# Lepage equivalents and the Variational Bicomplex

D.J. Saunders

###### Abstract

We show how to construct, for a Lagrangian of arbitrary order, a Lepage
equivalent satisfying the closure property: that the Lepage equivalent
vanishes precisely when the Lagrangian is null. The construction uses a
homotopy operator for the horizontal differential of the variational
bicomplex. A choice of symmetric linear connection on the manifold of
independent variables, and a global homotopy operator constructed using that
connection, may then be used to extend any global Lepage equivalent to one
satisfying the closure property.

In the second part of the paper we investigate the rôle of vertical
endomorphisms in constructing such Lepage equivalents. These endomorphisms may
be used directly to construct local homotopy operators. Together with a
symmetric linear connection they may also be used to construct global vertical
tensors, and these define infinitesimal nonholonomic projections which in turn
may be used to construct Lepage equivalents. We conjecture that these global
vertical tensors may also be used to define global homotopy operators.

MSC: 58A10, 58A20, 83D05

Keywords: Jet bundle, Poincaré–Cartan form, Lepage equivalent of a Lagrangian,
variational bicomplex

## Dedication

In the Notes to Chapter 5 of [15], Peter Olver wrote about the variational
complex and the variational bicomplex ‘It is hoped that these methods will
inspire further research in the geometric theory of the calculus of
variations’. A few years later [16] he wrote ‘In the geometric theory of the
calculus of variations in mechanics, the Cartan form, which first arose as the
integrand in Hilbert’s invariant integral, plays a ubiquitous role’. Lepage
equivalents are generalizations of Cartan forms, and I hope that this paper
will be a small contribution to Peter’s project.

##  1 Introduction

In recent years there has been a revival of interest in the ‘fundamental
Lepage equivalent’ of a Lagrangian, a differential form on a jet bundle which
(as with any such Lepage equivalent) provides a geometrical construction
leading to the Euler–Lagrange equations of the corresponding variational
problem, but which has the additional property that it is closed precisely
when the Lagrangian is null [17, 23]. The original formulation of the
fundamental Lepage equivalent was given for first order Lagrangians (in [12],
and then independently in [2]). Although expressed in local coordinates, the
form is in fact invariant under changes of coordinates, and so is a global
geometric object. There had, however, been been no similar construction for
higher order Lagrangians.

A construction for Lagrangians of arbitrary order has now been proposed in
[25], giving a Lepage form of order no greater than $4k-2$ for a Lagrangian of
order $k$. The construction is again given in local coordinates, but now there
is no guarantee that it will be defined globally. In addition, if the original
Lagrangian happens to be first order, the new Lepage form will in general be
of second order and will differ from the original, first order, fundamental
Lepage equivalent.

In the first part of this paper, after giving some background on the different
types of Lepage equivalent, we propose a new method of constructing a
fundamental Lepage equivalent for a Lagrangian of arbitrary order by using
homotopy operators for the horizontal differential in the variational
bicomplex on the infinite jet manifold. This has the disadvantage that any
bound on the order of the resulting Lepage form, although necessarily finite,
will depend on the number of independent variables. On the other hand, the
choice of a symmetric linear connection will allow the construction of a
global form, and in the case of a first order Lagrangian the result will be
independent of any connection and we recover the classic fundamental Lepage
equivalent.

In the second part of the paper we explore the potential for clarifying this
construction by using ‘vertical endomorphisms’ on jet bundles, tensorial
objects depending on a closed differential form, which are related to the
canonical isomorphism between the tangent space at any point of an affine
space, and the vector space on which the affine space is modelled [18]. We
recall how local homotopy operators for the horizontal differential can be
constructed from these vertical endomorphisms, and then we show how a
symmetric linear connection can be used to remove the dependence on the
differential form to produce a globally-defined, fully tensorial object. (A
related but technically different approach has been described in [3].) Such a
‘vertical tensor’ can be used to give an infinitesimal rigidity to
nonholonomic jet bundles, allowing the construction of a global Lepage
equivalent for a Lagrangian of arbitrary order. Finally we offer a conjecture
regarding how these vertical tensors, together with a covariant version of the
horizontal differential, might be used to construct a global homotopy operator
for the ordinary horizontal differential.

##  2 Notation

We adopt a modified version of the notation used in [19]. We let $\pi:E\to M$
be a fibred manifold with $\dim M=m$ and $\dim E=m+n$. The $k$-jet manifold of
$\pi$ will be denoted $J^{k}\pi$ with projections $\pi_{k}:J^{k}\pi\to M$,
$\pi_{k,0}:J^{k}\pi\to E$ and $\pi_{k,l}:J^{k}\pi\to J^{l}\pi$ where $l<k$. A
typical element of $J^{k}\pi$ will be denoted $j^{k}_{p}\phi$. We use similar
notation for jets of the cotangent bundle $\tau:T^{*}M\to M$.

We let $\mathfrak{X}(J^{k}\pi)$ denote the module of vector fields on
$J^{k}\pi$, and $\Omega^{r}(J^{k}\pi)$ the module of $r$-forms.

Regarding the jet bundle $\pi_{k-1}:J^{k-1}\pi\to M$ as the starting bundle,
we shall let $(\pi_{k-1})_{1}:J^{1}\pi_{k-1}\to M$ denote its first jet
bundle, and we let $\mathrm{i}_{1,k-1}:J^{k}\pi\to J^{1}\pi_{k-1}$ be the
canonical inclusion. There is also an intermediate submanifold
$\widehat{J}^{k}\pi\subset J^{1}\pi_{k-1}$, the semiholonomic manifold [14],
with a canonical symmetrization projection
$\mathrm{p}_{k}:\widehat{J}^{k}\pi\to J^{k}\pi$.

We also use the infinite jet bundle $\pi_{\infty}:J^{\infty}\pi\to M$ where
$J^{\infty}\pi$, although infinite dimensional, is a Fréchet manifold and so
is reasonably well behaved. We let $\Omega^{r}$ (without specifying a
manifold) denote the module of $r$-forms on $J^{\infty}\pi$ of globally finite
order, so that each such form is projectable to some $J^{k}\pi$.

Any differential form $\omega\in\Omega^{r}$ can be decomposed uniquely into
its contact components

$\omega=\omega^{(0)}+\omega^{(1)}+\cdots+\omega^{(p)}+\cdots+\omega^{(r)}$

where if $r>m$ then $\omega^{(p)}=0$ for $p<r-m$. We let
$\Omega^{p,q}\subset\Omega^{r}$ with $p+q=r$ denote the submodule of
$p$-contact $r$-forms. In a similar way, a differential form
$\omega\in\Omega^{r}(J^{k}\pi)$ on a finite order jet manifold may be
decomposed into contact components, but these will normally be defined on
$J^{k+1}\pi$ rather than on $J^{k}\pi$.

When using coordinates, we take fibred coordinates $(x^{i},u^{\alpha})$ on $E$
over base coordinates $(x^{i})$ on $M$. Jet coordinates will be denoted
$(u^{\alpha}_{i})$ on $J^{1}\pi$ and $(u^{\alpha}_{i},u^{\alpha}_{(ij)})$ on
$J^{2}\pi$ with parentheses denoting symmetrization because
$u^{\alpha}_{(ji)}$ is the same coordinate as $u^{\alpha}_{(ij)}$. For this
reason we use the symbol $\\#(ij)$ to equal $1$ when $i=j$ and to equal $2$
when $i\neq j$, in order to avoid double counting during summation. (We use
the standard summation convention for repeated indices.)

On higher order jet manifolds this notation becomes unwieldy and we write
$(u^{\alpha}_{I})$ instead, where $I\in\mathbb{N}^{m}$ is a multi-index with
$I(i)$ giving the number of copies of the index $i$, so that this notation
automatically takes care of symmetrization. We let $1_{i}$ denote the multi-
index with a single $1$ in the $i$-th position; $\lvert
I\rvert=\sum_{i=1}^{n}I(i)$ is the length of $I$, and $I!=I(1)!I(2)!\cdots
I(m)!$ is its factorial. Any summation involving multi-indices will be
indicated explicitly, including the zero multi-index where appropriate.

Sometimes we need to use a mixed notation, and converting to or from multi-
index notation requires coefficients to be adjusted. If $F(J)$ is some object
depending on the multi-index $J$ then

$\sum_{\lvert J\rvert=r+1}\frac{\lvert
J\rvert!}{J!}F(J)=\sum_{i=1}^{m}\sum_{\lvert I\rvert=r}\frac{\lvert
I\rvert!}{I!}F(I+1_{i})$

where the quotient $\lvert J\rvert!/J!$ is the ‘weight’ of the multi-index
$J$.

We use notation

$\theta^{\alpha}=du^{\alpha}-u^{\alpha}_{j}dx^{j}\,,\qquad\theta^{\alpha}_{i}=du^{\alpha}_{i}-u^{\alpha}_{(ij)}dx^{j}\,,\qquad\theta^{\alpha}_{I}=du^{\alpha}_{I}-u^{\alpha}_{I+1_{j}}dx^{j}$

for local contact $1$-forms, and

$\omega_{0}=dx^{1}\wedge\cdots\wedge
dx^{m}\,,\qquad\omega_{i}=i_{\partial/\partial
x^{i}}\,\omega_{0}=(-1)^{i-1}dx^{1}\wedge\cdots\wedge\widehat{dx^{i}}\wedge\cdots\wedge
dx^{m}$

(where the circumflex indicates an omitted factor) for local forms horizontal
over $M$. Local total derivatives, dual to the local contact forms, will be
denoted $d_{i}$ and are given explicitly as

$\frac{\partial}{\partial x^{i}}+u^{\alpha}_{i}\frac{\partial}{\partial
u^{\alpha}}\,,\qquad\frac{\partial}{\partial
x^{i}}+u^{\alpha}_{i}\frac{\partial}{\partial
u^{\alpha}_{i}}+u^{\alpha}_{(ij)}\frac{\partial}{\partial
u^{\alpha}_{j}}\,,\qquad\frac{\partial}{\partial
x^{i}}+\sum_{I}u^{\alpha}_{I+1_{i}}\frac{\partial}{\partial
u^{\alpha}_{I}}\,.$

In the finite order case they are vector fields along the map $\pi_{k,k-1}$
rather than on a single jet manifold. We also use the symbol $\partial_{i}$ to
indicate $\partial/\partial x^{i}$ as a vector field along the map $\pi_{k}$.

On a nonholonomic jet manifold we need to distinguish between the two levels
of jet coordinates, and we use juxtaposition, with a dot to indicate when a
particular index is missing. So on $J^{1}\pi_{1}$ the coordinates are
$(x^{i},u^{\alpha}_{\cdot\cdot},u^{\alpha}_{i\cdot},u^{\alpha}_{\cdot
j},u^{\alpha}_{ij})$ and on $J^{1}\pi_{k-1}$ they are
$(x^{i},u^{\alpha}_{I\cdot},u^{\alpha}_{Ij})$.

Finally we note that $\pi_{1,0}:J^{1}\pi\to E$ is an affine bundle, modelled
on the vector bundle $\pi^{*}T^{*}M\otimes V\pi$, so that the vertical bundle
$V\pi_{1,0}$ is canonically isomorphic to
$\pi_{1}^{*}T^{*}M\otimes\pi_{1,0}^{*}V\pi$; the inverse of this isomorphism
may be regarded as a tensor field

$S=\partial_{i}\otimes\theta^{\alpha}\otimes\frac{\partial}{\partial
u^{\alpha}_{i}}\,,$

a section of the bundle $\pi_{1}^{*}TM\otimes T^{*}J^{1}\pi\otimes TJ^{1}\pi$
over $J^{1}\pi$. We shall call this the _first order vertical tensor_.

