# Semi-discrete Lagrangian 2-forms and the Toda hierarchy

Duncan Sleigh  School of Mathematics, University of Leeds  Mats Vermeeren
Corresponding author. Email<EMAIL_ADDRESS>Department of
Mathematical Sciences, Loughborough University

###### Abstract

We present a variational theory of integrable differential-difference
equations (semi-discrete integrable systems). This is an extension of the
ideas known by the names “Lagrangian multiforms” and “Pluri-Lagrangian
systems”, which have previously been established in both the fully discrete
and fully continuous cases. The main feature of these ideas is to capture a
hierarchy of commuting equations in a single variational principle. Semi-
discrete Lagrangian multiforms provide a new way to relate differential-
difference equations and PDEs. We discuss this relation in the context of the
Toda lattice, which is part of an integrable hierarchy of differential-
difference equations, each of which involves a derivative with respect to a
continuous variable and a number of lattice shifts. We use the theory of semi-
discrete Lagrangian multiforms to derive PDEs in the continuous variables of
the Toda hierarchy, which hold as a consequence of the differential-difference
equations, but do not involve any lattice shifts. As a second example, we
briefly discuss the semi-discrete potential KdV equation, which is related to
the Volterra lattice.

##  1 Introduction

Integrable systems can be described in many different ways, but some of the
most important notions of integrability are formulated in the language of
Hamiltonian dynamics. These ideas go back at least as far as Liouville, but a
similar Lagrangian description of integrability is much more recent. It was
first proposed in the context of integrable lattice equations, where all
independent variables are discrete (see e.g. [3, 10, 11]). Later it was
developed in the fully continuous case as well, describing families of
commuting ODEs or PDEs (see e.g. [21, 22, 27]).

In the case of a family of ODEs, each equation is given its own independent
variable. The euclidean space spanned by all these variables is called multi-
time. The assumption that the ODEs commute means that solutions can be
understood as functions of multi-time, rather than functions of a single time
variable. In the case of PDEs, the members of an integrable family typically
share their space variables, but again they are each given their own time
variable. In this case, multi-time is spanned by both the common space-
variables and the individual time variables. In the fully discrete case,
multi-time is a lattice $\mathbb{Z}^{N}$ instead of a continuous euclidean
space.

The variational formulation of integrability has been presented in two subtly
different ways, under the names “Lagrangian multiforms” and “Pluri-Lagrangian
systems”. It involves a differential form on multi-time. If we are dealing
with ODEs, this is a 1-form. If we are considering a hierarchy of PDEs, it is
a $d$-form, where $d$ is the number of independent variables of each
individual member of the hierarchy. We can integrate this $d$-form over any
orientable $d$-dimensional submanifold of multi-time. The variational
principle requires that all such action integrals are critical. We can recover
the usual action of one member of the hierarchy by taking the submanifold to
be a coordinate (hyper)plane, but many other choices are possible. Hence the
complete set of “multi-time Euler-Lagrange equations” is larger than the set
set of Euler-Lagrange equations of the actions of each individual equation.
The additional equations can be thought of as compatibility conditions between
the coefficients of the Lagrangian $d$-form, in a similar sense to how
vanishing Poisson brackets are a compatibility condition between the
Hamiltonians of a Liouville integrable system. Hence a suitably chosen
Lagrangian $d$-form can describe an integrable hierarchy in a consistent way.

Connections have been established between the Lagrangian multiform approach
and classical topics in integrable systems such as Hamiltonian structures [21,
25], variational symmetries [16, 17, 19], and Lax pairs [18].

In the present work we extend the theory of Lagrangian multiforms to the semi-
discrete case, where some of the independent variables are continuous but
others discrete. An application to semi-discrete systems was proposed in one
of the early works on Lagrangian multiforms [28], but a systematic development
of this case has not been carried out before. Our main example will be the
hierarchy consisting of the Toda lattice and its symmetries, which together
form the Toda hierarchy. The first two equations of this hierarchy are

$\displaystyle q_{11}=\exp(\overline{q}-q)-\exp(q-\underline{q}),$
$\displaystyle q_{2}=q_{1}^{2}+\exp(\overline{q}-q)+\exp(q-\underline{q}),$

where subscripts denote derivatives with respect to the continuous independent
variables $t_{1}$ and $t_{2}$, and the bar and underline denote lattice shifts
in opposite directions. When considering only these two equations, we can take
$q$ to be a function of $\mathbb{Z}\times\mathbb{R}^{2}$, so our multi-time is
described by one discrete and two continuous variables. When considering
additional members of the hierarchy, with time variables $t_{3},\ldots,t_{N}$,
our multi-time will be $\mathbb{Z}\times\mathbb{R}^{N}$.

A continuous Lagrangian 1-form for the Toda hierarchy was given in [16]. In
that description, the elements of the configuration space are vectors
describing the positions of all particles. In particular, the discrete
direction is not treated as an independent variable, so the multi-time in this
case is $\mathbb{R}^{N}$. In this work we present a semi-discrete Lagrangian
2-form for the Toda hierarchy, in which the lattice position is a discrete
independent variable, i.e. multi-time is $\mathbb{Z}\times\mathbb{R}^{N}$. The
semi-discrete components of this 2-form are closely related to the
aforementioned 1-form, but the doubly continuous components are a new feature.
From these doubly continuous components we will derive PDEs that hold on each
single lattice site. These PDEs do not involve any lattice shifts, but hold as
a consequence of the lattice equations of the Toda hierarchy.

The task at hand is to introduce Lagrangian 2-form theory in the setting of a
semi-discrete multi-time $\mathbb{Z}\times\mathbb{R}^{N}$. This will require
us to define the notions of a semi-discrete differential form and of a semi-
discrete submanifold of $\mathbb{Z}\times\mathbb{R}^{N}$. The central
principle of semi-discrete Lagrangian multiform theory can then be formulated
in terms of the action integrals obtained by integrating a semi-discrete
2-form over an arbitrary semi-discrete surface within
$\mathbb{Z}\times\mathbb{R}^{N}$. Although in the example of the Toda
hierarchy the discrete direction has the interpretation of space and the
continuous variables can be thought of as times, this interpretation plays no
role on the general theory.

The plan for the paper is as follows. In Section 2 we introduce the notions of
semi-discrete manifolds and semi-discrete differential forms. In Section 3 the
theory of semi-discrete Lagrangian multiforms is developed. In Section 4 we
derive a semi-discrete Lagrangian 2-form for the Toda lattice and study its
implications. In Section 5 we briefly present a second example: the semi-
discrete potential KdV hierarchy, which is closely related to the Volterra
lattice. We close the paper with a few concluding thoughts and an appendix
containing the computations required to generalise the theory to higher semi-
discrete forms.

##  2 Semi-discrete geometry

In this section we present the necessary concepts of semi-discrete geometry.
For ease of presentation we assume throughout the main text that there is only
one discrete dimension. However, all concepts can be extended to a context
with several discrete directions, as is discussed in the appendix.

A semi-discrete surface in $\mathbb{Z}\times\mathbb{R}^{N}$ is a collection of
surfaces and curves in $\mathbb{R}^{N}$, which are each assigned a value of
$\mathbb{Z}$. A possible intuition is that the curves represent the locations
where the surface “jumps” to a different value of $\mathbb{Z}$. An example is
shown in Figure 1. This intuition is limited, however, because the semi-
discrete surfaces that have the most obvious dynamical meaning consist
entirely of lines. If $\mathbb{Z}\times\mathbb{R}^{N}$ is the space of
independent variables $(k,t_{1},\ldots,t_{N})$ of the Toda hierarchy, then the
$n$-th Toda equation can be considered on the subspace with
$t_{1},\ldots,t_{n-1},t_{n+1},\ldots,t_{N}$ fixed, which is a semi-discrete
surface consisting of a line in the $t_{n}$-direction at each lattice site.
Furthermore, this is the semi-discrete surface which we would integrate over
in the variational principle for the $n$-th Toda equation by itself. To obtain
a variational description of the hierarchy as a whole, i.e. a semi-discrete
Lagrangian 2-form, more general semi-discrete surfaces will be needed.

To be precise, we have the following definition:

Figure 1: Visualisation of the space $\mathbb{Z}\times\mathbb{R}^{2}$ and a
semi-discrete surface inside of it. The 2-dimensional elements of the semi-
discrete surface are shown in black and the 1-dimensional elements in white.
Note that we have oriented the 1-dimensional elements opposite to the
boundaries of the 2-dimensional elements, so that these cancel each other when
taking the boundary of the semi-discrete surface.

######  Definition 1.

  1. (a)

A $d$-dimensional semi-discrete submanifold $S$ of
$\mathbb{Z}\times\mathbb{R}^{N}$ is pair of disjoint unions

$S=\left(\bigsqcup_{k\in\mathbb{Z}}S_{k}^{d-1},\bigsqcup_{k\in\mathbb{Z}}S_{k}^{d}\right)$

where $S_{k}^{d-1}$ is a disjoint union of oriented $(d-1)$-dimensional
submanifolds of $\mathbb{R}^{N}$ and $S_{k}^{d}$ is a disjoint union of
oriented $d$-dimensional submanifolds of $\mathbb{R}^{N}$.

The disjoint unions $S_{k}^{d-1}$ and $S_{k}^{d}$ represent the continuous and
semi-discrete elements of the semi-discrete surface at discrete position
$k\in\mathbb{Z}$. They are defined as disjoint unions of submanifolds, rather
than submanifolds themselves, to allow for overlapping elements and count
their multiplicity.

  2. (b)

If $d=2$ we speak of a semi-discrete surface and if $d=3$ of a semi-discrete
volume.

  3. (c)

The boundary $\partial S$ of the $d$-dimensional semi-discrete submanifold $S$
is a $(d-1)$-dimensional semi-discrete submanifold given by

$\partial S=\left(\bigsqcup_{k\in\mathbb{Z}}-\partial
S_{k}^{d-1},\bigsqcup_{k\in\mathbb{Z}}\left(\partial S_{k}^{d}\sqcup
S_{k}^{d-1}\sqcup-S_{k+1}^{d-1}\right)\right),$

where the minus sign denotes a change of orientation.

Note that the sign conventions are chosen to ensure that the boundary of a
boundary is empty (modulo disjoint unions of two copies of the same manifold
with opposite orientations).

######  Remark 2.

To make sure a semi-discrete surface looks like a discretisation of a smooth
surface, we could require that the $S_{k}^{d-1}$ and $S_{k}^{d}$ do not
contain overlapping elements and that the corresponding subset

$\left(\bigcup_{k}\\{k\\}\times
S_{k}^{d}\right)\cup\left(\bigcup_{k}[k,k+1]\times
S_{k}^{d-1}\right)\subset\mathbb{R}\times\mathbb{R}^{N}$

is a topological manifold. However, this restriction is not needed in any of
the following. A related, but more useful, restriction would be to consider
only semi-discrete submanifolds without boundary. Just like in the classical
calculus of variations, we will impose that variations vanish on the boundary
of the submanifold on which the action is defined. Considering submanifolds
without boundary would remove this condition. Examples of semi-discrete
submanifolds without boundary include:

  * •

An unbounded continuous manifold at one lattice site: $S_{k}^{d-1}=\emptyset$,
$S_{k}^{d}=\emptyset$ for $k\neq k_{0}$, and $S_{k_{0}}^{d}$ a $d$-dimensional
manifold without boundary.

  * •

Copies of the same $(d-1)$-dimensional manifold in all lattice locations:
$S_{k}^{d-1}$ independent of $k$ and $S_{k}^{d}=\emptyset$.

  * •

The semi-discrete surface shown in Figure 1, assuming it is continued in a
suitable way beyond the edges of the image.

