{\xi}}}{\partial{p}}-\frac{\partial{L_{\xi}}}{\partial{q_{1}}}$

and

$\displaystyle\frac{\partial{H_{\xi}}}{\partial{q}}$
$\displaystyle=p\frac{\partial{w_{\xi}}}{\partial{q}}-\frac{\partial{L_{\xi}}}{\partial{q}}-\frac{\partial{L_{\xi}}}{\partial{q_{\xi}}}\frac{\partial{w_{\xi}}}{\partial{q}}$
$\displaystyle=\left(p-\frac{\partial{L_{\xi}}}{\partial{q_{\xi}}}\right)\frac{\partial{w_{\xi}}}{\partial{q}}-\frac{\partial{L_{\xi}}}{\partial{q}}\,.$

On solutions to the multi-time Euler-Lagrange equations this gives

$\frac{\partial{H_{\xi}}}{\partial{p}}=w_{\xi}\qquad\text{and}\qquad\frac{\partial{H_{\xi}}}{\partial{q}}=-\partial_{\xi}p\,.\qed$

Written in terms of the Hamiltonians, Equation (4.15) becomes

$p\partial_{[\xi,\nu]}q-H_{[\xi,\nu]}(q,p)=\partial_{\xi}\left(p\partial_{\nu}q-H_{\nu}(q,p)\right)-\partial_{\nu}\left(p\partial_{\xi}q-H_{\xi}(q,p)\right)\,,$

which simplifies to

$\displaystyle-H_{[\xi,\nu]}(q,p)$
$\displaystyle=(\partial_{\xi}p)(\partial_{\nu}q)-\partial_{\xi}H_{\nu}(q,p)-(\partial_{\nu}p)(\partial_{\xi}q)+\partial_{\nu}H_{\xi}(q,p)$
$\displaystyle=-\frac{\partial{H_{\xi}}}{\partial{q}}\frac{\partial{H_{\nu}}}{\partial{p}}-\\{H_{\xi},H_{\nu}\\}+\frac{\partial{H_{\nu}}}{\partial{q}}\frac{\partial{H_{\xi}}}{\partial{p}}+\\{H_{\nu},H_{\xi}\\}$
$\displaystyle=-\\{H_{\xi},H_{\nu}\\}\,.$

Hence Equation (4.15) is a Lagrangian version of the Poisson relations between
the integrals of the system (4.4). In summary, we have:

######  Theorem 4.11.

Let $\mathcal{L}$ be of the form (4.16) and $H_{\xi}$ the corresponding
Hamiltonian functions. The following are equivalent:

  1. (i)

$\mathrm{d}\mathcal{L}=0$ on solutions,

  2. (ii)

Equation (4.15) holds:
$L_{[\xi,\nu]}=\partial_{\xi}L_{\nu}-\partial_{\nu}L_{\xi}$,

  3. (iii)

$H_{[\xi,\nu]}=\\{H_{\xi},H_{\nu}\\}$.

###  4.4 Example: Kepler problem

The Kepler problem, governed by the Hamiltonian

$H(q,p)=\frac{1}{2}p^{2}-\frac{1}{|q|}$

on $T^{*}\mathbb{R}^{3}$, is superintegrable. Rather than the obvious $SO(3)$
rotational symmetry, it actually possesses a symmetry group isomorphic to
$SO(4)$. The conserved quantities are the angular momentum $\ell=q\times p$
and the Runge-Lenz vector

$A=p\times\ell-\frac{q}{|q|}\,,$

see for example [16, 17]

A Lax pair for the Kepler problem was proposed in [4] as a restriction of KdV
flows (cf. also [40] for an explicit form of this Lax representation).
However, this Lax pair does not seem suitable for deriving the integrals of
the Kepler problem. A simpler, but at the same time more powerful Lax
representation is given as follows. Define matrices $\boldsymbol{L}_{\alpha}$
and $\boldsymbol{M}$ by

$\displaystyle\boldsymbol{L}_{\alpha}=\left(\begin{array}[]{cc}q^{T}\alpha
p&p^{T}\alpha p\\\ -q^{T}\alpha q&-p^{T}\alpha
q\end{array}\right)\,,\qquad\boldsymbol{M}=\left(\begin{array}[]{cc}0&|q|^{-3}\\\
-1&0\end{array}\right)\,,$ (4.21)

where $\alpha$ is an arbitrary $3\times 3$ matrix, which plays the role of the
spectral parameter. Note that $\boldsymbol{M}$ does not depend on this matrix
spectral parameter. An elementary calculation shows that:

######  Proposition 4.12.

The equations of motion of the Kepler problem,

$\dot{q}=p\,,\qquad\dot{p}=-\frac{1}{|q|^{3}}\,,$

follow from the Lax equation
$\dot{\boldsymbol{L}}_{\alpha}=[\boldsymbol{M},\boldsymbol{L}_{\alpha}]$.

It follows from the Lax equation that any expression of the form

$\operatorname{tr}(\boldsymbol{L}_{\alpha}\boldsymbol{L}_{\beta}\boldsymbol{L}_{\gamma}\cdots)$

is an integral of the Kepler problem, for arbitrary choices of 3$\times$3
matrices $\alpha,\beta,\gamma,\cdots$. In particular,
$\textrm{tr}(\boldsymbol{L}_{\alpha})$, with $\alpha$ an arbitrary skew-
symmetric matrix, yields the angular momentum vector $\ell$ as integral.
Curiously, the Hamiltonian $H$ follows from the quantity
$\operatorname{tr}(\boldsymbol{M}\boldsymbol{L}_{\alpha})$ with
$\alpha=\boldsymbol{1}$ taken to be the identity matrix. In fact, this
quantity is by itself not an integral of the motion, but instead we have

$\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{tr}(\boldsymbol{M}\boldsymbol{L}_{\alpha})=\frac{\mathrm{d}}{\mathrm{d}t}\left(-\frac{3}{|q|}\right)\,,$

where the right-hand side stems from the derivative of the matrix
$\boldsymbol{M}$. Thus, we can deduce that

$\textrm{tr}(\boldsymbol{M}\boldsymbol{L}_{\alpha})+\frac{3}{|q|}=-2H$

is an integral, which is the Hamiltonian up to a factor. It remains an open
problem if the Runge-Lenz vector $A$ arises from the Lax representation
(4.21).

Using the same construction as for Hamiltonians in involution [43], we can
construct a Lagrangian 1-form from these conserved quantities:

$\displaystyle L_{1}=\frac{1}{2}|q_{1}|^{2}+\frac{1}{|q|}\,,$ $\displaystyle
L_{2}=q_{1}\cdot q_{2}-\ell_{x}=q_{1}\cdot q_{2}-(q\times
q_{1})\cdot\hat{x}\,,$ $\displaystyle L_{3}=q_{1}\cdot
q_{3}-\ell_{y}=q_{1}\cdot q_{3}-(q\times q_{1})\cdot\hat{y}\,,$ $\displaystyle
L_{4}=q_{1}\cdot q_{4}-\ell_{z}=q_{1}\cdot q_{4}-(q\times
q_{1})\cdot\hat{z}\,,$ $\displaystyle L_{5}=q_{1}\cdot q_{5}-A_{x}=q_{1}\cdot
q_{5}-|q_{1}|^{2}(q\cdot\hat{x})+(q_{1}\cdot\hat{x})(q_{1}\cdot
q)+\frac{q\cdot\hat{x}}{|q|}\,,$ $\displaystyle L_{6}=q_{1}\cdot
q_{6}-A_{y}=q_{1}\cdot
q_{6}-|q_{1}|^{2}(q\cdot\hat{y})+(q_{1}\cdot\hat{y})(q_{1}\cdot
q)+\frac{q\cdot\hat{y}}{|q|}\,,$ $\displaystyle L_{7}=q_{1}\cdot
q_{7}-A_{z}=q_{1}\cdot
q_{7}-|q_{1}|^{2}(q\cdot\hat{z})+(q_{1}\cdot\hat{z})(q_{1}\cdot
q)+\frac{q\cdot\hat{z}}{|q|}\,,$

where $A=(A_{x},A_{y},A_{z})$, $\ell=(\ell_{x},\ell_{y},\ell_{z})$ and
$\\{\hat{x},\hat{y},\hat{z}\\}$ is the Cartesian orthonormal basis. The multi-
time Euler-Lagrange equations are given by $Q_{i}=0$ for $1\leq i\leq 7$,
where

$\displaystyle Q_{1}=q_{11}+\frac{q}{|q|^{3}}\,,$ $\displaystyle
Q_{2}=q_{2}-\hat{x}\times q\,,$ $\displaystyle Q_{3}=q_{3}-\hat{y}\times q\,,$
$\displaystyle Q_{4}=q_{4}-\hat{z}\times q\,,$ $\displaystyle
Q_{5}=q_{5}-2(q\cdot\hat{x})q_{1}+(q_{1}\cdot
q)\hat{x}+(q_{1}\cdot\hat{x})q\,,$ $\displaystyle
Q_{6}=q_{6}-2(q\cdot\hat{y})q_{1}+(q_{1}\cdot
q)\hat{y}+(q_{1}\cdot\hat{y})q\,,$ $\displaystyle
Q_{7}=q_{7}-2(q\cdot\hat{z})q_{1}+(q_{1}\cdot
q)\hat{z}+(q_{1}\cdot\hat{z})q\,.$

By cross-differentiating we can verify that
$\partial_{\xi_{2}},\ldots,\partial_{\xi_{7}}$ (and hence
$\xi_{2},\ldots,\xi_{7}$) satisfy the following Lie algebra relations:

$[\xi_{i},\xi_{j}]$ | $j=2$ | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---
$i=2$ | 0 | $-\xi_{4}$ | $\xi_{3}$ | 0 | $-\xi_{7}$ | $\xi_{6}$
3 |  | 0 | $-\xi_{2}$ | $\xi_{7}$ | 0 | $-\xi_{5}$
4 |  |  | 0 | $-\xi_{6}$ | $\xi_{5}$ | 0
5 |  |  |  | 0 | $2H\xi_{4}+2\ell_{z}\xi_{1}$ | $-2H\xi_{3}-2\ell_{y}\xi_{1}$
6 |  |  |  |  | 0 | $2H\xi_{2}+2\ell_{x}\xi_{1}$
7 |  |  |  |  |  | 0

If we replace $\xi_{5},\xi_{6},\xi_{7}$ by

$\displaystyle\nu_{5}$
$\displaystyle=\frac{\xi_{5}}{\sqrt{-2H}}+\frac{A_{x}\xi_{1}}{\sqrt{-2H}}\,,$
$\displaystyle\nu_{6}$
$\displaystyle=\frac{\xi_{6}}{\sqrt{-2H}}+\frac{A_{y}\xi_{1}}{\sqrt{-2H}}\,,$
$\displaystyle\nu_{7}$
$\displaystyle=\frac{\xi_{7}}{\sqrt{-2H}}+\frac{A_{z}\xi_{1}}{\sqrt{-2H}}\,,$

we can recover from this table the standard Lie algebra relations of
$\mathfrak{so}(4)$. Hence it is natural to define the Lagrangian 1-form
$\mathcal{L}$ on $\mathbb{R}\times SO(4)$ by
$\partial_{\xi_{i}}\lrcorner\,\mathcal{L}=L_{i}$.

Using the properties of the triple product, the coefficients
$P_{ij}=\partial_{\xi_{j}}\lrcorner\,\partial_{\xi_{i}}\lrcorner\,\mathrm{d}\mathcal{L}$
are obtained by an elementary calculation. We find for example

$\displaystyle P_{23}$
$\displaystyle=\partial_{\xi_{2}}L_{3}-\partial_{\xi_{3}}L_{2}-[\partial_{\xi_{2}},\partial_{\xi_{3}}]\lrcorner\,L$
$\displaystyle=q_{12}(q_{3}-\hat{y}\times q)+q_{1}(q_{32}-\hat{y}\times
q_{2})$ $\displaystyle\qquad-q_{13}(q_{2}-\hat{x}\times
q)-q_{1}(q_{23}-\hat{x}\times q_{3})$
$\displaystyle\qquad+q_{1}(q_{23}-q_{32}-\hat{z}\times q)$
$\displaystyle=(q_{12}-\hat{x}\times q_{1})(q_{3}-\hat{y}\times
q)-(q_{13}-\hat{y}\times q_{1})(q_{2}-\hat{x}\times q)\,,$

which is a double zero on solutions. In general we find that for $2\leq
i,j\leq 7$,

$\displaystyle P_{1i}$
$\displaystyle=\partial_{\xi_{1}}L_{i}-\partial_{\xi_{i}}L_{1}-[\partial_{\xi_{1}},\partial_{\xi_{i}}]\lrcorner\,L$
$\displaystyle=Q_{1}Q_{i}$

and

$\displaystyle P_{ij}$
$\displaystyle=\partial_{\xi_{i}}L_{j}-\partial_{\xi_{j}}L_{i}-[\partial_{\xi_{i}},\partial_{\xi_{j}}]\lrcorner\,L$
$\displaystyle=Q_{j}\partial_{\xi_{1}}Q_{i}-Q_{i}\partial_{\xi_{1}}Q_{j}\,,$

so all $P_{ij}$ have a double zero on solutions.

On the Hamiltonian side one can check that indeed $H$ is in involution with
each of the components of $\ell$ and $A$. In addition, we have the following
relations

$\displaystyle\\{\ell_{i},\ell_{j}\\}=-\epsilon_{ijk}\ell_{k}\,,$
$\displaystyle\\{A_{i},\ell_{j}\\}=-\epsilon_{ijk}A_{k}\,,$
$\displaystyle\\{A_{i},A_{j}\\}=2\epsilon_{ijk}H\ell_{k}\,,$
$\displaystyle\\{H,\cdot\\}=0\,,$

where $\epsilon_{ijk}$ is the totally anti-symmetric tensor with
$\epsilon_{ijk}=1$ (see e.g. [16, Sec. 9–7], but beware that we use the
opposite sign convention for the Poisson bracket). These Poisson brackets
reflect the commutation relations between the corresponding vector fields, as
found above. There are several possible choices of three independent integrals
in involution, which make the Kepler problem into a Liouville integrable
system, for example $(H_{1},\ell_{i},|\ell|^{2})$, $(H_{1},\ell_{i},A_{i})$,
or $(H_{1},\ell_{i},|A|^{2})$.

###  4.5 Example: Calogero-Moser system

The Calogero-Moser (CM) system is governed by the Hamiltonian

$H(q,p)=\frac{1}{2}|p|^{2}+\frac{1}{2}\sum_{\beta=1}^{n}\sum_{\alpha=1}^{\beta-1}\frac{1}{(q^{\alpha}-q^{\beta})^{2}}\,,$

see for example [9, 26, 29]. Introducing the notation

$F(q)=\left(\sum_{\alpha\neq 1}(q^{1}-q^{\alpha})^{-3},\ldots,\sum_{\alpha\neq
n}(q^{n}-q^{\alpha})^{-3}\right)$

we can write this as

$H(q,p)=\frac{1}{2}|p|^{2}+\frac{1}{2}F(q)\cdot q\,.$

Noting that $\frac{\partial{}}{\partial{q}}(F(q)\cdot q)=-2F(q)$ we find the
equations of motion $\ddot{q}=F(q)$

The CM system possesses a sequence of conserved quantities $I_{j}$, containing
$H$ as $I_{2}$, which are pairwise in involution. In addition, there exist
conserved quantities $K_{j}$, which make the system superintegrable [45].
These conserved quantities can be constructed from the system’s Lax pair,
consisting of the $n\times n$ matrices $\boldsymbol{L}$ and $\boldsymbol{M}$
with entries

$\displaystyle\boldsymbol{L}$
$\displaystyle=\begin{pmatrix}p_{1}&\frac{i}{q_{1}-q_{2}}&\ldots&\frac{i}{q_{1}-q_{n}}\\\
\frac{i}{q_{2}-q_{1}}&p_{2}&\ldots&\frac{i}{q_{2}-q_{n}}\\\
\vdots&\vdots&\ddots&\vdots\\\
\frac{i}{q_{n}-q_{1}}&\frac{i}{q_{n}-q_{2}}&\ldots&p_{n}\\\ \end{pmatrix}\,,$
$\displaystyle\boldsymbol{M}$ $\displaystyle=\begin{pmatrix}-\sum_{k\neq
1}\frac{i}{(q_{1}-q_{k})^{2}}&\frac{i}{(q_{1}-q_{2})^{2}}&\ldots&\frac{i}{(q_{1}-q_{n})^{2}}\\\
\frac{i}{(q_{2}-q_{1})^{2}}&-\sum_{k\neq
2}\frac{i}{(q_{2}-q_{k})^{2}}&\ldots&\frac{i}{(q_{2}-q_{n})^{2}}\\\
\vdots&\vdots&\ddots&\vdots\\\
\frac{i}{(q_{n}-q_{1})^{2}}&\frac{i}{(q_{n}-q_{2})^{2}}&\ldots&-\sum_{k\neq
n}\frac{i}{(q_{1}-q_{k})^{2}}\\\ \end{pmatrix}\,,$

and the additional matrix $\boldsymbol{N}=\mathrm{diag}(q^{1},\ldots,q^{n})$.
They are

$I_{j}=\frac{1}{j}\operatorname{tr}(\boldsymbol{L}^{j})$

and

$K_{j}=\operatorname{tr}(\boldsymbol{N}\boldsymbol{L}^{j-1})\operatorname{tr}(\boldsymbol{L})-\operatorname{tr}(\boldsymbol{L}^{j})\operatorname{tr}(\boldsymbol{N})\,.$

Denote $e=(1,\ldots,1)$. As a minimal example to illustrate our framework, we
consider only three non-commuting Hamiltonians,

$\displaystyle I_{2}(q,p)=H(q,p)\,,$ $\displaystyle I_{1}(q,p)=e\cdot p\,,$
$\displaystyle K_{2}(q,p)=(q\cdot p)(e\cdot p)-\left(|p|^{2}+F(q)\cdot
q\right)(e\cdot q)\,.$

We have $\\{I_{2},K_{1}\\}=0$, $\\{I_{2},K_{2}\\}=0$, and
$\\{I_{1},K_{2}\\}=2nH-I_{1}^{2}$, hence we are dealing with nonlinear Poisson
relations.