##  3 Background

Many of the results in the geometrical calculus of variations can be described
in terms of _source forms_ and _Lepage equivalents_ (see [13] for a useful
summary and historical references).

A source form is a form $\varepsilon\in\Omega^{m+1}(J^{l}\pi)$ with the
properties that it is horizontal over $E$, and maximally horizontal over $M$,
so that in coordinates it appears as
$\varepsilon=\varepsilon_{\alpha}\theta^{\alpha}\wedge\omega_{0}$. The zero
set of a source form is a submanifold of $J^{l}\pi$ representing a family of
partial differential equations; if
$\lambda=L\,\omega_{0}\in\Omega^{m}(J^{k}\pi)$ is a horizontal $m$-form, a
Lagrangian, then it gives rise to a source form
$\varepsilon_{\lambda}\in\Omega^{m+1}(J^{2k}\pi)$, the _Euler–Lagrange form_
of $\lambda$, incorporating the Euler–Lagrange equations of the variational
problem defined by $\lambda$:

$\varepsilon_{\lambda}=\sum_{\lvert I\rvert=0}^{k}(-1)^{\lvert
I\rvert}d_{I}\biggl{(}\frac{\partial L}{\partial
u^{\alpha}_{I}}\biggr{)}\theta^{\alpha}\wedge\omega_{0}\,.$

A Lepage form is a form $\vartheta\in\Omega^{m}(J^{l}\pi)$ with the property
that $(d\vartheta)^{(1)}$, the $1$-contact component of its exterior
derivative, is a source form. A Lepage equivalent of a Lagrangian
$\lambda\in\Omega^{m}(J^{k}\pi)$ is a Lepage form
$\vartheta_{\lambda}\in\Omega^{m}(J^{l}\pi)$ with $l\geq k$ such that the
difference $\vartheta_{\lambda}-\pi_{l,k}^{*}\lambda$ is a contact form; the
corrresponding source form $(d\vartheta_{\lambda})^{(1)}$ is then just the
Euler–Lagrange form $\varepsilon_{\lambda}$. Different Lepage equivalents of
the same Lagrangian give the same Euler–Lagrange form.

If $m=1$ then each Lagrangian $\lambda$ gives rise to a unique globally-
defined Lepage equivalent $\vartheta_{\lambda}\in\Omega^{1}(J^{2k-1}\pi)$, the
Cartan form of the Lagrangian. However, complications arise when $m\geq 2$,
and these concern both existence and uniqueness. Clearly if
$\vartheta_{\lambda}$ is a Lepage equivalent of $\lambda$ then so is
$\vartheta_{\lambda}+d\psi+\omega$ where $\omega$ is at least $2$-contact, and
in fact the converse is true: if $\vartheta_{\lambda}$,
$\vartheta_{\lambda}^{\prime}$ are both Lepage equivalents of $\lambda$ then
$\theta_{\lambda}^{\prime}-\theta_{\lambda}=d\psi+\omega$.

As far as existence is concerned, if we initially consider forms which are at
most $1$-contact then locally

$\vartheta_{\lambda}=L\,\omega_{0}+\sum_{\lvert J\rvert=0}^{k-1}\sum_{\lvert
K\rvert=0}^{k-\lvert J\rvert-1}\frac{(-1)^{\lvert J\rvert}(J+K+1_{j})!\lvert
J\rvert!\lvert K\rvert!}{(\lvert J\rvert+\lvert
K\rvert+1)!J!K!}d_{J}\biggl{(}\frac{\partial L}{\partial
u^{\alpha}_{J+K+1_{j}}}\biggr{)}\theta^{\alpha}_{K}\wedge\omega_{j}$

is a Lepage equivalent, known as the _principal Lepage equivalent_ of
$\lambda$. When $k=1$ this is just the Poincaré–Cartan form of $\lambda$,

$\vartheta_{\lambda}=L\,\omega_{0}+\frac{\partial L}{\partial
u^{\alpha}_{j}}\theta^{\alpha}\wedge\omega_{j}$

and is defined globally; it is the unique Lepage equivalent of $\lambda$ which
is both at most $1$-contact and also horizontal over $E$. When $k=2$ we obtain

$\vartheta_{\lambda}=L\,\omega_{0}+\biggl{(}\biggl{(}\frac{\partial
L}{\partial u^{\alpha}}-\frac{1}{\\#(ij)}d_{i}\biggl{(}\frac{\partial
L}{\partial
u^{\alpha}_{(ij)}}\biggr{)}\biggr{)}\theta^{\alpha}+\frac{1}{\\#(ij)}\frac{\partial
L}{\partial u^{\alpha}_{(ij)}}\theta^{\alpha}_{i}\biggr{)}\wedge\omega_{j}$

which again, perhaps surprisingly, is invariant under a fibred change of
coordinates $\tilde{x}=\tilde{x}^{j}(x^{i})$,
$\tilde{u}^{\beta}=\tilde{u}^{\beta}(x^{i},u^{\alpha})$ and is therefore also
defined globally. For $k\geq 3$, however, there is no such Lepage equivalent
invariant under coordinate changes [11]; choices need to be made in order to
obtain a globally defined form. Several authors (see, for instance, [9, 10])
have used connections of various kinds for this purpose

Another approach [18] has been to ‘pretend’ that the $k$-th order Lagrangian
is really first order by using a tubular neighbourhood of $J^{k}\pi$ in
$J^{1}\pi_{k-1}$ and ‘spreading out’ the Lagrangian using the neigbourhood’s
projection. By repeating this process, a global Lepage equivalent may be
constructed. In fact only infinitesimal projections are needed, mapping
$T_{J^{k}\pi}J^{1}\pi_{k-1}$ to $TJ^{k}\pi$, and we shall see in Section 7
that a symmetric linear connection on $M$ determines a suitable family of
projections. In the second order case, it may be seen that only the
restriction of the projection to the semiholonomic manifold
$\widehat{J}^{2}\pi$ is needed, explaining why symmetrization projection
$\mathrm{p}_{2}$ may be used to give a global Lepage equivalent in this case.

The Lepage equivalents described so far have all been at most $1$-contact.
There have, however, been important examples of Lepage equivalents involving
higher contact terms. One such, defined for a nonvanishing first order
Lagrangian, is the _Carathéodory form_ [4]

$\vartheta_{\lambda}=\frac{1}{L^{m-1}}\bigwedge_{j=1}^{m}\biggl{(}L\,dx^{j}+\frac{\partial
L}{\partial u^{\alpha}_{j}}\theta^{\alpha}\biggr{)}\,;$

this decomposable form is again defined globally and indeed is invariant, not
just under a fibred change of coordinates, but under a general change
$\tilde{x}=\tilde{x}^{j}(x^{i},u^{\alpha})$,
$\tilde{u}^{\beta}=\tilde{u}^{\beta}(x^{i},u^{\alpha})$. A similar form for a
nonvanishing second order Lagrangian,

$\vartheta_{\lambda}=\frac{1}{L^{m-1}}\bigwedge_{j=1}^{m}\biggl{(}L\,dx^{j}+\biggl{(}\frac{\partial
L}{\partial u^{\alpha}_{j}}-\frac{1}{\\#(ij)}d_{i}\biggl{(}\frac{\partial
L}{\partial
u^{\alpha}_{(ij)}}\biggr{)}\biggr{)}\theta^{\alpha}+\frac{1}{\\#(ij)}\frac{\partial
L}{\partial u^{\alpha}_{(ij)}}\theta^{\alpha}_{i}\biggr{)}$

was described in [16] (see also [6]).

The Lepage equivalent of particular interest in the present paper, again
involving higher contact terms, is the _fundamental Lepage equivalent_ of a
first order Lagrangian [2, 12]

$\vartheta_{\lambda}=\sum_{p=0}^{\min\\{m,n\\}}\frac{1}{(p!)^{2}}\,\frac{\partial^{p}L}{\partial
u^{\alpha_{1}}_{j_{1}}\cdots\partial
u^{\alpha_{p}}_{j_{p}}}\,\theta^{\alpha_{1}}\wedge\cdots\wedge\theta^{\alpha_{p}}\wedge\omega_{j_{1}\cdots
j_{p}}\,.$

This satisfies the _closure property_ , that $d\vartheta_{\lambda}=0$
precisely when the Lagrangian is null: that is, when the Euler–Lagrange form
$\varepsilon_{\lambda}$ is zero. (Of course any individual form
$\vartheta_{\lambda}$ is either closed or not closed; the closure property
applies to the procedure mapping $\lambda$ to $\vartheta_{\lambda}$.) Once
again this form is defined globally, and indeed is invariant under a general
change of coordinates $\tilde{x}=\tilde{x}^{j}(x^{i},u^{\alpha})$,
$\tilde{u}^{\beta}=\tilde{u}^{\beta}(x^{i},u^{\alpha})$ [7]. The content of
the closure property lies in the requirement that $d\vartheta_{\lambda}=0$
when $\lambda$ is null; for any Lepage equivalent $\vartheta_{\lambda}$ it is
obvious that the converse holds, that $\lambda$ is null when
$d\vartheta_{\lambda}=0$.

In the next Section we consider how the construction of the fundamental Lepage
equivalent might be generalised for higher order Lagrangians.

##  4 The closure property

The question of whether it possible to find a procedure for constructing a
Lepage equivalent which satisfies the closure property, although solved in
1977 for first order Lagrangians, has been an open problem for higher order
Lagrangians (see [17, 23] and the references therein). An original solution to
this problem was given in [25], using the Vainberg–Tonti Lagrangian of a
source form $\varepsilon$, the horizontal $m$-form $\lambda_{\varepsilon}$
obtained locally in coordinates from
$\varepsilon=\varepsilon_{\alpha}\theta^{\alpha}\wedge\omega_{0}$ by the
fibred homotopy operator

$\lambda_{\varepsilon}=u^{\alpha}\int_{0}^{1}\varepsilon_{\alpha}(x^{i},tu^{\beta}_{I})dt\,.$

Typically $\lambda_{\varepsilon}$ has the same order as $\varepsilon$. If in
fact $\varepsilon=\varepsilon_{\lambda}$, so that the source form is the
Euler–Lagrange form of a given Lagrangian $\lambda$, then the Vainberg–Tonti
Lagrangian $\lambda_{\varepsilon_{\lambda}}$ and the pullback of $\lambda$
have the same Euler–Lagrange equations so that they differ by $h(d\alpha)$ for
some horizontal $(m-1)$-form $\alpha$. Then, taking
$\vartheta_{\lambda_{\varepsilon_{\lambda}}}$ to be the principal Lepage
equivalent of the Vainberg–Tonti Lagrangian in the given coordinates and
writing
$\vartheta^{\mathrm{F}}=\vartheta_{\lambda_{\varepsilon_{\lambda}}}+d\alpha$
we find that, to within pullbacks,

$(d\vartheta^{\mathrm{F}})^{(1)}=d\vartheta_{\lambda_{\varepsilon_{\lambda}}}^{(1)}=\varepsilon_{\lambda_{\varepsilon_{\lambda}}}=\varepsilon_{\lambda}$

so that $\vartheta^{\mathrm{F}}$ is a source form, and that

$h(\vartheta^{\mathrm{F}})=h(\vartheta_{\lambda_{\varepsilon_{\lambda}}})+h(d\alpha)=\lambda_{\varepsilon_{\lambda}}+h(d\alpha)=\lambda$

so that $\vartheta^{\mathrm{F}}$ is a Lepage equivalent of $\lambda$, and
finally that

$d\vartheta^{\mathrm{F}}=d\vartheta_{\lambda_{\varepsilon_{\lambda}}}$

so that if $\lambda$ is a null Lagrangian then $\varepsilon_{\lambda}=0$ and
therefore $d\vartheta^{\mathrm{F}}=0$.