At the other extreme, a simple example of a semi-discrete submanifold in
$\mathbb{Z}\times\mathbb{R}^{2}$ with a very large boundary can be constructed
by intersecting the inclined plane $\\{(t_{0},t_{1},t_{0})\mid
t_{0},t_{1}\in\mathbb{R}\\}\subset\mathbb{R}^{3}$ with
$\mathbb{Z}\times\mathbb{R}^{2}$. We get:

  * •

$S_{k}^{d-1}=\\{(t_{1},k)\mid t_{1}\in\mathbb{R}\\}$ and
$S_{k}^{d}=\emptyset$. It consists only of lines, and it is contained within
its boundary, which is given by

$\partial S=\left(\emptyset,\bigsqcup_{k\in\mathbb{Z}}\left(S_{k}^{d-1}\cup-
S_{k+1}^{d-1}\right)\right).$

This surface will not be of interest in the variational principle, since any
variation that vanishes on the boundary vanishes on the whole surface.

The semi-discrete space $\mathbb{Z}\times\mathbb{R}^{N}$ will be our space of
independent variables, which we call multi-time. We consider semi-discrete
fields $q:\mathbb{Z}\times\mathbb{R}^{N}\mapsto Q$, taking values in some
configuration space $Q$. Often we will have $Q=\mathbb{R}$ or $Q=\mathbb{C}$.
When there is no risk of confusion we will write $q^{[n]}$ or simply $q$ for
$q(n,t_{1},\ldots,t_{N})$. To denote partial derivatives of $q$ we will use a
multi-index notation. A multi-index is an $N$-tuple $I=(i_{1},\ldots,i_{N})$
of non-negative integers. We define

$q_{I}=\frac{\partial{{}^{i_{1}}}}{\partial{t_{1}^{i_{1}}}}\ldots\frac{\partial{{}^{i_{N}}}}{\partial{t_{N}^{i_{N}}}}q,$

so that each entry of $I$ states the number of derivatives to be taken with
respect to the corresponding time variable. We will use the notations $It_{j}$
and $I\setminus t_{j}$ to raise or lower an entry of $I$, i.e.

$\displaystyle It_{j}=(i_{1},\ldots,i_{j}+1,\ldots,i_{N}),$ $\displaystyle
I\setminus t_{j}=(i_{1},\ldots,i_{j}-1,\ldots,i_{N})\qquad\text{if }i_{j}>0.$

We write $I\not\ni t_{j}$ if $i_{j}=0$. By $\mathcal{T}$ we denote the shift
operator: $\mathcal{T}q^{[n]}=q^{[n+1]}$, i.e.

$\mathcal{T}q(n,t_{1},\ldots,t_{N})=q(n+1,t_{1},\ldots,t_{N}).$

We denote by $\mathcal{Q}$ the set of all semi-discrete fields. We are
interested in functions of the semi-discrete fields that are autonomous (only
depend on $\mathbb{Z}\times\mathbb{R}^{N}$ through $q\in\mathcal{Q}$) and
local in the sense that $f\\!\left[q^{[n]}\right]$ depends on
$q^{[n+k]}=q(n+k,t_{1},\ldots,t_{N})$ and its derivatives for a finite number
of $k\in\mathbb{Z}$. Since $f$ is assumed to be autonomous, the shift operator
acts on it as $\mathcal{T}f[q]=f[\mathcal{T}q]$.

######  Definition 3.

  1. (a)

A $\mathcal{Q}$-dependent semi-discrete $d$-form on
$\mathbb{Z}\times\mathbb{R}^{N}$ is a pair

$\mathcal{L}[q]=\left(\mathcal{L}^{d-1}[q]\,,\,\mathcal{L}^{d}[q]\right)$

consisting of a $(d-1)$-form and a $d$-form, with coefficients that are
functions of $\mathcal{Q}$ in the sense explained above.

  2. (b)

The semi-discrete integral of $\mathcal{L}$ over a $d$-dimensional semi-
discrete submanifold $S$ is given by

$\int_{S}\mathcal{L}[q]=\sum_{k}\int_{S^{d-1}_{k}}\mathcal{L}^{d-1}\\!\left[q^{[k]}\right]+\sum_{k}\int_{S^{d}_{k}}\mathcal{L}^{d}\\!\left[q^{[k]}\right],$

where the integral over a disjoint union of submanifolds is understood as the
sum of the integrals over each of the submanifolds.

  3. (c)

The exterior derivative of
$\mathcal{L}[q]=\left(\mathcal{L}^{d-1}[q],\mathcal{L}^{d}[q]\right)$ is a
$\mathcal{Q}$-dependent semi-discrete $(d+1)$-form defined by

$\mathbbm{d}\mathcal{L}=\left(\Delta(\mathcal{L}^{d})-\mathsf{d}\mathcal{L}^{d-1}\,,\,\mathsf{d}\mathcal{L}^{d}\right),$
(2.1)

where $\Delta=\mathsf{id}-\mathcal{T}^{-1}$ is the backward difference
operator.

The signs in Equation (2.1) are chosen such that we get the usual alternating
expressions in terms of the coefficients of $\mathcal{L}$. Indeed, if we write

$\mathcal{L}=\left(\sum_{i_{1}<\ldots<i_{d-1}}L_{0i_{1}\ldots
i_{d-1}}\,\mathsf{d}t_{i_{1}}\wedge\ldots\wedge\mathsf{d}t_{i_{d-1}}\,,\sum_{i_{1}<\ldots<i_{d}}L_{i_{1}\ldots
i_{d}}\,\mathsf{d}t_{i_{1}}\wedge\ldots\wedge\mathsf{d}t_{i_{d}}\right)$

then we find

$\displaystyle\mathbbm{d}\mathcal{L}$
$\displaystyle=\left(\sum_{i_{1}<\ldots<i_{d}}\left(\Delta L_{i_{1}\ldots
i_{d}}+\sum_{\alpha=1}^{d}(-1)^{\alpha}\operatorname{D}_{i_{\alpha}}L_{0i_{1}\ldots\widehat{i_{\alpha}}\ldots
i_{d}}\right)\mathsf{d}t_{i_{1}}\wedge\ldots\wedge\mathsf{d}t_{i_{d}}\,,\right.$
$\displaystyle\qquad\left.\sum_{i_{1}<\ldots<i_{d+1}}\sum_{\alpha=1}^{d+1}(-1)^{\alpha-1}\operatorname{D}_{i_{\alpha}}L_{i_{1}\ldots\widehat{i_{\alpha}}\ldots
i_{d+1}}\,\mathsf{d}t_{i_{1}}\wedge\ldots\wedge\mathsf{d}t_{i_{d+1}}\right),$

where

$\operatorname{D}_{i}=\frac{\partial}{\partial
t_{i}}+\sum_{I}q_{It_{i}}\frac{\partial}{\partial q_{I}}$

is the total derivative with respect to $t_{i}$.

Our definition of the exterior derivative is justified by the following semi-
discrete version of Stokes’ theorem.

######  Theorem 4.

Let $S$ be a $(d+1)$-dimensional semi-discrete submanifold of
$\mathbb{Z}\times\mathbb{R}^{N}$ and $\mathcal{L}$ a $\mathcal{Q}$-dependent
semi-discrete $d$-form. There holds

$\int_{S}\mathbbm{d}\mathcal{L}=\int_{\partial S}\mathcal{L}.$

###### Proof.

Using our definitions and the smooth Stokes theorem we find

$\displaystyle\int_{S}\mathbbm{d}\mathcal{L}$
$\displaystyle=\sum_{k}\int_{S^{d}_{k}}\left(\Delta\mathcal{L}^{d}\\!\left[q^{[k]}\right]-\mathsf{d}\mathcal{L}^{d-1}\\!\left[q^{[k]}\right]\right)+\sum_{k}\int_{S^{d+1}_{k}}\mathsf{d}\mathcal{L}^{d}\\!\left[q^{[k]}\right]$
$\displaystyle=\sum_{k}\left(\int_{S^{d}_{k}}\mathcal{L}^{d}\\!\left[q^{[k]}\right]-\int_{S^{d}_{k}}\mathcal{L}^{d}\\!\left[q^{[k-1]}\right]-\int_{\partial
S^{d}_{k}}\mathcal{L}^{d-1}\\!\left[q^{[k]}\right]\right)+\sum_{k}\int_{\partial
S^{d+1}_{k}}\mathcal{L}^{d}\\!\left[q^{[k]}\right]$
$\displaystyle=-\sum_{k}\int_{\partial
S^{d}_{k}}\mathcal{L}^{d-1}\\!\left[q^{[k]}\right]+\sum_{k}\left(\int_{\partial
S^{d+1}_{k}}\mathcal{L}^{d}\\!\left[q^{[k]}\right]+\int_{S^{d}_{k}}\mathcal{L}^{d}\\!\left[q^{[k]}\right]-\int_{S^{d}_{k+1}}\mathcal{L}^{d}\\!\left[q^{[k]}\right]\right)\\!\\!\\!\\!\\!\\!$
$\displaystyle=\int_{\partial S}\mathcal{L}.\qed$

We close this section with an important lemma about commuting the shift
operator and the total derivative with partial derivatives.

######  Lemma 5.

There holds

$\displaystyle\frac{\partial{\mathcal{T}^{k}f}}{\partial{q_{I}}}=\mathcal{T}^{k}\frac{\partial{f}}{\partial{\mathcal{T}^{-k}q_{I}}},$
(2.2)
$\displaystyle\frac{\partial{(\operatorname{D}_{j}f)}}{\partial{q_{I}}}=\frac{\partial{f}}{\partial{q_{I\setminus
t_{j}}}}+\operatorname{D}_{j}\frac{\partial{f}}{\partial{q_{I}}},$ (2.3)

where ${I\setminus t_{j}}$ is the multi-index obtained from $I$ be reducing
the $j$-th entry by one, and the term containing it is taken to be zero if
$I\not\ni t_{j}$.

###### Proof.

Equation (2.2) is a direct consequence of the fact that $\mathcal{T}$ acts on
all instances of $q$ in the expression to the right of it. To derive Equation
(2.3) we calculate

$\displaystyle\frac{\partial{(\operatorname{D}_{j}f)}}{\partial{q_{I}}}$
$\displaystyle=\frac{\partial{}}{\partial{q_{I}}}\left(\sum_{J}\frac{\partial{f}}{\partial{q_{J}}}q_{Jt_{j}}\right)=\sum_{J}\frac{\partial{f}}{\partial{q_{I}\partial
q_{J}}}q_{Jt_{j}}+\frac{\partial{f}}{\partial{q_{I\setminus
t_{j}}}}=\operatorname{D}_{j}\frac{\partial{f}}{\partial{q_{I}}}+\frac{\partial{f}}{\partial{q_{I\setminus
t_{j}}}}.\qed$

##  3 Semi-discrete Lagrangian multiforms

In the continuous Lagrangian multiform theory, the central object is a
Lagrangian $d$-from, which is integrated over arbitrary $d$-dimensional
submanifolds of multi-time. In the following, this role will be played by a
$\mathcal{Q}$-dependent semi-discrete 2-form
$\mathcal{L}[q]=\left(\mathcal{L}^{1}[q],\mathcal{L}^{2}[q]\right)$ with

$\mathcal{L}^{1}[q]=\sum_{j>0}L_{0j}[q]\,\mathsf{d}t_{j}$

and

$\mathcal{L}^{2}[q]=\sum_{j>i>0}L_{ij}[q]\,\mathsf{d}t_{i}\wedge\mathsf{d}t_{j}.$

If $j<i$ we define $L_{ij}=-L_{ji}$.

######  Definition 6.

We say that a semi-discrete field $q:\mathbb{Z}\times\mathbb{R}^{N}\to Q$ is
critical for $\mathcal{L}[q]$ if for every semi-discrete surface $S$ it
satisfies

$\frac{\partial{}}{\partial{\varepsilon}}\bigg{|}_{\varepsilon=0}\int_{S}\mathcal{L}[q+\varepsilon
v]=0$ (3.1)

for any field $v$ that vanishes (along with all of its derivatives) at the
boundary of $S$.