Let $\mathfrak{h}$ be the space of functions of $I_{1},I_{2},K_{2}$. As in
Section 4.2.1 we define $\mathcal{L}$ by

$f\lrcorner\,\mathcal{L}=q_{1}\cdot\partial_{f}q-f$

for $f\in\mathfrak{h}$. In particular, setting $\partial_{1}=\partial_{H}$,
$\partial_{2}=\partial_{I_{1}}$, and $\partial_{3}=\partial_{K_{2}}$ (such
that $\partial_{1}$ corresponds to the flow of the Calogero-Moser equation
itself), we have

$\displaystyle
L_{1}:=H\lrcorner\,\mathcal{L}=\frac{1}{2}|q_{1}|^{2}-\frac{1}{2}F(q)\cdot
q\,,$ $\displaystyle
L_{2}:=I_{1}\lrcorner\,\mathcal{L}=q_{1}\cdot(q_{2}-e)\,,$ $\displaystyle
L_{3}:=K_{2}\lrcorner\,\mathcal{L}=q_{1}\cdot q_{3}-(q\cdot q_{1})(e\cdot
q_{1})+\left(|q_{1}|^{2}+F(q)\cdot q\right)(e\cdot q)\,.$

Note that these are just three components of the 1-tensor $\mathcal{L}$ on the
infinite-dimensional space $\tilde{\mathfrak{g}}$. The multi-time Euler-
Lagrange equations yield

$\displaystyle q_{11}=F(q)\,,$ $\displaystyle q_{2}=e\,,$ $\displaystyle
q_{3}=(q\cdot q_{1})e+(e\cdot q_{1})q-2(e\cdot q)q_{1}\,.$

The coefficients of $\mathrm{d}\mathcal{L}$ corresponding to the generators
$\partial_{1}$, $\partial_{2}$, and $\partial_{3}$ are

$\displaystyle\partial_{1}L_{2}-\partial_{2}L_{1}$
$\displaystyle=(q_{2}-e)\cdot(q_{11}-F(q))\,,$
$\displaystyle\partial_{1}L_{3}-\partial_{3}L_{1}$
$\displaystyle=(q_{11}-F(q))\cdot(q_{3}+2(e\cdot q)q_{1}-(q\cdot
q_{1})e-(e\cdot q_{1})q)\,,$

and

$\displaystyle\partial_{2}L_{3}-\partial_{3}L_{2}+\\{I_{1},K_{2}\\}\lrcorner\,\mathcal{L}$
$\displaystyle=q_{12}\cdot(q_{3}+2(e\cdot q)q_{1}-(q\cdot q_{1})e-(e\cdot
q_{1})q)$ $\displaystyle\quad-(q_{2}-e)\cdot\left(q_{13}+2q_{11}(e\cdot
q)+q_{1}(e\cdot q_{1})-e|q_{1}|^{2}-e(q\cdot q_{11})-q(e\cdot q_{11})\right)$
$\displaystyle\quad+(q_{2}-e)\cdot\left(2(e\cdot
q)(q_{11}-F(q))-(q\cdot(q_{11}-F(q)))e-(e\cdot(q_{11}-F(q)))q\right)$
$\displaystyle\quad+q_{1}(q_{32}-q_{23}+\partial_{\\{I_{1},K_{2}\\}}q)\,,$

where the last term vanishes because it equals
$-q_{1}([\partial_{I_{1}},\partial_{K_{2}}]-\partial_{\\{I_{1},K_{2}\\}})q=0$.
Hence we see that all three coefficients attain a double zero on solutions to
the multi-time Euler-Lagrange equations.

######  Remark 4.13.

Looking only at $H$, $I_{1}$ and $K_{2}$, one may argue that we can linearise
the Poisson relations by rescaling the Hamiltonians, in which case we would
not need the extension discussed in Section 4.2.1 and we could give this small
system a multiform structure on a finite-dimensional Lie group. Indeed, taking
the following particular combinations

$\displaystyle H(q,p)=\frac{1}{2}|p|^{2}+\frac{1}{2}F(q)\cdot q\,,$
$\displaystyle I(q,p)=\frac{1}{2}I_{1}(q,p)^{2}=\frac{1}{2}(e\cdot p)^{2}\,,$
$\displaystyle K(q,p)=\frac{K_{2}(q,p)}{I_{1}}=q\cdot
p-\left(|p|^{2}+F(q)\cdot q\right)\frac{e\cdot q}{e\cdot p}\,,$

the Poisson brackets between pairs of these functions are given by
$\\{H,I\\}=\\{H,K\\}=0$ and $\\{I,K\\}=2I-2nH$ so that the corresponding
vector fields form a Lie algebra. However, this seems rather specific to the
small set of functions $H$, $I_{1}$ and $K_{2}$. We have not been able to
establish such a linearisation procedure in general for a large collection of
conserved quantities $I_{j}$, $K_{j}$ for the CM model.

##  5 Lagrangian 2-forms on Lie groups

So far we have been dealing with Lagrangian 1-forms, which describe ODEs. Now
we turn our attention to field theory, which in the simplest case is described
by Lagrangian 2-forms.

Consider a first order Lagrangian 2-form $\mathcal{L}[v]$ on a Lie group $G$
defined by

$\partial_{\xi_{j}}\lrcorner\,\partial_{\xi_{i}}\lrcorner\,\mathcal{L}[v]=L_{ij}(v,v_{1},\ldots,v_{n})\,,$

where $v_{i}=\partial_{\xi_{i}}v$ are the derivatives of the field $v:G\to Q$
along a basis vector $\xi_{i}$ of the Lie algebra of $G$. In this case the
multi-time Euler-Lagrange equations will be different from their counterparts
for commuting flows. We adopt a similar approach as in Theorem 4.1 to obtain
the following.

######  Theorem 5.1.

Let $\mathcal{L}$ be a 2-form depending on the first jet of $v$. The
variational principle of Definition 3.12 is equivalent to

$\displaystyle\frac{\partial{L_{jk}}}{\partial{v_{j}}}+\frac{\partial{L_{ki}}}{\partial{v_{i}}}=0\,,$
(5.1a) $\displaystyle\frac{\partial{L_{jk}}}{\partial{v_{\ell}}}=0\qquad\text{
for }\ell\neq j,k\,,$ (5.1b)
$\displaystyle\frac{\delta_{jk}{L_{jk}}}{\delta{v}}+\sum_{\ell}\mathrm{C}_{jk}^{\ell}p_{\ell}=0\,,$
(5.1c)

where $p_{j}=\frac{\partial{L_{ij}}}{\partial{v_{i}}}$, which is well-defined
(i.e. independent of $i$) in view of Equation (5.1a), and

$\frac{\delta_{jk}{}}{\delta{v}}=\frac{\partial{}}{\partial{v}}-\operatorname{D}_{j}\frac{\partial{}}{\partial{v_{j}}}-\operatorname{D}_{k}\frac{\partial{}}{\partial{v_{k}}}\,.$

###### Proof.

Fix $i<j<k$ and consider the function
$P_{ijk}[v]=\partial_{\xi_{k}}\lrcorner\,\partial_{\xi_{j}}\lrcorner\,\partial_{\xi_{i}}\lrcorner\,\mathrm{d}\mathcal{L}[v]$
of the free jet bundle. Then

$\displaystyle P_{ijk}$
$\displaystyle=\operatorname{D}_{i}(\partial_{\xi_{k}}\lrcorner\,\partial_{\xi_{j}}\lrcorner\,\mathcal{L})-\partial_{\xi_{k}}\lrcorner\,\partial_{[\xi_{i},\xi_{j}]}\lrcorner\,\mathcal{L}+\circlearrowleft_{ijk}$
$\displaystyle=\operatorname{D}_{i}L_{jk}+\sum_{\ell}\mathrm{C}_{jk}^{\ell}L_{i\ell}+\circlearrowleft_{ijk}\,,$

where $\circlearrowleft_{ijk}$ denotes all terms obtained by cyclicly
permuting $i,j,k$ in the previous terms.

The vertical exterior derivative of $P_{ijk}$ (which gives the multi-time
Euler-Lagrange equations using Lemma 3.13) can be expanded into a sum over
index-strings:

$\delta P_{ijk}=\sum_{\mathfrak{I}}A^{\mathfrak{I}}\delta v_{\mathfrak{I}}$

with

$A^{\mathfrak{I}}=\frac{\partial{P_{ijk}}}{\partial{v_{\mathfrak{I}}}}\,.$

Using Lemma 3.14 we can project to the quotiented jet bundle and write $\delta
P_{ijk}$ as a sum over independent $\delta v_{I}$, where $I$ ranges over the
set of multi-indices:

$\delta P_{ijk}=\sum_{I}B^{I}\delta v_{I}\,,$

where the $B^{I}$ are given by Equation (3.15). This allows us to find the
multi-time Euler-Lagrange equations in terms of the $L_{ij}$ as follows.

We have

$B^{ii}=\widetilde{\frac{\partial{P_{ijk}}}{\partial{v_{ii}}}}=\frac{\partial{L_{jk}}}{\partial{v_{i}}}\,.$

Setting this equal to zero, we obtain Equation (5.1b). Similarly, the
equations given by $B^{jj}$ and $B^{kk}$ give the same equation with the
indices permuted.

For $\ell\not\in\\{i,j,k\\}$, the coefficients $B^{i\ell}$ (in the case where
$\ell>i$) and $B^{\ell i}$ (in the case where $\ell<i$) are given by

$\widetilde{\frac{\partial{P_{ijk}}}{\partial{v_{\ell
i}}}}=\frac{\partial{L_{jk}}}{\partial{v_{\ell}}}\,.$

Setting this to zero is again equivalent to Equation (5.1b).

We next consider

$B^{ij}=\widetilde{\frac{\partial{P_{ijk}}}{\partial{v_{ij}}}}+\widetilde{\frac{\partial{P_{ijk}}}{\partial{v_{ji}}}}=\frac{\partial{L_{ki}}}{\partial{v_{i}}}+\frac{\partial{L_{jk}}}{\partial{v_{j}}}\,,$

which leads to Equation (5.1a), as do $B^{jk}=0$ and $B^{ik}=0$. Modulo
equation (5.1a) we can define
$p_{j}=\frac{\partial{L_{ij}}}{\partial{v_{i}}}$.