This procedure therefore satisfies the closure property. It is not, though, a
generalisation of the fundamental Lepage equivalent for first order
Lagrangians, because it is always at most $1$-contact, whereas the Fundamental
Lepage equivalent is obtained from the Poincaré–Cartan form by adding higher
contact terms.

We shall, instead, define an alternative procedure which uses homotopy
operators for the horizontal differential of the variational bicomplex to add
the higher contact terms. Recall that this bicomplex is defined for forms of
globally finite order on the infinite jet manifold $J^{\infty}\pi$, as shown
in the diagram below (note that the squares with vertical arrows labelled
$\pi_{\infty}^{*}$ commute, whereas those with vertical arrows labelled
$d_{\mathrm{v}}$ anticommute.) The rows and columns are all locally exact, and
indeed all the $d_{\mathrm{h}}$ rows apart from the first are globally exact
[1, 20, 21, 22, 24]. Any Lagrangian $\lambda\in\Omega^{m}(J^{k}\pi)$ will have
a pullback $\pi_{\infty,k}^{*}\lambda\in\Omega^{0,m}$ on $J^{\infty}\pi$ which
for simplicity we shall continue to denote by $\lambda$ without the pullback
map.

${0}$${0}$${0}$${0}$${0}$${\mathbb{R}}$${\Omega^{0}(M)}$${\Omega^{1}(M)}$${\Omega^{2}(M)}$${\cdots}$${\Omega^{m}(M)}$${0}$${\mathbb{R}}$${\Omega^{0,0}}$${\Omega^{0,1}}$${\Omega^{0,2}}$${\cdots}$${\Omega^{0,m}}$${0}$${\Omega^{1,0}}$${\Omega^{1,1}}$${\Omega^{1,2}}$${\cdots}$${\Omega^{1,m}}$${0}$${\Omega^{2,0}}$${\Omega^{2,1}}$${\Omega^{2,2}}$${\cdots}$${\Omega^{2,m}}$${\vdots}$${\vdots}$${\vdots}$${\vdots}$$\scriptstyle{d}$$\scriptstyle{\pi_{\infty}^{*}}$$\scriptstyle{d}$$\scriptstyle{\pi_{\infty}^{*}}$$\scriptstyle{\pi_{\infty}^{*}}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{\pi_{\infty}^{*}}$$\scriptstyle{d_{\mathrm{h}}}$$\scriptstyle{d_{\mathrm{v}}}$$\scriptstyle{d_{\mathrm{h}}}$$\scriptstyle{d_{\mathrm{v}}}$$\scriptstyle{d_{\mathrm{v}}}$$\scriptstyle{d_{\mathrm{h}}}$$\scriptstyle{d_{\mathrm{h}}}$$\scriptstyle{d_{\mathrm{v}}}$$\scriptstyle{d_{\mathrm{h}}}$$\scriptstyle{d_{\mathrm{v}}}$$\scriptstyle{d_{\mathrm{h}}}$$\scriptstyle{d_{\mathrm{v}}}$$\scriptstyle{d_{\mathrm{v}}}$$\scriptstyle{d_{\mathrm{h}}}$$\scriptstyle{d_{\mathrm{h}}}$$\scriptstyle{d_{\mathrm{v}}}$$\scriptstyle{d_{\mathrm{h}}}$$\scriptstyle{d_{\mathrm{v}}}$$\scriptstyle{d_{\mathrm{h}}}$$\scriptstyle{d_{\mathrm{v}}}$$\scriptstyle{d_{\mathrm{v}}}$$\scriptstyle{d_{\mathrm{h}}}$$\scriptstyle{d_{\mathrm{h}}}$$\scriptstyle{d_{\mathrm{v}}}$

The variational bicomplex

Let $\vartheta_{\lambda}$ denote the pullback to $J^{\infty}\pi$ of any local
Lepage equivalent of $\lambda$ which is at most $1$-contact, so that
$\vartheta_{\lambda}^{(1)}=\vartheta_{\lambda}-\lambda\in\Omega^{1,m-1}$, and
let $P$ denote any local homotopy operator for the $d_{\mathrm{h}}$ rows
(apart from the first) of the variational bicomplex. Define the _extension of
$\vartheta_{\lambda}$ by $P$_ to be the $m$-form defined locally by

$\displaystyle\vartheta^{\mathrm{F}}$
$\displaystyle=\vartheta_{\lambda}+\sum_{p=1}^{m-1}(-Pd_{\mathrm{v}})^{p}\vartheta_{\lambda}^{(1)}$
$\displaystyle=\lambda+\vartheta_{\lambda}^{(1)}-(Pd_{\mathrm{v}})\vartheta_{\lambda}^{(1)}+(Pd_{\mathrm{v}})^{2}\vartheta_{\lambda}^{(1)}-\cdots+(-Pd_{\mathrm{v}})^{m-1}\vartheta_{\lambda}^{(1)}$
$\displaystyle\in\Omega^{0,m}\oplus\Omega^{1,m-1}\oplus\Omega^{2,m-2}\oplus\Omega^{3,m-3}\oplus\cdots\oplus\Omega^{m,0}\,,$

so that $\vartheta^{\mathrm{F}}$ is another Lepage equivalent of $\lambda$. We
shall show that this method of constructing $\vartheta^{\mathrm{F}}$ satisfies
the closure property, by diagram chasing.

Suppose that $\lambda$ is a null Lagrangian, so that

$0=\varepsilon_{\lambda}=(d\vartheta^{\mathrm{F}})^{(1)}=d_{\mathrm{v}}\vartheta_{\lambda}^{(0)}+d_{\mathrm{h}}\vartheta_{\lambda}^{(1)}=d_{\mathrm{v}}\lambda+d_{\mathrm{h}}\vartheta_{\lambda}^{(1)}\,.$

Then

$\displaystyle(d\vartheta^{\mathrm{F}})^{(2)}$
$\displaystyle=d_{\mathrm{v}}(\vartheta^{\mathrm{F}(1)})+d_{\mathrm{h}}(\vartheta^{\mathrm{F}(2)})$
$\displaystyle=d_{\mathrm{v}}\vartheta_{\lambda}^{(1)}-d_{\mathrm{h}}Pd_{\mathrm{v}}\vartheta_{\lambda}^{(1)}$
$\displaystyle=Pd_{\mathrm{h}}d_{\mathrm{v}}\vartheta_{\lambda}^{(1)}=-Pd_{\mathrm{v}}d_{\mathrm{h}}\vartheta_{\lambda}^{(1)}=Pd_{\mathrm{v}}d_{\mathrm{v}}\lambda=0$

using the homotopy property $d_{\mathrm{h}}\circ P+P\circ
d_{\mathrm{h}}=\operatorname{id}$, and in a similar way

$\displaystyle(d\vartheta^{\mathrm{F}})^{(p+1)}$
$\displaystyle=d_{\mathrm{v}}(\vartheta^{\mathrm{F}(p)})+d_{\mathrm{h}}(\vartheta^{\mathrm{F}(p+1)})$
$\displaystyle=d_{\mathrm{v}}(\vartheta^{\mathrm{F}(p)})-d_{\mathrm{h}}Pd_{\mathrm{v}}(\vartheta^{\mathrm{F}(p)})$
$\displaystyle=Pd_{\mathrm{h}}d_{\mathrm{v}}(\vartheta^{\mathrm{F}(p)})=-Pd_{\mathrm{v}}d_{\mathrm{h}}(\vartheta^{\mathrm{F}(p)})=Pd_{\mathrm{v}}d_{\mathrm{v}}(\vartheta^{\mathrm{F}(p-1)})=0$

for $2\leq p\leq m-1$, where the penultimate equality arises recursively from

$d_{\mathrm{h}}(\vartheta^{\mathrm{F}(p)})+d_{\mathrm{v}}(\vartheta^{\mathrm{F}(p-1)})=(d\vartheta^{\mathrm{F}})^{(p)}=0\,.$

Thus $d\vartheta^{\mathrm{F}}=(d\vartheta^{\mathrm{F}})^{(m+1)}$, and we may
see that this maximal contact component also vanishes by traversing the
diagram in the opposite direction. For $2\leq p\leq m$ we have

$\displaystyle d_{\mathrm{h}}d_{\mathrm{v}}(\vartheta^{\mathrm{F}(p)})$
$\displaystyle=-d_{\mathrm{h}}d_{\mathrm{v}}Pd_{\mathrm{v}}(\vartheta^{\mathrm{F}(p-1)})$
$\displaystyle=d_{\mathrm{v}}d_{\mathrm{h}}Pd_{\mathrm{v}}(\vartheta^{\mathrm{F}(p-1)})$
$\displaystyle=d_{\mathrm{v}}d_{\mathrm{v}}(\vartheta^{\mathrm{F}(p-1)})-(d_{\mathrm{v}}P)d_{\mathrm{h}}d_{\mathrm{v}}(\vartheta^{\mathrm{F}(p-1)})$
$\displaystyle=-(d_{\mathrm{v}}P)d_{\mathrm{h}}d_{\mathrm{v}}(\vartheta^{\mathrm{F}(p-1)})\,;$

but

$d_{\mathrm{h}}d_{\mathrm{v}}(\vartheta^{\mathrm{F}(1)})=d_{\mathrm{h}}d_{\mathrm{v}}\vartheta_{\lambda}^{(1)}=-d_{\mathrm{v}}d_{\mathrm{h}}\vartheta_{\lambda}^{(1)}=d_{\mathrm{v}}d_{\mathrm{v}}\lambda=0$

so that $d_{\mathrm{h}}d_{\mathrm{v}}(\vartheta^{\mathrm{F}(m)})=0$. As
$d_{\mathrm{h}}:\Omega^{m+1,0}\to\Omega^{m+1,1}$ is injective by exactness, we
see finally that
$(d\vartheta^{\mathrm{F}})^{(m+1)}=d_{\mathrm{v}}(\vartheta^{\mathrm{F}(m)})=0$.
We shall describe suitable local homotopy operators for $d_{\mathrm{h}}$,
constructed using vertical endomorphisms, in the next Section; by using them
we obtain the following result.

######  Theorem 1.

Let $\lambda$ be the pullback to $J^{\infty}\pi$ of a Lagrangian of any order,
and let $\vartheta_{\lambda}$ be the pullback to $J^{\infty}\pi$ of any local
Lepage equivalent of $\lambda$ which is at most $1$-contact. A local homotopy
operator $P$ then defines a local Lepage equivalent $\vartheta^{\mathrm{F}}$
which is an extension of $\vartheta_{\lambda}$ and which satisfies the closure
property, that $d\vartheta^{\mathrm{F}}=0$ precisely when $\lambda$ is null.