Equation (3.1) can also be written as

$\delta\int_{S}\mathcal{L}[q]=\int_{S}\delta\mathcal{L}[q]=0$

where $\delta$ is the Gateaux derivative in a direction to be specified (the
arbitrary $v$ in Definition 6). This $\delta$ can also be understood as the
vertical exterior derivative in the variational bicomplex (see for example [2]
or [22, Appendix A]).

######  Definition 7.

The 1- and 2-dimensional continuous variational derivatives of a function $P$,
with respect to $q_{I}$, are defined as

$\displaystyle\frac{\delta_{i}{P}}{\delta{q_{I}}}=\sum_{\alpha\in\mathbb{N}}(-1)^{\alpha}\operatorname{D}_{i}^{\alpha}\frac{\partial{P}}{\partial{q_{It_{i}^{\alpha}}}},$
$\displaystyle\frac{\delta_{ij}{P}}{\delta{q_{I}}}=\sum_{\alpha,\beta\in\mathbb{N}}(-1)^{\alpha+\beta}\operatorname{D}_{i}^{\alpha}\operatorname{D}_{j}^{\beta}\frac{\partial{P}}{\partial{q_{It_{i}^{\alpha}t_{j}^{\beta}}}},$

where $\mathbb{N}=\\{0,1,2,\ldots\\}$. The semi-discrete variational
derivatives of a function $P$, with respect to $q_{I}$, are defined as

$\displaystyle\frac{\delta_{0}{P}}{\delta{q_{I}}}=\frac{\partial{}}{\partial{q_{I}}}\sum_{n\in\mathbb{N}}\mathcal{T}^{-n}P,$
(3.2a)
$\displaystyle\frac{\delta_{0i}{P}}{\delta{q_{I}}}=\frac{\delta_{i}{}}{\delta{q_{I}}}\sum_{n\in\mathbb{N}}\mathcal{T}^{-n}P,$
(3.2b)
$\displaystyle\frac{\delta_{0ij}{P}}{\delta{q_{I}}}=\frac{\delta_{ij}{}}{\delta{q_{I}}}\sum_{n\in\mathbb{N}}\mathcal{T}^{-n}P.$
(3.2c)

To give a few examples, denoting $\bar{q}=\mathcal{T}q$ and
$\underline{q}=\mathcal{T}^{-1}q$, we have

$\displaystyle\frac{\delta_{0i}{q_{t_{i}}^{2}}}{\delta{q}}$
$\displaystyle=-2\operatorname{D}_{i}q_{t_{i}}=-2q_{t_{i}t_{i}},$
$\displaystyle\frac{\delta_{0ij}{\bar{q}_{t_{i}}^{2}}}{\delta{q}}$
$\displaystyle=-2\operatorname{D}_{i}q_{t_{i}}=-2q_{t_{i}t_{i}},$
$\displaystyle\frac{\delta_{0i}{q\bar{q}}}{\delta{q}}$
$\displaystyle=\bar{q}+\underline{q},$
$\displaystyle\frac{\delta_{0i}{q\underline{q}}}{\delta{q}}$
$\displaystyle=\underline{q}.$

The last example highlights the fact that there are no positive shifts in the
right hand sides of Equations (3.2).

######  Remark 8.

The familiar variational derivative $\frac{\delta_{i}{}}{\delta{q}}$ is part
of an exact complex, satisfying
$\frac{\delta_{i}{}}{\delta{q}}\circ\operatorname{D}_{i}=0$. This property
fails for some of the variational derivatives of Definition 7. For example, we
have
$\big{(}\frac{\delta_{i}{}}{\delta{q_{i}}}\circ\operatorname{D}_{i}\big{)}q=1\neq
0$. The analogous discrete property also fails, for example
$\big{(}\frac{\delta_{0}{}}{\delta{q}}\circ\Delta\big{)}q=1\neq 0$. We still
use the term “variational derivative” because these are the expressions that
we encounter in the calculus of variations in multi-time.

######  Proposition 9.

The following are equivalent:

  1. (i)

The field $q$ is critical.

  2. (ii)

$\delta\mathbbm{d}\mathcal{L}=0$.

  3. (iii)

For all multi-indices $I$ and all $n$ there holds

$\frac{\delta_{0ij}{P_{0ij}}}{\delta{q_{I}^{[n]}}}=0\qquad\text{and}\qquad\frac{\delta_{ijk}{P_{ijk}}}{\delta{q_{I}^{[n]}}}=0.$
(3.3)

###### Proof.

Assume that $q$ is critical. Consider an arbitrary semi-discrete volume $V$
and integrate $\mathcal{L}$ over its boundary. By Theorem 4 we have

$\int_{\partial V}\mathcal{L}=\int_{V}\mathbbm{d}\mathcal{L}.$

Since $q$ is critical, infinitesimal variations of the left hand side must
vanish. It follows that

$\int_{V}\delta\mathbbm{d}\mathcal{L}=\delta\int_{V}\mathbbm{d}\mathcal{L}=0.$

Since $V$ is an arbitrary volume, it follows that
$\delta\mathbbm{d}\mathcal{L}=0$.

Following the above steps in reverse, we can see that if
$\delta\mathbbm{d}\mathcal{L}=0$, then the variational principle is satisfied
on all semi-discrete surfaces that are the boundary of some semi-discrete
volume. One can show that this implies that the variational principle is
satisfied on all semi-discrete surfaces. To do this, it is sufficient to
observe that the variational principle can be restricted without loss of
generality to variations with an arbitrarily small support, and that every
discrete surface is _locally_ the boundary of some semi-discrete volume.

To prove that (ii) and (iii) are equivalent, we will show that Equation (3.3)
holds for all multi-indices $I$ if and only if

$\frac{\partial{P_{0ij}}}{\partial{q_{I}^{[n]}}}=0\qquad\text{and}\qquad\frac{\partial{P_{ijk}}}{\partial{q_{I}^{[n]}}}=0$
(3.4)

for all multi-indices $I$ and all $n$. This establishes the claimed
equivalence, because the left hand sides of Equation (3.4) are the
coefficients of $\delta\mathbbm{d}\mathcal{L}$. The implication from Equation
(3.4) to Equation (3.3) follows immediately from the definition of the
variational derivatives. To prove the opposite implication, observe that we
can write a partial derivative in terms of variational derivatives:

$\frac{\partial{P_{0ij}}}{\partial{q_{I}^{[n]}}}=\sum_{\alpha,\beta\in\\{0,1\\}}\left(\operatorname{D}_{i}^{\alpha}\operatorname{D}_{j}^{\beta}\frac{\delta_{0ij}{P_{0ij}}}{\delta{q_{It_{i}^{\alpha}t_{j}^{\beta}}^{[n]}}}-\mathcal{T}^{-1}\operatorname{D}_{i}^{\alpha}\operatorname{D}_{j}^{\beta}\frac{\delta_{0ij}{P_{0ij}}}{\delta{q_{It_{i}^{\alpha}t_{j}^{\beta}}^{[n+1]}}}\right)$

and

$\frac{\partial{P_{ijk}}}{\partial{q_{I}^{[n]}}}=\sum_{\alpha,\beta,\gamma\in\\{0,1\\}}\operatorname{D}_{i}^{\alpha}\operatorname{D}_{j}^{\beta}\operatorname{D}_{k}^{\gamma}\frac{\delta_{ijk}{P_{ijk}}}{\delta{q_{It_{i}^{\alpha}t_{j}^{\beta}t_{k}^{\gamma}}^{[n]}}}.\qed$

Property $(ii)$ of Proposition 9 will be useful later on, because it is
satisfied if the coefficients of $\mathbbm{d}\mathcal{L}$ are products of two
factors that vanish on the equations of motion (or are sums of such products).
Hence, if we construct a semi-discrete 2-form such that
$\mathbbm{d}\mathcal{L}$ attains such a “double zero” on solutions to a set of
equations, then it is guaranteed that this set of equations implies the multi-
time Euler-Lagrange equations. The equivalence between $(i)$ and $(iii)$ will
be used in the proof of the following theorem.

######  Theorem 10.

A field is critical if and only if all of the following multi-time Euler-
Lagrange equations hold for all $n\in\mathbb{Z}$:

$\displaystyle\frac{\delta_{ij}{L_{ij}}}{\delta{q_{I}^{[n]}}}=0$
$\displaystyle\forall I\not\ni t_{i},t_{j},$ (3.5a)
$\displaystyle\frac{\delta_{ij}{L_{ij}}}{\delta{q_{It_{j}}^{[n]}}}-\frac{\delta_{ik}{L_{ik}}}{\delta{q_{It_{k}}^{[n]}}}=0$
$\displaystyle\forall I\not\ni t_{i},$ (3.5b)
$\displaystyle\frac{\delta_{ij}{L_{ij}}}{\delta{q_{It_{i}t_{j}}^{[n]}}}+\frac{\delta_{jk}{L_{jk}}}{\delta{q_{It_{j}t_{k}}^{[n]}}}+\frac{\delta_{ki}{L_{ki}}}{\delta{q_{It_{k}t_{i}}^{[n]}}}=0$
$\displaystyle\forall I,$ (3.5c)
$\displaystyle\frac{\delta_{ij}{L_{ij}}}{\delta{q_{It_{j}}^{[n]}}}+\frac{\delta_{0i}{L_{0i}}}{\delta{q_{I}^{[n]}}}=0$
$\displaystyle\forall I\not\ni t_{i},$ (3.5d)
$\displaystyle\frac{\delta_{ij}{L_{ij}}}{\delta{q_{It_{i}t_{j}}^{[n]}}}-\frac{\delta_{0j}{L_{0j}}}{\delta{q_{It_{j}}^{[n]}}}+\frac{\delta_{0i}{L_{0i}}}{\delta{q_{It_{i}}^{[n]}}}=0$
$\displaystyle\forall I.$ (3.5e)

If $n$ is such that $L_{ij}$ does not depend on $q_{I}^{[n]}$ for any $I$,
then it follows from (3.5d) and (3.5e) that

$\displaystyle\frac{\delta_{0i}{L_{0i}}}{\delta{q_{I}^{[n]}}}=0$
$\displaystyle\forall I\not\ni t_{i},$ (3.5f)
$\displaystyle\frac{\delta_{0j}{L_{0j}}}{\delta{q_{It_{j}}^{[n]}}}-\frac{\delta_{0i}{L_{0i}}}{\delta{q_{It_{i}}^{[n]}}}=0$
$\displaystyle\forall I.$ (3.5g)

###### Proof of Theorem 10.

We write $q$ for $q^{[n]}$, hence $\mathcal{T}^{m}q=q^{[m+n]}$. Using Lemma 5
we find, for any multi-index $J$,

$\displaystyle\frac{\partial{P_{0ij}}}{\partial{q_{J}}}$
$\displaystyle=\frac{\partial{L_{ij}}}{\partial{q_{J}}}-\mathcal{T}^{-1}\frac{\partial{L_{ij}}}{\partial{\mathcal{T}q_{J}}}-\operatorname{D}_{i}\frac{\partial{L_{0j}}}{\partial{q_{J}}}-\frac{\partial{L_{0j}}}{\partial{q_{J\setminus
t_{i}}}}+\operatorname{D}_{j}\frac{\partial{L_{0i}}}{\partial{q_{J}}}+\frac{\partial{L_{0i}}}{\partial{q_{J\setminus
t_{j}}}}.$

