Variational Sequences, Representation Sequences and Applications in Physics

Variational Sequences, Representation Sequences
and Applications in Physics\star\star$\star$This paper is a contribution to
the Special Issue on Analytical Mechanics and Differential Geometry in honour
of Sergio Benenti. The full collection is available at
http://www.emis.de/journals/SIGMA/Benenti.html

Marcella PALESE ${}^{{\dagger}^{1}}$, Olga ROSSI
${}^{{\dagger}^{2}{\dagger}^{3}}$, Ekkehart WINTERROTH
${}^{{\dagger}^{1}{\dagger}^{4}}$ and Jana MUSILOVÁ ${}^{{\dagger}^{5}}$

M. Palese, O. Rossi, E. Winterroth and J. Musilová

${}^{{\dagger}^{1}}$ Department of Mathematics, University of Torino, via C.
Alberto 10, 10123 Torino, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>
${}^{{\dagger}^{2}}$ Department of Mathematics, Faculty of Science, University
of Ostrava,
${}^{{\dagger}^{2}}$ Ostrava, Czech Republic<EMAIL_ADDRESS>
${}^{{\dagger}^{3}}$ Department of Mathematics and Statistics, La Trobe
University, Melbourne, Australia

${}^{{\dagger}^{4}}$ Lepage Research Institute, 783 42 Slatinice, Czech
Republic

${}^{{\dagger}^{5}}$ Institute of Theoretical Physics and Astrophysics,
Masaryk University Brno, Czech Republic<EMAIL_ADDRESS>
Received September 22, 2015, in final form April 26, 2016; Published online
May 02, 2016

This paper is a review containing new original results on the finite order
variational sequence and its different representations with emphasis on
applications in the theory of variational symmetries and conservation laws in
physics.

fibered manifold; jet space; Lagrangian formalism; variational sequence;
variational derivative; cohomology; symmetry; conservation law

55N30; 55R10; 58A12; 58A20; 58E30; 70S10

##  1 Introduction

After Élie Cartan, who in 1922 introduced differential geometric methods into
the calculus of variations using the celebrated Cartan form to formulate the
variational principle [10], Théophile Lepage in his pioneer work [73, 74]
initiated the study of the calculus of variations for field theories (multiple
integrals) as a theory of differential forms and their exterior differential
modulo contact forms (congruences). The main advantage of such an approach
evidently is a possibility of framing the calculus of variations within the de
Rham sequence of differential forms on a manifold and thus using the de Rham
cohomology as an effective tool to study relations between the image and
kernel of various differential operators appearing in the calculus of
variations.

The attempt to understand global structures of the calculus variations
motivated the use of $C$-spectral sequences, and gave rise to the
constructions of variational sequences and bicomplexes. This direction was
initiated in 1950’s by Dedecker [14, 15, 16, 17, 18], and continued in the
work of Horndeski [41], Gelfand and Dikii [31], Olver and Shakiban [79],
Dedecker and Tulczyjew [19], Tulczyjew [96, 97, 98], Takens [93], Anderson
[2], Anderson and Duchamp [3], Dickey [20], Vinogradov [99, 100, 101], and
others. A main motivation for this research was

  * •

to find solution of the inverse problem of the calculus of variations which
means to find local and global constraints for a system of ordinary or partial
differential equations to come from a variational principle as Euler–Lagrange
equations, and to construct a Lagrangian,

  * •

to determine the local and global structure of variationally trivial
Lagrangians, i.e., Lagrangians giving rise to identically zero Euler–Lagrange
expressions (the kernel of the Euler–Lagrange operator),

  * •

to understand the form and origin of different “canonically looking”
expressions appearing in the calculus of variations, as for example

$\displaystyle\frac{\partial L}{\partial y^{\sigma}}-d_{j}\frac{\partial
L}{\partial y^{\sigma}_{j}}+d_{j}d_{k}\frac{\partial L}{\partial
y^{\sigma}_{jk}}-\cdots,\qquad\frac{\partial L}{\partial
y^{\sigma}_{p}}-d_{j}\frac{\partial L}{\partial
y^{\sigma}_{jp}}+d_{j}d_{k}\frac{\partial L}{\partial
y^{\sigma}_{jkp}}-\cdots,$ $\displaystyle\frac{\partial E_{\sigma}}{\partial
q^{\nu}_{j}}-\sum_{k=j}^{s}(-1)^{k}\binom{k}{j}\frac{d^{k-j}}{dt^{k-j}}\frac{\partial
E_{\nu}}{\partial
q^{\sigma}_{k}},\qquad\sum_{l=0}^{s-j-1}(-1)^{j+l}\binom{j+l}{l}\frac{d^{l}}{dt^{l}}\frac{\partial
E_{\sigma}}{\partial q^{\nu}_{j+l+1}},$

  * •

to understand local and global phenomena typical for variational problems
given by local data, as, for instance examples of global Euler–Lagrange forms
which do not admit a global Lagrangian; the best known examples are the
Galilean relativistic mechanics, or Chern–Simons field theories (where a
global Lagrangian exists but is not gauge-invariant).

The theory was first formulated on infinite order jets. After several attempts
to obtain an alternative construction using finite order structures (see,
e.g., [3]), the finite order variational sequence was discovered by Krupka 25
years ago [49]. The inspiration originated from the Lepage’s idea of a
‘congruence’, indicating that there should be a close relationship between the
Euler–Lagrange operator and the exterior derivative of differential forms, and
resulted in the concept of a finite order (cohomological) exact sequence of
forms, in which the Euler–Lagrange mapping is included as one arrow.

Compared with the other approaches, the concept of Krupka’s variational
sequence is conceptually very simple due to the following reasons:

  * •

it uses only finite order jets,

  * •

it appears as a quotient sequence of the de Rham sequence.

This makes it quite easily accessible to potential users, since factorization
is a natural, straightforward, commonly understood, and well-mastered
technique. Moreover, the factorization procedure can be incorporated into
intrinsic operators which allow to avoid tedious and technically difficult
calculations.

It should be stressed, however, that here is an additional benefit of the
setting on finite order jets: Namely, it provides precise results on the order
of the local and global objects of interest.

The aim of this paper is to provide the reader a comprehensive source of the
current theory of the finite order variational sequences as it stands 25 years
after its beginnings. While for the variational bicomplex theory there is a
standard reference the (unpublished) book by Anderson [2], the results on the
finite order variational sequence up to now have not been collected, revised
and completed in a self-contained article. However, our aim is not merely to
provide a revised exposition of known results. Apart from putting known
results into a unified treatment, we add a number of original results in order
to develop the theory and make it complete. Our aim is also to stimulate the
use and applications of the variational sequence in physics. As a motivation
we mention a few applications, some of which are interesting and not widely
known or even are original here.

The paper is organized as follows: Section 2 introduces the variational
sequence as a quotient sequence of the de Rham sequence, developed by Krupka
[49, 51, 52, 53] (see also [36, 37, 54, 55]). Section 3 is the core of the
paper, and deals with the problem of representation of the variational
sequence by differential forms. In the variational sequence, the objects are
classes of local differential forms. However (as pointed out already by
Krupka) the classes can be represented by (global) differential forms:
Lagrangians or Lepage $n$-forms (where $n$ is the number of independent
variables in the variational problem), Euler–Lagrange forms, Helmholtz forms.
This property initiated the search of representation operators and
representations with help of differential forms. We present two different
representation sequences: the Takens representation sequence based on the so-
called interior Euler operator [2] which assigns to every class in the $q$-th
column of the variational sequence a source form of degree $q$ (a Lagrangian
if $q=n$, a dynamical form if $q=n+1$, a Helmholtz-like form if $q=n+2$), and
the Lepage representation sequence. The latter is a sequence of Lepage forms
(Lepage equivalents of source forms), such that the morphisms are just the
exterior derivatives. The existence of such a sequence clarifies the
relationship of the variational morphisms to the exterior derivative operator.
Moreover, it transfers the questions on the structure of the kernel and image
of all the variational morphisms just to application of the Poincaré lemma. A
complete representation of the variational sequence by source forms (of any
degree) was given by Musilová and Krbek [43, 44, 45] and independently by
Krupka and his collaborators [61, 63, 103]. Next, in Section 3 we recall the
concept of Lepage form ([46, 61, 64, 72] and introduce the Lepage
representation of the variational sequence. As a new result concerning Lepage
$n$-forms, we present an intrinsic formula for the Cartan form (Theorem 3.8,
formula (3.6)). Remarkably, our formula explicitly shows that (and explains
why) the higher-order Cartan form generically is not global (this fact was a
surprise when discovered and demonstrated by a counterexample in 1980’s [40],
and up to now has not been well understood). In the subsection on Lepage forms
of higher degrees (and any order) we present for the first time properties of
general Lepage forms, and namely we give a general intrinsic formula for a
Lepage $(n+k)$-form of any order. We also find a higher degree generalization
of the famous Cartan form. The techniques based on the use of the interior
Euler operator and the residual operator not only lead to elegant formulas and
easy proofs; they also enable to avoid lengthy, tedious, and sometimes even
impossible calculations in coordinates. The results then help us to find a
complete correct Lepage representation of the variational sequence. The last
remark in Section 3 concerns the variational order of objects in the
variational sequence. We explain how due to the finite order setting it is
possible to make (local and global) conclusions on a precise order of
Lagrangians and other objects in the variational sequence, distinguishing it
from alternative constructions on infinite order jets.

In Section 4 we are concerned with the Lie derivative in the variational
sequence. We explore the important property that the Lie derivative of
differential forms with respect to prolongations of projectable vector fields
preserves the contact structure. This suggests that a Lie derivative of
classes of forms, a variational Lie derivative, can be correctly defined as
the equivalence class of the standard Lie derivative of forms and represented
by forms. New results in this section are then contained in subsequent
theorems generalizing the first variation formula to forms of any degree.

The last section of the paper is devoted to selected applications of the
variational sequence theory. We recall the fundamental well-known applications
to the inverse variational problem and to variationally trivial Lagrangians.
Apart from that we emphasize some other possible applications which have not
yet been fully explored – like, for instance, properties of Helmholtz forms.
We include also examples of variational problems defined by local data, and on
symmetries and conservation laws, emphasizing cohomological questions and
local and global phenomena.

For a reader interested in a wider context of the theory of variational
sequences, we refer to the survey works [54, 102], and to the book [55].

##  2 The variational sequence

###  2.1 Jet bundles and contact forms

Fibred manifolds and their jet prolongations represent a convenient
mathematical framework for mechanics and field theories, suitable for
investigation of Lagrangian systems of different orders, and one or many
independent variables within a sufficiently general and unified geometric
framework. For an excellent exposition of the jet bundle geometry we refer to
the book [91].

In what follows, we shall consider a smooth fibred manifold
$\pi\colon\boldsymbol{Y}\to\boldsymbol{X}$, with $\dim\boldsymbol{X}=n$ and
$\dim\boldsymbol{Y}=n+m$, and its $r$-jet prolongations $\pi_{r}\colon
J^{r}\boldsymbol{Y}\to\boldsymbol{X}$, where $r\geq 1$. For $n=1$ the manifold
$\boldsymbol{Y}$ is a space of events for mechanical systems of $m$ degrees of
freedom, and local sections of $\pi$ are graphs of curves, so usually
$\boldsymbol{X}={\mathbb{R}}$ (meaning time, or just a parameter for the
curves). If $n>1$ then $\boldsymbol{X}$ has the meaning of a physical space or
spacetime, and local sections of $\pi$ are physical fields over the manifold
$\boldsymbol{X}$. Then $J^{r}\boldsymbol{Y}$ is a manifold of points
$j^{r}_{x}\gamma$, the equivalence classes of local sections $\gamma$ of $\pi$
with the same value at $x$ and the same partial derivatives at $x$ up to the
order $r$. Thus $J^{r}\boldsymbol{Y}$ serves as a domain for functions,
differential forms, vector fields, and other objects depending on higher
derivatives (up to the $r$-th order).

Due to the affine bundle structure of $\pi_{r+1,r}\colon
J^{r+1}\boldsymbol{Y}\to J^{r}\boldsymbol{Y}$, we have a natural splitting
$J^{r}\boldsymbol{Y}\times_{J^{r-1}\boldsymbol{Y}}T^{*}J^{r-1}\boldsymbol{Y}=J^{r}\boldsymbol{Y}\times_{J^{r-1}\boldsymbol{Y}}(T^{*}\boldsymbol{X}\oplus
V^{*}J^{r-1}\boldsymbol{Y})$ which induces natural splittings into horizontal
and vertical parts of projectable vector fields, forms, and of the exterior
differential on $J^{r}\boldsymbol{Y}$.

A differential $q$-form $\omega$ on $J^{r}\boldsymbol{Y}$ is called contact if
$J^{r}\gamma^{*}\omega=0$ for every local section $\gamma$ of $\pi$. Contact
forms on $J^{r}\boldsymbol{Y}$ form an ideal in the exterior algebra, called
the contact ideal of order $r$. The contact ideal is generated by contact
$1$-forms and their exterior derivatives [50]. A differential $q$-form
$\omega$ on $J^{r}\boldsymbol{Y}$ is called horizontal, or $0$-contact, if
$\xi\rfloor\omega=0$ for every vector field $\xi$ on $J^{r}\boldsymbol{Y}$,
vertical with respect to the projection onto $\boldsymbol{X}$.

From the definitions it follows that every $q$-form for $q>n$ is contact, and
that every form $\omega$ on $J^{r}\boldsymbol{Y}$, if lifted to
$J^{r+1}\boldsymbol{Y}$, is a sum of a unique horizontal and contact form.
This splitting can be further refined if the concept of a $k$-contact form for
$k\geq 1$ is introduced. A contact form $\omega$ is called $k$-contact if for
every vertical $\xi$ the contraction $\xi\rfloor\omega$ is $(k-1)$-contact.
Then, as proved in [48], every $q$-form $\omega$ on $J^{r}\boldsymbol{Y}$,
$q\geq 1$, admits the unique decomposition

$\displaystyle\quad\pi^{*}_{r+1,r}\omega=\sum_{k=0}^{q}p_{k}\omega$ (2.1)

into a sum of $k$-contact forms, $0\leq k\leq q$. For the horizontal
($0$-contact) operator often the notation $h=p_{0}$ is used. The form
$p_{k}\omega$ is called the $k$-contact component of $\omega$. Note that if
$q\geq n+1$ then $\omega$ is contact, and it is at least $(q-n)$-contact,
i.e., the contact components $p_{0},p_{1},\dots,p_{q-n-1}$ of $\omega$ are
equal to zero. If, moreover, $p_{q-n}\omega=0$, we speak about a strongly
contact form [49].

An important special case of the decomposition (2.1) concerns closed forms and
their local primitives, obtained due to the Poincaré lemma, by means of the
Poincaré homotopy operator $\mathcal{P}$. Since every form $\omega$ can be
locally expressed as $\omega=\mathcal{P}d\omega+d\mathcal{P}\omega$, equation
$\omega=d\rho$ has a local solution $\rho=\mathcal{P}\omega$. Applying,
however, the decomposition into contact components we can see that there
arises a modified homotopy operator, denoted by $\mathcal{A}$, and called the
contact homotopy operator [48], which is adapted to the decomposition (2.1)
and defined by $\mathcal{A}p_{0}\omega=0$,
$\mathcal{A}p_{k}\omega=p_{k-1}\mathcal{P}\omega$, $k\geq 1$. It satisfies
$\pi_{r+1,r}^{*}\omega=\mathcal{A}d(\pi_{r+1,r}^{*}\omega)+d\mathcal{A}(\pi_{r+1,r}^{*}\omega)$,
and moreover, if (locally) $\omega=d\rho$, then

$\displaystyle\quad\pi^{*}_{r+1,r}\rho=\mathcal{A}(\pi_{r+1,r}^{*}\omega)=\sum_{k=0}^{q}\mathcal{A}p_{k+1}\omega,$

hence $p_{k}\rho=\mathcal{A}p_{k+1}\omega$. Compared to $\mathcal{P}$, the
operator $\mathcal{A}$ concerns vertical curves (curves in the fibres over
$\boldsymbol{X}$) only.

Notice that the decomposition (2.1) applied to $\omega=d\rho$ induces a
splitting of the exterior derivative $d$ to $\pi_{r+1,r}^{*}d=d_{H}+d_{V}$
where the horizontal derivative $d_{H}=hd$, and the vertical derivative
$d_{V}$ is the contact part of $\pi_{r+1,r}^{*}d$.