We can also consider a global version of this result, noting that the diagram
chasing above would apply equally to global operators as it does to local
ones. We have seen that additional structures, such as connections or
nonholonomic projections, are needed to construct a global Lepage equivalent
when the order of the Lagrangian is greater than two. A global homotopy
operator for the horizontal differential on $J^{\infty}\pi$ has also been
found [1, Theorem 5.56] and again this uses a symmetric linear connection on
the base manifold $M$.

######  Theorem 2.

Let $\lambda$ be the pullback to $J^{\infty}\pi$ of a Lagrangian of any order,
and let $\vartheta_{\lambda}$ be the pullback to $J^{\infty}\pi$ of any global
Lepage equivalent of $\lambda$ which is at most $1$-contact, constructed using
additional data as appropriate. A global homotopy operator $P$, such as the
one described in [1] using a symmetric linear connection, then defines a
global Lepage equivalent $\vartheta^{\mathrm{F}}$ which is an extension of
$\vartheta_{\lambda}$ and which satisfies the closure property, that
$d\vartheta^{\mathrm{F}}=0$ precisely when $\lambda$ is null.

We remark that in fact there is no requirement for the homotopy operators in
each term to be the same, and we could generalise the formula to

$\vartheta^{\mathrm{F}}=\vartheta_{\lambda}+\sum_{p=1}^{m-1}(-1)^{p}(P_{p+1}d_{\mathrm{v}}P_{p}d_{\mathrm{v}}\cdots
P_{2}d_{\mathrm{v}})\vartheta_{\lambda}^{(1)}$

where $P_{p}$ is a homotopy operator for the $p$-contact row of the
variational bicomplex.

##  5 Vertical endomorphisms

The most basic example of a ‘vertical endomorphism’ is the almost tangent
structure on a tangent manifold $TM$. This is simply a tensorial expression of
the isomorphism between a vector space and its tangent space at any point,
applied to the tangent spaces to a manifold, and may be regarded as a $1$-form
taking values in the sub-bundle of $TTM\to TM$ containing the vertical
vectors. A similar object may be defined using a more complicated procedure on
a higher order tangent manifold $T^{k}M$ [5], now giving a $1$-form taking its
values in the sub-bundle of vertical vectors in $TT^{k}M\to T^{k}M$.

Vertical endomorphisms $S^{\eta}$ on jet manifolds $J^{k}\pi$, where
$\eta\in\Omega^{1}(M)$ is a closed $1$-form, were defined in [18]. The
construction started with a point $j^{k}_{p}\phi\in J^{k}\pi$ and a tangent
vector $\xi$ at $j^{k-1}_{p}\phi\in J^{k-1}\pi$ vertical over $M$. Any such
vector may be represented by a $1$-parameter family of local sections
$\phi_{t}$ where $\phi_{0}=\phi$ and $\xi$ is the tangent vector at $t=0$ to
the curve $t\mapsto j^{k-1}_{p}\phi_{t}$. Given a function $f$ on $M$ defined
in a neighbourhood of $p$, the _vertical lift_ of $\xi$ to $j^{k}_{p}\phi$ in
the direction specified by $df$ then used the $1$-parameter family of local
sections $\psi_{t}:q\mapsto\phi_{tf(q)}(q)$ to define a curve
$j^{k}_{p}\psi_{t}$ in $J^{k}\pi$ and therefore a tangent vector at
$j^{k}_{p}\phi$. The vertical endomorphism $S^{\eta}$ at any point
$j^{k}_{p}\phi\in J^{k}\pi$ was then defined by starting with any tangent
vector in $T_{j^{k}_{p}\phi}J^{k}\pi$, projecting it to
$T_{j^{k-1}_{p}\phi}J^{k-1}\pi$, taking the vertical representative using the
contact structure, and then applying the vertical lift (using any function $f$
satisfying $f(p)=0$ and $df=\eta$ in a neighbourhood of $p$) to give a new
tangent vector in $T_{j^{k}_{p}\phi}J^{k}\pi$. It may be shown that this
construction is well defined, and so independent of the choices of $\phi_{t}$
and $f$, and that it gives a tensor field
$S^{\eta}\in\Omega^{1}(J^{k}\pi)\otimes\mathfrak{X}(J^{k}\pi)$ expressed in
coordinates111The numerical coefficient given in [18, eqn 3.4] and repeated in
[19] after definition 6.5.6 is incorrect as it does not take account of the
use of an individual index $i$ in a multi-index formula. as

$S^{\eta}=\sum_{\lvert J\rvert+\lvert K\rvert\leq
k-1}\frac{(J+K+1_{i})!}{J!\,K!\,(\lvert K\rvert+1)}\frac{\partial^{\lvert
K\rvert}\eta_{i}}{\partial
x^{K}}\,\theta^{\alpha}_{J}\otimes\frac{\partial}{\partial
u^{\alpha}_{J+K+1_{i}}}\,.$

It is evident from this formula that, when acting on forms, the operators
$S^{\eta}$ on $J^{k}\pi$ and on $J^{l}\pi$ with $l>k$ are related by the
pullback map $\pi_{l,k}^{*}$, so that we may define a similar operator acting
on forms on $J^{\infty}\pi$ without ambiguity.

Given local coordinates $(x^{i})$ on $U\subset M$, we write $S^{i}$ rather
than $S^{dx^{i}}$ for the operators on $U^{\infty}=\pi_{\infty}^{-1}(U)$.
These local operators have the rather simpler coordinate description

$S^{i}=\sum_{\lvert
I\rvert=0}^{\infty}\bigl{(}I(i)+1\bigr{)}\theta^{\alpha}_{I}\otimes\frac{\partial}{\partial
u^{\alpha}_{I+1_{i}}}$

and may be used to construct local homotopy operators for the horizontal
differential $d_{\mathrm{h}}$ on $J^{\infty}\pi$. One such homotopy operator,
involving an ordering of the coordinates $x^{i}$, was given in [22]. Other
homotopy operators, not using such an ordering, may be constructed from two
different repeated actions of $S^{i}$ on forms: these are

$\tilde{S}^{J}=i_{S^{j_{1}}\circ S^{j_{2}}\circ\cdots\circ
S^{j_{r}}}\,,\qquad\hat{S}^{J}=i_{S^{j_{1}}}\circ
i_{S^{j_{2}}}\circ\cdots\circ i_{S^{j_{r}}}$

where $\lvert J\rvert=r$ and $J=1_{j_{1}}+1_{j_{2}}+\cdots+1_{j_{r}}$. Note
that the first action is a derivation, whereas the second is not if $r>1$; the
multi-index notation is justified because operators $S^{i}$ and $S^{j}$
commute. The following result was obtained in [8, Theorem 1].

######  Proposition 3.

Define the differential operators
$\tilde{P},\hat{P}:\Omega^{p,q}(U^{\infty})\to\Omega^{p,q-1}(U^{\infty})$,
with $p\geq 1$ and $1\leq q\leq m$, by
$\tilde{P}(\omega)=i_{d/dx^{i}}\bigl{(}\tilde{P}^{i}(\omega)\bigr{)}$,
$\hat{P}(\omega)=i_{d/dx^{i}}\bigl{(}\hat{P}^{i}(\omega)\bigr{)}$ where

$\displaystyle\tilde{P}^{i}(\omega)$
$\displaystyle=\sum_{I=0}^{\infty}\frac{(-1)^{\lvert I\rvert}(m-q)!\lvert
I\rvert!}{p(m-q+\lvert I\rvert+1)!I!}d_{I}\tilde{S}^{I+1_{i}}\omega\,,$
$\displaystyle\hat{P}^{i}(\omega)$
$\displaystyle=\sum_{I=0}^{\infty}\frac{(-1)^{\lvert I\rvert}(m-q)!\lvert
I\rvert!}{p^{\lvert I\rvert+1}(m-q+\lvert
I\rvert+1)!I!}d_{I}\hat{S}^{I+1_{i}}\omega\,.$

Then both $\tilde{P}$ and $\hat{P}$ are homotopy operators for
$d_{\mathrm{h}}$.

In general the operators $\tilde{P}$ and $\hat{P}$ are different, although
they are equal when acting on forms projectable to $J^{1}\pi$, and also when
acting on forms in $\Omega^{1,q}(U^{\infty})$. In the latter case, writing the
operator as $P$, [8, Theorem 2] gives

$\omega-d_{\mathrm{h}}P\omega=\theta^{\alpha}\wedge\sum_{\lvert
I\rvert=0}^{\infty}(-1)^{\lvert I\rvert}d_{I}\bigl{(}i_{\partial/\partial
u^{\alpha}_{I}}\omega\bigr{)}$

for any $\omega\in\Omega^{1,m}$, so that $\omega-d_{\mathrm{h}}P\omega$ is a
source form. If we write $\vartheta_{\lambda}=\lambda-Pd_{\mathrm{v}}\lambda$
then

$(d\vartheta_{\lambda})^{(1)}=d_{\mathrm{v}}(\vartheta_{\lambda}^{(0)})+d_{\mathrm{h}}(\vartheta_{\lambda}^{(1)})=(d_{\mathrm{v}}\lambda)-d_{\mathrm{h}}P(d_{\mathrm{v}}\lambda)$

so that in particular $(d\vartheta_{\lambda})^{(1)}$ is a source form. Thus
$\vartheta_{\lambda}$ is a Lepage form, and it is clearly a local Lepage
equivalent of $\lambda$. In coordinates with $\lambda=L\,\omega_{0}$

$\vartheta_{\lambda}=L\,\omega_{0}+\sum_{\lvert J\rvert,\lvert
K\rvert=0}^{\infty}\frac{(-1)^{\lvert J\rvert}(J+K+1_{j})!\,\lvert
J\rvert!\,\lvert K\rvert!}{(\lvert J\rvert+\lvert
K\rvert+1)!\,J!\,K!}d_{J}\biggl{(}\frac{\partial L}{\partial
u^{\alpha}_{J+K+1_{j}}}\biggr{)}\theta_{K}\wedge\omega_{j}\,,$

so that it is the pullback to $J^{\infty}\pi$ of the principal Lepage
equivalent of $\lambda$. We obtain the above formula from those for $S$ and
$P$ by using the multi-index Leibniz’ rule and the identity for weighted sums
of binomial coefficients

$\sum_{0\leq K\leq I}\frac{(-1)^{\lvert K\rvert}I!}{(\lvert
K\rvert+p+1)K!(I-K)!}=\frac{p!\lvert I\rvert!}{(\lvert I\rvert+p+1)!}$

obtained by first evalating the integral $\int_{0}^{1}x^{p}(x-1)^{r}dx$ in two
different ways, and then using the Vandermonde identity for the convolution of
scalar binomial coefficients.