Next, we find

$\begin{split}B^{i}&=\widetilde{\frac{\partial{P_{ijk}}}{\partial{v_{i}}}}+\sum_{\ell<m}\mathrm{C}_{\ell
m}^{i}\widetilde{\frac{\partial{P_{ijk}}}{\partial{v_{m\ell}}}}\\\
&=\frac{\partial{L_{jk}}}{\partial{v}}+\operatorname{D}_{i}\frac{\partial{L_{jk}}}{\partial{v_{i}}}+\operatorname{D}_{j}\frac{\partial{L_{ki}}}{\partial{v_{i}}}+\operatorname{D}_{k}\frac{\partial{L_{ij}}}{\partial{v_{i}}}+\sum_{\ell}\left(\mathrm{C}_{ij}^{\ell}\frac{\partial{L_{k\ell}}}{\partial{v_{i}}}+\mathrm{C}_{jk}^{\ell}\frac{\partial{L_{i\ell}}}{\partial{v_{i}}}+\mathrm{C}_{ki}^{\ell}\frac{\partial{L_{jk}}}{\partial{v_{i}}}\right)\\\
&\qquad+\mathrm{C}_{ij}^{i}\frac{\partial{L_{ki}}}{\partial{v_{j}}}+\mathrm{C}_{ik}^{i}\frac{\partial{L_{jk}}}{\partial{v_{k}}}+\mathrm{C}_{jk}^{i}\frac{\partial{L_{ki}}}{\partial{v_{k}}}\,.\end{split}$

By applying equations (5.1b) and (5.1a) to the above expression, we obtain
that

$B^{i}=\frac{\delta_{jk}{L_{jk}}}{\delta{v}}+\sum_{\ell}\mathrm{C}_{jk}^{j}p_{\ell}\,,$

which we set equal to zero to obtain Equation (5.1c). The same equation
follows from $B^{j}=0$ and $B^{k}=0$.

The equations $B^{\emptyset}=0$ and $B^{\ell}=0$ for $\ell\notin\\{i,j,k\\}$
are consequences of Equations (5.1b)–(5.1c). ∎

######  Remark 5.2 (On the stepped surface approach).

In the commutative case $G=\mathbb{R}^{N}$ the multi-time Euler-Lagrange
equations can be obtained by a stepped surface approach [39]. This consists in
approximating any given surface by a stepped surface, i.e. a piecewise flat
surface where each flat piece is tangent to two coordinate directions. On a
flat piece tangent to $\partial_{t_{i}}$ and $\partial_{t_{j}}$ the action
integral only sees the coefficient $L_{ij}$ of the Lagrangian two-form. This
leads to the Euler-Lagrange equation
$\frac{\delta_{ij}{L_{ij}}}{\delta{v_{I}}}=0$. The boundary terms that occur
where different flat pieces meet, lead to the other multi-time Euler-Lagrange
equations.

In the non-commutative case this approach breaks down. If $\xi_{i}$ and
$\xi_{j}$ do not commute, then there may not exist any surface that is tangent
to $\partial_{\xi_{i}}$ and $\partial_{\xi_{j}}$. Indeed, Frobenius’ theorem
indicates that such a surface only exists if
$[\partial_{\xi_{i}},\partial_{\xi_{j}}]$ lies in the span of
$\partial_{\xi_{i}}$ and $\partial_{\xi_{j}}$. Hence we cannot in general
reproduce the essential property that individual pieces of a stepped surface
only see one coefficient of the Lagrangian 2-form.

Irrespective of whether we are in the commutative or non-commutative case, the
integral of a (Lagrangian) 2-form over a 2-dimensional surface is well-
defined. Given a surface one can always endow it with local coordinates, say
$x$ and $y$, and use the coordinate functions to pull back the 2-form to a
subset of $\mathbb{R}^{2}$. This is the canonical way of defining the integral
of a differential form. What we cannot do in general, is use the differential
operators $\partial_{x}$ and $\partial_{y}$ as basis elements of the Lie
algebra $\mathfrak{g}$, because $\partial_{x}$ and $\partial_{y}$ will not in
general be left-invariant under $G$.

###  5.1 Example on $SE(2)$

In this subsection we consider an example of a Lagrangian 2-form with the Lie
group $SE(2)$ as multi-time. We parameterise $SE(2)$ by $(x,y,\theta)$, such
that its multiplication is given by Equation (3.3). Since we have global
coordinates, we could take as multi-time the Euclidean space spanned by
$\partial_{x},\partial_{y},\partial_{\theta}$, but in order to illustrate the
non-commutative aspects of Lagrangian multiform theory, we consider a basis of
left-invariant vector fields on $SE(2)$, given by

$\displaystyle\partial_{1}=\cos\theta\partial_{x}+\sin\theta\partial_{y}\,,$
$\displaystyle\partial_{2}=-\sin\theta\partial_{x}+\cos\theta\partial_{y}\,,$
$\displaystyle\partial_{3}=\partial_{\theta}\,.$

They satisfy

$[\partial_{i},\partial_{j}]=\sum_{k=1}^{3}\mathrm{C}_{ij}^{k}\partial_{k}$

with

$\mathrm{C}_{32}^{1}=\mathrm{C}_{13}^{2}=-1,\qquad\mathrm{C}_{23}^{1}=\mathrm{C}_{31}^{2}=+1,$

and all other structure constants equal to zero.

We consider a Lagrangian 2-form $\mathcal{L}[v]$ which describes harmonic
functions and the rotational symmetry of the notion of harmonicity. It is
defined by $\partial_{j}\lrcorner\,\partial_{i}\lrcorner\,\mathcal{L}=L_{ij}$
with

$\displaystyle L_{12}=\frac{1}{2}v_{1}^{2}+\frac{1}{2}v_{2}^{2}\,,$
$\displaystyle
L_{13}=v_{2}v_{3}-y(\theta)v_{1}v_{2}-\frac{1}{2}x(\theta)(v_{1}^{2}-v_{2}^{2})\,,$
$\displaystyle
L_{23}=-v_{1}v_{3}-x(\theta)v_{1}v_{2}+\frac{1}{2}y(\theta)(v_{1}^{2}-v_{2}^{2})\,,$

where $x(\theta)=x\cos\theta+y\sin\theta$ and
$y(\theta)=-x\sin\theta+y\cos\theta$. Note that $L_{13}$ and $L_{23}$ are non-
autonomous and that we have

$\displaystyle\partial_{3}x(\theta)=y(\theta)\,,$
$\displaystyle\partial_{1}x(\theta)=1\,,$
$\displaystyle\partial_{2}x(\theta)=0\,,$
$\displaystyle\partial_{3}y(\theta)=-x(\theta)\,,$
$\displaystyle\partial_{1}y(\theta)=0\,,$
$\displaystyle\partial_{2}y(\theta)=1\,.$

The multi-time Euler-Lagrange equations of type (5.1a) are trivially
satisfied. Those of type (5.1b) are

$\displaystyle
0=\frac{\delta_{13}{L_{13}}}{\delta{v_{2}}}=v_{3}-y(\theta)v_{1}+x(\theta)v_{2}\,,$
$\displaystyle
0=\frac{\delta_{23}{L_{23}}}{\delta{v_{1}}}=-v_{3}-x(\theta)v_{2}+y(\theta)v_{1}\,,$

so we find the multi-time Euler-Lagrange equation

$v_{3}=-x(\theta)v_{2}+y(\theta)v_{1}\,.$ (5.2)

Differentiating this equation by $\partial_{2}$ and $\partial_{1}$, we have,
respectively,

$\displaystyle v_{32}=y(\theta)v_{12}+v_{1}-x(\theta)v_{22}\,,$ $\displaystyle
v_{31}=y(\theta)v_{11}-v_{2}-x(\theta)v_{21}\,.$

Since
$v_{ij}=\partial_{j}\partial_{i}v=\partial_{i}\partial_{j}v+[\partial_{j},\partial_{i}]v=v_{ji}+\sum_{k}\mathrm{C}_{ji}^{k}v_{k}$
it follows that

$\displaystyle
v_{23}=(y(\theta)v_{12}+v_{1}-x(\theta)v_{22})-v_{1}=y(\theta)v_{12}-x(\theta)v_{22}\,,$
(5.3) $\displaystyle
v_{13}=(y(\theta)v_{11}-v_{2}-x(\theta)v_{21})+v_{2}=-x(\theta)v_{12}+y(\theta)v_{11}\,.$
(5.4)

There are three multi-time Euler-Lagrange equations of type (5.1c). The first
one is

$0=\frac{\delta_{12}{L_{12}}}{\delta{v}}=-v_{11}-v_{22}\,,$ (5.5)

because all the relevant structure constants are zero. The second one is

$\displaystyle
0=\frac{\delta_{13}{L_{13}}}{\delta{v}}+\mathrm{C}_{13}^{2}\frac{\partial{L_{12}}}{\partial{v_{1}}}$
$\displaystyle=\frac{\delta_{13}{L_{13}}}{\delta{v}}-\frac{\partial{L_{12}}}{\partial{v_{1}}}$
$\displaystyle=(-v_{23}+y(\theta)v_{12}+x(\theta)v_{11}+v_{1})-v_{1}$
$\displaystyle=(-v_{23}+y(\theta)v_{12}-x(\theta)v_{22})+x(\theta)(v_{22}+v_{11})\,,$

which is a consequence of equations (5.3) and (5.5). The third one is

$\displaystyle
0=\frac{\delta_{23}{L_{23}}}{\delta{v}}+\mathrm{C}_{23}^{1}\frac{\partial{L_{21}}}{\partial{v_{2}}}$
$\displaystyle=\frac{\delta_{23}{L_{23}}}{\delta{v}}-\frac{\partial{L_{21}}}{\partial{v_{2}}}$
$\displaystyle=(v_{13}+x(\theta)v_{12}+y(\theta)v_{22}+v_{2})-v_{2}$
$\displaystyle=(v_{13}+x(\theta)v_{12}-y(\theta)v_{11})+y(\theta)(v_{11}+v_{22})\,,$

which is a consequence of equations (5.4) and (5.5).

The coefficient $P_{123}$ of exterior derivative of $\mathcal{L}$ is

$\displaystyle P_{123}$
$\displaystyle=\operatorname{D}_{1}L_{23}-\sum_{\ell}\mathrm{C}_{12}^{\ell}L_{\ell
3}+\circlearrowleft_{123}$
$\displaystyle=\operatorname{D}_{1}L_{23}+\circlearrowleft_{123}$
$\displaystyle=(v_{11}+v_{22})(-v_{3}+y(\theta)v_{1}-x(\theta)v_{2})+v_{1}(v_{13}-v_{31})+v_{2}(v_{23}-v_{32})\,.$

Due to the Lie algebra structure, the last two terms cancel against each
other. The remaining term is a double zero on the multi-time Euler-Lagrange
equations.