The splitting of vector fields then arises with help of the total derivative
operator. If we denote $(x^{i},y^{\sigma})$ local fibred coordinates on
$\boldsymbol{Y}$, and $(x^{i},y^{\sigma}_{J})$ where $J$ is a multiindex,
$0\leq|J|\leq r$, the associated coordinates on $J^{r}\boldsymbol{Y}$, then
the $i$-th total (formal) derivative takes the form

$\displaystyle d_{i}=\frac{\partial}{\partial
x^{i}}+\sum_{|J|=0}^{r}y^{\sigma}_{Ji}\frac{\partial}{\partial
y^{\sigma}_{J}}.$

Considered as a vector field along the projection $\pi_{r+1,r}$, the total
derivative induces a splitting of vector fields to a horizontal and vertical
part as follows: Letting

$\displaystyle\xi=\xi^{i}\frac{\partial}{\partial
x^{i}}+\sum_{|J|=0}^{r}\xi^{\sigma}_{J}\frac{\partial}{\partial
y^{\sigma}_{J}}$

be a vector field on $J^{r}\boldsymbol{Y}$, in order to obtain the splitting
$\xi=\xi_{H}+\xi_{V}$, we put

$\displaystyle\xi_{H}=\xi^{i}d_{i},\qquad\xi_{V}=\xi-\xi_{H}=\sum_{|J|=0}^{r}\big{(}\xi^{\sigma}_{J}-y^{\sigma}_{Ji}\xi^{i}\big{)}\frac{\partial}{\partial
y^{\sigma}_{J}}.$

Note that both $\xi_{H}$ and $\xi_{V}$ are vector fields along the projection
$\pi_{r+1,r}$ rather than ‘ordinary’ vector fields on $J^{r}\boldsymbol{Y}$.
If $\Xi$ is a projectable vector field on $\boldsymbol{Y}$, we write
$J^{r}\Xi_{V}=(J^{r}\Xi)_{V}$ (this can be viewed as a definition of
prolongation of $\Xi_{V}$ which is a vector field along the projection
$\pi_{1,0}$).

Recall that ${\mathcal{C}}(\pi_{r})$, the contact distribution or Cartan
distribution of order $r$, is locally annihilated by contact $1$-forms

$\displaystyle\omega^{\sigma}_{J}=dy^{\sigma}_{J}-y^{\sigma}_{Jj}dx^{j},\qquad
1\leq\sigma\leq m,$

or, equivalently, locally generated by the vector fields $d_{i}$ along
$\pi_{r,r-1}$ and $\partial/\partial y^{\sigma}_{J}$ where $|J|=r$. The Cartan
distribution on $J^{r}\boldsymbol{Y}$ is not completely integrable, hence the
ideal generated by contact $1$-forms of order $r$ is not closed. Its closure
is then the contact ideal of order $r$ (the ideal of all contact forms on
$J^{r}\boldsymbol{Y}$).

In what follows, we shall need on $J^{r}\boldsymbol{Y}$ the ideal generated by
lifts of contact $1$-forms of order $1$, whose elements take the form
$\pi_{r,1}^{*}\omega\land\eta$ where $\omega\in{\mathcal{C}}^{0}(\pi_{1})$
(the annihilator of $\mathcal{C}(\pi_{1})$) and $\eta$ is of order $r$. Since
locally they are expressed as $\omega^{\sigma}\land\eta_{\sigma}$, where
$\omega^{\sigma}=dy^{\sigma}-y^{\sigma}_{j}dx^{j}$, $1\leq\sigma\leq m$, are
the basic local contact $1$-forms of order $1$, we say that forms belonging to
this ideal are $\omega^{\sigma}$-generated.

In the calculus of variations and the theory of differential equations on
fibred manifolds, the fundamental role is played by the sheaf
$\Lambda^{r}_{n,\boldsymbol{X}}$ of horizontal ($0$-contact) $n$-forms on
$J^{r}\boldsymbol{Y}$, the elements of which are called Lagrangians of order
$r$, and the sheaf $\Lambda^{r}_{n+1,1,\boldsymbol{Y}}$ of $1$-contact
$(n+1)$-forms on $J^{r}\boldsymbol{Y}$, horizontal with respect to the
projection onto $\boldsymbol{Y}$, the elements of which are called dynamical
forms of order $r$. By means of variation of the action defined by a
Lagrangian $\lambda$ one obtains a distinguished dynamical form $E_{\lambda}$,
called the Euler–Lagrange form of $\lambda$ [46]; the components of
$E_{\lambda}$ in every fibred chart are the Euler–Lagrange expressions of the
Lagrangian $\lambda$. Remarkably, if $\lambda$ is of order $r$ then
$E_{\lambda}$ is of order $\leq 2r$.

The mapping assigning to every Lagrangian its Euler–Lagrange form is called
the Euler–Lagrange mapping.

A dynamical form $E$ is called globally variational if there exists a
Lagrangian $\lambda$ such that (possibly up to a jet projection)
$E=E_{\lambda}$. $E$ is called locally variational if every point in the
domain of $E$ has a neighbourhood $U$ where $E$ is variational [48].

Dynamical forms are often called source forms, as originally introduced by
Takens [93]. However, we shall follow Krupková and Prince [70] who extended
the concept of source form to forms of all degrees $q=n+k$ where $k\geq 1$:

######  Definition 2.1.

Let $k\geq 1$. By a source form of degree $n+k$ we shall mean a
$\omega^{\sigma}$-generated $k$-contact $(n+k)$-form.

For $k=1$, dynamical forms are $\omega^{\sigma}$-generated $1$-contact
$(n+1)$-forms, hence, indeed, source forms of degree $n+1$. Source forms of
degree $n+2$ are called Helmholtz-like forms [69]. For the calculus of
variations the most important examples of source forms of degree $n+1$ are
Euler–Lagrange forms, and of source forms of degree $n+2$ the Helmholtz forms.

###  2.2 The variational sequence

Now we recall the finite order variational sequence as introduced by Krupka in
[49]. The concept uses the sheaf theory (standard references are, e.g., [8,
104]). However, for a first understanding, a concrete use of the variational
sequence, and calculations it is not necessary to be familiar with all the
rather difficult mathematical theory. In place of a ‘sheaf’ of forms in this
context one can roughly imagine a family of modules of local differential
forms defined on the open subsets of the corresponding manifold.

In what follows, we denote by ${\mathbb{R}}_{\boldsymbol{Y}}$ the constant
sheaf over ${\mathbb{R}}$, and by $\Lambda^{r}_{q}$ the sheaf of $q$-forms on
$J^{r}\boldsymbol{Y}$. We let $\Lambda^{r}_{0,c}=\\{0\\}$, and
$\Lambda^{r}_{q,c}$ be the sheaf of contact $q$-forms, if $q\leq n$, or the
sheaf of strongly contact $q$-forms, if $q>n$, on $J^{r}\boldsymbol{Y}$. We
set

$\displaystyle\Theta^{r}_{q}=\Lambda^{r}_{q,c}+\big{(}d\Lambda^{r}_{q-1,c}\big{)},$

where $d\Lambda^{r}_{q-1,c}$ is the image of $\Lambda^{r}_{q-1,c}$ by the
exterior derivative $d$, and $(d\Lambda^{r}_{q-1,c})$ denotes the sheaf
generated by the presheaf $d\Lambda^{r}_{q-1,c}$. We note that
$\Theta^{r}_{q}=0$ for $q>\operatorname{corank}{\cal C}(\pi_{r})$.

The subsequence

$\displaystyle 0\to\Theta^{r}_{1}\to\Theta^{r}_{2}\to\Theta^{r}_{3}\to\cdots$

of the de Rham sequence

$\displaystyle
0\to{\mathbb{R}}_{\boldsymbol{Y}}\to\Lambda^{r}_{0}\to\Lambda^{r}_{1}\to\Lambda^{r}_{2}\to\Lambda^{r}_{3}\to\cdots$

is an exact sequence of soft sheaves. The quotient sequence

$\displaystyle
0\to{\mathbb{R}}_{\boldsymbol{Y}}\to\Lambda^{r}_{0}\to\Lambda^{r}_{1}/\Theta^{r}_{1}\to\Lambda^{r}_{2}/\Theta^{r}_{2}\to\Lambda^{r}_{3}/\Theta^{r}_{3}\to\cdots$
(2.2)

is called the variational sequence of order $r$. The main theorem due to
Krupka [49] then states that the variational sequence is an acyclic resolution
of the constant sheaf ${\mathbb{R}}_{\boldsymbol{Y}}$ (meaning that the
sequence is locally exact with the exception of
${\mathbb{R}}_{\boldsymbol{Y}}$). Hence, due to the abstract de Rham theorem,
the cohomology groups of the cochain complex of global sections of the
variational sequence are identified with the de Rham cohomology groups
$H^{q}_{\rm dR}\boldsymbol{Y}$ of the manifold $\boldsymbol{Y}$.

It should be stressed that objects in the variational sequence are elements of
the quotient sheaves $\Lambda^{r}_{q}/\Theta^{r}_{q}$, i.e., they are
equivalence classes of local $r$-th order differential $q$-forms. We denote by
$[\rho]\in\Lambda^{r}_{q}/\Theta^{r}_{q}$ the class of
$\rho\in\Lambda^{r}_{q}$. Morphisms in the variational sequence are then by
construction quotients of the exterior derivative $d$. We denote

$\displaystyle{\cal E}_{q}\colon\
\Lambda^{r}_{q}/\Theta^{r}_{q}\to\Lambda^{r}_{q+1}/\Theta^{r}_{q+1},$

so that

$\displaystyle{\cal E}_{q}([\rho])=[d\rho],$

and ${\cal E}_{q}([\rho])=[d\rho]=0$ means that there exists
$[\eta]\in\Lambda^{r}_{q-1}/\Theta^{r}_{q-1}$ such that $[\rho]={\cal
E}_{q-1}([\eta])=[d\eta]$. If, moreover, $H^{q}_{\rm
dR}\boldsymbol{Y}=\\{0\\}$ then the class $[\eta]$ has a global
representative.

The meaning of this abstract construction for the calculus of variations
becomes clear in view of the following theorem:

######  Theorem 2.2 (Krupka [49, 54]).

The sheaf $\Lambda^{r}_{n}/\Theta^{r}_{n}$ is isomorphic with a subsheaf of
the sheaf of Lagrangians $\Lambda^{r+1}_{n,\boldsymbol{X}}$, the sheaf
$\Lambda^{r}_{n+1}/\Theta^{r}_{n+1}$ is isomorphic with a subsheaf of the
sheaf of source forms $\Lambda^{2r+1}_{n+1,1,\boldsymbol{Y}}$, and the
quotient mapping ${\cal
E}_{n}\colon\Lambda^{r}_{n}/\Theta^{r}_{n}\to\Lambda^{r}_{n+1}/\Theta^{r}_{n+1}$
is the Euler–Lagrange mapping.

By this theorem, the elements of the quotient sheaf
$\Lambda^{r}_{n}/\Theta^{r}_{n}$ have the meaning of Lagrangians. The kernel
of ${\cal E}_{n}$ then consists of null-Lagrangians (variationally trivial
Lagrangians), and its image, which, by exactness is the kernel of the next
morphism,

$\displaystyle{\cal E}_{n+1}\colon\
\Lambda^{r}_{n+1}/\Theta^{r}_{n+1}\to\Lambda^{r}_{n+2}/\Theta^{r}_{n+2},$

represents variational equations. Therefore ${\cal E}_{n+1}$ is called
Helmholtz mapping. The name refers to the famous “Helmholtz conditions” [39],
necessary and sufficient conditions for differential equations to be
variational (as they stand). Remarkably, the Helmholtz morphism, discovered
within the variational sequence theory, has not been known in the classical
calculus of variations.

The above theorem also suggests the idea that all classes in the variational
sequence could be representable by differential forms with a clear place in
the calculus of variations. So far there have been noticed two such
representation sequences: the “source form representation” which we call
Takens representation, and the “Lepage form representation”. In the former the
sheaves contain familiar differential forms which appear in the calculus of
variations, like Lagrangians and Euler–Lagrange forms, and their
generalizations, like, for example, Helmholtz forms. The morphisms are then
the corresponding variational mappings, like, e.g., the Euler–Lagrange mapping
sending Lagrangians to Euler–Lagrange forms, or the Helmholtz mapping, sending
dynamical forms to Helmholtz forms. The latter representation has as objects
Lepage forms, which are extensions of the “familiar forms” mentioned above by
higher contact components in such a way that the morphisms are the exterior
derivatives, so that in the Lepage form representation the variational
sequence “morally” (up to the orders of individual subsheaves) becomes a
subsequence of the de Rham sequence. We shall deal with the representation
sequences below.

In what follows, we shall denote

$\displaystyle{\cal V}^{r}_{0}=\Lambda^{r}_{0},\qquad{\cal
V}^{r}_{q}=\Lambda^{r}_{q}/\Theta^{r}_{q},$

and the variational sequence of order $r$ will be shortly written in the form

$\displaystyle 0\to{\mathbb{R}}_{\boldsymbol{Y}}\to{\cal V}^{r}_{*}.$

##  3 Representation sequences

###  3.1 The interior Euler operator

Let us introduce an operator which plays an essential role in the
representation theory for the variational sequence. It was introduced to the
calculus of variations within the variational bicomplex theory in [2, 4], and
adapted to the finite order situation of the variational sequence in [43, 44,
45, 61]. This operator, called interior Euler operator, and denoted by
$\mathcal{I}$, reflects in an intrinsic way the procedure of getting a
distinguished representative of a class
$[\rho]\in\Lambda^{r}_{q}/\Theta^{r}_{q}$ for $q>n$ by applying to $\rho$ the
operator $p_{q-n}$ and a factorization by $\Theta^{r}_{q}$.

Let $(x^{i},y^{\sigma})$ denote local fibred coordinates on $\boldsymbol{Y}$,
and $(x^{i},y^{\sigma}_{J})$ the associated coordinates on
$J^{r}\boldsymbol{Y}$, where $J$ is a multiindex, $0\leq|J|\leq r$,
$J=(j_{1},\dots,j_{p})$ with $1\leq j_{1}\leq j_{2}\leq\cdots\leq j_{p}\leq n$
and $p\leq r$.

Let $k\geq 1$ and $q=n+k$. Recall that if $\rho\in\Lambda^{r}_{n+k}$, we have

$\displaystyle\pi_{r+1,r}^{*}\rho=p_{k}\rho+p_{k+1}\rho+\cdots+p_{k+n}\rho.$

Following [45] we set

$\displaystyle\mathcal{I}(\rho)=\frac{1}{k}\omega^{\sigma}\wedge\sum_{|J|=0}^{r}(-1)^{|J|}d_{J}\big{(}\partial/\partial
y^{\sigma}_{J}\rfloor p_{k}\rho\big{)},$ (3.1)

where $d_{J}=d_{j_{1}}\dots d_{j_{p}}$ for $|J|=p$. It can be shown by means
of the partition of unity arguments that this formula defines a global form
$\mathcal{I}(\rho)$.

The operator $\mathcal{I}\colon\Lambda^{r}_{n+k}\to\Lambda^{2r+1}_{n+k}$ has
the following properties:

  1. (1)

for every $\rho\in\Lambda^{r}_{n+k}$, $\mathcal{I}(\rho)$ is a source form of
degree $n+k$,

  2. (2)

$\mathcal{I}(p_{k}\rho)=\mathcal{I}(\rho)$,

  3. (3)

$\mathcal{I}(\rho)$ belongs to the same class as $\pi_{2r+1,r}^{*}\rho$, i.e.,
$\pi_{{2r+1},{r}}^{*}\rho-\mathcal{I}(\rho)\in{\Theta}_{n+k}^{2r+1}$,

  4. (4)

$\mathcal{I}^{2}=\mathcal{I}$, up to a canonical projection; precisely,
$\mathcal{I}^{2}(\rho)=\pi_{{4r+3},{2r+1}}^{*}\mathcal{I}(\rho)$,

  5. (5)

$\operatorname{Ker}\mathcal{I}={\Theta}_{n+k}^{r}$.

By construction, $\mathcal{I}(\rho)$ is a $k$-contact form, so there is a
question about the difference between $\mathcal{I}(\rho)$ and $p_{k}\rho$. It
turns out that the difference can be expressed intrinsically by means of an
operator $\mathcal{R}$ which we shall call residual operator, as follows

$\displaystyle p_{k}\rho=\mathcal{I}(\rho)+p_{k}dp_{k}\mathcal{R}(\rho)$ (3.2)

(up to appropriate canonical projections). By the properties (2) and (4) of
$\mathcal{I}$ we can see that

$\displaystyle\mathcal{I}(p_{k}dp_{k}\mathcal{R}(\rho))=0.$ (3.3)

${\cal{R}}(\rho)$ is by definition a local strongly contact $(n+k-1)$-form.
Note that, contrary to ${\cal{I}}(\rho)$, the form ${\cal{R}}(\rho)$ need not
be unique and need not be global.

Moreover, $\mathcal{I}(\rho)$ is a source form, so there is a question about
its relationship with other source forms in the same class $[\rho]$. To
clarify this it is useful to introduce the concept of a canonical source form
as a source form $\rho$ such that (up to projection) $\rho=\mathcal{I}(\rho)$
[70].