We can also use the homotopy operators $P$ to give a simple proof of the
result mentioned earlier, that the difference between two Lepage equivalents
for the same Lagrangian is the sum of a closed form and a form which is at
least $2$-contact. Let $\vartheta_{\lambda}$ and
$\vartheta^{\prime}_{\lambda}$ be two Lepage equivalents for the Lagrangian
$\lambda$, and put
$\vartheta=\vartheta_{\lambda}-\vartheta^{\prime}_{\lambda}$, so that
$\vartheta^{(0)}=0$. As $\vartheta_{\lambda}$ and
$\vartheta^{\prime}_{\lambda}$ give rise to the same Euler–Lagrange form, we
see that

$d_{\mathrm{h}}(\vartheta^{(1)})+d_{\mathrm{v}}(\vartheta^{(0)})=(d\vartheta)^{(1)}=0$

so that $d_{\mathrm{h}}(\vartheta^{(1)})=0$ and therefore locally
$\vartheta^{(1)}=d_{\mathrm{h}}P(\vartheta^{(1)})$. Then

$\vartheta^{(1)}=dP(\vartheta^{(1)})-d_{\mathrm{v}}P(\vartheta^{(1)})$

where $d_{\mathrm{v}}P(\vartheta^{(1)})\in\Omega^{2,m-2}$, so that

$\displaystyle\vartheta$
$\displaystyle=dP(\vartheta^{(1)})+\bigl{(}\vartheta^{(2)}-d_{\mathrm{v}}P(\vartheta^{(1)})\bigr{)}+\cdots+\vartheta^{(m)}$
$\displaystyle\in
d\Omega^{1,m-2}\oplus\Omega^{2,m-2}\oplus\cdots\oplus\Omega^{m,0}\,.$

Finally in this Section we apply these operators to a first order Lagrangian
$\lambda=L\,\omega_{0}$. As $d_{\mathrm{v}}\lambda=d\lambda$ is also first
order, we see that

$\lambda-Pd_{\mathrm{v}}\lambda=L\,\omega_{0}+S^{i}\biggl{(}\frac{\partial
L}{\partial u^{\alpha}}\theta^{\alpha}+\frac{\partial L}{\partial
u^{\alpha}_{j}}\theta^{\alpha}_{j}\biggr{)}\omega_{i}=L\,\omega_{0}+\frac{\partial
L}{\partial u^{\alpha}_{i}}\theta^{\alpha}\wedge\omega_{i}\,,$

the local expression of the Poincaré–Cartan form. This is also first order,
and we then see that each successive operator $P^{i}$ is simply a multiple of
$S^{i}$. We obtain

$(-Pd_{\mathrm{v}})^{p}\lambda=\frac{1}{(p!)^{2}}\frac{\partial^{p}L}{\partial
u^{\alpha_{1}}_{i_{1}}\cdots\partial
u^{\alpha_{p}}_{i_{p}}}\theta^{\alpha_{1}}\wedge\cdots\wedge\theta^{\alpha_{p}}\wedge\omega_{i_{1}\cdots
i_{p}}\,,$

showing that $\sum_{p=0}^{m}(-Pd_{\mathrm{v}})^{p}\lambda$ gives the local
expression of the standard fundamental Lepage equivalent of a first order
Lagrangian.

##  6 Connections and vertical tensors

We have seen that, for a first order Lagrangian, the fundamental Lepage
equivalent may be constructed locally using the homotopy operators $S^{i}$,
and also that it is a global object which may be constructed using the first
order vertical tensor $S$. These are two different facets of the same
construction and they arise because, in the first order case, the formulation
of a vertical endomorphism $S^{\eta}$ does not in fact require the $1$-form
$\eta$ to be closed. From the coordinate description

$S^{\eta}=\eta_{i}\theta^{\alpha}\otimes\frac{\partial}{\partial
u^{\alpha}_{i}}$

it is clear that at any point $j^{1}_{p}\phi\in J^{1}\pi$ the value of
$S^{\eta}$ depends only on the cotangent vector $\eta|_{p}$ and not on the
values of $\eta$ at any other points. Thus, given any cotangent vector
$\eta|_{p}\in T^{*}_{p}M$, we may choose a closed $1$-form $\zeta$ in a
neighbourhood of $p$ satisfying $\zeta|_{p}=\eta|_{p}$, for example the form
given in coordinates centred on $p$ by $\zeta=d(\eta_{i}(p)x^{i})$, and put

$S^{\eta}|_{j^{1}_{p}\phi}=S^{\zeta}|_{j^{1}_{p}\phi}\in(T^{*}J^{1}\pi\otimes
TJ^{1}\pi)_{j^{1}_{p}\phi}\,.$

Doing this at each point of $J^{1}\pi$ gives a well defined vertical
endomorphism $S^{\eta}$ for an arbitrary $1$-form $\eta\in\Omega^{1}(M)$, and
it is clear that the mapping $\eta\mapsto S^{\eta}$ is just that given by the
vertical tensor $S$.

The same approach will not work directly for higher order vertical
endomorphisms. For example, the coordinate description of $S^{\eta}$ on
$J^{2}\pi$ is

$S^{\eta}=\eta_{i}\theta^{\alpha}\otimes\frac{\partial}{\partial
u^{\alpha}_{i}}+\frac{1}{\\#(ij)}\frac{\partial\eta_{i}}{\partial
x^{j}}\theta^{\alpha}\otimes\frac{\partial}{\partial
u^{\alpha}_{(ij)}}+\frac{2}{\\#(ij)}\eta_{i}\theta^{\alpha}_{j}\otimes\frac{\partial}{\partial
u^{\alpha}_{(ij)}}$

and at any point $j^{2}_{p}\phi\in J^{2}\pi$ the value of $S^{\eta}$ depends,
not just on the cotangent vector $\eta_{p}$, but also on the derivative of the
$1$-form $\eta$ at $p$.

We can, however, deal with this problem by supposing that we are given a
symmetric linear connection $\nabla$ on $M$; the infinitesimal parallel
translation defined by $\nabla$ will then provide enough information to
specify the derivative of $\eta$. The vertical endomorphism defined by the
$1$-form $\eta$ (not necessarily closed) and the connection $\nabla$ will be
given in coordinates as

$S^{\eta}_{\nabla}=\eta_{i}\theta^{\alpha}\otimes\frac{\partial}{\partial
u^{\alpha}_{i}}+\frac{1}{\\#(hj)}\eta_{i}\Gamma^{i}_{hj}\theta^{\alpha}\otimes\frac{\partial}{\partial
u^{\alpha}_{(hj)}}+\frac{2}{\\#(ij)}\eta_{i}\theta^{\alpha}_{j}\otimes\frac{\partial}{\partial
u^{\alpha}_{(ij)}}$

where $\Gamma^{i}_{hj}$ are the connection coefficients of $\nabla$, so that
the mapping $\eta\to S^{\eta}_{\nabla}$ will define a second order vertical
tensor $S_{\nabla}$, a section of the bundle $\pi_{2}^{*}TM\otimes
T^{*}J^{2}\pi\otimes TJ^{2}\pi$ over $J^{2}\pi$.

Formally, as the $1$-form $\eta$ is a section of the cotangent bundle
$\tau:T^{*}M\to M$, we regard the connection $\nabla$ as a linear Ehresmann
connection $\Gamma:T^{*}M\to J^{1}\tau$, a section of the jet bundle
$\tau_{1,0}:J^{1}\tau\to T^{*}M$, so that the connection coefficients
$\Gamma^{i}_{hj}$ are just the jet coordinates of $\Gamma$. (Of course the
connection $\nabla$ also defines a linear Ehresmann connection on the tangent
bundle, but there the jet coordinates are $-\Gamma^{i}_{hj}$.) For each
$j^{2}_{p}\phi\in J^{2}\pi$ we may choose a closed $1$-form $\zeta$ in a
neighbourhood of $p$ satisfying $j^{1}_{p}\zeta=\Gamma(\eta|_{p})$, for
example the form given in coordinates centred on $p$ by
$\zeta=d\bigl{(}\eta_{i}(p)x^{i}+\tfrac{1}{2}\eta_{i}(p)\Gamma^{i}_{hj}(p)x^{h}x^{j}\bigr{)}$,
and put

$S_{\nabla}^{\eta}|_{j^{2}_{p}\phi}=S^{\zeta}|_{j^{2}_{p}\phi}\in(T^{*}J^{2}\pi\otimes
TJ^{2}\pi)_{j^{2}_{p}\phi}\,.$

Doing this at each point of $J^{2}\pi$ now gives a well defined vertical
endomorphism $S_{\nabla}^{\eta}$ for an arbitrary $1$-form
$\eta\in\Omega^{1}(M)$, and so we can construct a second order vertical tensor
with coordinate expression

$S_{\nabla}=\partial_{i}\otimes\biggl{(}\theta^{\alpha}\otimes\frac{\partial}{\partial
u^{\alpha}_{i}}+\frac{1}{\\#(hj)}\Gamma^{i}_{hj}\theta^{\alpha}\otimes\frac{\partial}{\partial
u^{\alpha}_{(hj)}}+\frac{2}{\\#(ij)}\theta^{\alpha}_{j}\otimes\frac{\partial}{\partial
u^{\alpha}_{(ij)}}\biggr{)}\,.$ (6.1)

A similar procedure may be carried out for higher order vertical
endomorphisms, but requires the use of semiholonomic jets to allow for
symmetrization. For example, in the third order case we use the connection map
$\Gamma:T^{*}M\to J^{1}\tau$, regarded as a bundle morphism $\tau\to\tau_{1}$
over the identity on $M$, and its prolongation $j^{1}\Gamma:J^{1}\tau\to
J^{1}\tau_{1}$. The composition $j^{1}\Gamma\circ\Gamma$ then takes its values
in the semiholonomic manifold $\widehat{J}^{2}\tau\subset J^{1}\tau_{1}$ [19,
Section 5.3], so that if $\mathrm{p}_{2}:\widehat{J}^{2}\tau\to J^{2}\tau$ is
the symmetrization projection then we may use

$\Gamma_{2}=\mathrm{p}_{2}\circ j^{1}\Gamma\circ\Gamma:T^{*}M\to J^{2}\tau$

as the map which allows us to specify the first and second derivatives at $p$
of the closed local $1$-form $\zeta$ by setting
$j^{2}_{p}\zeta=\Gamma_{2}(\eta|_{p})$.

More generally, we construct the maps $\Gamma_{l}$ recursively. Suppose we
have the map $\Gamma_{l-1}:T^{*}M\to J^{l-1}\tau$, and that it is a section of
$\tau_{l-1,0}$ with the property that $j^{1}\Gamma_{l-1}\circ\Gamma$ takes its
values in the semiholonomic manifold $\widehat{J}^{l}\tau\subset
J^{1}\tau_{l-1}$, so that we may set

$\Gamma_{l}=\mathrm{p}_{l}\circ j^{1}\Gamma_{l-1}\circ\Gamma:T^{*}M\to
J^{l}\tau\,.$

We note first that

$\displaystyle\tau_{l,l-1}\circ\Gamma_{l}$
$\displaystyle=(\tau_{l-1})_{1,0}\circ i_{1,l-1}\circ\Gamma_{l}$
$\displaystyle=(\tau_{l-1})_{1,0}\circ i_{1,l-1}\circ\mathrm{p}_{l}\circ
j^{1}\Gamma_{l-1}\circ\Gamma$ $\displaystyle=(\tau_{l-1})_{1,0}\circ
j^{1}\Gamma_{l-1}\circ\Gamma$
$\displaystyle=\Gamma_{l-1}\circ\tau_{1,0}\circ\Gamma$
$\displaystyle=\Gamma_{l-1}\,,$

so that

$\displaystyle\tau_{l,0}\circ\Gamma_{l}=\tau_{l-1,0}\circ\tau_{l,l-1}\circ\Gamma_{l}=\tau_{l-1,0}\circ\Gamma_{l-1}=\operatorname{id}_{T^{*}M}$

and therefore that $\Gamma_{l}$ is a section of $\tau_{l,0}:J^{l}\tau\to
T^{*}M$. We must also check that $j^{1}\Gamma_{l}\circ\Gamma$ takes its values
in the semiholonomic manifold $\widehat{J}^{l+1}\tau$, the submanifold of
$J^{1}\tau_{l}$ given by equality of the two maps $j^{1}\tau_{l,l-1}$ and
$i_{1,l-1}\circ(\tau_{l})_{1,0}$ to $J^{1}\tau_{l-1}$ [19, Section 6.2]; but
at any point $j^{1}_{p}\omega\in J^{1}\tau$ we know that