Hence

$\displaystyle\frac{\delta_{0ij}{P_{0ij}}}{\delta{q_{J}}}$
$\displaystyle=\sum_{m,\alpha,\beta\in\mathbb{N}}(-1)^{\alpha+\beta}\operatorname{D}_{i}^{\alpha}\operatorname{D}_{j}^{\beta}\frac{\partial{\mathcal{T}^{-m}P_{0ij}}}{\partial{q_{Jt_{i}^{\alpha}t_{j}^{\beta}}}}$
$\displaystyle=\sum_{m,\alpha,\beta\in\mathbb{N}}(-1)^{\alpha+\beta}\operatorname{D}_{i}^{\alpha}\operatorname{D}_{j}^{\beta}\left(\mathcal{T}^{-m}\frac{\partial{L_{ij}}}{\partial{\mathcal{T}^{m}q_{Jt_{i}^{\alpha}t_{j}^{\beta}}}}-\mathcal{T}^{-m-1}\frac{\partial{L_{ij}}}{\partial{\mathcal{T}^{m+1}q_{Jt_{i}^{\alpha}t_{j}^{\beta}}}}\right)$
$\displaystyle\qquad-\sum_{m,\alpha,\beta\in\mathbb{N}}(-1)^{\alpha+\beta}\operatorname{D}_{j}^{\beta}\left(\operatorname{D}_{i}^{\alpha+1}\frac{\partial{\mathcal{T}^{-m}L_{0j}}}{\partial{q_{Jt_{i}^{\alpha}t_{j}^{\beta}}}}+\operatorname{D}_{i}^{\alpha}\frac{\partial{\mathcal{T}^{-m}L_{0j}}}{\partial{q_{Jt_{i}^{\alpha-1}t_{j}^{\beta}}}}\right)$
$\displaystyle\qquad+\sum_{m,\alpha,\beta\in\mathbb{N}}(-1)^{\alpha+\beta}\operatorname{D}_{i}^{\alpha}\left(\operatorname{D}_{j}^{\beta+1}\frac{\partial{\mathcal{T}^{-m}L_{0i}}}{\partial{q_{Jt_{i}^{\alpha}t_{j}^{\beta}}}}+\operatorname{D}_{j}^{\beta}\frac{\partial{\mathcal{T}^{-m}L_{0i}}}{\partial{q_{Jt_{i}^{\alpha}t_{j}^{\beta-1}}}}\right)$
$\displaystyle=\sum_{\alpha,\beta\in\mathbb{N}}(-1)^{\alpha+\beta}\operatorname{D}_{i}^{\alpha}\operatorname{D}_{j}^{\beta}\frac{\partial{L_{ij}}}{\partial{q_{Jt_{i}^{\alpha}t_{j}^{\beta}}}}$
$\displaystyle\qquad-\sum_{m,\beta\in\mathbb{N}}(-1)^{\beta}\operatorname{D}_{j}^{\beta}\frac{\partial{\mathcal{T}^{-m}L_{0j}}}{\partial{q_{Jt_{i}^{-1}t_{j}^{\beta}}}}+\sum_{m,\alpha\in\mathbb{N}}(-1)^{\alpha}\operatorname{D}_{i}^{\alpha}\frac{\partial{\mathcal{T}^{-m}L_{0i}}}{\partial{q_{Jt_{i}^{\alpha}t_{j}^{-1}}}}$
$\displaystyle=\frac{\delta_{ij}{L_{ij}}}{\delta{q_{J}}}-\frac{\delta_{0j}{L_{0j}}}{\delta{q_{Jt_{i}^{-1}}}}+\frac{\delta_{0i}{L_{0i}}}{\delta{q_{Jt_{j}^{-1}}}},$
(3.6)

where $Jt_{i}^{-1}=J\setminus t_{i}$ denotes the multi-index obtained for $J$
by reducing the $i$-th entry by one if it is positive, and any term containing
$Jt_{i}^{-1}$ is taken to be zero if the $i$-th entry of $J$ is zero.
Similarly, we find

$\frac{\delta_{ijk}{P_{ijk}}}{\delta{q_{J}}}=\frac{\delta_{ij}{L_{ij}}}{\delta{q_{Jt_{k}^{-1}}}}+\frac{\delta_{jk}{L_{jk}}}{\delta{q_{Jt_{i}^{-1}}}}+\frac{\delta_{ki}{L_{ki}}}{\delta{q_{Jt_{j}^{-1}}}}.$
(3.7)

By Proposition 9 it follows that the semi-discrete field is critical if and
only if the expressions (3.6)–(3.7) equal zero for all $J$. Considering these
equations for different types of multi-indices, we find equations
(3.5a)–(3.5e):

  * •

From Equation (3.6), with $J=I\not\ni t_{i},t_{j}$, we obtain Equation (3.5a).
If $J=It_{j}$ with $I\not\ni t_{i}$ we find Equation (3.5d), and if
$J=It_{i}t_{j}$ we find Equation (3.5e)

  * •

From Equation (3.7) with $J=It_{k}$, $I\not\ni t_{i},t_{j}$, we obtain once
again Equation (3.5a). If $J=It_{j}t_{k}$ with $I\not\ni t_{i}$ we find
Equation (3.5b), and if $J=It_{i}t_{j}t_{k}$ we find Equation (3.5c). If
$J\not\ni t_{i},t_{j},t_{k}$, then (3.7) vanishes identically. ∎

######  Remark 11.

A semi-discrete Lagrangian 2-form $\mathcal{L}$ on
$\mathbb{Z}\times\mathbb{R}^{N}$ can be reduced to a continuous Lagrangian
1-form $\mathcal{M}$ on $\mathbb{R}^{N}$ by summing over the lattice sites.
Let us illustrate this for the case of a periodic lattice,
$\mathcal{T}^{n}q=q$. A continuous 1-form is obtained from the 1-form part of
the semi-discrete 2-form $\mathcal{L}=(\mathcal{L}^{1},\mathcal{L}^{2})$:

$\mathcal{M}=\sum_{\alpha=0}^{n-1}\mathcal{L}^{1}[q^{[\alpha]}].$

Fix a curve $\gamma$ in $\mathbb{R}^{N}$ and consider the semi-discrete
surface of integration $S=(S_{k}^{1},S_{k}^{2})$ with $S_{k}^{1}=\gamma$ and
$S_{k}^{2}=\emptyset$. The semi-discrete integral of $\mathcal{L}$ over $S$ is

$\sum_{\alpha=0}^{n-1}\int_{\gamma}L[q^{[\alpha]}]=\int_{\gamma}\mathcal{M}.$

Hence every action integral of the continuous 1-form $\mathcal{M}$ is also an
action for the semi-discrete 2-form $\mathcal{L}$, so every solution to the
variational problem for the 2-form $\mathcal{L}$ is also a solution to the
variational problem for the 1-form $\mathcal{M}$.

##  4 Toda lattice

The Toda lattice [23] is an integrable model consisting of $k$ particles on a
line, with nearest-neighbour interaction. The deviation from equilibrium of
one of the particles is given by $q=q^{[n]}=q(n,t_{1},t_{2},\ldots)$. We use a
bar-notation for shifts:

$\overline{q}=\mathcal{T}q=q(n+1,t_{1},t_{2},\ldots),\qquad\underline{q}=\mathcal{T}^{-1}q=q(n-1,t_{1},t_{2},\ldots).$

For derivatives of $q$ we use the subscript notations:

$q_{i}=q_{t_{i}}=\frac{\partial{q}}{\partial{t_{i}}},\qquad
q_{ij}=q_{t_{i}t_{j}}=\frac{\partial{{}^{2}q}}{\partial{t_{i}\partial
t_{j}}}.$

The Toda lattice and the next two members of its hierarchy are given by

$\displaystyle q_{11}=\exp(\overline{q}-q)-\exp(q-\underline{q}),$ (4.1a)
$\displaystyle q_{2}=q_{1}^{2}+\exp(\overline{q}-q)+\exp(q-\underline{q}),$
(4.1b) $\displaystyle
q_{3}=q_{1}^{3}+(2q_{1}+\underline{q}_{1})\exp(q-\underline{q})+(2q_{1}+\overline{q}_{1})\exp(\overline{q}-q),$
(4.1c)

where either open-ended or periodic boundary conditions can be used. In
Section 4.1 we will sketch a systematic construction of this hierarchy. A
continuous Lagrangian 1-form for this hierarchy is known, where the
configuration is represented by a vector in $\mathbb{R}^{k}$ containing the
positions of all particles. This ignores the physical intuition behind the
system, where we think of the particles on a discrete lattice in space. To
capture this, along with the continuous time evolution, we develop a semi-
discrete 2-form for the Toda lattice.

Before we get started, let us think about whether lattice shifts could be
eliminated form the system (4.1a)–(4.1b). By considering these equations as a
linear system for the two exponential terms, we find an equivalent system

$\displaystyle\exp(\bar{q}-q)=\frac{1}{2}(q_{2}+q_{11}-q_{1}^{2}),$ (4.2a)
$\displaystyle\exp(q-\underline{q})=\frac{1}{2}(q_{2}-q_{11}-q_{1}^{2}).$
(4.2b)

This shows that we can eliminate one of the lattice shifts from Equations
(4.1). In fact, Equations (4.2) can be understood as an NLS-type system

$\displaystyle U_{2}=U_{11}+2U^{2}V,\qquad V_{2}=-V_{11}-2UV^{2},$

with variables $U=\exp(\overline{q})$ and $V=\exp(q)$ [1]. This observation
relates the Toda lattice to a system of integrable PDEs by promoting a lattice
shift to an additional dependent variable. It is far from obvious if this
additional variable can be eliminated to obtain one scalar PDE. As we will see
below, the Lagrangian multiform will provide a solution to this problem.

Our construction is inspired on the known continuous 1-form, which can be
obtained for example from the discrete-time Toda lattice using a continuum
limit [24], or from the variational symmetries of the system [16].
Alternatively, it could be obtained from the Hamiltonian formulation of the
hierarchy (see e.g. [20]) using the methods of [25]. Here we adapt the latter
approach to yield a semi-discrete 2-form.

###  4.1 Hamiltonian formulation

The geometric structure of the Toda lattice is usually presented in Flaschka
variables [4]

$a=\exp(\bar{q}-q),\qquad b=q_{1},$

with the Poisson brackets

$\\{a,b\\}=a,\qquad\\{a,\bar{b}\\}=-a.$

We consider Hamilton functions

$H_{i}=\sum_{\alpha\in\mathbb{Z}}\mathcal{T}^{\alpha}h_{i}=\ldots+\underline{h_{i}}+h_{i}+\overline{h_{i}}+\ldots,$

where

$\displaystyle h_{1}$ $\displaystyle=\frac{1}{2}b^{2}+a,$ (4.3a)
$\displaystyle h_{2}$ $\displaystyle=\frac{1}{3}b^{3}+a(b+\bar{b}),$ (4.3b)
$\displaystyle h_{3}$
$\displaystyle=\frac{1}{4}b^{4}+a(b^{2}+\bar{b}^{2})+ab\bar{b}+a\bar{a}+\frac{1}{2}a^{2},\
\ldots$ (4.3c)

Note that the subscripts on $h$ and $H$ are labels, not derivatives. These
Hamilton functions can be obtained from the usual Lax formulation of the Toda
lattice, $\frac{\partial{L}}{\partial{t_{1}}}=[B,L]$, by taking
$H_{i}=\frac{1}{i}\mathrm{tr}{L^{i}}$ (see [4], also [20, Chapter 3]). We
choose to write them as a sum
$H_{i}=\sum_{\alpha\in\mathbb{Z}}\mathcal{T}^{\alpha}h_{i}$ in such a way that
$h_{i}$ does not contain any negative shifts of $a$ or $b$.

The corresponding equations of motion are

$\displaystyle
a_{i}=\\{H_{i},a\\}=\left(\frac{\partial{H_{i}}}{\partial{\bar{b}}}-\frac{\partial{H_{i}}}{\partial{b}}\right)a,$
(4.4a) $\displaystyle
b_{i}=\\{H_{i},b\\}=\Delta\left(\frac{\partial{H_{i}}}{\partial{a}}a\right).$
(4.4b)

In the original coordinates, the equations of motion are

$\displaystyle q_{i}=Q_{i}:=\frac{\partial{H_{i}}}{\partial{b}},$ (4.5a)
$\displaystyle
q_{1i}=B_{i}:=\Delta\left(\frac{\partial{H_{i}}}{\partial{a}}a\right),$ (4.5b)

where the subscripts on $Q$ and $B$ are labels, not derivatives.