Solutions to the multi-time Euler-Lagrange equations (5.2)–(5.5) are harmonic
functions with their rotations parameterised by $\theta$. For example, we
could take

$v(x,y,\theta)=\exp(x(\theta))\sin(y(\theta))\,.$

Its derivatives are

$\displaystyle\partial_{1}v=\exp(x(\theta))\sin(y(\theta))=v\,,$
$\displaystyle\partial_{2}v=\exp(x(\theta))\cos(y(\theta))\,,$
$\displaystyle\partial_{3}v=y(\theta)v_{1}-x(\theta)v_{2}\,,$

and similarly for higher derivatives. It is an easy calculation to see that
$v$ satisfies the Euler-Lagrange equations.

###  5.2 Infinite hierarchy example

In this section, we construct an example of a multi-time with non-commuting
vector fields by realising the idea initially proposed in [27]. We use the
approach of [11] and [12] to produce an example of a generalisation of the
AKNS hierarchy [1] with non-commuting flows.

Let us first recall the main ingredients of the description in the commutative
case, i.e. the case where the multi-time is simply $\mathbb{R}^{\mathbb{N}}$.
In this case, the Lagrangian $2$-form of interest is of the form

$\mathcal{L}=\sum_{m<n}L^{mn}\,\mathrm{d}t_{m}\wedge\mathrm{d}t_{n}$

and the Lagrangian coefficients $L^{mn}$ are most conveniently assembled into
a _generating Lagrangian multiform_ , which is the following formal series in
$\lambda^{-1}$, $\mu^{-1}$,

$L(\lambda,\mu)=\sum_{m,n=0}^{\infty}L^{mn}\,\lambda^{-m-1}\mu^{-n-1}\,,$
(5.6)

where $L^{mn}=-L^{nm}$. In [12] an expression for a large class of generating
Lagrangian multiforms was introduced. In the present work, we focus on the
following choice

$L(\lambda,\mu)=\operatorname{tr}\left(\varphi(\mu)^{-1}\partial_{\lambda}\varphi(\mu)J-\varphi(\lambda)^{-1}\partial_{\mu}\varphi(\lambda)J\right)-\frac{1}{\lambda-\mu}\operatorname{tr}\left(Q(\lambda)Q(\mu)-J^{2}\right)$
(5.7)

and will explain how to extend it to a non-commutative setting. A few
definitions are in order. The matrix $J$ is a constant element of the
underlying Lie algebra $\mathfrak{g}$. The object $\varphi(\lambda)$ is an
element of the group with (matrix) Lie algebra
$\mathfrak{g}\otimes\lambda^{-1}\mathbb{C}[[\lambda^{-1}]]$, that is a matrix
of the form

$\varphi(\lambda)=\mathbf{1}+\sum_{n\geq 1}\varphi^{(n)}\lambda^{-n}\,,$ (5.8)

and

$Q(\lambda)=\varphi(\lambda)\,J\,\varphi(\lambda)^{-1}\,.$

The group element $\varphi(\lambda)$, or the algebra element $Q(\lambda)$,
contains the dynamical variables (fields) of the hierarchy and defines the
phase space. The generating derivation $\partial_{\lambda}$ is given by the
formal series

$\partial_{\lambda}=\sum_{n=0}^{\infty}\partial_{t_{n}}\lambda^{-n-1}\,.$

One can check that (5.7) is of the form (5.6) and derive the Lagrangian
coefficients $L^{mn}$ in terms of the dynamical variables. If we choose
$\mathfrak{g}=\mathfrak{sl}(2)$ and take $J=-i\sigma_{3}$, (5.7) produces all
the coefficients for a Lagrangian multiform of the AKNS hierarchy [11, 12].
Other choices of $\mathfrak{g}$ would produce multicomponent generalisations
of the AKNS hierarchy.

The multi-time Euler-Lagrange equation associated to (5.7) takes the form of a
generating Lax equation,

$\displaystyle\partial_{\lambda}Q(\mu)=\left[\frac{1}{\lambda-\mu}Q(\lambda),Q(\mu)\right]\,.$
(5.9)

This is the central equation obtained in [15], written here in generating form
(see also [28]). Of course, an important result is that the flows of the
hierarchy commute, meaning that the generating vector fields satisfy
$[\partial_{\lambda},\partial_{\mu}]=0$ or, in other words that
$[\partial_{t_{n}},\partial_{t_{m}}]=0$ for all $n,m\geq 0$.

In this construction, there is a freedom in choosing the matrix $J$ defining
the hierarchy. The effects of choosing another element of the underlying Lie
algebra have been studied in detail in [12]. With this is mind, the idea
promoted in [27] consists in attaching a hierarchy of vector fields to each
possible choice of matrix $J$ and to consider the resulting hierarchies
assembled in a single, enlarged hierarchy with not necessarily commuting
vector fields. With the tools developed in the present paper, we are in a
position to realise this idea in the context of Lagrangian multiforms as
follows.

Let us consider the generators $J_{1},\ldots,J_{N}$ of a matrix Lie algebra
$\mathfrak{g}$, satisfying

$[J_{a},J_{b}]=\sum_{c}\mathrm{C}_{ab}^{c}J_{c}\,.$ (5.10)

As in Equation (5.8), we denote by $\varphi$ an element of the group
corresponding to the half loop algebra
$\mathfrak{g}\otimes\lambda^{-1}\mathbb{C}[[\lambda^{-1}]]$ and define

$Q_{a}(\lambda)=\varphi(\lambda)\,J_{a}\,\varphi(\lambda)^{-1}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ a=1,\dots,N\,.$

These elements $Q_{a}(\lambda)$ contain the phase space variables of our
hierarchy. To each $a\in\\{1,\dots,N\\}$, we associate a generating vector
field

$\partial_{\lambda,a}=\sum_{n=0}^{\infty}\partial_{n,a}\lambda^{-n-1}\,,$
(5.11)

which acts on our phase space. Note that we wrote $\partial_{n,a}$ on purpose
and not $\partial_{t_{n,a}}$ since in general these vector fields do not
commute and cannot be thought of as derivatives with respect to time
coordinates $t_{n,a}$. Instead, we will construct a hierarchy where these
generating vector fields satisfy the Lie algebra relations

$[\partial_{\lambda,a},\partial_{\mu,b}]=\sum_{c}\mathrm{C}_{ab}^{c}\frac{\partial_{\lambda,c}-\partial_{\mu,c}}{\lambda-\mu}\,.$
(5.12)

This should be understood as the generating form for the following algebra
relations

$[\partial_{n,a},\partial_{m,b}]=\sum_{c}\mathrm{C}_{ab}^{c}\,\partial_{n+m,c}\,,\qquad
n,m\geq 0\,.$ (5.13)

We can now define a generalisation of (5.7) to the present context as follows.
Define the Lagrangian 2-form $\mathcal{L}$ by
$\partial_{\mu,b}\lrcorner\,\partial_{\lambda,a}\lrcorner\,\mathcal{L}=L_{ab}(\lambda,\mu)$
where

$L_{ab}(\lambda,\mu)=\operatorname{tr}\left(\varphi(\mu)^{-1}\partial_{\lambda,a}\varphi(\mu)J_{b}-\varphi(\lambda)^{-1}\partial_{\mu,b}\varphi(\lambda)J_{a}\right)-\frac{1}{\lambda-\mu}\operatorname{tr}\left(Q_{a}(\lambda)Q_{b}(\mu)-J_{a}J_{b}\right).$
(5.14)

The following lemma ensures that $L_{ab}(\lambda,\mu)$ can be expanded as in
(5.6),

$L_{ab}(\lambda,\mu)=\sum_{m,n=0}^{\infty}L_{ab}^{mn}\,\lambda^{-m-1}\mu^{-n-1}\,.$
(5.15)

######  Lemma 5.3.

The expression
$\operatorname{tr}\left(Q_{a}(\lambda)Q_{b}(\mu)-J_{a}J_{b}\right)$ in (5.14)
is divisible by $\lambda^{-1}-\mu^{-1}$

###### Proof.

We have

$\displaystyle\operatorname{tr}\left(Q_{a}(\lambda)Q_{b}(\mu)-J_{a}J_{b}\right)$
$\displaystyle=\operatorname{tr}\left(\varphi(\lambda)J_{a}\varphi(\lambda)^{-1}\varphi(\mu)J_{b}\varphi(\mu)^{-1}-J_{a}J_{b}\right)$
$\displaystyle=\operatorname{tr}\left(\left(\varphi(\mu)^{-1}\varphi(\lambda)-\mathbf{1}\right)J_{a}\varphi(\lambda)^{-1}\varphi(\mu)J_{b}\right.$
$\displaystyle\qquad\qquad\left.+J_{a}(\varphi(\lambda)^{-1}\varphi(\mu)-\mathbf{1})J_{b}\right)\,.$

Now it remains to show that $(\varphi(\mu)^{-1}\varphi(\lambda)-\mathbf{1})$
and $(\varphi(\lambda)^{-1}\varphi(\mu)-\mathbf{1})$ are divisible by
$\lambda^{-1}-\mu^{-1}$. We can expand $\varphi(\lambda)^{-1}\varphi(\mu)$ as

$\varphi(\lambda)^{-1}\varphi(\mu)=\sum_{n\geq
0}\sum_{j=0}^{n}F_{n,j}\lambda^{-j}\mu^{-(n-j)}\,,$ (5.16)

where $F_{0,0}=\mathbf{1}$. Setting $\lambda=\mu$ in equation (5.16) we find
that for all $n\geq 1$