######  Proposition 3.1.

Every source form is equivalent with a canonical source form $($of possibly
higher order$)$. More precisely, letting
$\rho=\omega^{\sigma}\land\eta_{\sigma}$ be a source $(n+k)$-form it holds
$($up to projection$)$

$\displaystyle\rho=k\mathcal{I}(\rho)-(k-1)\omega^{\sigma}\land\mathcal{I}(\eta_{\sigma})$

and

$\displaystyle\rho-\mathcal{I}(\rho)=p_{k}dp_{k}\mathcal{R}(\rho)=\frac{k-1}{k}\omega^{\sigma}\land
p_{k-1}dp_{k-1}\mathcal{R}(\eta_{\sigma}).$

As a consequence, for $k\geq 2$,

$\displaystyle\mathcal{I}(\omega^{\sigma}\land\eta_{\sigma})=\mathcal{I}(\omega^{\sigma}\land\mathcal{I}(\eta_{\sigma})).$

###### Proof.

The first assertion is just the property (3) of $\mathcal{I}$. Next two
formulas easily follow from the definitions of $\mathcal{I}$ and $\mathcal{R}$
and the fact that $\rho$ is the source form, i.e.,
$\pi_{r+1,r}^{*}\rho=p_{k}\rho$. Indeed, for
$\rho=\omega^{\sigma}\land\eta_{\sigma}$ we have

$\displaystyle\mathcal{I}(\omega^{\sigma}\land\eta_{\sigma})=\frac{1}{k}\omega^{\sigma}\land\sum_{|J|=0}^{r}(-1)^{|J|}d_{J}(\partial/\partial
y^{\sigma}_{J}\rfloor(\omega^{\nu}\land\eta_{\nu}))$
$\displaystyle\hphantom{\mathcal{I}(\omega^{\sigma}\land\eta_{\sigma})}{}=\frac{1}{k}\omega^{\sigma}\\!\land\\!\bigg{(}\eta_{\sigma}-\omega^{\nu}\land\\!\sum_{|J|=0}^{r}\\!(-1)^{|J|}d_{J}(\partial/\partial
y^{\sigma}_{J}\rfloor\eta_{\nu})\bigg{)}=\frac{1}{k}\omega^{\sigma}\land\eta_{\sigma}+\frac{k\\!-\\!1}{k}\omega^{\sigma}\land\mathcal{I}(\eta_{\sigma}).$

This formula then yields

$\displaystyle
p_{k}dp_{k}\mathcal{R}(\rho)=\rho-\mathcal{I}(\rho)=\frac{k-1}{k}\omega^{\sigma}\land\bigl{(}\eta_{\sigma}-\mathcal{I}(\eta_{\sigma})\bigr{)}=\frac{k-1}{k}\omega^{\sigma}\land
p_{k-1}dp_{k-1}\mathcal{R}(\eta_{\sigma}),$

as desired.

Finally, applying $\mathcal{I}$ to the first formula we get
$(k-1)\mathcal{I}(\rho)=(k-1)\mathcal{I}(\omega^{\sigma}\land\mathcal{I}(\eta_{\sigma}))$,
which provides the last formula for $k\geq 2$. ∎

For dynamical forms Proposition 3.1 gives the following important uniqueness
result [70].

######  Corollary 3.2.

For $k=1$, every source form is a canonical source form.

Note that this can be seen also directly from the definition of $\mathcal{I}$:
Indeed, if $k=1$ and $\rho$ is a $\omega^{\sigma}$-generated $1$-contact
$(n+1)$-form then in (3.1) the contractions give horizontal $n$-forms with the
same components as those of $\rho$, and the summation over $|J|$ reduces to
one term, $|J|=0$. Hence,
$\mathcal{I}(\rho)=\omega^{\sigma}\land(\partial/\partial
y^{\sigma}\rfloor\rho)=\rho$.

For the description of the image of $\mathcal{I}$ we refer also to [63].

######  Example 3.3.

We shall show the computations of $\mathcal{I}(\rho)$ and $\mathcal{R}(\rho)$
for a second-order 1-contact 2-form $\rho$ in mechanics ($n=1$, $k=1$, $r=2$).

Consider local fibred coordinates $(t,q^{\sigma})$ on $\boldsymbol{Y}$ and the
corresponding associated coordinates $(t,q^{\sigma},q^{\sigma}_{j})$, $1\leq
j\leq r$, $1\leq\sigma\leq m$, on $J^{r}\boldsymbol{Y}$, and denote $d_{t}$
the total derivative operator and
$\omega^{\sigma}_{j}=d{q}_{j}^{\sigma}-{q}_{j+1}^{\sigma}dt$ the local basic
contact $1$-forms, and put $\omega^{\sigma}=\omega^{\sigma}_{0}$. The
calculation of the image of $\rho$ by the interior Euler operator is direct,
using (3.1). We have

$\displaystyle p_{1}\rho=A^{0}_{\sigma}\omega^{\sigma}\wedge
dt+A^{1}_{\sigma}{\omega}^{\sigma}_{1}\wedge
dt+A^{2}_{\sigma}\omega^{\sigma}_{2}\wedge dt,$

hence

$\displaystyle{\cal{I}}(\rho)=\omega^{\sigma}\wedge\sum_{j=0}^{r=2}(-1)^{j}d_{t}^{j}\left(\frac{\partial}{\partial
q^{\sigma}_{j}}\rfloor
p_{1}\rho\right)=\big{(}A^{0}_{\sigma}-d_{t}A^{1}_{\sigma}+d_{t}^{2}A^{2}_{\sigma}\big{)}\omega^{\sigma}\wedge
dt.$

The residual term is then

$\displaystyle
p_{1}dp_{1}{\cal{R}}(\rho)=\pi^{*}_{5,3}p_{1}\rho-{\cal{I}}(\rho)=\big{(}d_{t}A_{\sigma}^{1}-d_{t}^{2}A^{2}_{\sigma}\big{)}\omega^{\sigma}\wedge
dt+A^{1}_{\sigma}\omega^{\sigma}_{1}\wedge
dt+A_{\sigma}^{2}\omega^{\sigma}_{2}\wedge dt.$

Let us compute $\mathcal{R}(\rho)$. Since $\mathcal{R}(\rho)$ is a contact
$1$-form of order two, it holds
$\mathcal{R}(\rho)=C_{\sigma}^{0}\omega^{\sigma}+C_{\sigma}^{1}\omega^{\sigma}_{1}+C_{\sigma}^{2}\omega^{\sigma}_{2}=p_{1}\mathcal{R}(\rho)$.
The $1$-contact part of the exterior derivative is
$p_{1}d\mathcal{R}(\rho)=-d_{t}C_{\sigma}^{0}\omega^{\sigma}\land
dt-(d_{t}C_{\sigma}^{1}+C^{0}_{\sigma})\omega^{\sigma}_{1}\land
dt-(d_{t}C_{\sigma}^{2}+C^{1}_{\sigma})\omega^{\sigma}_{2}\land dt-
C_{\sigma}^{2}\omega^{\sigma}_{3}\land dt$. Comparing this with the formula
above yields

$\displaystyle C^{2}_{\sigma}=0,\qquad
d_{t}C_{\sigma}^{2}+C^{1}_{\sigma}=-A^{2}_{\sigma},\qquad
d_{t}C_{\sigma}^{1}+C^{0}_{\sigma}=-A^{1}_{\sigma},\qquad-
d_{t}C_{\sigma}^{0}=d_{t}A_{\sigma}^{1}-d_{t}^{2}A^{2}_{\sigma}.$

From here we immediately see that $C^{1}_{\sigma}=-A^{2}_{\sigma}$ and
$C^{0}_{\sigma}=-A^{1}_{\sigma}+d_{t}A_{\sigma}^{2}$, and that the equation
$-d_{t}C_{\sigma}^{0}=d_{t}A_{\sigma}^{1}-d_{t}^{2}A^{2}_{\sigma}$ is
satisfied identically. So finally,

$\displaystyle\mathcal{R}(\rho)=\big{(}d_{t}A_{\sigma}^{2}-A^{1}_{\sigma}\big{)}\omega^{\sigma}-A_{\sigma}^{2}\omega^{\sigma}_{1}.$

We can easily verify that
$\pi^{*}_{5,3}p_{1}\rho={\cal{I}}(\rho)+p_{1}dp_{1}{\cal{R}}(\rho)$.

###  3.2 Takens representation of the variational sequence

Once we have the interior Euler operator, we can obtain a sequence of sheaves
of differential forms (rather than of classes of differential forms), such
that both the objects and the morphisms have a straightforward interpretation
in the calculus of variations. Such a representation of the variational
sequence, introduced by Krbek and Musilová (see [43, 44, 45]) basically
concerns source forms $($of all degrees$)$, therefore we call it in honour of
Takens the Takens representation.

The motivation for the representation comes from the following observations:

$\bullet$ If $q\leq n$ then for every $q$-form $\rho$ of order $r$, the
contact decomposition (2.1) takes the form

$\displaystyle\pi_{r+1,r}^{*}\rho=h\rho+p_{1}\rho+\dots+p_{q}\rho,$

and $\Theta^{r}_{q}=\Lambda^{r}_{q,c}+(d\Lambda^{r}_{q-1,c})$, where we have
sheaves of contact forms. This means that given a class
$[\rho]\in\mathcal{V}^{r}_{q}=\Lambda^{r}_{q}/\Theta^{r}_{q}$, one has
$\rho_{1}\sim\rho_{2}$ iff $\rho_{1}-\rho_{2}=\vartheta+d\eta$ where
$\vartheta$ is a contact $q$-form and $\eta$ is a contact $(q-1)$-form, i.e.,

$\displaystyle h\rho_{1}=h\rho_{2}.$

Hence, every class $[\rho]$ is completely determined by a unique horizontal
form, and, in particular, if $q=n$, by a Lagrangian.

Summarizing, for $q\leq n$ the operator of horizontalization,

$\displaystyle h\colon\
\Lambda^{r}_{q}\to\Lambda^{r+1}_{q,\boldsymbol{X}}\subset\Lambda^{r+1}_{q},\qquad\rho\to
h\rho,$

induces a representation mapping,

$\displaystyle R_{q}\colon\ {\mathcal{V}}^{r}_{q}\to\Lambda^{r+1}_{q},\qquad
R_{q}([\rho])=h\rho.$

$\bullet$ If $q=n+k$ for $k>1$ then for every $q$-form $\rho$ of order $r$,
the contact decomposition (2.1) takes the form

$\displaystyle\pi_{r+1,r}^{*}\rho=p_{k}\rho+p_{k+1}\rho+\dots+p_{q}\rho,$

and $\Theta^{r}_{q}=\Lambda^{r}_{q,c}+(d\Lambda^{r}_{q-1,c})$, where we have
sheaves of strongly contact forms (with the only exception of
$\Lambda^{r}_{n,c}$ which is a sheaf of contact forms). This means that given
a class $[\rho]\in\mathcal{V}^{r}_{n+k}=\Lambda^{r}_{n+k}/\Theta^{r}_{n+k}$,
one has $\rho_{1}\sim\rho_{2}$ iff $\rho_{1}-\rho_{2}=\vartheta+d\eta$ where
$\vartheta$ is a strongly contact $(n+k)$-form (i.e., such that
$p_{k}\vartheta=0$) and $\eta$ is a strongly contact $(n+k-1)$-form
($p_{k-1}\eta=0$). Thus $p_{k}\rho_{1}=p_{k}\rho_{2}+p_{k}dp_{k}\eta\neq
p_{k}\rho_{2}$. However, $\rho_{1}-\rho_{2}\in\Theta^{r}_{n+k}$, so that

$\displaystyle\mathcal{I}(\rho_{1})=\mathcal{I}(\rho_{2}).$

Hence, every class $[\rho]\in\mathcal{V}^{r}_{n+k}$ is completely determined
by a unique canonical source form.

If we denote by $\mathcal{I}\Lambda^{r}_{{n+k}}$ the image of
$\Lambda^{r}_{n+k}$ by $\mathcal{I}$, we can see that the interior Euler
operator

$\displaystyle\mathcal{I}\colon\
\Lambda^{r}_{n+k}\to\mathcal{I}\Lambda^{r}_{n+k}\subset\Lambda^{2r+1}_{n+k},\qquad\rho\to\mathcal{I}(\rho),$

induces a representation mapping,

$\displaystyle R_{n+k}\colon\
{\mathcal{V}}^{r}_{n+k}\to\Lambda^{2r+1}_{n+k},\qquad
R_{n+k}([\rho])=\mathcal{I}(\rho).$

For $k=1$ there is no other source form representing a class
$[\rho]\in\mathcal{V}^{r}_{n+1}$, so that every class is completely determined
by a unique dynamical form. If $[\rho]=\mathcal{E}_{n}(\mu)=[d\mu]$ then
$\mathcal{I}(d\mu)=\mathcal{I}(dh\mu)$ is the Euler–Lagrange form of the
Lagrangian $h\mu$. Note that this argument proves the uniqueness of the
Euler–Lagrange form. At the same time Proposition 3.1 shows us that for $k\geq
2$ the representing canonical source form is no longer a unique source form in
the class $[\rho]$. An explicit important example of nonuniquness for $k=2$ is
realized by Helmholtz forms, and will be discussed below in Section 5.1.

$\bullet$ If $q=0$ we define $R_{0}$ be the identity mapping.

Now, by means of the representation mappings $R_{q}$ we construct to the
variational sequence $0\to{\mathbb{R}}_{\boldsymbol{Y}}\to\mathcal{V}^{r}_{*}$
a representation sequence

$\displaystyle 0\to{\mathbb{R}}_{\boldsymbol{Y}}\to
R_{0}(\mathcal{V}^{r}_{0})\to R_{1}(\mathcal{V}^{r}_{1})\to
R_{2}(\mathcal{V}^{r}_{2})\to\cdots$

shortly denoted by $0\to{\mathbb{R}}_{\boldsymbol{Y}}\to
R_{*}(\mathcal{V}^{r}_{*})$, and called the Takens representation of
$0\to{\mathbb{R}}_{\boldsymbol{Y}}\to\mathcal{V}^{r}_{*}$. We also denote by

$\displaystyle E_{q}\colon\ R_{q}(\mathcal{V}^{r}_{q})\to
R_{q+1}(\mathcal{V}^{r}_{q+1}),\qquad q\geq 1,$

the morphisms in the representation sequence. Hence, the definition of the
$E_{q}$’s follows from the commutativity of the diagrams

$\displaystyle\begin{CD}\cdots
@>{\mathcal{E}_{q-1}}>{}>\mathcal{V}_{q}^{r}@>{\mathcal{E}_{q}}>{}>\mathcal{V}_{q+1}^{r}@>{\mathcal{E}_{q+1}}>{}>\cdots\\\
@V{{R_{q}}}V{}V@V{{R_{q+1}}}V{}V\\\ \cdots
@>{E_{q-1}}>{}>R_{q}(\mathcal{V}^{r}_{q})@>{E_{q}}>{}>R_{q+1}(\mathcal{V}_{q+1}^{r})@>{E_{q+1}}>{}>\cdots\end{CD}$
(3.4)

and we can immediately see that the following theorem holds true:

######  Theorem 3.4.

The representation sequence $0\to{\mathbb{R}}_{\boldsymbol{Y}}\to
R_{*}(\mathcal{V}^{r}_{*})$ is exact.