An elementary calculation shows that

$\\{H_{i},H_{j}\\}=\sum_{\alpha\in\mathbb{Z}}\mathcal{T}^{\alpha}(B_{i}Q_{j}-Q_{i}B_{j}).$
(4.6)

It is well-known that the $H_{i}$ are in Poisson involution, so this sum must
be zero. It follows that the summand can be written as a difference:

$B_{i}Q_{j}-Q_{i}B_{j}=\Delta F_{ij}$ (4.7)

for some $F_{ij}$.

###  4.2 Semi-discrete 2-form

Following the construction of Lagrangian 1-forms from Hamiltonians in
involution [25], we find a continuous Lagrangian 1-form for the Toda lattice
with coefficients

$L_{j}=\sum_{\alpha}\mathcal{T}^{\alpha}(q_{1}q_{j}-h_{j}),$

where the first few $h_{j}$ are given in Equation (4.3). We now look for a
semi-discrete 2-form that reduces to this 1-form by the method of Remark 11.
This motivates the choice

$L_{0j}=q_{1}q_{j}-h_{j},$ (4.8)

but does not guide our choice of coefficients $L_{ij}$.

We want to construct coefficients $L_{ij}$ such that the exterior derivative
of the semi-discrete 2-form

$\left(\sum_{j}L_{0j}\,\mathsf{d}t_{j}\ ,\
\sum_{i<j}L_{ij}\,\mathsf{d}t_{i}\wedge\mathsf{d}t_{j}\right)$

vanishes on solutions. Furthermore, in light of Proposition 9, we would like
it to attain a double zero on solutions. The following fact, which can be
thought of as a local version of Equation (4.6), will come in useful.

######  Lemma 12.

On the equations of motion (4.5), there holds

$\displaystyle\operatorname{D}_{i}h_{j}=-B_{j}q_{i}+Q_{j}q_{1i}+\Delta\left(\frac{\partial{H_{j}}}{\partial{a}}a\mathcal{T}q_{i}+\sum_{\alpha\geq
1}\sum_{\beta=1}^{\alpha}\mathcal{T}^{\beta}\left(\frac{\partial{(\mathcal{T}^{-\alpha}h_{j})}}{\partial{a}}a_{i}+\frac{\partial{(\mathcal{T}^{-\alpha}h_{j})}}{\partial{b}}b_{i}\right)\right).$

###### Proof.

Since $h_{j}$ does not contain any negative shifts, we can write

$\displaystyle\operatorname{D}_{i}h_{j}$ $\displaystyle=\sum_{\alpha\geq
0}\left(\frac{\partial{h_{j}}}{\partial{\mathcal{T}^{\alpha}a}}\mathcal{T}^{\alpha}a_{i}+\frac{\partial{h_{j}}}{\partial{\mathcal{T}^{\alpha}b}}\mathcal{T}^{\alpha}b_{i}\right)$
$\displaystyle=\sum_{\alpha\geq
0}\mathcal{T}^{\alpha}\left(\frac{\partial{\mathcal{T}^{-\alpha}h_{j}}}{\partial{a}}a_{i}+\frac{\partial{\mathcal{T}^{-\alpha}h_{j}}}{\partial{b}}b_{i}\right)$
$\displaystyle=\sum_{\alpha\geq
0}\left(\frac{\partial{\mathcal{T}^{-\alpha}h_{j}}}{\partial{a}}a_{i}+\frac{\partial{\mathcal{T}^{-\alpha}h_{j}}}{\partial{b}}b_{i}\right)+\sum_{\alpha\geq
1}\sum_{\beta=1}^{\alpha}\Delta\left(\mathcal{T}^{\beta}\left(\frac{\partial{\mathcal{T}^{-\alpha}h_{j}}}{\partial{a}}a_{i}+\frac{\partial{\mathcal{T}^{-\alpha}h_{j}}}{\partial{b}}b_{i}\right)\right).$

Using once more that $h_{j}$ does not contain any negative shifts, it follows
that

$\operatorname{D}_{i}h_{j}=\frac{\partial{H_{j}}}{\partial{a}}a(\mathcal{T}q_{i}-q_{i})+\frac{\partial{H_{j}}}{\partial{b}}q_{1i}+\sum_{\alpha\geq
1}\sum_{\beta=1}^{\alpha}\Delta\left(\mathcal{T}^{\beta}\left(\frac{\partial{\mathcal{T}^{-\alpha}h_{j}}}{\partial{a}}a_{i}+\frac{\partial{\mathcal{T}^{-\alpha}h_{j}}}{\partial{b}}b_{i}\right)\right).$

To finish the proof, observe that the first term in the right hand side is
equal to

$\frac{\partial{H_{j}}}{\partial{a}}a(\mathcal{T}q_{i}-q_{i})=-\Delta\left(\frac{\partial{H_{j}}}{\partial{a}}a\right)q_{i}+\Delta\left(\frac{\partial{H_{j}}}{\partial{a}}a\mathcal{T}q_{i}\right)$

and use Equations (4.5). ∎

The computation in the proof of Theorem 13 below, which uses Lemma 12, shows
that the exterior derivative $\mathbbm{d}\mathcal{L}$ attains a double zero on
solutions to the Toda hierarchy if we set

$\begin{split}L_{ij}&=\frac{\partial{H_{i}}}{\partial{a}}a\mathcal{T}q_{j}+\sum_{\alpha\geq
1}\sum_{\beta=1}^{\alpha}\mathcal{T}^{\beta}\left(\frac{\partial{(\mathcal{T}^{-\alpha}h_{i})}}{\partial{a}}a_{j}+\frac{\partial{(\mathcal{T}^{-\alpha}h_{i})}}{\partial{b}}b_{j}\right)\\\
&\quad-\frac{\partial{H_{j}}}{\partial{a}}a\mathcal{T}q_{i}-\sum_{\alpha\geq
1}\sum_{\beta=1}^{\alpha}\mathcal{T}^{\beta}\left(\frac{\partial{(\mathcal{T}^{-\alpha}h_{j})}}{\partial{a}}a_{i}+\frac{\partial{(\mathcal{T}^{-\alpha}h_{j})}}{\partial{b}}b_{i}\right)-F_{ij},\end{split}$
(4.9)

where $F_{ij}$ is as in Equation (4.7).

######  Theorem 13.

Let $\mathcal{L}$ be the semi-discrete 2-form with coefficients given by
Equations (4.8) and (4.9). There holds $\delta\mathbbm{d}\mathcal{L}=0$ on
solutions to the Toda hierarchy, hence the Toda hierarchy implies the multi-
time Euler-Lagrange equations.

###### Proof.

Equation (2.1) states that $\mathbbm{d}\mathcal{L}$ has coefficients
$P_{ijk}=\operatorname{D}_{i}L_{jk}-\operatorname{D}_{j}L_{ik}+\operatorname{D}_{k}L_{ij}$
and

$\displaystyle P_{0ij}$ $\displaystyle=\Delta
L_{ij}-\operatorname{D}_{i}L_{0j}+\operatorname{D}_{j}L_{0i}$
$\displaystyle=-q_{1i}q_{j}+q_{1j}q_{i}+\operatorname{D}_{i}h_{j}-\operatorname{D}_{j}h_{i}+\Delta
L_{ij}.$

Using Lemma 12 we find

$\displaystyle P_{0ij}$
$\displaystyle=-q_{1i}q_{j}+q_{1j}q_{i}-B_{j}q_{i}+Q_{j}q_{1i}+B_{i}q_{j}-Q_{i}q_{1j}-\Delta
F_{ij}$
$\displaystyle=-(q_{1i}-B_{i})(q_{j}-Q_{j})+(q_{1j}-B_{j})(q_{i}-Q_{i}).$
(4.10)

Hence $P_{0ij}$ attains a double zero on solutions.

In addition, we have

$\displaystyle\Delta P_{ijk}$ $\displaystyle=\operatorname{D}_{i}\Delta
L_{jk}-\operatorname{D}_{j}\Delta L_{ik}+\operatorname{D}_{k}\Delta L_{ij}$
$\displaystyle=\operatorname{D}_{i}P_{0jk}-\operatorname{D}_{j}P_{0ik}+\operatorname{D}_{k}P_{0ij},$

which also attains a double zero on solutions. Therefore, on solutions,

$\displaystyle\frac{\partial{\Delta
P_{ijk}}}{\partial{q_{I}^{[n]}}}=\frac{\partial{P_{ijk}}}{\partial{q_{I}^{[n]}}}-\frac{\partial{\mathcal{T}^{-1}P_{ijk}}}{\partial{q_{I}^{[n]}}}=0$

for all $i,j,k,I$, and $n$. Using Lemma 5, this becomes

$\displaystyle\frac{\partial{P_{ijk}}}{\partial{q_{I}^{[n]}}}-\mathcal{T}^{-1}\frac{\partial{P_{ijk}}}{\partial{q_{I}^{[n+1]}}}=0.$
(4.11)

For every $I$ such that, for some $n$, $q^{[n]}_{I}$ appears in $P_{ijk}$, we
let $n_{max}$ be the maximum $n$ such that $q^{[n]}_{I}$ appears in $P_{ijk}$.
Then (4.11) tells us that, on solutions,

$\displaystyle\frac{\partial{P_{ijk}}}{\partial{q_{I}^{[n_{max}]}}}=0.$

It then follows inductively from (4.11) that

$\displaystyle\frac{\partial{P_{ijk}}}{\partial{q_{I}^{[n]}}}=0$

for all $I$ and $n$, or equivalently that $\delta P_{ijk}=0$ on solutions.
Hence, on solutions to the Toda hierarchy, $\delta\mathbbm{d}\mathcal{L}=0$.
Finally, by Proposition 9 this means that the multi-time Euler-Lagrange
equations are consequences of the Toda hierarchy. ∎

###  4.3 Explicit calculations

Using the Hamiltonians $h_{1},h_{2}$ from Equation (4.3), we find

$\displaystyle L_{01}=\frac{1}{2}q_{1}^{2}-\exp(\overline{q}-q)$
$\displaystyle
L_{02}=q_{1}q_{2}-\frac{1}{3}q_{1}^{3}-(q_{1}+\overline{q}_{1})\exp(\overline{q}-q)$

Theorem 13 gives us the coefficient

$L_{12}=-(b+\overline{b})a\overline{q}_{1}-a\overline{b}_{1}+a\overline{q}_{2}-F_{12},$

where $F_{12}$ should satisfy

$\displaystyle\Delta F_{12}$ $\displaystyle=B_{1}Q_{2}-Q_{1}B_{2}$
$\displaystyle=(\Delta\underline{a})(b^{2}+a+\underline{a})-b\Delta((b+\underline{b})\underline{a})$
$\displaystyle=a^{2}-\underline{a}^{2}-b\bar{b}a+b\underline{b}\underline{a}$
$\displaystyle=\Delta(a^{2}-\overline{b}ba).$

Hence for $L_{12}$ we could take

$\displaystyle L_{12}$
$\displaystyle=a\overline{q}_{2}-(q_{1}+\overline{q}_{1})a\overline{q}_{1}-a\overline{q}_{11}-a^{2}+\overline{q}_{1}q_{1}a$
$\displaystyle=-a(\overline{q}_{1}^{2}+\overline{q}_{11}-\overline{q}_{2}+a)$
$\displaystyle=-\left(\frac{1}{2}(\overline{q}_{1}^{2}+\overline{q}_{11}-\overline{q}_{2})+a\right)^{2}+\frac{1}{4}(\overline{q}_{1}^{2}+\overline{q}_{11}-\overline{q}_{2})^{2}.$

From Equation (4.2b) we see that the first term attains a double zero on
solutions, hence we obtain an equivalent semi-discrete two-form if we leave it
out and take

$L^{\prime}_{12}=\frac{1}{4}(\overline{q}_{1}^{2}+\overline{q}_{11}-\overline{q}_{2})^{2}.$

Note that the factorisation (4.10) is valid for $L_{12}$. With
$L^{\prime}_{12}$ we would get a different expression for $P_{012}$, which
also attains a double zero on solutions.