$\sum_{j=0}^{n}F_{n,j}=0\,.$

Hence

$\displaystyle\varphi(\lambda)^{-1}\varphi(\mu)$ $\displaystyle=\sum_{n\geq
0}\sum_{j=0}^{n}F_{n,j}(\lambda^{-j}\mu^{-(n-j)}-\mu^{n})+\mathbf{1}$
$\displaystyle=\sum_{n\geq
0}\sum_{j=0}^{n}F_{n,j}\mu^{-(n-j)}(\lambda^{-j}-\mu^{-j})+\mathbf{1}\,.$

Similarly we also find that $(\varphi(\mu)^{-1}\varphi(\lambda)-\mathbf{1})$
is divisible by $\lambda^{-1}-\mu^{-1}$. ∎

The multi-time Euler Lagrange equations for $\mathcal{L}$ so defined produce a
non-commutative version of the generating Lax equation (5.9), controlled by
the Lie algebra with Lie bracket given by (5.10). Indeed, we have the
following

######  Proposition 5.4.

The multi-time Euler Lagrange equations for $\mathcal{L}$ take the form

$\begin{split}\partial_{\lambda,a}Q_{b}(\mu)&=\left[\frac{Q_{a}(\lambda)-Q_{a}(\mu)}{\lambda-\mu},Q_{b}(\mu)\right]\\\
&=\frac{1}{\lambda-\mu}\left(\left[Q_{a}(\lambda),Q_{b}(\mu)\right]-\sum_{c}\mathrm{C}_{ab}^{c}Q_{c}(\mu)\right)\,.\end{split}$
(5.17)

###### Proof.

Using the fact that partial derivatives of a trace are calculated as
$\frac{\partial{}}{\partial{B^{T}}}\operatorname{tr}(ABC)=CA$, we find

$\frac{\delta{L_{ab}(\lambda,\mu)}}{\delta{\varphi(\mu)^{T}}}=\left[J_{b},\varphi(\mu)^{-1}\partial_{\lambda,a}\varphi(\mu)\right]\varphi(\mu)^{-1}+\frac{1}{\lambda-\mu}\varphi(\mu)^{-1}\left[Q_{a}(\lambda),Q_{b}(\mu)\right]$

and

$p_{b}(\mu):=\frac{\partial{L_{ab}(\lambda,\mu)}}{\partial{(\partial_{\lambda,a}\varphi(\mu)^{T})}}=J_{b}\varphi(\mu)^{-1}\,.$

The multi-time Euler-Lagrange equations of types (5.1a) and (5.1b) are
trivially satisfied. Equation (5.1c) reads

$0=\frac{\delta{L_{ab}(\lambda,\mu)}}{\delta{\varphi(\mu)}}+\sum_{c}\frac{\mathrm{C}_{ab}^{c}}{\lambda-\mu}p_{c}(\mu)\,.$
(5.18)

Hence we find

$\displaystyle 0$
$\displaystyle=\left[J_{b},\varphi(\mu)^{-1}\partial_{\lambda,a}\varphi(\mu)\right]\varphi(\mu)^{-1}+\frac{1}{\lambda-\mu}\varphi(\mu)^{-1}\left[Q_{a}(\lambda),Q_{b}(\mu)\right]-\sum_{c}\frac{\mathrm{C}_{ab}^{c}}{\lambda-\mu}J_{c}\varphi(\mu)^{-1}$
$\displaystyle=\left[J_{b},\varphi(\mu)^{-1}\partial_{\lambda,a}\varphi(\mu)\right]\varphi(\mu)^{-1}+\frac{1}{\lambda-\mu}\varphi(\mu)^{-1}\left(\left[Q_{a}(\lambda),Q_{b}(\mu)\right]-\sum_{c}\mathrm{C}_{ab}^{c}Q_{c}(\mu)\right)$
$\displaystyle=\left[J_{b},\varphi(\mu)^{-1}\partial_{\lambda,a}\varphi(\mu)\right]\varphi(\mu)^{-1}+\frac{1}{\lambda-\mu}\varphi(\mu)^{-1}\left[Q_{a}(\lambda)-Q_{a}(\mu),Q_{b}(\mu)\right]\,,$
(5.19)

where the second term can be expanded in negative powers of $\lambda$ and
$\mu$ because $Q_{a}(\lambda)-Q_{a}(\mu)$ is divisible by
$\lambda^{-1}-\mu^{-1}$. As a consequence we find

$\displaystyle\partial_{\lambda,a}Q_{b}(\mu)$
$\displaystyle=\partial_{\lambda,a}\varphi(\mu)J_{b}\varphi(\mu)^{-1}-\varphi(\mu)J_{b}\varphi(\mu)^{-1}\partial_{\lambda,a}\varphi(\mu)\varphi(\mu)^{-1}$
$\displaystyle=\varphi(\mu)\left[\varphi(\mu)^{-1}\partial_{\lambda,a}\varphi(\mu),J_{b}\right]\varphi(\mu)^{-1}$
$\displaystyle=\frac{1}{\lambda-\mu}\left[Q_{a}(\lambda)-Q_{a}(\mu),Q_{b}(\mu)\right]$
(5.20)

as claimed. ∎

Equation (5.17) is a non-commutative generalisation of the generating Lax
equation (5.9) in the sense that the latter is obtained in the special case
where we consider only one generator $J_{a}$ (which would be proportional to
$\sigma_{3}$ in the historical example of [15]). In our more general context,
we have $N$ copies of (5.9), obtained when $b=a$, which are coupled with each
other by the remaining equations when $b\neq a$.

A direct calculation shows that (5.17) implies

$[\partial_{\lambda,a},\partial_{\mu,b}]Q_{c}(\nu)=\sum_{d}\mathrm{C}_{ab}^{d}\frac{\partial_{\lambda,d}-\partial_{\mu,d}}{\lambda-\mu}\,Q_{c}(\nu)\,,$

so that we have a realisation of the Lie algebra (5.12) as desired. In fact,
the calculations show the stronger result that the following deformation of
the usual zero curvature equations for a hierarchy holds. Let us introduce

$V_{a}(\lambda,\nu)=\frac{Q_{a}(\lambda)}{\lambda-\nu}\,.$ (5.21)

The formal series expansion in this expression is understood as

$V_{a}(\lambda,\nu)=\sum_{n=0}^{\infty}\frac{1}{\lambda^{n+1}}V_{a}^{(n)}(\nu)\,,\qquad\text{where}\leavevmode\nobreak\
\leavevmode\nobreak\ V_{a}^{(n)}(\nu)=\sum_{k=0}^{n}\nu^{k}Q_{a}^{(n-k)}\,.$
(5.22)

This defines the Lax matrices $V_{a}^{(n)}(\nu)$ of the hierarchy, which are
polynomials in $\nu$, and in terms of which we have:

######  Proposition 5.5.

The following non-commutative zero curvature equation, in generating form,
holds:

$\partial_{\lambda,a}V_{b}(\mu,\nu)-\partial_{\mu,b}V_{a}(\lambda,\nu)+[V_{b}(\mu,\nu),V_{a}(\lambda,\nu)]=\sum_{c}\mathrm{C}_{ab}^{c}\frac{V_{c}(\lambda,\nu)-V_{c}(\mu,\nu)}{\lambda-\mu}\,.$
(5.23)

The coefficient of $\lambda^{-n-1}\mu^{-m-1}$ gives the set of zero curvature
equations for the non-commutative hierarchy,

$\partial_{n,a}V_{b}^{(m)}(\nu)-\partial_{m,b}V_{a}^{(n)}(\nu)+[V_{b}^{(m)}(\nu),V_{a}^{(n)}(\nu)]=-\sum_{c}\mathrm{C}_{ab}^{c}V_{c}^{(m+n)}(\nu)\,,$
(5.24)

where $n,m\geq 0$.

###### Proof.

Equation (5.23) is obtained by direct calculation from Equations (5.17) and
(5.21).

The coefficient of $\lambda^{-n-1}\mu^{-m-1}$ in Equation (5.23) gives
Equation (5.24). On the left hand side this is obvious, whereas on right hand
side we use

$\displaystyle\frac{1}{\lambda-\mu}\left(V_{c}(\lambda,\nu)-V_{c}(\mu,\nu)\right)$
$\displaystyle=\frac{\lambda^{-1}\mu^{-1}}{\mu^{-1}-\lambda^{-1}}\sum_{n=0}^{\infty}\left(\lambda^{-n-1}-\mu^{-n-1}\right)V_{c}^{(n)}(\nu)$
$\displaystyle=\sum_{n=0}^{\infty}\left(-\lambda^{-n-1}\mu^{-1}-\lambda^{-n}\mu^{-2}-\ldots-\lambda^{-1}\mu^{-n-1}\right)V_{c}^{(n)}(\nu)$
$\displaystyle=-\sum_{m,n=0}^{\infty}\lambda^{-n-1}\mu^{-m-1}V_{c}^{(m+n)}(\nu)\,.\qed$

######  Remark 5.6.

The terminology “non-commutative zero curvature equations” is used to suggest
that our Equations (5.23) and (5.24) generalise the usual zero curvature
equations in the commutative case. The latter have the interpretation of
encoding the flatness of a (Lax) connection or, equivalently, the
commutativity of the components of a covariant derivative. On the other hand,
Equations (5.23) and (5.24) are not meant to be interpreted as such a flatness
condition. However, they do possess a beautiful interpretation in terms of
(generating) “covariant vector fields”, defined as

${\cal D}_{\lambda,a}(\nu)\equiv\partial_{\lambda,a}-V_{a}(\lambda,\nu)\,.$

We do _not_ call these “covariant derivatives” because the vector fields
$\partial_{n,a}$ should not be thought of as derivatives with respect to time
variables $t_{n,a}$, as already advocated above. Assuming the Lie algebra
relations (5.12) hold, (5.23) is equivalent to

$[{\cal D}_{\lambda,a}(\nu),{\cal
D}_{\mu,b}(\nu)]=\sum_{c}\frac{\mathrm{C}_{ab}^{c}}{\lambda-\mu}\left({\cal
D}_{\lambda,c}(\nu)-{\cal D}_{\mu,c}(\nu)\right)\,.$ (5.25)

This is the generalisation to our (generating) non-commutative context of the
following well-known fact. If one has commuting vector fields, say
$[\partial_{x},\partial_{t}]=0$, and a (Lax) connection $U\,dx+V\,dt$, then
the flatness or zero curvature condition is equivalent to the commutativity of
the covariant derivatives, $[\partial_{x}-U,\partial_{t}-V]=0$. In our
setting, with Lie algebra relations (5.12), the “non-commutative zero
curvature equations” (5.23) are equivalent to the relations (5.25) on the
“covariant vector fields” ${\cal D}_{\lambda,a}(\nu)$ which realise once again
the Lie algebra structure.