Indeed, by definition of $E_{q}$ and $\mathcal{E}_{q}$,

$\displaystyle E_{q}\circ
R_{q}([\rho])=R_{q+1}\circ\mathcal{E}_{q}([\rho])=R_{q+1}([d\rho]),$

and for every $\eta\in R_{q}(\mathcal{V}_{q}^{r})$ we have
$\eta=R_{q}([\rho])$ for some $[\rho]\in\mathcal{V}^{r}_{q}$. Hence

$\displaystyle(E_{q+1}\circ E_{q})(\eta)=(E_{q+1}\circ E_{q}\circ
R_{q})([\rho])=(E_{q+1}\circ R_{q+1}\circ\mathcal{E}_{q})([\rho])$
$\displaystyle\hphantom{(E_{q+1}\circ E_{q})(\eta)}{}=(E_{q+1}\circ
R_{q+1})([d\rho])=(R_{q+2}\circ\mathcal{E}_{q+1}([d\rho])=R_{q+2}([dd\rho])=0.$

The representation sequence can be easily understood, and explicit formulas
for the morphisms $E_{q}$ and the forms $E_{q}(\eta)\in
R_{q+1}(\mathcal{V}^{r}_{q+1})$ can be easily obtained, if we write the
diagram (3.4) in the explicit form

$\displaystyle\begin{CD}\cdots
@>{\mathcal{E}_{n-1}}>{}>\Lambda_{n}^{r}/\Theta_{n}^{r}@>{\mathcal{E}_{n}}>{}>\Lambda_{n+1}^{r}/\Theta_{n+1}^{r}@>{\mathcal{E}_{n+1}}>{}>\Lambda_{n+2}^{r}/\Theta_{n+2}^{r}@>{\mathcal{E}_{n+2}}>{}>\cdots\\\
@V{{R_{n}}}V{}V@V{{R_{n+1}}}V{}V@V{{R_{n+2}}}V{}V\\\ \cdots
@>{E_{n-1}}>{}>\Lambda_{n,\boldsymbol{X}}^{r+1}@>{E_{n}}>{}>\Lambda_{n+1,1,\boldsymbol{Y}}^{r+1}@>{E_{n+1}}>{}>\mathcal{I}\Lambda_{n+2}^{r+1}@>{}>{}>\cdots\end{CD}$

where $R_{q}$, $1\leq q\leq n$, has the meaning of horizontalization $h$ on
classes of order $r$, and $R_{n+k}$ is the operator $\mathcal{I}$ acting on
$p_{k}\rho$ where $\rho$ is of order $r$. Then since elements of the sheaves
$R_{q}(\mathcal{V}^{r}_{q})$ are functions for $q=0$, horizontal forms for
$1\leq q\leq n$, and canonical source forms for $q\geq n+1$, we have

$\displaystyle E_{0}(f)=hdf,\qquad E_{q}(h\rho)=hdh\rho,\qquad 1\leq q\leq
n-1,$ $\displaystyle E_{n}(h\rho)=\mathcal{I}(dh\rho),\qquad
E_{n+k}(\mathcal{I}(\rho))=\mathcal{I}(d(\mathcal{I}(\rho))),\qquad k\geq 1.$

In particular, for $q=n$ we can write in coordinates $h\rho={L}\omega_{0}$,
where $\omega_{0}=dx^{1}\wedge\cdots\wedge dx^{n}$, and then

$\displaystyle
E_{n}(h\rho)=R_{n+1}([d\rho])={\cal{I}}(d\rho)={\cal{I}}(dh\rho)=\sum_{|J|=0}^{r}(-1)^{|J|}d_{J}\left(\frac{\partial{L}}{\partial
y^{\sigma}_{J}}\right)\omega^{\sigma}\wedge\omega_{0}.$

Hence, $E_{n}\colon R_{n}(\mathcal{V}^{r}_{n})\to
R_{n+1}(\mathcal{V}^{r}_{n+1})$ is the Euler–Lagrange mapping, assigning to
every Lagrangian $\lambda=h\rho$ its Euler–Lagrange form
$E_{\lambda}=\mathcal{I}(d\lambda)$. The next morphism, $E_{n+1}\colon
R_{n+1}(\mathcal{V}^{r}_{n+1})\to R_{n+2}(\mathcal{V}^{r}_{n+2})$ is the
Helmholtz mapping, assigning to every dynamical form $\varepsilon$ its
canonical Helmholtz form $H_{\varepsilon}=\mathcal{I}(d\varepsilon)$. Note
that the preceding morphism to the Euler–Lagrange morphism, $E_{n-1}\colon
R_{n-1}(\mathcal{V}^{r}_{n-1})\to R_{n}(\mathcal{V}^{r}_{n})$, assigns to
every horizontal $(n-1)$-form $\varphi$ (resp., if $\dim X=n=1$, to a function
$f$) a Lagrangian $\lambda=hd\varphi$ (resp. $\lambda=hdf$). This is so-called
null-Lagrangian (also called variationally trivial Lagrangian), due to the
fact that its Euler–Lagrange form is by exactness of the sequence equal to
zero.

More generally, for $q\geq 1$, elements of $R_{q}(\mathcal{V}^{r}_{q})$
belonging to the kernel of the variational morphism $E_{q}$ are called
variationally trivial.

We say that two elements $\eta_{1}$, $\eta_{2}$ of
$R_{q}(\mathcal{V}^{r}_{q})$ are $($variationally$)$ equivalent if their
difference is variationally trivial, i.e.,
$\eta_{2}-\eta_{1}\in\operatorname{Ker}E_{q}$. Thus, two Lagrangians are
equivalent iff they differ by a null Lagrangian, $hd\rho$, two dynamical forms
are equivalent iff they differ by a locally variational form, and two
Helmholtz-like forms are equivalent iff they differ by a Helmholtz form.
Generally, two canonical source forms are equivalent iff they differ by
$\mathcal{I}(d(\mathcal{I}\rho))$.

######  Example 3.5.

As an illustration of the use of the Takens representation of the variational
sequence let us compute the Euler–Lagrange operator with help of the operator
$\mathcal{I}$. Indeed, given a Lagrangian $\lambda$, the Euler–Lagrange form
of $\lambda$ arises either as the image of $\lambda$ by the morphism $E_{n}$,
or, by application of $\mathcal{I}$ to the exterior derivative of $\lambda$
(remind the formula $E_{n}(\lambda)={\cal{I}}(d\lambda)$ coming from the
commutativity relation $E_{n}\circ{\cal{I}}={\cal{I}}\circ d$).

Consider the Lagrangian of a free quantum particle moving in one dimension.
The quantum mechanical state of such a system is, in the coordinate
representation, described by a wave function
$\phi(t,x)\colon{\mathbb{R}}\times{\mathbb{R}}\to\mathbb{C}$, and the
Lagrangian is

$\displaystyle\lambda=\left(-\frac{\hbar^{2}}{4m}(\phi_{x}\phi^{*}_{x})+\frac{{\rm
i}\hbar}{4}(\phi^{*}\phi_{t}-\phi\phi^{*}_{t})\right)\omega_{0},\qquad\omega_{0}=dt\wedge
dx.$

Denoting $v(t,x)=\operatorname{Re}\phi(t,x)$ and
$w(t,x)=\operatorname{Im}\phi(t,x)$ we obtain

$\displaystyle\lambda=\left(-\frac{\hbar^{2}}{4m}\big{(}v_{x}^{2}+w_{x}^{2}\big{)}-\frac{\hbar}{2}(vw_{t}-wv_{t})\right)\omega_{0},$
(3.5)

which is a first-order Lagrangian for the fibred manifold
$\pi\colon{\mathbb{R}}^{2}\times{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$ (hence
$n=2$, $m=2$, $r=1$) where $\pi$ is the canonical projection, with fibred
coordinates
$(x^{1},x^{2},y^{1},y^{2},y^{1}_{1},y^{1}_{2},y^{2}_{1},y^{2}_{2})=(t,x,v,w,v_{t},v_{x},w_{t},w_{x})$.
In this notation, the generating contact $1$-forms on
$J^{2}({\mathbb{R}}^{2}\times{\mathbb{R}}^{2})$ read as follows:
$\omega^{1}=dv-v_{t}dt-v_{x}dx$, $\omega^{2}=dw-w_{t}dt-w_{x}dx$,
$\omega^{1}_{t}=dv_{t}-v_{tt}dt-v_{tx}dx$, $\omega^{1}_{x}=dv_{x}-v_{xt}dt-
v_{xx}dx$, $\omega^{2}_{t}=dw_{t}-w_{tt}dt-w_{tx}dx$,
$\omega^{2}_{x}=dw_{x}-w_{xt}dt-w_{xx}dx$. Now, $p_{1}d\lambda$ is the
following 1-contact 3-form on $J^{2}({\mathbb{R}}^{2}\times{\mathbb{R}}^{2})$:

$\displaystyle
p_{1}d\lambda=\left(-\frac{\hbar}{2}w_{t}dv+\frac{\hbar}{2}v_{t}dw+\frac{\hbar}{2}wdv_{t}-\frac{\hbar}{2}vdw_{t}-\frac{\hbar^{2}}{2m}v_{x}dv_{x}-\frac{\hbar^{2}}{2m}w_{x}dw_{x}\right)\omega_{0}.$

Calculating ${\cal{I}}(d\lambda)$ using formula (3.1) we obtain

$\displaystyle{\cal{I}}(d\lambda)=\omega^{1}\wedge\left[-\frac{\hbar}{2}w_{t}-d_{t}\left(\frac{\hbar}{2}w\right)-d_{x}\left(-\frac{\hbar^{2}}{2m}v_{x}\right)\right]\omega_{0}$
$\displaystyle\hphantom{{\cal{I}}(d\lambda)=}{}+\omega^{2}\wedge\left[\frac{\hbar}{2}v_{t}-d_{t}\left(-\frac{\hbar}{2}v\right)-d_{x}\left(-\frac{\hbar^{2}}{2m}w_{x}\right)\right]\omega_{0}$
$\displaystyle\hphantom{{\cal{I}}(d\lambda)}{}=\left[\left(-\hbar
w_{t}+\frac{\hbar^{2}}{2m}v_{xx}\right)\omega^{1}+\left(\hbar
v_{t}+\frac{\hbar^{2}}{2m}w_{xx}\right)\omega^{2}\right]\wedge\omega_{0}.$

The dynamical $3$-form ${\cal{I}}(d\lambda)$ is the Euler–Lagrange form of
(3.5). This means that

$\displaystyle\varepsilon_{1}=\frac{\hbar^{2}}{2m}v_{xx}-\hbar
w_{t},\qquad\varepsilon_{2}=\frac{\hbar^{2}}{2m}w_{xx}+\hbar v_{t},$

are the Euler–Lagrange expressions of the Lagrangian (3.5), and equations
$\varepsilon_{1}\circ J^{2}\gamma=0$, $\varepsilon_{2}\circ J^{2}\gamma=0$ are
the Euler–Lagrange equations for extremals $\gamma$ of (3.5). Denoting
$\varepsilon=\varepsilon_{1}+{\rm i}\varepsilon_{2}$ we obtain the
Euler–Lagrange equation

$\displaystyle\left(\frac{\hbar^{2}}{2m}\phi_{xx}+{\rm
i}\hbar\phi_{t}\right)\circ J^{2}\gamma=0,$

which, indeed, is the Schrödinger equation.

###  3.3 Lepage $\boldsymbol{n}$-forms and the first variation formula

Lepage $n$-forms were introduced by Krupka in 1973 [46] (see also [48]) in
order to establish foundations of the higher-order calculus of variations in
jet bundles. They are fundamental for a geometric formulation of the intrinsic
first variation formula, and of coordinate free, global Lagrangian and
Hamiltonian mechanics and field theory. Combined with the concepts of
invariant variational functionals they provide geometric formulations of
Noether theorems, as well as geometric integration methods based on
symmetries.

Here we remind only some basic properties of Lepage $n$-forms, explored in the
variational sequence theory. For more details and applications we refer the
reader to the survey papers [54, 59, 60, 67, 76], and the book [55].

As above, $\pi\colon\boldsymbol{Y}\to\boldsymbol{X}$ is a fibred manifold,
$n=\dim\boldsymbol{X}$, and $m=\dim\boldsymbol{Y}-n$.

######  Definition 3.6.

Let $r\geq 0$. A $n$-form $\rho$ on $J^{r}\boldsymbol{Y}$ is called a Lepage
$n$-form of order $r$ if for every $\pi_{r,0}$-vertical vector field $\xi$ on
$J^{r}\boldsymbol{Y}$

$\displaystyle h(\xi\rfloor d\rho)=0.$

Since $\rho$ is an $n$-form on $J^{r}\boldsymbol{Y}$ its horizontal component
$h\rho$ is a Lagrangian on $J^{r+1}\boldsymbol{Y}$.

Notice that

  * •

every $n$-form on $\boldsymbol{Y}$ is a Lepage $n$-form. The corresponding
Lagrangian is then defined on $J^{1}\boldsymbol{Y}$, and it is a polynomial of
degree $n$ in the first derivatives111This fact is of particular importance in
Hamiltonian mechanics and field theory, or in dealing with variational forces
and energy-momentum tensors [38, 54, 68].,

  * •

every closed $n$-form on $J^{r}\boldsymbol{Y}$ is a Lepage $n$-form. The
corresponding Lagrangian is a null-Lagrangian (giving rise to the zero
Euler–Lagrange form).

The structure of Lepage $n$-forms is characterized as follows:

######  Theorem 3.7 (Krupka [46]).

The following conditions are equivalent:

  1. $(1)$

$\rho$ is a Lepage $n$-form of order $r$,

  2. $(2)$

$p_{1}d\rho$ is a dynamical form, i.e., $p_{1}d\rho=\mathcal{I}(d\rho)$,

  3. $(3)$

$\pi_{r+1,r}^{*}d\rho=E+F$, where $E$ is a dynamical form, and $F$ is at least
$2$-contact,

  4. $(4)$

$\pi_{r+1,r}^{*}\rho=\theta_{h\rho}+d\nu+\mu=\theta_{h\rho}+p_{1}d\nu+\eta$,
where $\theta_{h\rho}$ is the Cartan form of the Lagrangian $h\rho$, $\nu$ is
a contact $(n-1)$-form, and $\mu$, resp. $\eta$ is at least $2$-contact.

It should be stressed that for $n\geq 2$ and $r\geq 3$ the Cartan form is
generically not global. However, the above theorem states that it can be
“globalized” by adding $p_{1}d\nu$. Indeed, in the decomposition of
$\pi_{r+1,r}^{*}\rho$ in (4), $\theta_{h\rho}+p_{1}d\nu$ and $\eta$ are global
(being the at most $1$-contact and the at least $2$-contact part of $\rho$,
respectively), while, in general, the decomposition $\theta_{h\rho}+p_{1}d\nu$
is not invariant under changes of fibred coordinates.

The name ‘Cartan form’ refers to É. Cartan, who introduced the form to the
classical calculus of variations [10]. Concerning different aspects of
generalization to many independent variables we refer, e.g., to [6, 9, 12, 18,
29, 33, 34, 46, 47, 78, 90]. Higher-order Cartan forms have been introduced
and studied by many authors; see, e.g., [12, 13, 21, 22, 30, 40, 48] to name
just a few.

Accounting the definition and properties of the operators $\mathcal{I}$ and
$\mathcal{R}$ we are able to find an intrinsic formula for the Cartan form:

######  Theorem 3.8.

Let $r\geq 0$, and $\rho\in\Lambda^{r}_{n}$. Then the Cartan form of the
Lagrangian $\lambda=h\rho$ takes the form

$\displaystyle\theta_{\lambda}=\lambda-p_{1}\mathcal{R}(d\lambda).$ (3.6)

Notice that, indeed, $\theta_{h\rho}$ is a Lepage $n$-form, equivalent with
$\rho$ (and $h\rho$), since by formulas (3.2) and (3.3)

$\displaystyle
p_{1}d\theta_{h\rho}=\mathcal{I}(d\theta_{h\rho})+p_{1}dp_{1}\mathcal{R}(d\theta_{h\rho})=\mathcal{I}(d\theta_{h\rho})+p_{1}dp_{1}\mathcal{R}(dh\rho)-p_{1}dp_{1}\mathcal{R}(dp_{1}\mathcal{R}(dh\rho))$
$\displaystyle\hphantom{p_{1}d\theta_{h\rho}}{}=\mathcal{I}(d\theta_{h\rho})+p_{1}dp_{1}\mathcal{R}(dh\rho)-p_{1}dp_{1}\mathcal{R}(dh\rho)+\mathcal{I}(dp_{1}\mathcal{R}(dh\rho))=\mathcal{I}(d\theta_{h\rho}),$

and, writing $\pi_{r+1,r}^{*}\rho=h\rho+\beta$ we can see that (for a proper
$s$)

$\displaystyle\pi_{s,r}^{*}\rho-\theta_{h\rho}=\beta+p_{1}\mathcal{R}(dh\rho)\in\Theta^{s}_{n}.$

The meaning of Lepage $n$-forms for the calculus of variations comes from the
fact that if $\rho$ is a Lepage $n$-form then the dynamical form
$p_{1}d\rho=E_{h\rho}$ is the Euler–Lagrange form of the Lagrangian $h\rho$.
Indeed, the components of $p_{1}d\rho$ are the Euler–Lagrange expressions of
the Lagrangian $\lambda=h\rho$.

We stress that even though $\theta_{h\rho}$ need not be global, the local
forms do give rise to the global form $p_{1}d\theta_{h\rho}$. Indeed, formula
(3.6) yields

$\displaystyle p_{1}d\theta_{\lambda}=p_{1}d\lambda-
p_{1}dp_{1}\mathcal{R}(d\lambda)=\mathcal{I}(d\lambda),$

which, indeed, is a global form. (Note that we recovered the formula for the
Euler–Lagrange form of $\lambda$ in Takens representation of the variational
sequence.) Moreover, since for every Lepage form $\rho$,
$\pi_{r+1,r}^{*}p_{1}d\rho=p_{1}d\theta_{h\rho}$, we get another proof of
uniqueness of the Euler–Lagrange form.

As the horizontal parts of Lepage $n$-forms are Lagrangians, we can view a
Lepage $n$-form as an extension of a Lagrangian by a contact form, and we come
to the concept of Lepage equivalent of a Lagrangian [46]. Given a Lagrangian
$\lambda$, a Lepage equivalent of $\lambda$ is a Lepage form $\rho$ such that
$\lambda=h\rho$.

Remarkably, if $\lambda$ is of order $r$, then its Lepage equivalents are
generically of order $2r-1$, and the Euler–Lagrange form is of order $2r$.

Local existence of Lepage equivalents of $\lambda$ and their structure follows
immediately from the above theorem. Moreover, it is well known that every
Lagrangian admits a global Lepage equivalent. For $n=1$ and any $r$, or for
any $n$ and $r\leq 2$ a global Lepage equivalent of $\lambda$ is the Cartan
form. If $n=1$ (mechanics and higher-order mechanics) the Cartan form
$\theta_{\lambda}$ is the unique Lepage equivalent of $\lambda$. For $n>1$,
Lepage equivalent of $\lambda$ is no longer unique. Remarkably, apart from the
(globalized) Cartan form, there are also other distinguished global Lepage
equivalents of $\lambda$, as, e.g., the celebrated Carathéodory form ([9, 78]
for $r=1$, and [92] for $r=2$), which is invariant with respect to all (not
only fibred) coordinate transformations, and the Krupka–Betounes form (for
$r=1$) [6, 47], which has the property that $d\rho=0$ if and only if
$E_{h\rho}=0$ (i.e., the Lagrangian belongs to $\operatorname{Ker}E_{n}$).