$\displaystyle\bigl{(}j^{1}\tau_{l,l-1}\circ
j^{1}\Gamma_{l}\bigr{)}(j^{1}_{p}\omega)$
$\displaystyle=j^{1}(\tau_{l,l-1}\circ\Gamma_{l})(j^{1}_{p}\omega)$
$\displaystyle=j^{1}\Gamma_{l-1}(j^{1}_{p}\omega)$ and
$\displaystyle\bigl{(}i_{l-1,1}\circ(\tau_{l})_{1,0}\circ
j^{1}\Gamma_{l}\bigr{)}(j^{1}_{p}\omega)$
$\displaystyle=\bigl{(}i_{l-1,1}\circ(\tau_{l})_{1,0}\bigr{)}\bigl{(}j^{1}_{p}(\Gamma_{l}\circ\omega)\bigr{)}$
$\displaystyle=i_{l-1,1}\bigl{(}\Gamma_{l}(\omega(p))\bigr{)}$
$\displaystyle=j^{1}\Gamma_{l-1}\bigl{(}\Gamma(\omega(p))\bigr{)}\,,$

so that if $j^{1}_{p}\omega$ is in the image of $\Gamma$ then
$j^{1}_{p}\omega=\Gamma(\omega(p))$ and

$\bigl{(}j^{1}\tau_{l,l-1}\circ
j^{1}\Gamma_{l}\bigr{)}\bigl{(}\Gamma(\omega(p))\bigr{)}=j^{1}\Gamma_{l-1}\bigl{(}\Gamma(\omega(p))\bigr{)}=\bigl{(}i_{l-1,1}\circ(\tau_{l})_{1,0}\circ
j^{1}\Gamma_{l}\bigr{)}\bigl{(}\Gamma(\omega(p))\bigr{)}$

as required. We may therefore define
$\Gamma_{l+1}=\mathrm{p}_{l+1}\circ\Gamma_{l}\circ\Gamma$ and continue the
process. The recursion starts with $l=2$ and $\Gamma_{1}=\Gamma:T^{*}M\to
J^{1}\tau$, or even degenerately with $l=1$ and
$\Gamma_{0}=\operatorname{id}_{T^{*}M}:T^{*}M\to J^{0}\tau=T^{*}M$.

To find a coordinate expression for these maps, let $(x^{i},y_{j})$ be the
coordinates on $T^{*}M$, so that the jet coordinates on $J^{1}\tau$ are
$y_{ij}$ and on $J^{l}\tau$ are $y_{iJ}$. As the connection is linear and
symmetric, we see that $y_{ij}\circ\Gamma=y_{h}\Gamma^{h}_{ij}$ with
$y_{ji}\circ\Gamma=y_{ij}\circ\Gamma$, and in general if
$y_{iJ}\circ\Gamma_{l}=y_{h}\Gamma^{h}_{J+1_{i}}$ then

$y_{iJj}\circ
j^{1}\Gamma_{l}=y_{h}\frac{\partial\Gamma^{h}_{J+1_{i}}}{\partial
x^{j}}+y_{hj}\Gamma^{h}_{J+1_{i}}$

so that

$y_{iJj}\circ
j^{1}\Gamma_{l}\circ\Gamma=y_{g}\biggl{(}\frac{\partial\Gamma^{g}_{J+1_{i}}}{\partial
x^{j}}+\Gamma^{g}_{hj}\Gamma^{h}_{J+1_{i}}\biggr{)}\,;$

the coordinates $y_{iJ+1_{j}}\circ\Gamma_{l+1}$ may then be obtained by
symmetrization. In the degererate case, the coordinates of $\Gamma_{0}$ are of
course $\Gamma^{h}_{i}=\delta^{h}_{i}$.

We have, therefore, obtained the following result.

######  Theorem 4.

Let $\pi:E\to M$ be a fibred manifold, and let $\nabla$ be a symmetric linear
connection on $M$. On any jet manifold $J^{k}\pi$ there is a canonical
vertical tensor $S_{\nabla}$ defined in the following way. If
$\eta\in\Omega^{1}(M)$ and $j^{k}_{p}\phi\in J^{k}\pi$, let $\zeta$ be any
local closed $1$-form defined in a neighbourhood of $p$ satisfying
$j^{k-1}_{p}\zeta=\Gamma_{k-1}(\eta|_{p})$ (for example, a $1$-form defined
using a polynomial in coordinates $x^{i}$ centred on $p$) and put
$S^{\eta}_{\nabla}|_{j^{k}_{p}\phi}=S^{\zeta}|_{j^{k}_{p}\phi}$. Then
$S^{\eta}_{\nabla}|_{j^{k}_{p}\phi}$ is independent of the choice of $\zeta$.
The resulting map $j^{k}_{p}\phi\mapsto S^{\eta}_{\nabla}|_{j^{k}_{p}\phi}$ is
a vertical endomorphism depending at each point $j^{k}_{p}\phi$ only on the
cotangent vector $\eta|_{p}$ and so defines a vertical tensor $\eta\mapsto
S^{\eta}_{\nabla}$.

The coordinate expression of $S_{\nabla}$ is

$S_{\nabla}=\partial_{h}\otimes\sum_{\lvert J\rvert+\lvert K\rvert\leq
k-1}\frac{(J+K+1_{i})!}{J!\,K!\,(\lvert
K\rvert+1)}\Gamma^{h}_{K+1_{i}}\theta^{\alpha}_{J}\otimes\frac{\partial}{\partial
u^{\alpha}_{J+K+1_{i}}}\,,$

and combining the sums over the index $i$ and the multi-index $K$ in the usual
way then gives

$S_{\nabla}=\partial_{h}\otimes\sum_{\begin{subarray}{c}\lvert J\rvert+\lvert
K\rvert\leq k\\\ \lvert
K\rvert>0\end{subarray}}\frac{(J+K)!}{J!\,K!}\Gamma^{h}_{K}\theta^{\alpha}_{J}\otimes\frac{\partial}{\partial
u^{\alpha}_{J+K}}\,.$

A similar formula, without an upper bound on the length of the multi-indices,
may be used on $J^{\infty}\pi$ for the map $\pi_{\infty}^{*}T^{*}M\otimes
T^{*}J^{\infty}\pi\to TJ^{\infty}\pi$.

##  7 Infinitesimal nonholonomic projections

As mentioned earlier, two possible approaches to defining global Lepage
equivalents for higher order Lagrangians involve using either connections, or
tubular neighbourhoods of holonomic jet manifolds inside nonholonomic jet
manifolds. We have remarked that the latter approach really involves only the
infinitesimal projection defined by the tubular neighbourhood at points of the
holonomic submanifold, and we can now see that the existence of vertical
tensors allows the two approaches to be related: a symmetric linear connection
on the base manifold will define an infinitesimal nonholonomic projection
$TJ^{1}\pi_{k-1}\to TJ^{k}\pi$ for $k\geq 2$.

The simplest example is in the second order case, where
$\mathrm{i}_{1,1}:J^{2}\pi\to J^{1}\pi_{1}$ is the canonical inclusion. We
start with a point $j^{2}_{p}\phi\in J^{2}\pi$ and a tangent vector $\xi\in
T_{j^{2}_{p}\phi}J^{1}\pi_{1}$ which is vertical over $J^{1}\pi$, so that
$\xi\in V_{j^{2}_{p}\phi}(\pi_{1})_{1,0}$. We then apply the isomorphism

$\mathsf{A}:V(\pi_{1})_{1,0}\to(\pi_{1})_{1}^{*}T^{*}M\otimes(\pi_{1})_{1,0}^{*}V\pi_{1}$

arising from the affine structure of $(\pi_{1})_{1,0}:J^{1}\pi_{1}\to
J^{1}\pi$ (restricted to points of $J^{2}\pi$) and follow this by
$S_{\nabla}$, giving a map

$p_{\nabla}=S_{\nabla}\circ\mathsf{A}:V_{J^{2}\pi}(\pi_{1})_{1,0}\to
V\pi_{2,0}$

so that $p_{\nabla}(\xi)\in V_{j^{2}_{p}\phi}\pi_{2,0}$.

There are, of course, many possible extensions of $p_{\nabla}$ to a map
$T_{J^{1}\pi}J^{1}\pi_{1}\to TJ^{1}\pi$; but there is precisely one such
extension satisfying the requirement that $p_{\nabla}\circ
T\mathrm{i}_{1,1}=\operatorname{id}_{TJ^{2}\pi}$. We may see this by looking
at coordinate representations. At any point
$\mathrm{i}_{1,1}(j^{2}_{p}\phi)\in J^{1}\pi_{1}$

$\mathsf{A}\biggl{(}\frac{\partial}{\partial u^{\alpha}_{\cdot
j}}\biggr{)}=dx^{j}\otimes\frac{\partial}{\partial
u^{\alpha}}\,,\qquad\mathsf{A}\biggl{(}\frac{\partial}{\partial
u^{\alpha}_{ij}}\biggr{)}=dx^{j}\otimes\frac{\partial}{\partial
u^{\alpha}_{i}}\,,$

and composing with $S_{\nabla}$ as presented in formula (6.1) in the previous
Section gives

$p_{\nabla}\biggl{(}\frac{\partial}{\partial u^{\alpha}_{\cdot
j}}\biggr{)}=\frac{\partial}{\partial
u^{\alpha}_{j}}+\frac{1}{\\#(ik)}\Gamma^{j}_{ik}\frac{\partial}{\partial
u^{\alpha}_{(ik)}}\,,\qquad p_{\nabla}\biggl{(}\frac{\partial}{\partial
u^{\alpha}_{ij}}\biggr{)}=\frac{1}{\\#(ij)}\frac{\partial}{\partial
u^{\alpha}_{(ij)}}\,.$

(Nominally the image of, say,

$\frac{\partial}{\partial
u^{\alpha}_{ij}}\biggr{|}_{\mathrm{i}_{1,1}(j^{2}_{p}\phi)}\mapsto
dx^{j}|_{p}\otimes\frac{\partial}{\partial
u^{\alpha}_{i}}\biggr{|}_{j^{1}_{p}\phi}\in T^{*}_{p}M\otimes
T_{j^{1}_{p}\phi}J^{1}\pi$

is not directly in the domain of $S_{\nabla}$; but as $S_{\nabla}$
incorporates the projection $T\pi_{2,1}:TJ^{2}\pi\to TJ^{1}\pi$ we may
represent that image by an element of $T^{*}_{p}M\otimes
T_{j^{2}_{p}\phi}J^{2}\pi$ without ambiguity.) Noting now that