######  Remark 5.7.

There is a different way of using the generating zero curvature equation
(5.23) which was used in [27]. If we set $\lambda=\ell$, $\mu=\ell^{\prime}$,
$\nu=k$, $Q_{a}(\lambda)=R_{\ell}$,
$Q_{b}(\mu)=\widetilde{R}_{\ell^{\prime}}$, and
$[R_{\ell},\widetilde{R}_{\ell}]=\widehat{R}_{\ell}$, and if we multiply
(5.23) by $(\ell-k)(\ell^{\prime}-k)$ and require that it holds identically in
$k$, then we obtain the pair of equations (43) from [27]:

$\displaystyle\widetilde{\partial}_{\ell^{\prime}}R_{\ell}-\partial_{\ell}\widetilde{R}_{\ell^{\prime}}=-\frac{\widehat{R}_{\ell}-\widehat{R}_{\ell^{\prime}}}{\ell-\ell^{\prime}}\,,$
(5.26a)
$\displaystyle\ell^{\prime}\widetilde{\partial}_{\ell^{\prime}}R_{\ell}-\ell\partial_{\ell}\widetilde{R}_{\ell^{\prime}}+[R_{\ell},\widetilde{R}_{\ell^{\prime}}]+\frac{\ell^{\prime}\widehat{R}_{\ell}-\ell\widehat{R}_{\ell^{\prime}}}{\ell-\ell^{\prime}}=0\,.$
(5.26b)

In [27], the commutative analogue of (5.26) led to a connection with the Wess-
Zumino-Witten sigma model and it is an intriguing problem to cast (5.26) into
a non-commutative version of this connection.

######  Proposition 5.8 (Closure relation).

The form $\mathcal{L}$ with coefficients (5.14) satisfies the closure
relation. On the equations of motion (5.17), we have

$\partial_{\nu,c}\mathcal{L}_{ab}(\lambda,\mu)+\partial_{\mu,b}\mathcal{L}_{ca}(\nu,\lambda)+\partial_{\lambda,a}\mathcal{L}_{bc}(\mu,\nu)=0\,.$
(5.27)

The proof is given in Appendix A.

In the rest of this section we restrict our attention to
$\mathfrak{g}=\mathfrak{sl}(2)$ to illustrate how the usual (unreduced)
nonlinear Schrödinger equation of the AKNS hierarchy sits within our non-
commutative extension. It is enough to consider (5.24) for
$n,m\in\\{0,1,2\\}$. To avoid confusing notations on the vector fields that
would arise if we also used $a,b\in\\{1,2,3\\}$, we prefer to use the other
common choice of $a,b\in\\{x,y,z\\}$ for the labels associated to the basis
elements of $\mathfrak{sl}(2)$. Hence, instead of having vector fields such as
$\partial_{1,2}$ and $\partial_{2,1}$ in Equation (5.11), we will have
$\partial_{1,y}$ and $\partial_{2,x}$. This shows more clearly what labels the
level in the hierarchy and what labels the direction in the underlying Lie
algebra. Let us emphasise again that $x,y,z$ should not be thought of as
actual coordinates, since in general the vector fields $\partial_{n,x}$,
$\partial_{m,y}$ and $\partial_{k,z}$ do not commute.

As a basis of $\mathfrak{sl}(2)$ we choose

$J_{x}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}=\sigma_{x}\,,\qquad
J_{y}=\begin{pmatrix}0&-i\\\ i&0\end{pmatrix}=\sigma_{y}\,,\qquad
J_{z}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}=\sigma_{z}\,,$

which gives us $\mathrm{C}_{ab}^{c}=2i\epsilon_{abc}$ where $\epsilon_{abc}$
is the totally antisymmetric tensor with $\epsilon_{xyz}=1$. We write the
coefficients $\varphi^{(n)}$ of $\varphi(\lambda)$ in (5.8) as

$\varphi^{(n)}=\begin{pmatrix}A_{n}&B_{n}\\\ C_{n}&D_{n}\end{pmatrix}\,.$

Because we consider $\mathfrak{g}=\mathfrak{sl}(2)$, we have $\det\varphi=1$
which implies that $D_{1}+A_{1}=0$ and $D_{2}+A_{2}+A_{1}D_{1}-B_{1}C_{1}=0$.
We use this to eliminate $D_{1}$ and $D_{2}$. This gives us

$\displaystyle Q_{x}^{(0)}=\sigma_{1}\,,\qquad
Q_{x}^{(1)}=\begin{pmatrix}B_{1}-C_{1}&2A_{1}\\\
-2A_{1}&C_{1}-B_{1}\end{pmatrix}\,,$ $\displaystyle
Q_{x}^{(2)}=\begin{pmatrix}B_{2}-C_{2}-A_{1}(B_{1}+C_{1})&A_{1}^{2}-B_{1}^{2}+2A_{2}\\\
3A_{1}^{2}+2B_{1}C_{1}-C_{1}^{2}-2A_{2}&C_{2}-B_{2}+A_{1}(B_{1}+C_{1})\end{pmatrix}\,,$
$\displaystyle Q_{y}^{(0)}=\sigma_{2}\,,\qquad
Q_{y}^{(1)}=\begin{pmatrix}i(B_{1}+C_{1})&-2iA_{1}\\\
-2iA_{1}&-i(B_{1}+C_{1})\end{pmatrix}\,,$ $\displaystyle
Q_{y}^{(2)}=\begin{pmatrix}-i(A_{1}(B_{1}-C_{1})-B_{2}-C_{2})&-i(A_{1}^{2}+B_{1}^{2}+2A_{2})\\\
i(3A_{1}^{2}+2B_{1}C_{1}+C_{1}^{2}-2A_{2})&i(A_{1}(B_{1}-C_{1})-B_{2}-C_{2})\end{pmatrix}\,,$
$\displaystyle Q_{z}^{(0)}=\sigma_{3}\,,\qquad
Q_{z}^{(1)}=\begin{pmatrix}0&-2B_{1}\\\ 2C_{1}&0\end{pmatrix}\,,$
$\displaystyle Q_{z}^{(2)}=\begin{pmatrix}2B_{1}C_{1}&-2A_{1}B_{1}-2B_{2}\\\
-2A_{1}C_{1}+2C_{2}&-2B_{1}C_{1}\end{pmatrix}\,.$

We now spell out the content of (5.24) for the first three levels $n,m=0,1,2$
and write the equations they entail on the dynamical variables $A_{1}$,
$B_{1}$, $C_{1}$. The level $n=0$, $m=0$ just gives the Lie algebra relations
(5.10). At the level $n=0$, $m=1$, we have

$\partial_{0,a}Q_{b}^{(1)}+[Q_{b}^{(1)},\sigma_{a}]=-\sum_{c}\mathrm{C}_{ab}^{c}Q_{c}^{(1)}\,.$

Looking at the possible cases for $a,b$, this gives

$\displaystyle\partial_{0,x}A_{1}=C_{1}-B_{1}\,,$
$\displaystyle\partial_{0,x}B_{1}=-2A_{1}\,,$
$\displaystyle\partial_{0,x}C_{1}=2A_{1}\,,$ (5.28a)
$\displaystyle\partial_{0,y}A_{1}=-i(B_{1}+C_{1})\,,$
$\displaystyle\partial_{0,y}B_{1}=2iA_{1}\,,$
$\displaystyle\partial_{0,y}C_{1}=2iA_{1}\,,$ (5.28b)
$\displaystyle\partial_{0,z}A_{1}=0\,,$
$\displaystyle\partial_{0,z}B_{1}=2B_{1}\,,$
$\displaystyle\partial_{0,z}C_{1}=-2C_{1}\,.$ (5.28c)

In the standard AKNS case, at this first level we would only have (5.28c)
corresponding to $\sigma_{z}$, which is easily integrated. Here, to integrate
the full set of equations would require the use of the whole group $SL(2)$.
For the higher flows, the first few of which are described next, this becomes
even more complicated. One can imagine that a version of the usual dressing
method could be implemented which will require the use of the group
corresponding to the loop algebra
$\mathfrak{sl}(2)\otimes\lambda^{-1}\mathbb{C}[[\lambda^{-1}]]$ but, contrary
to the standard case, it is far from clear how explicitly the (seed) solutions
can be constructed. This task is beyond the scope of the present paper.