######  Example 3.9.

Let us illustrate the use of the Lepage representation of the variational
sequence on an example. Consider the quantum particle Lagrangian (3.5) in
Example 3.5 (including notations). Its Cartan form reads

$\displaystyle\theta_{\lambda}=L\omega_{0}+\frac{\partial L}{\partial
v_{t}}\omega^{1}\wedge dx+\frac{\partial L}{\partial w_{t}}\omega^{2}\wedge
dx-\frac{\partial L}{\partial v_{x}}\omega^{1}\wedge dt-\frac{\partial
L}{\partial w_{x}}\omega^{2}\wedge dt$
$\displaystyle\hphantom{\theta_{\lambda}}{}=\left(-\frac{\hbar^{2}}{4m}\big{(}v_{x}^{2}+w_{x}^{2}\big{)}-\frac{\hbar}{2}(vw_{t}-wv_{t})\right)\omega_{0}$
$\displaystyle\hphantom{\theta_{\lambda}=}{}+\frac{\hbar^{2}}{2m}v_{x}\omega^{1}\wedge
dt+\frac{\hbar^{2}}{2m}w_{x}\omega^{2}\wedge
dt+\frac{\hbar}{2}w\omega^{1}\wedge dx-\frac{\hbar}{2}v\omega^{2}\wedge dx.$

As we have seen, being a Lepage equivalent of the Lagrangian (3.5), the form
$\theta_{\lambda}$ quickly gives us the Euler–Lagrange form $E_{\lambda}$, and
hence the Euler–Lagrange equations:

$\displaystyle
E_{\lambda}=p_{1}d\theta_{\lambda}=\left(\frac{\hbar^{2}}{2m}v_{xx}-\hbar
w_{t}\right)\omega^{1}\wedge\omega_{0}+\left(\frac{\hbar^{2}}{2m}w_{xx}+\hbar
v_{t}\right)\omega^{2}\wedge\omega_{0}.$

We can see that, indeed, $E_{\lambda}$ equals to ${\cal{I}}(d\lambda)$
obtained in Example 3.5; this follows from the commutativity relation
$\operatorname{Lep}_{2}\circ E_{2}=d\circ\operatorname{Lep}_{1}$.

We note that the form $\theta_{\lambda}$ (or any other Lepage equivalent of
the Lagrangian) also gives local Noetherian conserved currents related with
symmetries of the Lagrangian and the Euler–Lagrange form (see the discussion
below).

With help of a Lepage equivalent of a Lagrangian the non-intrinsic procedure
of integration by parts in the first variation of the action function is
substituted by a geometric splitting with help of the Cartan formula for
decomposition of the Lie derivative. As a consequence, the integral first
variation formula appears also in the following intrinsic differential form
[46] (for more details see, e.g., [54]). Given a Lagrangian $\lambda$ of order
$r$, and a $\pi$-projectable vector field $\Xi$ on $\boldsymbol{Y}$ (a
“variation”), it holds

$\displaystyle L_{J^{r}\Xi}\lambda\equiv
L_{J^{2r}\Xi}h\rho=h\big{(}J^{2r-1}\Xi\rfloor
d\rho\big{)}+hd\big{(}J^{2r-1}\Xi\rfloor\rho\big{)}\equiv
hL_{J^{2r-1}\Xi}\rho,$ (3.7)

where $L_{J^{r}\Xi}$ denotes the Lie derivative along the $r$-jet prolongation
of $\Xi$, and $\rho$ is (any) Lepage equivalent of $\lambda$. Let us show that
the first term in the sum of (3.7) is the Euler–Lagrange term (carrying
information about extremals), and the second term is the Noether term,
carrying information about conservations laws.

We have $\lambda=h\rho$, and the Euler–Lagrange form is defined by
$E_{\lambda}=p_{1}d\varrho$. For the first differential form on the right-hand
side of (3.7) we obtain

$\displaystyle h\big{(}J^{2r-1}\Xi\rfloor
d\rho\big{)}=h\big{(}J^{2r}\Xi\rfloor\pi^{*}_{2r,2r-1}d\rho\big{)}=h\big{(}J^{2r}\Xi\rfloor(p_{1}d\rho+\mu)\big{)}=h\big{(}J^{2r}\Xi\rfloor(E_{\lambda})\big{)},$

where $\mu$ is at least 2-contact, hence $J^{2r}\Xi\rfloor\mu$ is contact and
$h(J^{2r}\Xi\rfloor\mu)=0$. Denote
$E_{\lambda}=E_{\sigma}\omega^{\sigma}\wedge\omega_{0}$ (i.e., $E_{\sigma}$
are the Euler–Lagrange expressions of $\lambda$),
$\omega_{i}=\frac{\partial}{\partial x^{i}}\rfloor\omega_{0}$, and
$\Xi=\xi^{i}\frac{\partial}{\partial
x^{i}}+\Xi^{\sigma}\frac{\partial}{\partial y^{\sigma}}$. Then

$\displaystyle h\big{(}J^{2r-1}\Xi\rfloor
d\rho\big{)}=h\big{(}J^{2r}\Xi\rfloor(E_{\sigma}\omega^{\sigma}\wedge\omega_{0})\big{)}=h\big{(}E_{\sigma}\Xi^{\sigma}\omega_{0}-E_{\sigma}\xi^{i}\omega^{\sigma}\wedge\omega_{i}\big{)}=E_{\sigma}\Xi^{\sigma}\omega_{0},$

showing that, indeed, $h(J^{2r-1}\Xi\rfloor d\rho)$ provides Euler–Lagrange
equations.

Let us turn to the second differential form in the sum of (3.7). Assume that
$\Xi$ is a Noether symmetry of the Lagrangian $\lambda$, meaning that
$L_{J^{r}\Xi}\lambda=0$. Then in the infinitesimal first variation formula
(3.7) the left-hand side vanishes, and along every extremal $\gamma$ of the
Lagrangian $\lambda$ the $n$-form $h(J^{2r-1}\Xi\rfloor d\rho)$ vanishes,
because the Euler–Lagrange equations $E_{\sigma}\circ J^{2r}\gamma=0$,
$1\leq\sigma\leq m$, hold true. Consequently, the $n$-form
$hd(J^{2r-1}\Xi\rfloor\rho)$ vanishes along extremals, which is the celebrated
Noether theorem. The $(n-1)$-form

$\displaystyle\Phi(\Xi)=J^{2r-1}\Xi\rfloor\rho$

is then the corresponding Noether current.

In the simplest case of first-order mechanics when
$\rho=\theta_{\lambda}=Ldt+\frac{\partial
L}{\partial\dot{q}^{\sigma}}\omega^{\sigma}$, we recover the well-known
formulas

$\displaystyle h\big{(}J^{1}\Xi\rfloor
d\theta_{\lambda}\big{)}=\left(\frac{\partial L}{\partial
q^{\sigma}}-\frac{d}{dt}\frac{\partial
L}{\partial\dot{q}^{\sigma}}\right)\Xi^{\sigma}dt,$
$\displaystyle\Phi(\Xi)=L\xi^{0}+\frac{\partial
L}{\partial\dot{q}^{\sigma}}(\Xi^{\sigma}-\dot{q}^{\sigma}\xi^{0})=-H\xi^{0}+p_{\sigma}\Xi^{\sigma}.$

If $\Xi$ is a Noether symmetry of $\lambda$, then the function $\Phi(\Xi)$ is
constant along extremals.

We shall return to the variational splitting of the Lie derivative in a more
general context in Section 4.

###  3.4 Lepage forms of higher degrees, and the Lepage representation
of the variational sequence

A motivation for introducing Lepage forms of higher degrees is the following
observation: If $\dim X=1$ then every Lagrangian $\lambda$ has a unique Lepage
equivalent, the Cartan form $\theta_{\lambda}$. The mapping
$\operatorname{Lep}_{1}\colon\Lambda^{r}_{1,\boldsymbol{X}}\ni\lambda\to\theta_{\lambda}\in\Lambda^{2r-1}_{1}$
thus relates the Euler–Lagrange mapping to the exterior derivative through the
relationship $E_{\lambda}=p_{1}d\theta_{\lambda}$. This suggests the idea to
extend $\operatorname{Lep}_{1}$ to dynamical forms, i.e., to
$\operatorname{Lep}_{2}\colon\Lambda^{s}_{2,\boldsymbol{Y}}\ni\varepsilon\to\alpha_{\varepsilon}\in\Lambda^{s}_{2}$,
so that we would have

$\displaystyle\begin{CD}\lambda @>{E_{1}}>{}>E_{\lambda}\\\
@V{{\operatorname{Lep}_{1}}}V{}V@V{}V{{\operatorname{Lep}_{2}}}V\\\
\theta_{\lambda}@>{d}>{}>d\theta_{\lambda}\end{CD}$

The concept of a Lepage equivalent of a dynamical form was introduced in [64]
for locally variational forms. In that situation one obtains an extension of
the Euler–Lagrange form to a closed form which (similarly as in the case of a
Lepage equivalent of a Lagrangian) is unique for $n=1$ and nonunique otherwise
(see [37, 38, 64, 66, 72]). Lepage equivalents of Euler–Lagrange forms became
important particularly in study of the inverse variational problem,
Hamiltonian theory, and geometric integration based on symmetries of the
equations [34, 55, 65, 66, 68, 71].

Having the variational sequence and its Takens representation allows to define
the concept of Lepage equivalent for any source form, and, in this way, to
link all the variational operators to the exterior derivative. Remarkably,
with Lepage forms one gets another representation of the variational sequence
where variational morphisms simplify to exterior derivatives. The immediate
benefit is the reformulation and solution of the problem of existence of
Lagrangians for given differential equations “as they stand”, the problem of
the structure of null Lagrangians, and the other inverse problems in the
variational sequence: by using Lepage forms these problems are reduced to
application of the Poincaré lemma.

The idea how to generalize the concept of Lepage form to higher degrees is
suggested by property (2) (or (3)) in Theorem 3.7, that the lowest contact
component of the exterior derivative of a Lepage form $\rho$ is the canonical
source form for $d\rho$:

######  Definition 3.10 ([45, 61]).

Let $k\geq 0$. A $(n+k)$-form $\rho$ on $J^{r}\boldsymbol{Y}$ is called Lepage
form if

$\displaystyle p_{k+1}d\rho=\mathcal{I}(d\rho).$ (3.8)

Lepage $(n+k)$-forms for $k>0$ have many similar properties as have Lepage
$n$-forms. First,

  * •

every $q$-form on $\boldsymbol{Y}$, $q\geq n$, is a Lepage form,

  * •

every closed $q$-form on $J^{r}\boldsymbol{Y}$, $q\geq n$, is a Lepage form.

Structure of Lepage forms can be derived by direct calculations in coordinates
from the definition. However, this procedure is quite lengthy and tedious.
Much more advantageously, we again explore the operator $\mathcal{R}$ to solve
the equation (3.8) with respect to $\rho$ in an intrinsic way. Then we obtain:

######  Theorem 3.11.

Equation (3.8) has the solution

$\displaystyle\pi_{r+1,r}^{*}\rho=\theta_{p_{k}\rho}+d\nu+\mu=\theta_{p_{k}\rho}+p_{k+1}d\nu+\eta,$
(3.9)

where

$\displaystyle\theta_{p_{k}\rho}=p_{k}\rho-p_{k+1}\mathcal{R}(dp_{k}\rho),$
(3.10)

$\nu$ is an arbitrary at least $(k+1)$-contact $(n+k-1)$-form, and $\mu$
$($resp. $\eta)$ is an arbitrary at least $(k+2)$-contact form.

For every choice of $\nu$ and $\mu$ $($resp. $\eta)$ the Lepage forms (3.9)
belong to the same variational class $[\rho]\in\mathcal{V}^{s}_{n+k}$ $($for a
proper $s)$, i.e., $\pi_{s,r}^{*}\rho-\theta_{p_{k}\rho}\in\Theta^{s}_{n+k}$.

###### Proof.

First we show that $\theta_{p_{k}\rho}$ given by (3.10) is a Lepage form,
i.e., satisfies $p_{k+1}d\theta_{p_{k}\rho}=\mathcal{I}(d\theta_{p_{k}\rho})$.
Using the same procedure as above for $k=0$, we obtain:

$\displaystyle
p_{k+1}d\theta_{p_{k}\rho}=\mathcal{I}(d\theta_{p_{k}\rho})+p_{k+1}dp_{k+1}\mathcal{R}(d\theta_{p_{k}\rho})$
$\displaystyle\hphantom{p_{k+1}d\theta_{p_{k}\rho}}{}=\mathcal{I}(d\theta_{p_{k}\rho})+p_{k+1}dp_{k+1}\mathcal{R}(dp_{k}\rho)-p_{k+1}dp_{k+1}\mathcal{R}(dp_{k+1}\mathcal{R}(dp_{k}\rho))=\mathcal{I}(d\theta_{p_{k}\rho}),$

since

$\displaystyle
p_{k+1}dp_{k+1}\mathcal{R}(dp_{k+1}\mathcal{R}(dp_{k}\rho))=p_{k+1}dp_{k+1}\mathcal{R}(dp_{k}\rho)-\mathcal{I}(dp_{k+1}\mathcal{R}(dp_{k}\rho))$
$\displaystyle\hphantom{p_{k+1}dp_{k+1}\mathcal{R}(dp_{k+1}\mathcal{R}(dp_{k}\rho))}{}=p_{k+1}dp_{k+1}\mathcal{R}(dp_{k}\rho),$

in view of
$\mathcal{I}(dp_{k+1}\mathcal{R}(dp_{k}\rho))=\mathcal{I}(p_{k+1}(dp_{k+1}\mathcal{R}(dp_{k}\rho)))=0$.
Hence, $\theta_{p_{k}\rho}$ is a solution of (3.8). Since the equation
concerns $d\rho$, it is clear that if $\rho$ is a solution then also
$\rho+d\nu$ is a solution, where $\nu$ is an arbitrary $(n+k-1)$-form.
Moreover, equation (3.8) allows to determine only the lowest, $(k+1)$-contact
component of $d\rho$. Thus, any solution $\rho$ is determined up to
$d\nu+\mu$, where $\mu$ is at least $(k+2)$-contact. In this way, so far we
have obtained $\rho=\theta_{p_{k}\rho}+d\nu+\mu$, where $\mu$ is at least
$(k+2)$-contact. We observe that $p_{k}\theta_{p_{k}\rho}=p_{k}\rho$, hence
$p_{k}d\nu$ must be equal to zero, meaning that $\nu$ is at least
$(k+1)$-contact.

Finally, we have to show that whatever the choice of $\nu$ and $\mu$, $\rho$
is equivalent with $\theta_{p_{k}\rho}$. This is, however, easily seen: if we
denote by $s$ the order of $\theta_{p_{k}\rho}$, and by $\beta$ the at least
$(k+1)$-contact component of $\rho$, we obtain
$\pi_{s,r}^{*}\rho-\theta_{p_{k}\rho}=p_{k}\rho+\beta-\theta_{p_{k}\rho}=\beta+p_{k+1}\mathcal{R}(dp_{k}\rho)$,
which is a strongly contact $(n+k)$-form, thus belonging to the kernel
$\Theta_{n+k}^{s}$. ∎

The $(n+k)$-form $\theta_{p_{k}\rho}$ is the higher-degree generalization of
the Cartan form. It is completely determined by its lowest contact
($k$-contact) component. Similarly as the Cartan form of degree $n$ it is
generically not global, since it is defined by means of the operator
$\mathcal{R}$. However, quite similarly as in the case of Lepage $n$-forms, it
can be “globalized” by adding a proper term $p_{k+1}d\nu$.

And similarly as in the familiar $k=0$ situation, $p_{k+1}d\theta_{p_{k}\rho}$
is global, and it holds

$\displaystyle p_{k+1}d\theta_{p_{k}\rho}=\mathcal{I}(dp_{k}\rho).$

Also in the higher degree situation, there is a distinguished case of $n=1$
(and arbitrary $r$) (mechanics). Then $\nu$ is a $k$-form and $\mu$ is a
$(k+1)$-form, hence they both are zero and we have:

######  Corollary 3.12.

Let $n=\dim\boldsymbol{X}=1$, $r\geq 0$. Then for every $k\geq 0$, the higher
degree Cartan form $\theta_{p_{k}\rho}$ is unique and hence global.

The higher-degree generalization of Theorem 3.7 now reads as follows:

######  Theorem 3.13.

Let $k\geq 0$. The following conditions are equivalent:

  1. $(1)$

$\rho$ is a Lepage $(n+k)$-form of order $r$,

  2. $(2)$

$\pi_{r+1,r}^{*}d\rho=\mathcal{I}(d\rho)+F$, where $F$ is at least
$(k+2)$-contact,

  3. $(3)$

$p_{k+1}d\mathcal{R}(p_{k+1}d\rho)=0$,

  4. $(4)$

$\pi_{r+1,r}^{*}\rho=\theta_{p_{k}\rho}+d\nu+\mu=\theta_{p_{k}\rho}+p_{k+1}d\nu+\eta$,
where $\theta_{p_{k}\rho}=p_{k}\rho-p_{k+1}\mathcal{R}(dp_{k}\rho)$, $\nu$ is
an at least $(k+1)$-contact $(n+k-1)$-form, and $\mu$, resp. $\eta$ is at
least $(k+2)$-contact.