Theorem 13 now implies that $q$ satisfies the Toda equations (4.1) if and only
if it is critical for the semi-discrete 2-form
$\mathcal{L}=\left(L_{01}\,\mathsf{d}t_{1}+L_{02}\,\mathsf{d}t_{2}\,,\,L^{\prime}_{12}\,\mathsf{d}t_{1}\wedge\mathsf{d}t_{2}\right)$.
Indeed we can recover the first Toda equation from the variational principle
by integrating $\mathcal{L}$ over the semi-discrete surface spanned by $t_{1}$
and the discrete direction (i.e. consisting of copies at each lattice site of
a line in the $t_{1}$-direction). Similarly, the second Toda equation can be
obtained using the semi-discrete surface spanned by $t_{2}$ and the discrete
direction. There are many other semi-discrete surfaces we could consider. Of
particular interest are those that consist only of the $(t_{1},t_{2})$-plane
at one single lattice site. The resulting Euler-Lagrange equation is the
subject of the following corollary.

######  Corollary 14.

The Toda hierarchy (4.1) implies the PDE

$\frac{1}{2}q_{22}-q_{11}q_{2}-2q_{12}q_{1}-\frac{1}{2}q_{1111}+3q_{1}^{2}q_{11}=0.$
(4.12)

###### Proof.

The PDE is obtained as the shifted multi-time Euler-Lagrange equation

$\mathcal{T}^{-1}\frac{\delta_{12}{L_{12}^{\prime}}}{\delta{\overline{q}}}=0.$

Hence, by virtue of Theorem 13, it is implied by the Toda hierarchy. ∎

This result indicates that PDE (4.12) is integrable in its own right, since
Equation (4.2) provides an auto-Bäcklund transformation for it. Indeed,
Equation (4.12) can be identified as an integrable Boussinesq-type equation.
In particular, it is equivalent to Equation (66) of [14] via $u=q_{1}$ and
$v=q_{2}$, and to Equation (1.2) of [7] via $u=2q_{1}$ and
$h=2q_{2}-2q_{1}^{2}$.

While it is not entirely surprising that a higher-order PDE can be obtained by
eliminating lattice shifts form Equations (4.1), doing this by direct
computation would be tedious. It is remarkable that from our semi-discrete
Lagrangian 2-form it follows immediately. This sheds a new light on the
observation that integrable PDEs are connected to differential-difference
equations [9, 8]. It is yet another indication that Lagrangian multiform
theory captures integrable hierarchies in a fundamental way.

Using the same methods as above we obtain

$\displaystyle
L_{03}=q_{1}q_{3}-\frac{1}{4}q_{1}^{4}-a(q_{1}^{2}+\overline{q}_{1}^{2}+q_{1}\overline{q}_{1})-a\overline{a}+\frac{1}{2}a^{2},$
$\displaystyle
L_{13}=-a\left(\overline{q}_{1}^{3}+2a\overline{q}_{1}+\overline{a}\overline{\overline{q}}_{1}+2\overline{q}_{1}\overline{q}_{11}+q_{1}\overline{q}_{11}-\overline{q}_{3}+aq_{1}-\overline{a}q_{1}\right),$

and

$\displaystyle L_{23}$
$\displaystyle=-a\Big{(}\overline{q}_{2}\left(q_{1}^{2}+\overline{q}_{1}^{2}+q_{1}\overline{q}_{1}+\underline{a}+a\right)+\overline{\overline{q}}_{2}\overline{a}+2\overline{q}_{1}\overline{q}_{12}+q_{1}\overline{q}_{12}-\overline{q}_{3}\left(q_{1}+\overline{q}_{1}\right)-\overline{q}_{13}$
$\displaystyle\qquad\qquad-
q_{1}^{2}\overline{q}_{1}^{2}-\underline{a}\overline{q}_{1}^{2}+2aq_{1}\overline{q}_{1}-\overline{a}q_{1}^{2}-a\overline{a}-\underline{a}\overline{a}-\underline{a}a-a^{2}\Big{)}.$

Again we can use the multi-time Euler-Lagrange equations to obtain a PDE at a
single lattice site:

######  Corollary 15.

The Toda hierarchy (4.1) implies the PDE

$q_{3}=-2q_{1}^{3}+3q_{1}q_{2}+q_{111}.$ (4.13)

###### Proof.

From the Euler-Lagrange equation
$\displaystyle\frac{\delta_{13}{L_{13}}}{\delta{\bar{q}}}=0$ we obtain

$q_{1}^{3}-3q_{1}q_{11}+6q_{1}a+q_{111}-q_{3}=0,$

which we can write using Equation (4.2a) as

$-2q_{1}^{3}+3q_{1}q_{2}+q_{111}-q_{3}=0.\qed$

At this stage, it is unclear whether Equation (4.13) by itself is integrable.
However, the system of Equations (4.12)–(4.13) is integrable in the sense of
existence of an auto-Bäcklund transformation, given by Equation (4.2): if $q$
solves both PDEs , then so does
$\bar{q}=q+\log\left(\frac{1}{2}(q_{2}+q_{11}-q_{1}^{2})\right)$, as can be
verified by a long but elementary calculation. A detailed investigation of
Equation (4.13), as part of the hierarchy of higher equations which can
presumably be obtained in an analogous way, is left for future work.

##  5 Semi-discrete potential KdV

As another example of a system of interacting particles on a line, we consider
the semi-discrete potential KdV equation. It belongs to the class of equations
studied by direct linearisation in [15] and appears as a semi-continuous limit
of the lattice potential KdV equation [26]. The semi-discrete potential KdV
equation and the second member of its hierarchy can be written as

$\displaystyle q_{1}=\frac{\alpha}{\alpha+\overline{q}-\underline{q}}-\beta,$
(5.1) $\displaystyle
q_{2}=\frac{-\alpha^{2}}{(\alpha+\overline{q}-\underline{q})^{2}}\left(\frac{1}{\alpha+\overline{\overline{q}}-q}+\frac{1}{\alpha+q-\underline{\underline{q}}}\right),$
(5.2)

for constants $\alpha$ and $\beta$. In [26] we find this hierarchy with
$\beta=1$, and with a different second equation which is a linear combination
of our Equations (5.1) and (5.2). Solutions of equations (5.1) and (5.2) give
critical points of the actions associated to the Lagrangians

$\displaystyle
L_{01}=q_{1}\overline{q}-\alpha\log(\alpha+\overline{q}-\underline{q}),$
$\displaystyle
L_{02}=q_{2}\overline{q}-\frac{\alpha^{2}}{(\alpha+\overline{q}-\underline{q})(\alpha+\overline{\overline{q}}-q)}.$

Notice that $L_{01}$ does not depend on $\beta$. Indeed, its Euler-Lagrange
equation is

$\overline{q}_{1}-\underline{q}_{1}=\frac{\alpha}{\alpha+\overline{\overline{q}}-q}-\frac{\alpha}{\alpha+q-\underline{\underline{q}}},$

which is implied by Equation (5.1) (regardless of the value of $\beta$) but
not equivalent to it.

To find a semi-discrete Lagrangian two-form we calculate
$\operatorname{D}_{1}L_{02}-\operatorname{D}_{2}L_{01}$ and write it as a
discrete derivative. To keep the length of our expressions under control we
will write

$v=\frac{1}{\alpha+\overline{q}-\underline{q}}.$

An elementary calculation shows that with

$L_{12}=\alpha^{2}\overline{v}^{2}\left(v\overline{\overline{q}}_{1}+\overline{\overline{v}}q_{1}\right)+\alpha^{2}\beta\overline{v}^{2}\left(\overline{\overline{v}}+v\right)-\alpha^{3}v\overline{v}^{2}\overline{\overline{v}}+\alpha\overline{v}q_{2}+\beta\overline{q}_{2}$

there holds

$\displaystyle\Delta
L_{12}-\operatorname{D}_{1}L_{02}+\operatorname{D}_{2}L_{01}$
$\displaystyle=\left(\overline{q}_{2}+\alpha^{2}\overline{v}^{2}\left(\overline{\overline{v}}+v\right)\right)\left(q_{1}-\alpha
v+\beta\right)$
$\displaystyle\qquad-\left(q_{2}+\alpha^{2}v^{2}\left(\overline{v}+\underline{v}\right)\right)\left(\overline{q}_{1}-\alpha\overline{v}+\beta\right),$

so the exterior derivative of the semi-discrete 2-form
$\mathcal{L}=\left(L_{01}\,\mathsf{d}t_{1}+L_{02}\,\mathsf{d}t_{2},L_{12}\,\mathsf{d}t_{1}\wedge\mathsf{d}t_{2}\right)$
attains a double zero on solutions to the semi-discrete potential KdV
hierarchy. We can check that its Euler-Lagrange equations are equivalent to
this hierarchy. For example, we have

$\frac{\delta_{12}{L_{12}}}{\delta{\overline{q}_{2}}}=-\frac{\delta_{01}{L_{01}}}{\delta{\overline{q}}}\quad\Rightarrow\quad
q_{1}=\alpha v-\beta$

and

$\frac{\delta_{12}{L_{12}}}{\delta{\overline{q}_{1}}}=\frac{\delta_{02}{L_{02}}}{\delta{\overline{q}}}\quad\Rightarrow\quad
q_{2}=-\alpha^{2}v^{2}\left(\overline{v}+\underline{v}\right).$

Note that these equations are stronger than the Euler-Lagrange equations of
$L_{01}$ and $L_{02}$ individually.

######  Remark 16.

The semi-discrete potential KdV hierarchy is closely related to the Volterra
hierarchy. Its leading equation is given by

$a_{1}=a(\overline{a}-\underline{a})$ (5.3)

and can be obtained from the semi-discrete potential KdV hierarchy by defining

$a=\alpha v\underline{v},$

see [5, Exercise 5.4]. The Volterra lattice (5.3) can be viewed as a
generalisation of the Lotka-Volterra predator-prey model to a chain of $n$
species, each of which is preyed upon by the next. As an integrable system it
first appeared in [12] and [6]. It is part of a hierarchy that can be obtained
by restricting the even-numbered flows of the Toda hierarchy to the manifold
defined by $b=0$. Alternatively, it can be obtained from the Toda lattice by a
Miura transformation [20, Chapter 4]. To our knowledge, no direct Lagrangian
description of the Volterra hierarchy is known.

##  6 Conclusions

We have presented the semi-discrete theory of Lagrangian multiforms, with the
Toda lattice as our leading example. While the main text considers only the
case of a single discrete independent variable, the general theory is
analogous and outlined in the appendix.

The ideas of this paper closely follow the multiform theory in the fully
discrete and continuous cases. Nevertheless, it led to an unexpected result:
the semi-discrete multiform formulation of the Toda hierarchy produces PDEs
which hold at a single lattice site as a consequence of the differential-
difference equations of the Toda hierarchy. This phenomenon showcases the
utility of the Lagrangian multiform approach in this context, and is a strong
motivation to develop the multiform formulation of other semi-discrete
hierarchies.

## Acknowledgements

MV acknowledges support by the Deutsche Forschungsgemeinschaft (DFG), project
number VE 1211/1-1.

We would like to thank Vladimir Novikov for his help to identify Equation
(4.12) as a Boussinesq-type equation, and Vincent Caudrelier and Frank Nijhoff
for inspiring discussions on many topics related to this work.