The next levels are studied in detail in Appendix B. They lead to

$\displaystyle\partial_{1,b}Q_{a}^{(1)}+[Q_{b}^{(2)},\sigma_{a}]=0\,,$ (5.29a)
$\displaystyle\partial_{2,a}Q_{b}^{(1)}-\partial_{1,a}Q_{b}^{(2)}+[Q_{b}^{(1)},Q_{a}^{(2)}]=0\,.$
(5.29b)

Equations (5.29) contain the required information to cast the (unreduced) NLS
equation into our non-commutative extension. As derived in Appendix B, with
$B_{1}=-\frac{1}{2}q$, $C_{1}=\frac{1}{2}r$, $A_{1}=\frac{1}{2}p$, we obtain

$\displaystyle\partial_{2,x}q-\frac{1}{2}\partial_{1,x}\partial_{1,z}q+q\partial_{1,x}p=0\,,$
(5.30a)
$\displaystyle\partial_{2,x}r-\frac{1}{2}\partial_{1,x}\partial_{1,z}r-r\partial_{1,x}p=0\,,$
(5.30b)
$\displaystyle\partial_{2,x}p-\frac{1}{2}\partial_{1,x}^{2}q+p\partial_{1,x}p-\frac{1}{2}(q+r)\partial_{1,x}q=0\,,$
(5.30c)

as well as similar equations for $\partial_{2,y}$

$\displaystyle\partial_{2,y}q-\frac{1}{2}\partial_{1,y}\partial_{1,z}q+q\partial_{1,y}p=0\,,$
(5.31a)
$\displaystyle\partial_{2,y}r+\frac{1}{2}\partial_{1,y}\partial_{1,z}r-r\partial_{1,y}p=0\,,$
(5.31b)
$\displaystyle\partial_{2,y}p-\frac{i}{2}\partial_{1,y}^{2}q+p\partial_{1,y}p+\frac{1}{2}(q-r)\partial_{1,y}q=0\,,$
(5.31c)

and for $\partial_{2,z}$

$\displaystyle\partial_{2,z}q-\frac{1}{2}\partial_{1,z}^{2}q+q^{2}r=0\,,$
(5.32a)
$\displaystyle\partial_{2,z}r+\frac{1}{2}\partial_{1,z}^{2}q-qr^{2}=0\,,$
(5.32b)
$\displaystyle\partial_{2,z}p-\frac{1}{2}r\partial_{1,z}q-\frac{1}{4}\partial_{1,z}(\partial_{1,x}+i\partial_{1,y})q+pqr=0\,.$
(5.32c)

When taken on their own, Equations (5.32a)-(5.32b) produce the unreduced NLS
system. This is seen by setting $2\partial_{2,z}=i\partial_{T}$,
$\partial_{1,z}=i\partial_{X}$ to obtain

$i\partial_{T}q+\partial_{X}^{2}q+2qrq=0\,,\qquad-i\partial_{T}r+\partial_{X}^{2}r-2rqr=0\,.$

The familiar nonlinear Schrödinger equation,

$iq_{T}+q_{XX}\pm 2|q|^{2}q=0,$

is obtained by applying the reduction $r=\pm q^{*}$. Equations (5.32a)–(5.32b)
are the lowest nonlinear ones in the usual AKNS hierarchy of vector fields
$\partial_{n,z}$ associated to $J_{z}=\sigma_{z}$. In our construction, it is
now part of the larger set of equations (5.30)-(5.32) which play the similar
role of being the lowest nonlinear equations in our non-commutative AKNS
hierarchy of vector fields $\partial_{n,a}$. All these equations, including
their higher counterparts contained in (5.24), derive from the multi-time
Euler Lagrange equations for our multiform $\mathcal{L}$. It is important to
note that, unlike the commutative case where one can consider (5.32a)-(5.32b)
without any reference to the entire hierarchy, it is not clear at this stage
whether the truncated system (5.30)-(5.32) can be studied as a set of
equations without the rest of the equations (5.24) in the hierarchy. This is
because of the (infinite dimensional) Lie algebra relations (5.13) on the
vector fields. If $a\neq b$, this may mean that the “differential
consequences” of (5.30)-(5.32) call upon higher equations not present in the
truncation. This is an intriguing new feature which is not present in the
usual case where $a=b=z$ and the flows of different levels commute. Its full
understanding requires to investigate the solutions of the non-commutative
AKNS hierarchy. As mentioned above, this is left for future work.

##  6 Conclusions

The main purpose of this work has been to extend the ideas of Lagrangian
multiforms and pluri-Lagrangian systems to apply to non-commuting flows. The
resulting theory of Lagrangian multiforms on Lie groups allows us to capture
the full symmetry group of a system in the variational description, no matter
if this group is abelian or not. To our knowledge, this is the first attempt
at a Lagrangian theory of Lie group actions on manifolds.

One application of the theory of Lagrangian multiforms on Lie groups is to
provide a variational description of superintegrable systems. On the other
hand, the development of this theory shows that the notion of Lagrangian
multiforms should not be constrained to the context of integrable systems.
Indeed, it can be applied to any Lagrangian system with symmetries and
captures both the dynamical system and its symmetries in a single variational
principle.

Some questions remain on the topic of non-commuting hierarchies of PDEs. These
include the proper interpretation of a differential form in generating form
$L(\lambda,\mu)$ and the derivation of exact solutions to the non-commutative
generalisation of the AKNS hierarchy, possibly using a version of the dressing
method. An additional topic for future research, given that previous results
have shown a remarkable similarity between the continuous and discrete
theories of Lagrangian multiforms, is to develop a theory discrete Lagrangian
multiforms for non-commuting maps and partial difference equations.

##### Acknowledgement

The majority of this work was carried out while MV was a research fellow at
the University of Leeds, supported by a Research Fellowship of the Deutsche
Forschungsgemeinschaft (project number VE 1211/1-1). FN is currently supported
by the EPSRC grant EP/W007290/1.

##### Declaration

The authors have no competing interests to declare that are relevant to the
content of this article.

## Appendix

##  Appendix A Proof of Proposition 5.8

The coefficients of $\mathrm{d}\mathcal{L}$ are generated by

$\partial_{\nu,c}\mathcal{L}_{ab}(\lambda,\mu)+\circlearrowleft\,,$

where $\circlearrowleft$ denotes the two terms obtained by cyclic permutations
of $(a,b,c)$ and $(\lambda,\mu,\nu)$. We write
$L_{ab}(\lambda,\mu)=K_{ab}(\lambda,\mu)-V_{ab}(\lambda,\mu)$ with

$\displaystyle
K_{ab}(\lambda,\mu)=\operatorname{tr}\left(\varphi(\mu)^{-1}\partial_{\lambda,a}\varphi(\mu)J_{b}-\varphi(\lambda)^{-1}\partial_{\mu,b}\varphi(\lambda)J_{a}\right)\,,$
$\displaystyle
V_{ab}(\lambda,\mu)=\frac{1}{\lambda-\mu}\operatorname{tr}\left(Q_{a}(\lambda)Q_{b}(\mu)-J_{a}J_{b}\right)\,,$

and analyse the contributions in turn. Using the Euler-Lagrange equation
(5.19) and the cyclic property of the trace repeatedly, we find

$\displaystyle\partial_{\nu,c}K_{ab}(\lambda,\mu)+\circlearrowleft$
$\displaystyle=\frac{1}{\lambda-\mu}\frac{1}{\nu-\mu}\operatorname{tr}\left(\left[Q_{c}(\nu)-Q_{c}(\mu),Q_{b}(\mu)\right]\left(Q_{a}(\lambda)-Q_{a}(\mu)\right)\right)+\circlearrowleft\,.$

Note that
$\operatorname{tr}\left(\left[Q_{c}(\nu),Q_{b}(\mu)\right]Q_{a}(\lambda)\right)$
and
$\operatorname{tr}\left(\left[Q_{c}(\mu),Q_{b}(\mu)\right]Q_{a}(\mu)\right)=\operatorname{tr}\left(\left[J_{c},J_{b}\right]J_{a}\right)$
are invariant under cyclic permutations and that
$\frac{1}{\lambda-\mu}\frac{1}{\nu-\mu}+\circlearrowleft=0$. Hence

$\displaystyle\partial_{\nu,c}K_{ab}(\lambda,\mu)+\circlearrowleft$
$\displaystyle=\frac{1}{\lambda-\mu}\frac{1}{\nu-\mu}\operatorname{tr}\left(-\left[Q_{c}(\mu),Q_{b}(\mu)\right]Q_{a}(\lambda)-\left[Q_{c}(\nu),Q_{b}(\mu)\right]Q_{a}(\mu)\right)+\circlearrowleft$
$\displaystyle=\frac{1}{\lambda-\mu}\frac{1}{\nu-\mu}\operatorname{tr}\left(Q_{a}(\lambda)\left[Q_{b}(\mu),Q_{c}(\mu)\right]+\left[Q_{a}(\mu),Q_{b}(\mu)\right]Q_{c}(\nu)\right)+\circlearrowleft\,.$
(A.1)

Similarly, we have

$\displaystyle\partial_{\nu,c}V_{ab}(\lambda,\mu)+\circlearrowleft$
$\displaystyle=\frac{1}{\lambda-\mu}\frac{1}{\nu-\lambda}\operatorname{tr}\left(\left[Q_{c}(\nu)-Q_{c}(\lambda),Q_{a}(\lambda)\right]Q_{b}(\mu)\right)$
$\displaystyle\qquad+\frac{1}{\lambda-\mu}\frac{1}{\nu-\mu}\operatorname{tr}\left(\left[Q_{c}(\nu)-Q_{c}(\mu),Q_{b}(\mu)\right]Q_{a}(\lambda)\right)+\circlearrowleft\,.$

Using again that
$\operatorname{tr}\left(\left[Q_{c}(\nu),Q_{b}(\mu)\right]Q_{a}(\lambda)\right)$
and
$\operatorname{tr}\left(\left[Q_{c}(\mu),Q_{b}(\mu)\right]Q_{a}(\mu)\right)$
are invariant under cyclic permutations, and that
$\frac{1}{\lambda-\mu}\frac{1}{\nu-\mu}+\circlearrowleft=0$, we find

$\displaystyle\partial_{\nu,c}V_{ab}(\lambda,\mu)+\circlearrowleft$