$\displaystyle T\mathrm{i}_{1,1}\biggl{(}\frac{\partial}{\partial
x^{i}}\biggr{)}$ $\displaystyle=\frac{\partial}{\partial x^{i}}$
$\displaystyle T\mathrm{i}_{1,1}\biggl{(}\frac{\partial}{\partial
u^{\alpha}_{i}}\biggr{)}$ $\displaystyle=\frac{\partial}{\partial
u^{\alpha}_{i\cdot}}+\frac{\partial}{\partial u^{\alpha}_{\cdot i}}$
$\displaystyle T\mathrm{i}_{1,1}\biggl{(}\frac{\partial}{\partial
u^{\alpha}}\biggr{)}$ $\displaystyle=\frac{\partial}{\partial
u^{\alpha}_{\cdot\cdot}}$ $\displaystyle
T\mathrm{i}_{1,1}\biggl{(}\frac{\partial}{\partial
u^{\alpha}_{(ij)}}\biggr{)}$ $\displaystyle=\frac{\partial}{\partial
u^{\alpha}_{ij}}$

we see that necessarily we must have

$p_{\nabla}\biggl{(}\frac{\partial}{\partial
x^{i}}\biggr{)}=\frac{\partial}{\partial x^{i}}\,,\qquad
p_{\nabla}\biggl{(}\frac{\partial}{\partial
u^{\alpha}_{\cdot\cdot}}\biggr{)}=\frac{\partial}{\partial
u^{\alpha}}\,,\qquad p_{\nabla}\biggl{(}\frac{\partial}{\partial
u^{\alpha}_{i\cdot}}\biggr{)}=-\frac{1}{\\#(jk)}\Gamma^{i}_{jk}\frac{\partial}{\partial
u^{\alpha}_{(jk)}}\,.$

A direct calculation confirms that these coordinate formulæ for $p_{\nabla}$
are invariant under fibred coordinate transformations on $\pi$ and their
prolongations to $J^{1}\pi_{1}$ and $J^{2}\pi$; we also see that the
restriction of $p_{\nabla}$ to the semiholonomic submanifold
$\widehat{J}^{2}\pi\subset J^{1}\pi_{1}$ is the symmetrization map
$\mathrm{p}_{2}$, as we would expect. Note that $p_{\nabla}$ is a bundle
morphism over $j^{1}\pi_{1,0}\to\pi_{2,1}$, not over
$(\pi_{1})_{1,0}\to\pi_{2,1}$.

${J^{1}\pi_{1}}$${J^{2}\pi}$${TJ^{1}\pi_{1}}$${TJ^{2}\pi}$${J^{1}\pi}$${J^{1}\pi}$${TJ^{1}\pi}$${TJ^{1}\pi}$$\scriptstyle{j^{1}\pi_{1,0}}$$\scriptstyle{(\pi_{1})_{1,0}}$$\scriptstyle{\pi_{2,1}}$$\scriptstyle{p_{\nabla}}$$\scriptstyle{Tj^{1}\pi_{1,0}}$$\scriptstyle{T\pi_{2,1}}$

We apply the same approach in the general case to obtain the infinitesimal
projection $p_{\nabla}:TJ^{1}\pi_{k-1}\to TJ^{k}\pi$, with coordinate
expressions

$\displaystyle p_{\nabla}\biggl{(}\frac{\partial}{\partial
u^{\alpha}_{Ij}}\biggr{)}$ $\displaystyle=\sum_{\lvert K\rvert>0}^{k-\lvert
I\rvert}\frac{(I+K)!}{I!\,K!}\Gamma^{j}_{K}\frac{\partial}{\partial
u^{\alpha}_{I+K}}$ obtained by composing $\mathsf{A}$ with $S_{\nabla}$, with
the necessary consequence that $\displaystyle
p_{\nabla}\biggl{(}\frac{\partial}{\partial u^{\alpha}_{J\cdot}}\biggr{)}$
$\displaystyle=(1-\lvert J\rvert)\frac{\partial}{\partial
u^{\alpha}_{J}}-\sum_{I+1_{j}=J}\biggl{(}\sum_{\lvert K\rvert=2}^{k-\lvert
I\rvert}\frac{(I+K)!}{I!\,K!}\Gamma^{j}_{K}\frac{\partial}{\partial
u^{\alpha}_{I+K}}\biggr{)}$ $\displaystyle
p_{\nabla}\biggl{(}\frac{\partial}{\partial x^{i}}\biggr{)}$
$\displaystyle=\frac{\partial}{\partial x^{i}}\,.$

This gives us the following result.

######  Theorem 5.

Let $\pi:E\to M$ be a fibred manifold, and let $\nabla$ be a symmetric linear
connection on $M$. For each nonholonomic jet manifold $J^{1}\pi_{k-1}$ there
is a unique infinitesimal projection $p_{\nabla}$ satisfying $p_{\nabla}\circ
T\mathrm{i}_{1,k-1}=\operatorname{id}_{J^{k}\pi}$ and
$p_{\nabla}|_{\widehat{J}^{k}\pi}=\mathrm{p}_{k}$, constructed by composing
the isomorphism

$\mathsf{A}:V(\pi_{k-1})_{1,0}\to(\pi_{k-1})_{1}^{*}T^{*}M\otimes(\pi_{k-1})_{1,0}^{*}V\pi_{k-1}$

arising from the affine structure of $(\pi_{k-1})_{1,0}:J^{1}\pi_{k-1}\to
J^{k-1}\pi$ (restricted to points of $J^{k}\pi$) with the vertical tensor
$S_{\nabla}$ on $J^{k}\pi$.

##  8 Homotopy operators for the horizontal differential

We have seen that homotopy operators for the horizontal differential play an
important part in the construction of Lepage equivalents satisfying the
closure condition (and, indeed, of at most $1$-contact Lepage equivalents in
general), and that locally such homotopy operators can be constructed using
vertical endomorphisms. We have also noted that global homotopy operators
(depending on a choice of a symmetric linear connection) have been shown to
exist, but the construction in [1] uses a quite different method, relating the
horizontal differential on forms to an operator acting on evolutionary vector
fields. It is therefore of some interest to see whether a global homotopy
operator can be constructed directly for differental forms by using vertical
differentials. I conjecture that this can be done, and offer a possible method
of doing so. The proposed formula has been checked for small values of the
parameters $p$, $q$ and $r$ (see the Appendix for an example calculation);
although the general result might be amenable to a direct calculation, there
may well be a more geometric method of approaching it.

There are three ingredients in the proposed formula, which mimics the local
formula described above. The vertical tensor $S_{\nabla}$, regarded as a map
$\Omega^{p,q}\to\mathfrak{X}(M)\otimes\Omega^{p,q}$, has already been
specified, and this can be iterated to give a map
$S_{\nabla}^{r}:\Omega^{p,q}\to\odot^{r}\mathfrak{X}(M)\otimes\Omega^{p,q}$,
where $\odot^{r}\mathfrak{X}(M)$ denotes the symmetric multivector fields on
$M$. We shall also need a covariant version of the horizontal differential,
which we shall denote $d_{\mathrm{h}\nabla}$; this will be a map
$\odot^{r}\mathfrak{X}(M)\otimes\Omega^{p,q}\to\odot^{r}\mathfrak{X}(M)\otimes\Omega^{p,q+1}$,
given on basis tensors by

$d_{\mathrm{h}\nabla}(X\otimes\omega)=\nabla X\wedge\omega+X\otimes
d_{\mathrm{h}}\omega$

and extended by multilinerarity, symmetry and skewsymmetry. The final
ingredient will be an operator
$\mathsf{C}:\odot^{r}\mathfrak{X}(M)\otimes\Omega^{p,q}\to\odot^{r-1}\mathfrak{X}(M)\otimes\Omega^{p,q-1}$
contracting a vector component with a form component, again taking advantage
of symmetry and skewsymmetry. The proposed homotopy operator is then
$P_{\nabla}:\Omega^{p,q}\to\Omega^{p,q-1}$ where

$P_{\nabla}\omega=\sum_{r=0}^{\infty}\frac{(-1)^{r}(m-q)!}{p(m-q+r+1)r!}\bigl{(}\mathsf{C}\circ
d_{\mathrm{h}\nabla}\bigr{)}^{r}\mathsf{C}\bigl{(}S_{\nabla}^{r+1}\omega\bigr{)}\,.$

##  9 Discussion

One of the features of the approach taken in this paper is that it combines
the use of finite and infinite jets. Variational problems are by theie nature
of finite order, and the various differential forms involved in their analysis
are normally defined on a finite order jet manifold. Indeed, as we have seen,
the properties of Lepage equivalents of first order and second order
Lagrangians are rather different from those of higher order Lagrangians.

On the other hand, the variational bicomplex is best considered on the
infinite jet manifold. In [1] a subcomplex called the Jacobian subcomplex
which is projectable to a finite order jet manifold is shown after lengthy
calculations to be locally exact; but no mention is made of a homotopy
operator acting on forms which are not $d_{\mathrm{h}}$-closed. The operators
$\hat{P}$ and $\tilde{P}$ described earlier, although acting on all the forms
on each finite order jet manifold, generally increase their order. It seems to
be the case that the complexity of ascertaining a bound on the order of the
forms obscures the homotopy structure of the problem, and indeed the potential
for a global solution. The alternative approach in [25], which involves a
single homotopy operator for the variational derivative (and thus,
essentially, for the vertical differential) avoids this problem, but then
cannot reduce to the classical fundamental Lepage equivalent for first order
Lagrangians; in addition, global versions are likely to be constrained by
topological considerations.

The investigations in the second half of the paper suggest that vertical
endomorphisms, when glued together as a vertical tensor using a symmetric
linear connection, could be a significant part of the geometry of the jet
bundle structure on a fibred manifold. If the conjecture that they define a
global homotopy operator for $d_{\mathrm{h}}$ is correct, then the simple
formula

$\vartheta_{\lambda,\nabla}=\sum_{p=0}^{m}(-P_{\nabla}d_{\mathrm{v}})^{p}\lambda$

will give a Lepage equivalent of the Lagrangian $\lambda$ satisfying the
closure property without the need for a separate choice of
$\vartheta_{\lambda}^{(1)}$ to start the recursion. There will, though, be the
question of whether the truncated form
$\lambda-(P_{\nabla}d_{\mathrm{v}})\lambda$ is the same as the Poincaré–Cartan
form constructed using the infinitesimal projections $p_{\nabla}$.

A final observation is that we have not explicitly addressed the question of
whether it is possible to find, for second order Lagrangians, a geometrical
construction of a Lepage equivalent satisfying the closure condition
independently of any connection, as can be done for the Poincaré–Cartan form
and the Carathéodory form. I suspect that this will not be the case.

## Acknowledgements

I should like to acknowledge correspondence with Nicoleta Voicu which
encouraged me to return to this topic after a number of years. Some results
from this paper were presented at a meeting in Torino in honour of Marco
Ferraris in June 2023, and at the International Summer School on Global
Analysis and Applications in Prešov in August 2023.