Again, in the decomposition of $\pi_{r+1,r}^{*}\rho$ in (4),
$\theta_{p_{k}\rho}+p_{k+1}d\nu$ and $\eta$ are global (being the at most
$(k+1)$-contact and the at least $(k+2)$-contact part of $\rho$,
respectively), while, in general, the decomposition
$\theta_{p_{k}\rho}+p_{k+1}d\nu$ is not invariant under changes of fibred
coordinates. And $p_{k+1}d\rho$ is unique (independent upon a choice of
$\rho$).

Also the concept of Lepage equivalent of a Lagrangian extends to
$(n+k)$-forms. Given a $k$-contact form $\sigma$, by a Lepage equivalent of
$\sigma$ we mean a Lepage form $\rho$ such that $p_{k}\rho=\sigma$. Theorem
3.13 then gives the structure of Lepage equivalents, and guarantees local
existence.

There remains to answer a question about existence of global Lepage
equivalents of higher degrees. The result is affirmative, and its proof is a
generalization of a proof for $n$-forms [48]. The idea is to show that there
exists a Lepage equivalent which is a global form on a closed submanifold of
$J^{2r+1}\boldsymbol{Y}$. Then, since the (local) form defines a soft sheaf of
forms it can be extended to a global form on the whole space.

Let us consider the canonical injection $\iota_{r+1,r}\colon
J^{2r+1}\boldsymbol{Y}\to J^{r+1}(J^{r}\boldsymbol{Y})$ defined by
$\iota_{r+1,r}(j^{2r+1}_{x}\gamma)$ $=j^{r+1}_{x}(J^{r}\gamma)$.

######  Theorem 3.14.

Let $\tilde{\phi}$ be a $(n+k)$-form such that $p_{l}\tilde{\phi}=0$, $l\geq
k+1$, e.g., $\tilde{\phi}$ is the $p_{k}$ component defined on
$J^{1}(J^{r}\boldsymbol{Y})$ of a $(n+k)$-form on $J^{r}\boldsymbol{Y}$. There
exists a $(n+k)$-form $\tilde{\eta}$ on $J^{r+1}(J^{r}\boldsymbol{Y})$ such
that

  1. $(1)$

$\tilde{\eta}$ is $(\pi_{r})_{r+1,0}$-horizontal and $\tilde{\eta}$ is a
$(k+1)$-contact $(n+k)$-form on $J^{r+1}(J^{r}\boldsymbol{Y})$;

  2. $(2)$

for every $(\pi_{r})_{r+1,0}$-vertical vector field $\boldsymbol{Z}$ on
$J^{r+1}(J^{r}\boldsymbol{Y})$ $($i.e., such that the projection of
$\boldsymbol{Z}$ on $J^{r}\boldsymbol{Y}$ is zero$)$

$\displaystyle\iota_{r+1,r}^{*}L_{\boldsymbol{Z}}((\pi_{r})^{*}_{r+1,1}\tilde{\phi}+\tilde{\eta})$

is a $(k+1)$-contact $(n+k)$-form on $J^{r+1}(J^{r}\boldsymbol{Y})$;

  3. $(3)$

the form $\iota_{r+1,r}^{*}((\pi_{r})^{*}_{r+1,1}\tilde{\phi}+\tilde{\eta})$
is a Lepage $(n+k)$-form.

###### Proof.

It is enough to prove uniqueness of $\tilde{\eta}$ when restricted to the
submanifold $\iota_{r+1,r}(J^{2r+1}\boldsymbol{Y})$. In fact, for $\rho$ a
$(n+k)$ form on $J^{r}\boldsymbol{Y}$, we can take
$\tilde{\eta}=-\mathcal{I}(p_{k}\rho)$ when $\tilde{\phi}=p_{k}\rho$. The
result follows by the uniqueness of $\mathcal{I}(p_{k}\rho)$ which implies
that it is a globally defined form on a closed subset of
$J^{2r+1}\boldsymbol{Y}$ (defined by the injection above), therefore since the
(local) form defines a soft sheaf of forms on $J^{2r+1}\boldsymbol{Y}$, then
$\tilde{\eta}$ can be globalized. ∎

The most important Lepage forms of degree $>n$ are Lepage equivalents of
canonical source forms. By the above, if $\sigma$ is a canonical source form
of degree $n+k$, $k\geq 1$, and order $r$, i.e., $\mathcal{I}(\sigma)$ is
$\pi_{2r+1,r}$-projectable, and $p_{k}\sigma=\sigma=\mathcal{I}(\sigma)$, then
all its local Lepage equivalents take the form

$\displaystyle\rho=\theta_{\sigma}+d\nu+\mu=\theta_{\sigma}+p_{k+1}d\nu+\eta,$
(3.11)

where

$\displaystyle\theta_{\sigma}=\sigma-p_{k+1}\mathcal{R}(d\sigma),$

$\nu$ is an arbitrary at least $(k+1)$-contact $(n+k-1)$-form, and $\mu$
(resp. $\eta$) is an arbitrary at least $(k+2)$-contact form. Moreover,

$\displaystyle p_{k+1}d\theta_{\sigma}=\mathcal{I}(d\sigma).$ (3.12)

In particular, if $\varepsilon$ is a dynamical form then
$p_{2}d\theta_{\varepsilon}$ is the canonical Helmholtz form
$H_{\varepsilon}=\mathcal{I}(d\varepsilon)$.

Comparing Lepage forms of different degrees yields the following results:

######  Theorem 3.15.

Let $q\geq n$. If $\alpha$ is a Lepage equivalent of $R_{q}([\rho])$ then
$d\alpha$ is a Lepage equivalent of $R_{q+1}([d\rho])$.

###### Proof.

Since $\alpha$ is a Lepage form, $d\alpha$ is Lepage, as trivially follows
from the definition. Denote $\sigma=R_{q}([\rho])$. If $q=n$, $\sigma$ is a
Lagrangian, hence $R_{n}([\rho])=h\rho$, and
$R_{n+1}([d\rho])=\mathcal{I}(d\rho)=\mathcal{I}(dh\rho)$. Thus
$\alpha=\theta_{h\rho}+d\nu+\mu$, and $d\alpha$ is a Lepage equivalent of
$p_{1}d\alpha$. However, by (3.12),
$p_{1}d\alpha=p_{1}d\theta_{h\rho}=\mathcal{I}(dh\rho)=R_{n+1}([d\rho])$. If
$q=n+k$, $k>1$, we have $R_{n+k}([\rho])=\mathcal{I}(\rho)$, and
$R_{n+k+1}([d\rho])=\mathcal{I}(d\rho)=\mathcal{I}(d\mathcal{I}(\rho))=\mathcal{I}(d\sigma)$.
The form $d\alpha$ is a Lepage equivalent of $p_{k+1}d\alpha$, which by (3.12)
is equal to $p_{k+1}d\theta_{\sigma}=\mathcal{I}(d\sigma)=R_{n+k+1}([d\rho])$.
∎

######  Theorem 3.16.

A closed Lepage equivalent of a canonical source form of degree $n+k$, $k\geq
1$, is locally equal to $d\alpha$ where $\alpha$ is a Lepage equivalent of a
canonical source form of degree $n+k-1$ if $k\geq 2$, resp. of a Lagrangian if
$k=1$.

###### Proof.

If $\beta$ is a closed Lepage equivalent of
$\sigma=p_{k}\beta=\mathcal{I}(\beta)$, then locally $\beta=d\alpha$ where
(with help of the contact homotopy operator), $\alpha=\mathcal{A}\beta$, and
$p_{k-1}\alpha=\mathcal{A}p_{k}\beta=\mathcal{A}\sigma$.

If $k=1$ then $p_{0}\alpha=h\alpha$ is a Lagrangian, and
$p_{1}d\alpha=p_{1}\beta=\mathcal{I}(\beta)$, so that
$\alpha=\mathcal{A}\beta$ is a Lepage equivalent of
$h\alpha=\mathcal{A}\sigma$.

If $k\geq 2$, set $\varepsilon=\mathcal{I}(\mathcal{A}\sigma)$. $\varepsilon$
is a canonical source form of degree $n+k-1$ representing the class
$[\mathcal{A}\sigma]$. Therefore (by the preceding theorem), if
$\alpha^{\prime}$ is a Lepage equivalent of $\mathcal{I}(\mathcal{A}\sigma)$
then $d\alpha^{\prime}$ is a Lepage equivalent of
$\mathcal{I}(d\varepsilon)=\mathcal{I}(d\mathcal{A}\sigma)=\mathcal{I}(dp_{k-1}\alpha)=\mathcal{I}(d\alpha)=\mathcal{I}(\beta)=\sigma$.
Now, both $d\alpha^{\prime}$ and $\beta$ are Lepage equivalents of $\sigma$,
hence we have $\beta=d\alpha^{\prime}+d\nu+\mu$, where $\nu$ is at least
$(k+1)$-contact and $\mu$ is at least $(k+2)$-contact, and since $d\beta=0$,
we get $d\mu=0$, i.e., locally $\mu=d\tau$ and $\tau=\mathcal{A}\mu$. Hence,
$\beta=d(\alpha^{\prime}+\nu+\tau)$, and
$\alpha=\mathcal{A}\beta=\alpha^{\prime}+\nu+\tau$ is a Lepage form (differing
from the Lepage $(k-1)$-form $\alpha^{\prime}$ by an at least $(k+1)$-contact
form), and it is a Lepage equivalent of
$p_{k-1}\alpha=p_{k-1}\alpha^{\prime}=\mathcal{I}(\mathcal{A}\sigma)$, as
required. Moreover, we note that since
$p_{k-1}\alpha=\mathcal{A}p_{k}\beta=\mathcal{A}\sigma$, it holds
$\mathcal{I}(\mathcal{A}\sigma)=\mathcal{A}\sigma=\mathcal{A}(\mathcal{I}\sigma)$.
∎

######  Corollary 3.17.

  1. $(1)$

For every canonical source form $\sigma$

$\displaystyle\mathcal{I}(\mathcal{A}\sigma)=\mathcal{A}\sigma=\mathcal{A}(\mathcal{I}\sigma).$

  2. $(2)$

If $\sigma$ is a canonical source $(n+k)$-form then $\mathcal{A}\sigma$ is a
canonical source $(n+k-1)$-form.

######  Corollary 3.18.

  1. $(1)$

If $\beta$ is a closed Lepage equivalent of a dynamical form $\varepsilon$
then $\mathcal{A}\beta$ is a Lepage equivalent of the Lagrangian
$\lambda=\mathcal{A}\varepsilon$.

  2. $(2)$

If $\beta$ is a closed Lepage equivalent of a canonical source form $\sigma$
then $\mathcal{A}\beta$ is a Lepage equivalent of the canonical source form
$\mathcal{A}\sigma$.

The above properties of Lepage forms suggest a new representation of the
variational sequence, based on Lepage equivalents of canonical source forms,
as follows: For every $k\geq 0$ we have a mapping $\operatorname{Lep}_{n+k}$
assigning to every Lagrangian $\lambda\in\Lambda^{r}_{n,\boldsymbol{X}}$, if
$k=0$, and every canonical source form
$\sigma\in\mathcal{I}(\Lambda^{r}_{n+k})$ if $k\geq 1$, the family
$\\{\rho\\}$ of its Lepage equivalents (3.11).

With help of Lepage mappings Theorem 3.15 can be reformulated as follows:

Theorem 3.15′. If $\operatorname{Lep}_{q}(R_{q}([\rho]))=\\{\alpha\\}$ then
$\operatorname{Lep}_{q+1}(R_{q+1}([d\rho]))=\\{d\alpha\\}$, or

$\displaystyle\\{d\operatorname{Lep}_{q}(R_{q}([\rho]))\\}=\\{\operatorname{Lep}_{q+1}(R_{q+1}([d\rho]))\\}.$

The Lepage maps then induce the following representation mappings:

$\displaystyle\tilde{R}_{q}=R_{q}\qquad\text{for}\quad 0\leq q\leq
n-1\qquad\text{and}\qquad\tilde{R}_{q}=\operatorname{Lep}_{q}\circ
R_{q}\qquad\text{for}\quad q\geq n.$

Note that for $q>n$ the images by $\tilde{R}_{q}$ of classes in
$\mathcal{V}^{r}_{q}$ are subsheaves of $\Lambda^{r}_{q}$.

With help of the representation mappings $\tilde{R}_{q}$ we construct to the
variational sequence $0\to{\mathbb{R}}_{\boldsymbol{Y}}\to\mathcal{V}^{r}_{*}$
a representation sequence

$\displaystyle
0\to{\mathbb{R}}_{\boldsymbol{Y}}\to\tilde{R}_{0}(\mathcal{V}^{r}_{0})\to\tilde{R}_{1}(\mathcal{V}^{r}_{1})\to\tilde{R}_{2}(\mathcal{V}^{r}_{2})\to\cdots$

shortly denoted by
$0\to{\mathbb{R}}_{\boldsymbol{Y}}\to\tilde{R}_{*}(\mathcal{V}^{r}_{*})$, and
called the Lepage representation of the variational sequence
$0\to{\mathbb{R}}_{\boldsymbol{Y}}\to\mathcal{V}^{r}_{*}$.

######  Theorem 3.19.

The representation sequence
$0\to{\mathbb{R}}_{\boldsymbol{Y}}\to\tilde{R}_{*}(\mathcal{V}^{r}_{*})$ is
exact, and every morphism
$\tilde{R}_{q}(\mathcal{V}^{r}_{q})\to\tilde{R}_{q+1}(\mathcal{V}^{r}_{q+1})$
for $q\geq n$, is the exterior derivative operator $d$ $($acting on classes of
Lepage forms$)$.

###### Proof.

Denote

$\displaystyle\tilde{E}_{q}\colon\
\tilde{R}_{q}(\mathcal{V}^{r}_{q})\to\tilde{R}_{q+1}(\mathcal{V}^{r}_{q+1}),\qquad
q\geq n.$

With the representation mapping $\tilde{R}_{q}$, Theorem 3.15 states that ‘if
$\tilde{R}_{q}([\rho])=\\{\alpha\\}$ then $\tilde{R}_{q+1}([d\rho])$
$=\\{d\alpha\\}$’, in other words, the exterior derivative operator $d$
extends in an obvious way to classes of Lepage forms; denoting the exterior
derivative of classes by the same symbol $d$, we can see that

$\displaystyle\tilde{E}_{q}(\\{\alpha\\})=\\{d\alpha\\}\equiv d\\{\alpha\\}.$

This proves that $\tilde{E}_{q}=d$, $q\geq n$.

In view of this result, and since $\tilde{R}_{q}=R_{q}$ for $0\leq q\leq n-1$,
and the Takens representation sequence is exact, it remains to check the
exactness in the arrow $E_{n-1}$. To this end, let us write the variational
sequence and its Lepage representation in form of the following diagram:

$\displaystyle\begin{CD}\cdots
@>{\mathcal{E}_{n-2}}>{}>\mathcal{V}_{n-1}^{r}@>{\mathcal{E}_{n-1}}>{}>\mathcal{V}_{n}^{r}@>{\mathcal{E}_{n}}>{}>\mathcal{V}_{n+1}^{r}@>{\mathcal{E}_{n+1}}>{}>\cdots\\\
@V{{R_{n-1}}}V{}V@V{{\tilde{R}_{n}}}V{}V@V{{\tilde{R}_{n+1}}}V{}V\\\ \cdots
@>{E_{n-2}}>{}>\Lambda_{n-1,\boldsymbol{X}}^{r}@>{E_{n-1}}>{}>\tilde{R}_{n}(\mathcal{V}^{r}_{n})@>{d}>{}>\tilde{R}_{n+1}(\mathcal{V}_{n+1}^{r})@>{d}>{}>\cdots\end{CD}$

Then for every
$\eta\in\tilde{R}_{n-1}(\mathcal{V}_{n-1}^{r})=\Lambda^{r}_{n-1,\boldsymbol{X}}$
we obtain

$\displaystyle(d\circ E_{n-1})(\eta)=(d\circ E_{n-1})(R_{n-1}([\rho]))=(d\circ
E_{n-1}\circ R_{n-1})([\rho])$ $\displaystyle\hphantom{(d\circ
E_{n-1})(\eta)}{}=(d\circ\tilde{R}_{n}\circ\mathcal{E}_{n-1})([\rho])=d(\tilde{R}_{n}([d\rho]))=d\operatorname{Lep}_{n}(R_{n}([d\rho])$
$\displaystyle\hphantom{(d\circ
E_{n-1})(\eta)}{}=\operatorname{Lep}_{n+1}(R_{n+1}([dd\rho])=0.$ ∎

It is worth to write all the three sequences within one scheme as follows:

$\displaystyle\begin{CD}\cdots
@>{\mathcal{E}_{n-1}}>{}>\Lambda_{n}^{r}/\Theta_{n}^{r}@>{\mathcal{E}_{n}}>{}>\Lambda_{n+1}^{r}/\Theta_{n+1}^{r}@>{\mathcal{E}_{n+1}}>{}>\Lambda_{n+2}^{r}/\Theta_{n+2}^{r}@>{\mathcal{E}_{n+2}}>{}>\cdots\\\
@V{{R_{n}}}V{}V@V{{R_{n+1}}}V{}V@V{{R_{n+2}}}V{}V\\\ \cdots
@>{E_{n-1}}>{}>\Lambda_{n,\boldsymbol{X}}^{r+1}@>{E_{n}}>{}>\Lambda_{n+1,1,\boldsymbol{Y}}^{r+1}@>{E_{n+1}}>{}>\mathcal{I}\Lambda_{n+2}^{r+1}@>{E_{n+2}}>{}>\cdots\\\
@V{{\operatorname{Lep}_{n}}}V{}V@V{{\operatorname{Lep}_{n+1}}}V{}V@V{{\operatorname{Lep}_{n+2}}}V{}V\\\
\cdots
@>{E_{n-1}}>{}>\operatorname{Lep}(\Lambda_{n,\boldsymbol{X}}^{r+1})@>{d}>{}>\operatorname{Lep}(\Lambda_{n+1,1,\boldsymbol{Y}}^{r+1})@>{d}>{}>\operatorname{Lep}(\mathcal{I}\Lambda_{n+2}^{r+1})@>{d}>{}>\cdots\end{CD}$

As we have seen, for $n=\dim X=1$ (mechanics), Lepage equivalent is unique for
every $k\geq 0$ (recall that this is the Cartan form of degree $n+k$). This
means that for every $k$,

$\displaystyle\operatorname{Lep}_{n+k}\colon\
\mathcal{I}(\Lambda^{*}_{n+k})\to\Lambda^{*}_{n+k},$

i.e., $\operatorname{Lep}_{n+k}(\mathcal{I}\rho)$ is an element of the sheaf
$\Lambda^{*}_{n+k}$ (rather than a family of elements of the sheaf), and $d$
is the “true” exterior derivative of differential forms. Hence “morally” (up
to orders of the sheaves) the Lepage representation sequence becomes a
subsequence of the de Rham sequence.