## Appendix: general semi-discrete multi-time EL equations

We consider a semi-discrete space $\mathbb{Z}^{M}\times\mathbb{R}^{N-M}$ of
independent coordinates $n_{1},\ldots,n_{M}$, $t_{M+1},\ldots,t_{N}$, and
dependent variable $q$. We define the shift operator $\mathcal{T}_{i}$ such
that

$\mathcal{T}_{i}q(n_{1},\ldots,n_{i},\ldots,n_{M},t_{M+1},\ldots,t_{N})=q(n_{1},\ldots,n_{i}+1,\ldots,n_{M},t_{M+1},\ldots,t_{N}).$
(6.1)

We will use the notation $\operatorname{D}_{i}$ for difference operator or the
total derivative, depending on whether $i$ represents a discrete or continuous
direction. For $1\leq i\leq M$ we define

$\operatorname{D}_{i}=\mathcal{T}_{i}-\textsf{id},$ (6.2)

where id represents the identity operator. This differs from the definition of
the discrete derivative $\Delta=\textsf{id}-\mathcal{T}^{-1}$ given in Section
2. The reasons for this difference are discussed at the end of this appendix.
For $M<i\leq N$ we define

$\operatorname{D}_{i}=\frac{\partial}{\partial
t_{i}}+\sum_{I}q_{It_{i}}\frac{\partial}{\partial q_{I}},$ (6.3)

where $I$ is an $N$-component multi-index $(i_{1},\ldots,i_{N})$ representing
shifts with respect to $n_{1},\ldots,n_{M}$ and derivatives with respect to
$t_{M+1},\ldots,t_{N}$. We shall also use the notation $I\alpha$ to mean
$(i_{1},\ldots,i_{\alpha}+1,\ldots,i_{N})$ and $I\setminus\alpha$ to mean
$(i_{1},\ldots,i_{\alpha}-1,\ldots,i_{N})$.

We introduce symbols $\textsf{d}n_{i}$, which are the discrete analogue of the
$\textsf{d}t_{i}$. They are the same as the $\Delta_{i}$ from [13]. In the
exterior algebra spanned by the $\textsf{d}n_{i}$ and $\textsf{d}t_{i}$ we
consider a $k$-form

$\mathcal{L}=\sum_{1\leq i_{1}<\ldots<i_{k}\leq N}L_{(i_{1}\ldots i_{k})}\
\textsf{d}n_{i_{1}}\wedge\ldots\wedge\textsf{d}n_{i_{j}}\wedge\textsf{d}t_{i_{j+1}}\wedge\ldots\wedge\textsf{d}t_{i_{k}},$
(6.4)

where $j$ is the largest integer such that $i_{j}\leq M$. We assume that each
$L_{(i_{1}\ldots i_{k})}$ depends on $q$ and shifts of $q$ in the
$n_{1},\ldots,n_{M}$ coordinates (without loss of generality, we shall assume
that there are no backward shifts), derivatives of $q$ in the
$t_{M+1},\ldots,t_{N}$ coordinates and combinations thereof. Even though
$\mathcal{L}$ is formally a $k$-form, only the $\mathsf{d}t_{i}$ have an
interpretation as differentials. The $\mathsf{d}n_{i}$ are formal symbols, so
the geometric interpretation of
$\textsf{d}n_{i_{1}}\wedge\ldots\wedge\textsf{d}n_{i_{j}}\wedge\textsf{d}t_{i_{j+1}}\wedge\ldots\wedge\textsf{d}t_{i_{k}}$
is a $(k-j)$-form. Hence, geometrically, Equation (6.4) is a differential form
of mixed type (a special case of which was considered in Definition 3), but
computationally it is treated in very close analogy to a proper differential
$k$-form. Using the definitions of [13] we find

$\mathbbm{d}\mathcal{L}=\sum_{1\leq i_{1}<\ldots<i_{k+1}\leq N}A^{i_{1}\ldots
i_{k+1}}\,\textsf{d}n_{i_{1}}\wedge\ldots\wedge\textsf{d}n_{i_{j}}\wedge\textsf{d}t_{i_{j+1}}\wedge\ldots\wedge\textsf{d}t_{i_{k+1}},$
(6.5)

where the $A^{i_{1}\ldots i_{k+1}}$ are given by

$A^{i_{1}\ldots
i_{k+1}}=\sum_{\alpha=1}^{k+1}(-1)^{(\alpha+1)}\operatorname{D}_{{i_{\alpha}}}L_{(i_{1},\ldots,i_{\alpha-1},i_{\alpha+1},\ldots,i_{k+1})}.$
(6.6)

For a fixed $i_{1},\ldots,i_{k+1}$, we shall write $L_{(\bar{\alpha})}$ to
denote $L_{(i_{1},\ldots,i_{\alpha-1},i_{\alpha+1},\ldots,i_{k+1})}$. A multi-
index denoted by $J$ is such that component $j_{i}=0$ whenever $i\neq
i_{1},\ldots,i_{k+1}$, i.e. $J$ represents shifts with respect to
$n_{i_{1}},\ldots,n_{i_{j}},t_{i_{j+1}},\ldots,t_{i_{k+1}}$. We define the
variational derivative with respect to $q_{I}$ acting on $L_{(\bar{\alpha})}$
and $A^{i_{1}\ldots i_{k+1}}$ respectively as

$\begin{split}&\frac{\delta L_{(\bar{\alpha})}}{\delta
q_{I}}=\sum_{\begin{subarray}{c}J\\\
j_{i_{\alpha}}=0\end{subarray}}(\mathcal{T}^{-1})_{J^{n}}(-\operatorname{D})_{J^{t}}\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{IJ}},\\\ &\frac{\delta A^{i_{1}\ldots
i_{k+1}}}{\delta
q_{I}}=\sum_{J}(\mathcal{T}^{-1})_{J^{n}}(-\operatorname{D})_{J^{t}}\frac{\partial
A^{i_{1}\ldots i_{k+1}}}{\partial q_{IJ}},\end{split}$ (6.7)

where $I$ is again an $N$ component multi-index $(i_{1},\ldots,i_{N})$
representing shifts with respect to $n_{1},\ldots,n_{M}$ and derivatives with
respect to $t_{M+1},\ldots,t_{N}$. We use the notation $I^{n}$ to denote only
the first $M$ components of $I$ that relate to shifts in the $n_{i}$
coordinates, and $I^{t}$ to denote the last $N-M$ components of $I$ that
relate to derivatives with respect to the $t_{i}$. Therefore,

$q_{I}=\mathcal{T}_{I^{n}}\operatorname{D}_{I^{t}}q=\mathcal{T}_{1}^{i_{1}}\ldots\mathcal{T}_{N}^{i_{M}}\operatorname{D}_{M+1}^{i_{M+1}}\ldots\operatorname{D}_{N}^{i_{N}}q.$
(6.8)

We define that a variational derivative with respect to $q_{I}$ is zero in the
case where any component of the multi-index $I$ is negative (we are only able
to do this because we have assumed that there are no backward shifts in our
Lagrangians). We note that in contrast to the variational derivative operators
defined in Section 3, for brevity of notation, we now omit an index on the
operator. For example, in this appendix we write
$\frac{\delta{L_{ij}}}{\delta{q_{I}}}$ instead of
$\frac{\delta_{ij}{L_{ij}}}{\delta{q_{I}}}$.

######  Theorem 17 (Multi-time Euler-Lagrange equations).

The function $q$ is a critical point of the $k$-form $\mathcal{L}$ as defined
in (6.4) if and only if for all $i_{1},\ldots i_{k+1}$ such that $1\leq
i_{1}<\ldots<i_{k+1}\leq N$, and for all $I$,

$\frac{\delta}{\delta q_{I}}A^{i_{1}\ldots i_{k+1}}=0,$ (6.9)

or equivalently,

$\sum_{\alpha=1}^{j}(-1)^{\alpha+1}\mathcal{T}_{\alpha}\frac{\delta
L_{(\bar{\alpha})}}{\delta q_{I\setminus
i_{\alpha}}}+\sum_{\alpha=j+1}^{k+1}(-1)^{\alpha+1}\frac{\delta
L_{(\bar{\alpha})}}{\delta q_{I\setminus i_{\alpha}}}=0,$ (6.10)

where $j$ is the largest integer such that $i_{j}\leq M$.

In order to prove that these are the multi-time EL equations, we will require
the following lemma:

######  Lemma 18.

Let $1\leq i_{1}<\ldots<i_{k+1}\leq N$ be fixed. For all multi-indices $I$,

$\frac{\partial A^{i_{1}\ldots i_{k+1}}}{\partial
q_{I}}=\sum_{\begin{subarray}{c}J\\\ j_{i}\leq
1\end{subarray}}(-\mathcal{T}^{-1})_{J^{n}}\operatorname{D}_{J^{t}}\frac{\delta
A^{i_{1}\ldots i_{k+1}}}{\delta q_{IJ}},$ (6.11)

where the summation is over all multi-indices $J$ such that $j_{i}=0$ whenever
$i\neq i_{1},\ldots,i_{k+1}$ and the non-zero $j_{i}$ are equal to 1.

###### Proof.

We first notice that the partial derivative on the left hand side of (6.11)
appears only once in the sum on the right hand side. We now need to show that
all other terms that appear on the right hand side of (6.11), sum to zero. We
note that all terms on the right hand side of (6.11) are of the form

$(\mathcal{T}^{-1})_{K^{n}}\operatorname{D}_{K^{t}}\frac{\partial
A^{i_{1}\ldots i_{k+1}}}{\partial q_{IK}}$ (6.12)

for some multi-index $K=(k_{1},\ldots,k_{N})$ which satisfies $k_{i}=0$
whenever $i\neq i_{1},\ldots,i_{k+1}$. Let $r$ be the number of non-zero
entries in $K$. We notice that the term (6.12) appears exactly once when
$|J|=0$, exactly ${r\choose 1}$ times with a factor of $-1$ when $|J|=1$,
exactly ${r\choose 2}$ times when $|J|=2$ etc… In total, this term appears
with a factor of $\sum_{i=0}^{r}(-1)^{i}{r\choose i}$. It can easily be seen
that this sum is zero by considering the binomial expansion of $(1-1)^{r}$. ∎

###### Proof of Theorem 17.

The first part of the proof of Proposition 9 of Section 3, showing that
criticality is equivalent to $\delta\mathbbm{d}\mathcal{L}=0$, immediately
extends to the present case. We note that the equations given by
$\delta\mathbbm{d}\mathcal{L}=0$ are equivalent to

$\frac{\partial}{\partial q_{I}}A^{i_{1}\ldots i_{k+1}}=0$ (6.13)

for all $I$ and all $1\leq i_{1}<\ldots<i_{k+1}\leq N$. Using Lemma 18, we see
that this is the case if and only if Equation (6.9) holds for all $I$ and all
$1\leq i_{1}<\ldots<i_{k+1}\leq N$. It remains to show that (6.10) and (6.9)
are equivalent expressions.