## Appendix: An example calculation

We consider the form $\omega=f^{i}_{\alpha m}dx^{m}\otimes\theta^{\alpha}_{i}$
where $p=q=1$ and the form is projectable to $J^{2}\pi$, so that the formula
is

$P_{\nabla}\omega=\sum_{r=0}^{\infty}\frac{(-1)^{r}(m-1)!}{(m+r)r!}\bigl{(}\mathsf{C}\circ
d_{\mathrm{h}\nabla}\bigr{)}^{r}\mathsf{C}\bigl{(}S_{\nabla}^{r+1}\omega\bigr{)}\,.$

We obtain

$d_{\mathrm{h}}\omega=(d_{l}f^{i}_{\alpha m})dx^{l}\wedge
dx^{m}\wedge\theta^{\alpha}_{i}+f^{i}_{\alpha m}dx^{l}\wedge
dx^{m}\wedge\theta^{\alpha}_{(il)}$

so that

$\displaystyle S_{\nabla}(d_{\mathrm{h}}\omega)$
$\displaystyle=(d_{l}f^{i}_{\alpha m})\partial_{i}\otimes dx^{l}\wedge
dx^{m}\wedge\theta^{\alpha}+f^{i}_{\alpha m}\Gamma^{k}_{il}\partial_{k}\otimes
dx^{l}\wedge dx^{m}\wedge\theta^{\alpha}$ $\displaystyle\qquad+f^{h}_{\alpha
m}\partial_{k}\otimes dx^{k}\wedge
dx^{m}\wedge\theta^{\alpha}_{h}+f^{k}_{\alpha m}\partial_{k}\otimes
dx^{h}\wedge dx^{m}\wedge\theta^{\alpha}_{h}\,,$
$\displaystyle\mathsf{C}S_{\nabla}(d_{\mathrm{h}}\omega)$
$\displaystyle=(d_{i}f^{i}_{\alpha
j})dx^{j}\wedge\theta^{\alpha}-(d_{j}f^{i}_{\alpha
i})dx^{j}\wedge\theta^{\alpha}+f^{i}_{\alpha
j}\Gamma^{k}_{ik}dx^{j}\wedge\theta^{\alpha}-f^{i}_{\alpha
k}\Gamma^{k}_{ij}dx^{j}\wedge\theta^{\alpha}$
$\displaystyle\qquad+m\,\omega-f^{i}_{\alpha
i}dx^{j}\wedge\theta^{\alpha}_{j}$ and $\displaystyle
S_{\nabla}^{2}(d_{\mathrm{h}}\omega)$
$\displaystyle=2\partial_{i}\otimes\partial_{l}\otimes\bigl{(}f^{i}_{\alpha
m}dx^{l}\wedge dx^{m}\wedge\theta^{\alpha}\bigr{)}\,,$
$\displaystyle\mathsf{C}S_{\nabla}^{2}(d_{\mathrm{h}}\omega)$
$\displaystyle=2m\partial_{i}\otimes\bigl{(}f^{i}_{\alpha
j}dx^{j}\wedge\theta^{\alpha}\bigr{)}-2\partial_{j}\otimes\bigl{(}f^{i}_{\alpha
i}dx^{j}\wedge\theta^{\alpha}\bigr{)}\,,$
$\displaystyle\tfrac{1}{2}d_{\mathrm{h}\nabla}\mathsf{C}S_{\nabla}^{2}d_{\mathrm{h}}\omega$
$\displaystyle=m\Gamma^{k}_{ih}\partial_{k}\otimes
dx^{h}\wedge\bigl{(}f^{i}_{\alpha
j}dx^{j}\wedge\theta^{\alpha}\bigr{)}+m\partial_{i}\otimes\bigl{(}(d_{k}f^{i}_{\alpha
j})dx^{k}\wedge dx^{j}\wedge\theta^{\alpha}\bigr{)}$
$\displaystyle\qquad+m\partial_{i}\otimes\bigl{(}f^{i}_{\alpha j}dx^{k}\wedge
dx^{j}\wedge\theta^{\alpha}_{k}\bigr{)}-\Gamma^{k}_{jh}\partial_{k}\otimes
dx^{h}\wedge\bigl{(}f^{i}_{\alpha i}dx^{j}\wedge\theta^{\alpha}\bigr{)}$
$\displaystyle\qquad-\partial_{j}\otimes\bigl{(}(d_{k}f^{i}_{\alpha
i})dx^{k}\wedge
dx^{j}\wedge\theta^{\alpha}\bigr{)}-\partial_{j}\otimes\bigl{(}f^{i}_{\alpha
i}dx^{k}\wedge dx^{j}\wedge\theta^{\alpha}_{k}\bigr{)}\,,$
$\displaystyle\tfrac{1}{2}\mathsf{C}d_{\mathrm{h}\nabla}\mathsf{C}S_{\nabla}^{2}(d_{\mathrm{h}}\omega)$
$\displaystyle=m\Gamma^{k}_{ik}f^{i}_{\alpha
j}dx^{j}\wedge\theta^{\alpha}-m\Gamma^{k}_{ij}dx^{j}\wedge f^{i}_{\alpha
k}\theta^{\alpha}$ $\displaystyle\qquad+m(d_{i}f^{i}_{\alpha
j})dx^{j}\wedge\theta^{\alpha}+m\omega-(d_{j}f^{i}_{\alpha
i})dx^{j}\wedge\theta^{\alpha}-f^{i}_{\alpha
i}dx^{j}\wedge\theta^{\alpha}_{j}\,.$

On the other hand,

$\displaystyle S_{\nabla}\omega$
$\displaystyle=\partial_{i}\otimes(f^{i}_{\alpha
m}dx^{m}\wedge\theta^{\alpha})\,,$ $\displaystyle\mathsf{C}S_{\nabla}\omega$
$\displaystyle=f^{i}_{\alpha i}\theta^{\alpha}\,,$ $\displaystyle
d_{\mathrm{h}}(\mathsf{C}S_{\nabla}\omega)$ $\displaystyle=(d_{j}f^{i}_{\alpha
i})dx^{j}\wedge\theta^{\alpha}+f^{i}_{\alpha
i}dx^{j}\wedge\theta^{\alpha}_{j}$

so that

$\displaystyle\tfrac{1}{2}\mathsf{C}d_{\mathrm{h}\nabla}\mathsf{C}S_{\nabla}^{2}(d_{\mathrm{h}}\omega)+d_{\mathrm{h}}(\mathsf{C}S_{\nabla}\omega)$
$\displaystyle=m\Gamma^{k}_{ik}f^{i}_{\alpha
j}dx^{j}\wedge\theta^{\alpha}-m\Gamma^{k}_{ij}dx^{j}\wedge f^{i}_{\alpha
k}\theta^{\alpha}$ $\displaystyle\qquad+m(d_{i}f^{i}_{\alpha
j})dx^{j}\wedge\theta^{\alpha}+m\omega\,.$

But from

$\displaystyle\mathsf{C}S_{\nabla}(d_{\mathrm{h}}\omega)$
$\displaystyle=(d_{i}f^{i}_{\alpha
j})dx^{j}\wedge\theta^{\alpha}+f^{i}_{\alpha
j}\Gamma^{k}_{ik}dx^{j}\wedge\theta^{\alpha}-f^{i}_{\alpha
k}\Gamma^{k}_{ij}dx^{j}\wedge\theta^{\alpha}+m\,\omega-
d_{\mathrm{h}}(\mathsf{C}S_{\nabla}\omega)$

we see that

$\tfrac{1}{2}\mathsf{C}d_{\mathrm{h}\nabla}\mathsf{C}S_{\nabla}^{2}(d_{\mathrm{h}}\omega)=m\,\mathsf{C}S_{\nabla}d_{\mathrm{h}}\omega-m(m-1)\omega+(m-1)d_{\mathrm{h}}\mathsf{C}S_{\nabla}\omega$

so that

$\omega=\biggl{(}\frac{1}{m-1}\mathsf{C}S_{\nabla}-\frac{1}{2m(m-1)}\mathsf{C}d_{\mathrm{h}\nabla}\mathsf{C}S_{\nabla}^{2}\biggr{)}d_{\mathrm{h}}\omega+d_{\mathrm{h}}\biggl{(}\frac{1}{m}\mathsf{C}S_{n}\omega\biggr{)}\,.$

## References

  * [1] I.M. Anderson: The variational bicomplex Technical Report, Utah State University 1989; available at http://www.math.usu.edu/~fg_mp/
  * [2] D. Betounes: Extension of the classical Cartan form Phys.Rev.D 29(4) (1984) 599–606
  * [3] D. Betounes: Global shift operators and the higher order calculus of variations J.Geom.Phys. 10(2) (1993) 185–201
  * [4] C. Carathéodory: Über die Variationsrechnung bei mehrfachen Integralen Acta Szeged Sect.Sci.Math. 4 (1929) 193–216
  * [5] M. Crampin, W. Sarlet and F. Cantrijn: Higher-order differential equations and higher-order Lagrangian mechanics Math.Proc.Camb.Phil.Soc. 99 (1986) 565–587
  * [6] M. Crampin and D.J. Saunders: The Hilbert–Carathéodory and Poincaré–Cartan forms for higher-order multiple-integral variational problems Houston J.Math. 30(3) (2004) 657–689
  * [7] M. Crampin and D.J. Saunders: On null Lagrangians Diff.Geom.Appl. 22 (2005) 131–146
  * [8] M. Crampin and D.J. Saunders: Homotopy operators for the variational bicomplex, representations of the Euler–Lagrange complex, and the Helmholtz–Sonin conditions Lobachvskii J.Math. 30(2) (2009) 107–123
  * [9] M. Ferraris and M. Francaviglia: Global formalisms in higher order calculus of variations In: Proc.Conf.Diff.Geom.Appl. 1983, (J.E. Purkyně Univ., Brno, 1984) 93–117
  * [10] P.L. García and M. Muñoz: On the geometrical structure of higher order variational calculus In: Modern developments in analytical mechanics Vol I, (Acta Acad.Sci.Taurinensis, 1982) 128–147
  * [11] M. Horák and I. Kolář: On the higher-order Poincaré–Cartan forms Czech.Math.J. 33(108) (1983) 467–475
  * [12] D. Krupka: A map associated to the Lepagian forms on the calculus of variations in fibred manifolds Czech.Math.J. 27(102) (1977) 114–118
  * [13] D. Krupka, O. Krupková and D.J. Saunders: The Cartan form and its generalizations in the calculus of variations Int.J.Geom.Meth.Mod.Phys. 7(4) (2010) 631–654
  * [14] P. Libermann: Introduction to the theory of semi-holonomic jets Arch.Math.(Brno) 33 (1997) 173–189
  * [15] P.J. Olver: Applications of Lie Groups to Differential Equations (Springer, 1986)
  * [16] P.J. Olver: Equivalence and the Cartan form Acta Appl.Math. 31 (1993) 99–136
  * [17] M. Palese, O. Rossi and F. Zanello: Geometric integration by parts and Lepage equivalents Diff.Geom.Appl. 81 (2022) 101866
  * [18] D.J. Saunders: An alternative approach to the Cartan form in Lagrangian field theories J.Phys.A.(Math.Gen.) 20 (1987) 339–349
  * [19] D.J. Saunders: The Geometry of Jet Bundles (Cambridge, 1989)
  * [20] F. Takens: A global version of the inverse problem of the calculus of variations J.Diff.Geom. 14 (1979) 543–562
  * [21] T. Tsujishita: On variation bicomplexes associated to differential equations Osaka J.Math. 19 (1982) 311–363
  * [22] W.M. Tulczyjew: The Euler–Lagrange resolution In: Lecture Notes in Mathematicsd 836: Differential geometric methods in mathematical physics, (Springer, 1980) 22–48
  * [23] Z. Urban and J. Volná: The fundamental Lepage form in two independent variables: a generalization using order-reducibility Mathematics 10(8) (2022) 1211
  * [24] A.M. Vinogradov: A spectral sequence associated with a non–linear differential equation, and algebro–geometric foundations of Lagrangian field theory with constraints Sov.Math.Dokl. 19 (1978) 144–148
  * [25] N. Voicu, S. Garoiu and B. Vasian: On the closure property of Lepage equivalents of Lagrangian Diff.Geom.Appl. 81 (2022) 101852

D.J. Saunders
Lepage Research Institute
Email<EMAIL_ADDRESS>