Finally we want to stress that in view of the two corollaries of Theorem 3.15
the contact homotopy operator $\mathcal{A}$ restricts to canonical source
forms. This means that one can obtain primitives (inverse images) not only of
closed Lepage forms in the Lepage representation, but also of canonical source
forms in the Takens representation. For a primitive of a variational dynamical
form $\varepsilon$ (i.e., a Lagrangian) one has
$\lambda=\mathcal{A}\varepsilon$ which in coordinates is the celebrated Tonti
Lagrangian [94, 95]. An analogous formula holds then, indeed, also for forms
of higher degrees: if $\sigma=E_{n+k}(\eta)$ then $\mathcal{A}\sigma=\eta$.

######  Example 3.20.

Let us write down explicit examples of Cartan forms of higher degrees in
mechanics ($n=1$). Denote local fibred coordinates on $\boldsymbol{Y}$ by
$(t,q^{\sigma})$ and the associated coordinates on $J^{3}\boldsymbol{Y}$ by
$(t,q^{\sigma},\dot{q}^{\sigma},\ddot{q}^{\sigma},\dddot{q}^{\sigma})$, and
let

$\displaystyle\omega^{\sigma}=dq^{\sigma}-\dot{q}^{\sigma}dt,\qquad\dot{\omega}^{\sigma}=\omega^{\sigma}_{1}=d\dot{q}^{\sigma}-\ddot{q}^{\sigma}dt,\qquad\ddot{\omega}^{\sigma}=\omega^{\sigma}_{2}=d\ddot{q}^{\sigma}-\dddot{q}^{\sigma}dt.$

By Theorem 3.11, formula (3.10) we have:

$\bullet$ For $\rho$ of degree $n=1$ and such that $h\rho=\lambda=Ldt$ is of
order $r=1$: $\mathcal{R}(d\lambda)=-\frac{\partial
L}{\partial\dot{q}^{\sigma}}\omega^{\sigma}$, hence we obtain

$\displaystyle\theta_{\lambda}=\lambda+\frac{\partial
L}{\partial\dot{q}^{\sigma}}\omega^{\sigma},$

which is the classical Cartan $1$-form.

$\bullet$ For $\rho$ of degree $n+1=2$ such that $p_{1}\rho$ is a second-order
dynamical form, i.e., $p_{1}\rho=\varepsilon=E_{\sigma}\omega^{\sigma}\land
dt$, the Cartan $2$-form is
$\theta_{\varepsilon}=\varepsilon-\mathcal{R}(d\varepsilon)$. To compute
$\mathcal{R}(d\varepsilon)$ we need the following formula for the residual
operator applied to a second-order $2$-contact $3$-form: denote

$\displaystyle\alpha=\sum_{i,j=0}^{2}A_{\sigma\nu}^{ij}\omega^{\sigma}_{i}\land\omega^{\nu}_{j}\land
dt,\qquad A_{\sigma\nu}^{ij}=-A_{\nu\sigma}^{ji}$

Then

$\displaystyle\mathcal{R}(\alpha)=\frac{1}{2}\left(A_{\sigma\nu}^{1j}-A_{\nu\sigma}^{j1}-\frac{d}{dt}\big{(}A_{\sigma\nu}^{2j}-A_{\nu\sigma}^{j2}\big{)}\right)\omega^{\sigma}\land\omega^{\nu}_{j}$
$\displaystyle\hphantom{\mathcal{R}(\alpha)=}{}-\frac{1}{2}\big{(}A_{\sigma\nu}^{2j}-A_{\nu\sigma}^{j2}\big{)}\omega^{\sigma}\land\omega^{\nu}_{j+1}+\frac{1}{2}\big{(}A_{\sigma\nu}^{2j}-A_{\nu\sigma}^{j2}\big{)}\omega^{\sigma}_{1}\land\omega^{\nu}_{j}$

(summation over $j=0,1,2$). Taking $\alpha=d\varepsilon$ we obtain

$\displaystyle\theta_{\varepsilon}=\varepsilon+\frac{1}{2}\Bigl{(}\frac{\partial
E_{\sigma}}{\partial\dot{q}^{\nu}}-\frac{d}{dt}\frac{\partial
E_{\sigma}}{\partial\ddot{q}^{\nu}}\Bigr{)}\omega^{\sigma}\land\omega^{\nu}-\frac{\partial
E_{\sigma}}{\partial\ddot{q}^{\nu}}\dot{\omega}^{\sigma}\land\omega^{\nu}.$

For a locally variational form $\varepsilon$ this is the Lepage equivalent of
$\varepsilon$, introduced in [64] (see also [65]).

$\bullet$ For $\rho$ of degree $n+2=3$ such that $p_{2}\rho$ is a Helmholtz-
like form denote
$p_{2}\rho=\eta=H_{\sigma\nu}^{0}\omega^{\sigma}\land\omega^{\nu}\land
dt+H_{\sigma\nu}^{1}\omega^{\sigma}\land\dot{\omega}^{\nu}\land
dt+H_{\sigma\nu}^{2}\omega^{\sigma}\land\ddot{\omega}^{\nu}\land dt$; the
Cartan $3$-form is $\theta_{\eta}=\eta-\mathcal{R}(d\eta)$. The formula for
the residual operator applied to a general second-order $3$-contact $4$-form

$\displaystyle\alpha=\sum_{i,j,k=0}^{2}A_{\sigma\nu\rho}^{ijk}\omega^{\sigma}_{i}\land\omega^{\nu}_{j}\land\omega^{\rho}_{k}\land
dt,\qquad
A_{\sigma\nu\rho}^{ijk}=-A_{\nu\sigma\rho}^{jik}=A_{\nu\rho\sigma}^{jki},$

reads

$\displaystyle\mathcal{R}(\alpha)=-\left(\big{\\{}A_{\sigma\nu\rho}^{1jk}\big{\\}}-\frac{d}{dt}\big{\\{}A_{\sigma\nu\rho}^{2jk}\big{\\}}\right)\omega^{\sigma}\land\omega^{\nu}_{j}\land\omega^{\rho}_{k}+2\big{\\{}A_{\sigma\nu\rho}^{2jk}\big{\\}}\omega^{\sigma}\land\omega^{\nu}_{j}\land\omega^{\rho}_{k+1}$
$\displaystyle\hphantom{\mathcal{R}(\alpha)=}{}-\big{\\{}A_{\sigma\nu\rho}^{2jk}\big{\\}}\omega^{\sigma}_{1}\land\omega^{\nu}_{j}\land\omega^{\rho}_{k},$

where
$\\{A_{\sigma\nu\rho}^{ijk}\\}=\frac{1}{3}(A_{\sigma\nu\rho}^{ijk}+A_{\rho\sigma\nu}^{kij}+A_{\nu\rho\sigma}^{jki})$,
and summation over $j,k=0,1,2$ applies. Now we take $\alpha=d\eta$, so that

$\displaystyle\big{\\{}A_{\sigma\nu\rho}^{000}\big{\\}}=\left\\{\frac{\partial
H^{0}_{\sigma\nu}}{\partial
q^{\rho}}\right\\},\qquad\big{\\{}A_{\sigma\nu\rho}^{001}\big{\\}}=\left\\{\frac{\partial
H^{0}_{\sigma\nu}}{\partial\dot{q}^{\rho}}+\frac{\partial
H^{1}_{\nu\rho}}{\partial
q^{\sigma}}\right\\},\qquad\big{\\{}A_{\sigma\nu\rho}^{011}\big{\\}}=\left\\{\frac{\partial
H^{1}_{\sigma\nu}}{\partial\dot{q}^{\rho}}\right\\},$
$\displaystyle\big{\\{}A_{\sigma\nu\rho}^{002}\big{\\}}=\left\\{\frac{\partial
H^{0}_{\sigma\nu}}{\partial\ddot{q}^{\rho}}+\frac{\partial
H^{2}_{\nu\rho}}{\partial
q^{\sigma}}\right\\},\qquad\\!\big{\\{}A_{\sigma\nu\rho}^{012}\big{\\}}=\left\\{\frac{\partial
H^{1}_{\sigma\nu}}{\partial\ddot{q}^{\rho}}-\frac{\partial
H^{2}_{\sigma\rho}}{\partial\dot{q}^{\nu}}\right\\},\qquad\\!\big{\\{}A_{\sigma\nu\rho}^{022}\big{\\}}=\left\\{\frac{\partial
H^{2}_{\sigma\nu}}{\partial\ddot{q}^{\rho}}\right\\},$

and $\\{A_{\sigma\nu\rho}^{ijk}\\}=0$ for $i,j,k\neq 0$, and obtain the Cartan
$3$-form of $\eta$ as follows:

$\displaystyle\theta_{\eta}=\eta-\mathcal{R}(d\eta)=\eta+\left(\big{\\{}A_{\sigma\nu\rho}^{100}\big{\\}}-\frac{d}{dt}\big{\\{}A_{\sigma\nu\rho}^{200}\big{\\}}\right)\omega^{\sigma}\land\omega^{\nu}\land\omega^{\rho}$
$\displaystyle\hphantom{\theta_{\eta}=}{}+2\left(\big{\\{}A_{\sigma\nu\rho}^{101}\big{\\}}-\big{\\{}A^{200}_{\sigma\nu\rho}\big{\\}}+\frac{1}{2}\big{\\{}A^{002}_{\sigma\nu\rho}\big{\\}}-\frac{d}{dt}\big{\\{}A_{\sigma\nu\rho}^{201}\big{\\}}\right)\omega^{\sigma}\land\omega^{\nu}\land\dot{\omega}^{\rho}$
$\displaystyle\hphantom{\theta_{\eta}=}{}+2\left(\big{\\{}A_{\sigma\nu\rho}^{102}\big{\\}}-\big{\\{}A^{201}_{\sigma\nu\rho}\big{\\}}-\frac{d}{dt}\big{\\{}A_{\sigma\nu\rho}^{202}\big{\\}}\right)\omega^{\sigma}\land\omega^{\nu}\land\ddot{\omega}^{\rho}-2\big{\\{}A^{202}_{\sigma\nu\rho}\big{\\}}\omega^{\sigma}\land\omega^{\nu}\land\dddot{\omega}^{\rho}$
$\displaystyle\hphantom{\theta_{\eta}=}{}+2\left(\big{\\{}A_{\sigma\nu\rho}^{021}\big{\\}}-\big{\\{}A_{\sigma\nu\rho}^{210}\big{\\}}\right)\omega^{\sigma}\land\dot{\omega}^{\nu}\land\dot{\omega}^{\rho}+2\left(\big{\\{}A_{\sigma\nu\rho}^{022}\big{\\}}-\big{\\{}A_{\sigma\nu\rho}^{202}\big{\\}}\right)\omega^{\sigma}\land\dot{\omega}^{\nu}\land\ddot{\omega}^{\rho}$

with the non-zero $A$’s as above (cf. [69] where a corresponding formula for
the variationally trivial case was obtained).

###  3.5 The variational order

A key issue of the Krupka variational sequence for the variational calculus,
making the main difference between his and other approaches ($C$-spectral
sequences, the variational bicomplex and infinite order variational sequences)
is that the variational sequence (2.2) fixes (and thus defines) the order of
variational problems in the sense of the following definition:

######  Definition 3.21.

Let $\eta\in\Lambda^{s}_{q}$ be a source form, or a Lepage form of degree
$q\geq 1$ defined on (an open subset of) $J^{s}\boldsymbol{Y}$. We say that
$\eta$ has the variational order $r$ if it comes from the variational sequence
of order $r$ (i.e., it is a Takens or Lepage representation of a variational
class $[\rho]\in\Lambda^{r}_{q}/\Theta^{r}_{q}$.

How to understand this concept? A variational class
$[\rho]\in\Lambda^{r}_{q}/\Theta^{r}_{q}$ concerns local differential forms of
degree $q$ and order $r$. Takens representation (and thus also Lepage
representation) is based on the horizontalization operator $h=p_{0}$ and the
interior Euler operator ${\cal I}$ which provide a differential form of
generally higher order. For example, if $r=1$ we have (for $q=n$) the
Lagrangian $\lambda=R_{n}([\rho])=h\rho$, (for $q=n+1$) the dynamical form
$\varepsilon=R_{n+1}([\rho])={\cal I}(\rho)$, (for $q=n+2$) the Helmholtz-like
form $H=R_{n+2}([\rho])={\cal I}(\rho)$, etc., all of order $2$. As given by
the structure of ${\cal I}$, generically, the components of the differential
$(n+k)$-forms ($k\geq 1$) arising from $\Lambda^{r}_{n+k}/\Theta^{r}_{n+k}$
are polynomials in the derivatives starting from the order $r+1$ up to the
order $2r$ in the linear term. Notice that such a polynomial behaviour is
known to be typical for functions known as “variational derivatives”. However
the point is that the “variational derivatives” are not just general
polynomials: the analysis of ${\cal I}$ shows that the coefficients of these
polynomials have a determined structure of derivatives of the components of
$\rho$ (which are of order $r$), with prescribed symmetrization/skew-
symmetrization rules in the indices, in other words they have to satisfy
certain symmetry and integrability conditions (“order reducibility
conditions”). For instance, a dynamical form $\varepsilon$ of order $2$ (which
defines a system of second order differential equations)222In the Takens
representation $\varepsilon$ is an element of the sheaf of differential forms
$\Lambda^{2}_{n+1}$. can be of variational order $1$ or $2$. In particular, if
$\varepsilon$ is locally variational (comes locally from a Lagrangian, i.e.,
its image by the Helmholtz morphism in the Takens representation sequence
vanishes) we obtain that if the variational order of $\varepsilon$ is $1$, we
have local Lagrangians arising as $\lambda=h\rho$ from first order $\rho$;
thus $\lambda$ is of order $2$, or of order $1$ if $h\rho$ is
$\pi_{2,1}$-projectable. If the variational order of $\varepsilon$ is
nontrivially $2$, we have local $3$rd order Lagrangians arising as
$\lambda=h\rho$ from second order $\rho$, or second order Lagrangians if
$h\rho$ is $\pi_{3,2}$-projectable (which, indeed, cannot be reduced to the
first order by extracting a variationally trivial Lagrangian). The same order
discussion concerns global Lagrangians if $H^{n}_{\rm
dR}(\boldsymbol{Y})=\\{0\\}$.

######  Example 3.22.

Lagrangians, dynamical forms and Helmholtz-like forms of variational order
one:

$\bullet$ Let $n=\dim\boldsymbol{X}=1$. Any
$\rho\in\Lambda^{1}_{1}/\Theta^{1}_{1}$ takes the form

$\displaystyle\rho=Adt+F_{\sigma}\omega^{\sigma}+C_{\sigma}d\dot{q}^{\sigma}\sim
Adt+C_{\sigma}d\dot{q}^{\sigma},$

so that

$\displaystyle\lambda=h\rho=Ldt,\qquad\text{where}\quad
L=A+C_{\sigma}\ddot{q}^{\sigma}.$

We can see that a generic Lagrangian of variational order one in mechanics is
of second order, affine in the second derivatives. Then, in particular,
reducible second-order Lagrangians (and, of course, first-order Lagrangians)
correspond to those classes $[\rho]$ which are generated by
$\pi_{1,0}$-horizontal forms.