The identities

$\frac{\partial}{\partial
q_{I}}\mathcal{T}_{i}=\mathcal{T}_{i}\frac{\partial}{\partial q_{I\setminus
i}}$ (6.14)

for $1\leq i\leq M$ and

$\frac{\partial}{\partial q_{I}}\operatorname{D}_{i}=\frac{\partial}{\partial
q_{I\setminus i}}+\operatorname{D}_{i}\frac{\partial}{\partial q_{I}}$ (6.15)

for $M<i\leq N$ tell us that

$\frac{\partial}{\partial q_{I}}A^{i_{1}\ldots
i_{k+1}}=\sum_{\alpha=1}^{j}(-1)^{\alpha+1}\bigg{(}\mathcal{T}_{{i_{\alpha}}}\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{I\setminus i_{\alpha}}}-\frac{\partial
L_{(\bar{\alpha})}}{\partial
q_{I}}\bigg{)}+\sum_{\alpha=j+1}^{k+1}(-1)^{\alpha+1}\bigg{(}\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{I\setminus
i_{\alpha}}}+\operatorname{D}_{{i_{\alpha}}}\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{I}}\bigg{)}$ (6.16)

so

$\begin{split}\frac{\delta}{\delta q_{I}}A^{i_{1}\ldots
i_{k+1}}&=\sum_{J}(\mathcal{T}^{-1})_{J^{n}}(-\operatorname{D})_{J^{t}}\frac{\partial}{\partial
q_{IJ}}A^{i_{1}\ldots i_{k+1}}\\\
&=\sum_{J}(T^{-1})_{J^{n}}(-\operatorname{D})_{J^{t}}\sum_{\alpha=1}^{j}(-1)^{\alpha+1}\bigg{(}\mathcal{T}_{{i_{\alpha}}}\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{IJ\setminus i_{\alpha}}}-\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{IJ}}\bigg{)}\\\
&\qquad+\sum_{J}(\mathcal{T}^{-1})_{J^{n}}(-\operatorname{D})_{J^{t}}\sum_{\alpha=j+1}^{k+1}(-1)^{\alpha+1}\bigg{(}\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{IJ\setminus
i_{\alpha}}}+\operatorname{D}_{{i_{\alpha}}}\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{IJ}}\bigg{)}.\end{split}$ (6.17)

For $1\leq\alpha\leq j$, whenever $j_{i_{\alpha}}\neq 0$ in this sum, so $J$
is of the form $Ki_{\alpha}$ for some multi-index $K$, then

$\pm(\mathcal{T}^{-1})_{J^{n}}(-\operatorname{D})_{J^{t}}\mathcal{T}_{{i_{\alpha}}}\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{IJ\setminus
i_{\alpha}}}=\pm(\mathcal{T}^{-1})_{K^{n}}(-\operatorname{D})_{K^{t}}\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{IK}}$ (6.18)

will appear in this sum. When $J=K$, the term

$\mp(\mathcal{T}^{-1})_{K^{n}}(-\operatorname{D})_{K^{t}}\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{IK}}$ (6.19)

will appear, so these two terms cancel. Similarly, for $j+1\leq\alpha\leq k$,
whenever $j_{i_{\alpha}}\neq 0$ in this sum, so $J$ is of the form
$Ki_{\alpha}$ for some multi-index $K$, then

$\pm(T^{-1})_{J^{n}}(-\operatorname{D})_{J^{t}}\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{IJ\setminus
i_{\alpha}}}=\mp(T^{-1})_{K^{n}}\operatorname{D}_{{i_{\alpha}}}(-\operatorname{D})_{K^{t}}\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{IK}}$ (6.20)

will appear in this sum. When $J=K$, the term

$\pm(T^{-1})_{K^{n}}(-\operatorname{D})_{K^{t}}\operatorname{D}_{{i_{\alpha}}}\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{IK}}$ (6.21)

will appear, so these two terms cancel. As a result, (6.17) simplifies to

$\begin{split}\frac{\delta}{\delta q_{I}}A^{i_{1}\ldots
i_{k+1}}&=\sum_{\alpha=1}^{j}\,\sum_{\begin{subarray}{c}J\\\
j_{i_{\alpha}}=0\end{subarray}}(-1)^{\alpha+1}(\mathcal{T}^{-1})_{J^{n}}(-\operatorname{D})_{J^{t}}\mathcal{T}_{i_{\alpha}}\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{IJ\setminus i_{\alpha}}}\\\
&\qquad+\sum_{\alpha=j+1}^{k+1}\,\sum_{\begin{subarray}{c}J\\\
j_{i_{\alpha}}=0\end{subarray}}(-1)^{\alpha+1}(\mathcal{T}^{-1})_{J^{n}}(-\operatorname{D})_{J^{t}}\frac{\partial
L_{(\bar{\alpha})}}{\partial q_{IJ\setminus i_{\alpha}}}\\\
&=\sum_{\alpha=1}^{j}(-1)^{\alpha+1}\mathcal{T}_{i_{\alpha}}\frac{\delta
L_{(\bar{\alpha})}}{\delta q_{I\setminus
i_{\alpha}}}+\sum_{\alpha=j+1}^{k+1}(-1)^{\alpha+1}\frac{\delta
L_{(\bar{\alpha})}}{\delta q_{I\setminus i_{\alpha}}}.\end{split}$ (6.22)

This shows that $\delta\mathbbm{d}\mathcal{L}=0$ is equivalent to

$\sum_{\alpha=1}^{j}(-1)^{\alpha+1}\mathcal{T}_{i_{\alpha}}\frac{\delta
L_{(\bar{\alpha})}}{\delta q_{I\setminus
i_{\alpha}}}+\sum_{\alpha=j+1}^{k+1}(-1)^{\alpha+1}\frac{\delta
L_{(\bar{\alpha})}}{\delta q_{I\setminus i_{\alpha}}}=0.\qed$

In this appendix, we defined the discrete derivarive
$\operatorname{D}_{i}=\mathcal{T}_{i}-\textsf{id}$. Alternatively, we could
have defined $\operatorname{D}_{i}=\textsf{id}-\mathcal{T}_{i}^{-1}$ (as we
did in Section 2) which would have led to the equivalent multi-time EL
equations

$\sum_{\alpha=1}^{j}(-1)^{\alpha+1}\frac{\delta\tilde{L}_{(\bar{\alpha})}}{\delta
q_{I}}+\sum_{\alpha=j+1}^{k+1}(-1)^{\alpha+1}\frac{\delta\tilde{L}_{(\bar{\alpha})}}{\delta
q_{I\setminus i_{\alpha}}}=0.$ (6.23)

We use $\tilde{L}$ to denote the Lagrangians because they are not the same as
the ones in (6.10). They are related by

$\tilde{L}_{\bar{\alpha}}=\prod_{\begin{subarray}{c}\beta=1\\\
\beta\neq\alpha\end{subarray}}^{j}\mathcal{T}_{i_{\beta}}^{-1}L_{\bar{\alpha}},$
(6.24)

i.e. by a shift in all discrete directions except the direction labelled by
$i_{\alpha}$ (if it is discrete). The equivalence of (6.10) and (6.23) can
then be seen by applying $\prod_{\beta=1}^{j}\mathcal{T}_{i_{\beta}}^{-1}$ to
(6.10) and re-labeling the multi-index $I$ to obtain (6.23).

We choose to present the general semi-discrete multi-time EL equations in the
form given in (6.10) in order to highlight the close connection between the
semi-discrete and continuous multi-time EL equations, with $I\setminus
i_{\alpha}$ appearing in both. Also, when the equations are presented in this
way, it is clear that they include the usual EL equations for each
$L_{\bar{\alpha}}$ (obtained by setting $I=i_{\alpha}$). On the other hand,
the multi-time EL equations that we presented in Section 3 are in the form
given in (6.23) in order to avoid the presence of shift operators in the
multi-time EL equations.

## References

  * Adler and Shabat [1997] Adler V. E. & Shabat A. B.  On a class of Toda chains.  Theoretical and Mathematical Physics, 111 : 647–657.  ISSN 0040-5779, 1573-9333, 1997.
  * Anderson [1992] Anderson I. M.  Introduction to the variational bicomplex.  In Gotay M., Marsden J. & Moncrief V., editors, _Mathematical Aspects of Classical Field Theory_ , pages 51–73. AMS, 1992.
  * Boll et al. [2014] Boll R., Petrera M. & Suris Y. B.  What is integrability of discrete variational systems? Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 470 : 20130550, 2014.
  * Flaschka [1974] Flaschka H.  The Toda lattice. II. existence of integrals.  Physical Review B, 9 : 1924, 1974.
  * Hietarinta et al. [2016] Hietarinta J., Joshi N. & Nijhoff F. W.  Discrete Systems and Integrability.  Volume 54. Cambridge university press, 2016.
  * Kac and van Moerbeke [1975] Kac M. & van Moerbeke P.  On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices.  Advances in Mathematics, 16 : 160–169, 1975.
  * Kupershmidt [1985] Kupershmidt B.  Mathematics of dispersive water waves.  Communications in Mathematical Physics, 99 : 51–73, 1985.
  * Levi [1981] Levi D.  Nonlinear differential difference equations as Bäcklund transformations.  Journal of Physics A: Mathematical and General, 14 : 1083, 1981.
  * Levi and Benguria [1980] Levi D. & Benguria R.  Bäcklund transformations and nonlinear differential difference equations.  Proceedings of the National Academy of Sciences, 77 : 5025–5027, 1980.
  * Lobb and Nijhoff [2009] Lobb S. & Nijhoff F.  Lagrangian multiforms and multidimensional consistency.  Journal of Physics A: Mathematical and Theoretical, 42 : 454013, 2009.
  * Lobb et al. [2009] Lobb S., Nijhoff F. & Quispel R.  Lagrangian multiform structure for the lattice KP system.  Journal of Physics A:Mathematical and Theoretical, 43 : 472002, 2009.
  * Manakov [1974] Manakov S.  Complete integrability and stochastization of discrete dynamical systems.  Zh. Exp. Teor. Fiz, 67 : 543–555, 1974.
  * Mansfield and Hydon [2008] Mansfield E. & Hydon P.  Difference forms.  Foundations of Computational Mathematics, 8 : 427–467, 2008.
  * Mikhailov et al. [2007] Mikhailov A. V., Novikov V. S. & Wang J. P.  On classification of integrable nonevolutionary equations.  Studies in Applied Mathematics, 118 : 419–457, 2007.
  * Nijhoff et al. [1983] Nijhoff F., Quispel G. & Capel H.  Direct linearization of nonlinear difference-difference equations.  Physics Letters A, 97 : 125–128, 1983.
  * Petrera and Suris [2017] Petrera M. & Suris Yu. B.  Variational symmetries and pluri-Lagrangian systems in classical mechanics.  Journal of Nonlinear Mathematical Physics, 24 (Sup. 1) : 121–145, 2017.
  * Petrera and Vermeeren [2021] Petrera M. & Vermeeren M.  Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs.  European Journal of Mathematics, 7 : 741–765, 2021.
  * Sleigh et al. [2019] Sleigh D., Nijhoff F. & Caudrelier V.  A variational approach to Lax representations.  Journal of Geometry and Physics, 142 : 66–79, 2019.
  * Sleigh et al. [2020] Sleigh D., Nijhoff F. & Caudrelier V.  Variational symmetries and Lagrangian multiforms.  Letters in Mathematical Physics, 110 : 805–826, 2020.
  * Suris [2003] Suris Yu. B.  The Problem of Integrable Discretization: Hamiltonian Approach.  Birkhäuser, 2003.
  * Suris [2013] Suris Yu. B.  Variational formulation of commuting Hamiltonian flows: Multi-time Lagrangian 1-forms.  Journal of Geometric Mechanics, 5 : 365–379, 2013.
  * Suris and Vermeeren [2016] Suris Yu. B. & Vermeeren M.  On the Lagrangian structure of integrable hierarchies.  In Bobenko A. I., editor, _Advances in Discrete Differential Geometry_ , pages 347–378. Springer, 2016.
  * Toda [1967] Toda M.  Vibration of a chain with nonlinear interaction.  Journal of the Physical Society of Japan, 22 : 431–436, 1967.
  * Vermeeren [2019] Vermeeren M.  Continuum limits of pluri-Lagrangian systems.  Journal of Integrable Systems, 4 : xyy020, 2019.
  * Vermeeren [2021] Vermeeren M.  Hamiltonian structures for integrable hierarchies of Lagrangian PDEs.  Open Communications in Nonlinear Mathematical Physics, 1, 2021.
  * Wiersma and Capel [1987] Wiersma G. L. & Capel H.  Lattice equations, hierarchies and Hamiltonian structures.  Physica A: Statistical Mechanics and its Applications, 142 : 199–244, 1987.
  * Xenitidis et al. [2011] Xenitidis P., Nijhoff F. & Lobb S.  On the Lagrangian formulation of multidimensionally consistent systems.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 467 : 3295–3317, 2011.
  * Yoo-Kong et al. [2011] Yoo-Kong S., Lobb S. & Nijhoff F.  Discrete-time Calogero–Moser system and Lagrangian 1-form structure.  Journal of Physics A: Mathematical and Theoretical, 44 : 365203, 2011.