If $n=\dim\boldsymbol{X}>1$, we denote $\omega_{i_{1}}=\partial/\partial
x^{i_{1}}\rfloor\omega_{0}$, $\omega_{i_{1}i_{2}}=\partial/\partial
x^{i_{2}}\rfloor\omega_{i_{1}}$, etc.; we have

$\displaystyle\rho\sim
A\omega_{0}+\sum_{k=1}^{n}C_{\sigma_{1}\dots\sigma_{k}}^{j_{1}\dots
j_{k},i_{1}\dots i_{k}}dy^{\sigma_{1}}_{j_{1}}\land\dots\land
dy^{\sigma_{k}}_{j_{k}}\land\omega_{i_{1}\dots i_{k}},$

where the coefficients $C_{\sigma_{1}\dots\sigma_{k}}^{j_{1}\dots
j_{k},i_{1}\dots i_{k}}$ are totally antisymmetric in the indices $i_{1}\dots
i_{k}$, and totally symmetric in the pairs
$(\sigma_{1},j_{1})\dots(\sigma_{k},j_{k})$. Now, $\lambda=h\rho=L\omega_{0}$,
where

$\displaystyle
L=A+\bar{C}_{\sigma_{1}}^{j_{1},i_{1}}y^{\sigma_{1}}_{j_{1}i_{1}}+\bar{C}_{\sigma_{1}\sigma_{2}}^{j_{1}j_{2},i_{1}i_{2}}y^{\sigma_{1}}_{j_{1}i_{1}}y^{\sigma_{2}}_{j_{2}i_{2}}+\cdots+\bar{C}_{\sigma_{1}\dots\sigma_{n}}^{j_{1}\dots
j_{n},i_{1}\dots i_{n}}y^{\sigma_{1}}_{j_{1}i_{1}}\cdots
y^{\sigma_{n}}_{j_{n}i_{n}}$

(here the $\bar{C}$’s are just corresponding multipliers of the $C$’s). Thus,
in field theory, a generic Lagrangian of variational order one is of second
order, polynomial of degree $n$ in the second derivatives, with coefficients
obeying the symmetry-antisymmetry relations described above. In particular,
reducible (and first-order) Lagrangians correspond to generating forms $\rho$
that are $\pi_{1,0}$-horizontal. The most famous Lagrangian of this kind is
the scalar curvature, giving rise to the Einstein equations.

$\bullet$ Let again $n=\dim\boldsymbol{X}=1$. Any
$\rho\in\Lambda^{1}_{2}/\Theta^{1}_{2}$ takes the form

$\displaystyle\rho\sim A^{1}_{\sigma}\omega^{\sigma}\land
dt+A^{2}_{\sigma}d\omega^{\sigma}+B^{1}_{\sigma\nu}\omega^{\sigma}\land
d\dot{q}^{\nu}+C_{\sigma\nu}d\dot{q}^{\sigma}\land d\dot{q}^{\nu}$
$\displaystyle\hphantom{\rho}{}\sim\omega^{\sigma}\land(A_{\sigma}dt+B_{\sigma\nu}d\dot{q}^{\nu})+C_{\sigma\nu}d\dot{q}^{\sigma}\land
d\dot{q}^{\nu},$

where $C_{\sigma\nu}=-C_{\nu\sigma}$. In Takens representation we get the
dynamical form $\mathcal{I}(\rho)=\varepsilon=E_{\sigma}\omega^{\sigma}\land
dt$, where

$\displaystyle
E_{\sigma}=A_{\sigma}+B_{\sigma\nu}\ddot{q}^{\nu}-\frac{d}{dt}(2C_{\sigma\nu}\ddot{q}^{\nu})=A_{\sigma}+\bar{B}_{\sigma\nu}\ddot{q}^{\nu}-2\frac{\partial
C_{\sigma\nu}}{\partial\dot{q}^{\rho}}\ddot{q}^{\nu}\ddot{q}^{\rho}-2C_{\sigma\nu}\dddot{q}^{\nu}.$

Thus a generic dynamical form of variational order one in mechanics is of
order three, affine in the third derivatives, and polynomial of degree two in
the second derivatives $($with a special form of coefficients as above$)$.
Second-order dynamical forms of variational order one correspond to generating
forms $\rho$ such that $C_{\sigma\nu}=0$, meaning that the corresponding
variational class is represented by a $\omega^{\sigma}$-generated form $\rho$
and the differential equations are affine in second derivatives:

$\displaystyle\rho\sim A_{\sigma}\omega^{\sigma}\land
dt+B_{\sigma\nu}\omega^{\sigma}\land d\dot{q}^{\nu},\qquad\text{i.e.,}\qquad
E_{\sigma}=A_{\sigma}+B_{\sigma\nu}\ddot{q}^{\nu}.$

In particular, if variational ($[d\rho]=0$, i.e., $[\rho]=[d\lambda]$), they
come from second-order Lagrangians of the form

$\displaystyle L=A+C_{\sigma}\ddot{q}^{\sigma},\qquad\frac{\partial
C_{\sigma}}{\partial\dot{q}^{\nu}}=\frac{\partial
C_{\nu}}{\partial\dot{q}^{\sigma}},$

that, indeed, are reducible (equivalent with first-order Lagrangians).

Finally, all first-order dynamical forms are of variational order one and they
arise from the class $[\rho]=[A_{\sigma}\omega^{\sigma}\land dt]$.

$\bullet$ Any $\rho\in\Lambda^{1}_{3}/\Theta^{1}_{3}$ takes the form

$\displaystyle\rho\sim A^{1}_{\sigma\nu}\omega^{\sigma}\land\omega^{\nu}\land
dt+A^{2}_{\sigma\nu}\omega^{\sigma}\land
d\omega^{\nu}+A^{3}_{\sigma\nu}d\dot{q}^{\sigma}\land d\omega^{\nu}$
$\displaystyle\hphantom{\rho\sim}{}+B^{1}_{\sigma\nu\rho}\omega^{\sigma}\land\omega^{\nu}\land
d\dot{q}^{\rho}+B^{2}_{\sigma\nu\rho}\omega^{\sigma}\land d\dot{q}^{\nu}\land
d\dot{q}^{\rho}+C_{\sigma\nu\rho}d\dot{q}^{\sigma}\land d\dot{q}^{\nu}\land
d\dot{q}^{\rho},$

where $A^{1}_{\sigma\nu}=-A^{1}_{\nu\sigma}$,
$A^{3}_{\sigma\nu}=-A^{3}_{\nu\sigma}$,
$B^{1}_{\sigma\nu\rho}=-B^{1}_{\nu\sigma\rho}$,
$B^{2}_{\sigma\nu\rho}=-B^{2}_{\sigma\rho\nu}$, and the $C$’s are totally
antisymmetric. Then the corresponding Helmholtz-like form becomes
$\mathcal{I}(\rho)=\eta=H^{1}_{\sigma\nu}\omega^{\sigma}\land\omega^{\nu}\land
dt+H^{2}_{\sigma\nu}\omega^{\sigma}\land\dot{\omega}^{\nu}\land
dt+H^{3}_{\sigma\nu}\omega^{\sigma}\land\ddot{\omega}^{\nu}\land dt$, where,
as expected, the components have a polynomial structure in second and higher
derivatives:

$\displaystyle
H^{1}_{\sigma\nu}=A^{1}_{\sigma\nu}+B^{1}_{\sigma\nu\rho}\ddot{q}^{\rho}+\frac{1}{4}\frac{d}{dt}\bigl{(}A^{2}_{\sigma\nu}-A^{2}_{\nu\sigma}-2\big{(}B^{2}_{\sigma\nu\rho}-B^{2}_{\nu\sigma\rho}\big{)}\ddot{q}^{\rho}\bigr{)}-\frac{1}{2}\frac{d^{2}}{dt^{2}}\big{(}A^{3}_{\sigma\nu}-3C_{\sigma\nu\rho}\ddot{q}^{\rho}\big{)}$
$\displaystyle\hphantom{H^{1}_{\sigma\nu}}{}=\alpha^{1}_{\sigma\nu}+\alpha^{2}_{\sigma\nu\rho}\ddot{q}^{\rho}+\alpha^{3}_{\sigma\nu\rho\lambda}\ddot{q}^{\rho}\ddot{q}^{\lambda}+\alpha^{4}_{\sigma\nu\rho\lambda\kappa}\ddot{q}^{\rho}\ddot{q}^{\lambda}\ddot{q}^{\kappa}+\alpha^{5}_{\sigma\nu\rho}\dddot{q}^{\rho}+\alpha^{6}_{\sigma\nu\rho\lambda}\ddot{q}^{\rho}\dddot{q}^{\lambda}+\frac{3}{2}C_{\sigma\nu\rho}q^{\rho}_{4},$
$\displaystyle
H^{2}_{\sigma\nu}=-\frac{1}{2}\big{(}A^{2}_{\sigma\nu}+A^{2}_{\nu\sigma}\big{)}+\big{(}B^{2}_{\sigma\nu\rho}+B^{2}_{\nu\sigma\rho}\big{)}\ddot{q}^{\rho},\qquad
H^{3}_{\sigma\nu}=A^{3}_{\sigma\nu}-3C_{\sigma\nu\rho}\ddot{q}^{\rho}.$

The requirement that the class $[\rho]$ is $\omega^{\sigma}$-generated gives
$A^{3}_{\sigma\nu}=0$ and $C_{\sigma\nu\rho}=0$, and simplifies the Helmholtz-
like forms to $H^{3}_{\sigma\nu}=0$, and

$\displaystyle
H^{1}_{\sigma\nu}=\alpha^{1}_{\sigma\nu}+\alpha^{2}_{\sigma\nu\rho}\ddot{q}^{\rho}+\alpha^{3}_{\sigma\nu\rho\lambda}\ddot{q}^{\rho}\ddot{q}^{\lambda}+\alpha^{5}_{\sigma\nu\rho}\dddot{q}^{\rho}.$

If, moreover, $\eta$ is a Helmholtz form (i.e., $[d\rho]=0$), related with a
$\omega^{\sigma}$-generated class then it corresponds to a second-order
dynamical form $\varepsilon=E_{\sigma}\omega^{\sigma}\land dt$, where
$E_{\sigma}=A_{\sigma}+B_{\sigma\nu}\ddot{q}^{\nu}$, and
$B_{\sigma\nu}=B_{\nu\sigma}$.

The above discussion illustrates that the range of applications of the
variational sequence is wider than that of corresponding infinite order
constructions or even of some other known finite order constructions.
Remarkably, it gives solution of the

Inverse order problem (order reduction problem): If a differential $q$-form
$\eta$ of order $s$ is $($locally/globally$)$ trivial, or
$($locally/globally$)$ variational $($its image under the corresponding
morphism $E_{q}$ vanishes$)$ what is the ‘minimal order’ of the
$($local/global$)$ preimage $(q-1)$-form $\mu$ such that $\eta=E_{q-1}(\mu)$?

Note that this question is answered by the variational sequence, but is not
answered, e.g., by the variational bicomplex, or even by Takens
representation(!), since $\eta$ is an element of a sheaf of differential forms
of order $s$ which contains forms of different variational orders.

######  Example 3.23.

Consider a dynamical form $\varepsilon$ (nontrivially) of order three over a
fibred manifold with $n=\dim\boldsymbol{X}=1$. $\varepsilon$ belongs to
$\Lambda^{3}_{2}$ and represents a system of 3rd order ordinary differential
equations. Takens sequence gives us that if locally variational, $\varepsilon$
comes from a local Lagrangian of order $3$. However, from the variational
sequence we can get information about the variational order of $\varepsilon$,
and hence a finer result about the order of the local Lagrangians. Namely, the
following cases may arise:

  * •

The variational order of $\varepsilon$ is (nontrivially) three. This means
that in the variational sequence $\varepsilon$ is represented by a class
$[\rho]\in\Lambda^{3}_{2}/\Theta^{3}_{2}$. If $\varepsilon$ is locally
variational, any minimal order Lagrangian is of order three (not further
reducible).

  * •

The variational order of $\varepsilon$ is two. This means that in the
variational sequence $\varepsilon$ is represented by a class
$[\rho]\in\Lambda^{2}_{2}/\Theta^{2}_{2}$. If locally variational, it has a
(local) second-order Lagrangian.

  * •

The variational order of $\varepsilon$ is one. This means that in the
variational sequence $\varepsilon$ is represented by a class
$[\rho]\in\Lambda^{1}_{2}/\Theta^{1}_{2}$. As follows from the previous
example, if $\varepsilon$ is locally variational then the Tonti Lagrangian
(which is of order three) is locally reducible to a second-order Lagrangian
affine in the second derivatives.

##  4 Lie derivative in the variational sequence

One of the most important features of the geometric formulation of the
calculus of variations in jet bundles is the fact that variations can be
described by Lie derivatives of differential forms with respect to
prolongations of projectable vector fields. A famous example is the first
variation formula which takes the integral form [46] (cf. [32])

$\displaystyle\int_{\Omega}J^{r}\gamma^{*}L_{J^{r}\Xi}\lambda=\int_{\Omega}J^{2r-1}\gamma^{*}\big{(}J^{2r-1}\Xi\rfloor
d\rho\big{)}+\int_{\Omega}J^{2r-1}\gamma^{*}d\big{(}J^{2r-1}\Xi\rfloor\rho\big{)},$

where $\rho$ is a Lepage equivalent of $\lambda$, $\Xi$ is a projectable
vector field on $\boldsymbol{Y}$ (“variation vector field”), and
$\Omega\subset\boldsymbol{X}$ is a compact connected $n$-dimensional
submanifold with boundary. It is, however, even more interesting (and
fundamental) that with use of the differential calculus in jet bundles, this
formula can be (as mentioned in Section 3.3), invariantly expressed in the
“infinitesimal form” as a formula for differential forms on the manifold
$J^{2r}\boldsymbol{Y}$ [46]

$\displaystyle L_{J^{r}\Xi}\lambda=h\big{(}J^{2r-1}\Xi\rfloor
d\rho\big{)}+hd\big{(}J^{2r-1}\Xi\rfloor\rho\big{)},$ (4.1)

and (with the account of $\lambda=h\rho$) it yields a remarkable property of
the Lie derivative in the calculus of variations:

$\displaystyle L_{J^{2r}\Xi}h\rho=hL_{J^{2r-1}\Xi}\rho.$ (4.2)

Moreover, as discovered in [62], the Lie derivative commutes with the
Euler–Lagrange operator, as the Lie derivative of the Euler–Lagrange form of
$\lambda$ is the Euler–Lagrange form of the transformed Lagrangian:

$\displaystyle L_{J^{2r}\Xi}E_{\lambda}=E_{L_{J^{r}\Xi}\lambda}.$ (4.3)

Such properties of the Lie derivative provide an elegant intrinsic formulation
(and an easy proof) of Noether theorem, Noether–Bessel-Hagen theorem [5, 77],
conservation laws, and relationship between symmetries of a Lagrangian and its
Euler–Lagrange form (for any order and any number of independent variables)
[48, 55].

Having in mind the variational sequence and its representations, there arises
a question about generalization of these properties to variational forms of
any degree.

First we recall results obtained in [56, 57, 58]. Let ${\cal C}_{r}$ denote
the contact ideal of order $r$.

######  Definition 4.1.

A vector field $Z$ on $J^{r}\boldsymbol{Y}$ is called a contact symmetry if it
is a symmetry of the contact ideal ${\cal C}_{r}$ (meaning that for every
contact form $\omega$ the Lie derivative $L_{Z}\omega$ is a contact form.

Contact symmetries were characterized in [56]. It is worth notice that for any
projectable vector field $\Xi$ on $\boldsymbol{Y}$ the $r$-jet prolongation
$Z=J^{r}\Xi$ is a contact symmetry, and conversely, if a contact symmetry $Z$
on $J^{r}Y$ is projectable onto $\boldsymbol{X}$ then $Z=J^{r}\Xi$ for some
$\Xi$ on $\boldsymbol{Y}$.

Let $k\geq 0$. As an immediate consequence of the definition it follows that
the Lie derivative of a $k$-contact form by a contact symmetry is an at least
$k$-contact form. This means, however, that the Lie derivatives by contact
symmetries preserve the sheaves $\Theta^{r}_{q}$.

######  Proposition 4.2.

Let $Z$ be a contact symmetry defined on $J^{r}\boldsymbol{Y}$. Then for all
$q\geq 1$, $L_{Z}\Theta^{r}_{q}\subset\Theta^{r}_{q}$.

Equivalently we can claim that for any two forms $\rho_{1}$, $\rho_{2}$
belonging to the same class $[\rho]$ in the variational sequence, the Lie
derivatives $L_{Z}\rho_{1}$, $L_{Z}\rho_{2}$ by any contact symmetry $Z$ are
also equivalent. Thus, following [57] we can define the Lie derivative ${\cal
L}_{Z}$ of a class

$\displaystyle{\cal L}_{Z}[\rho]=[L_{Z}\rho].$ (4.4)

Now one can easily prove that the property (4.3) extends for any contact
symmetry, and through the whole variational sequence, for any morphism ${\cal
E}_{k}\colon\Lambda^{r}_{k}/\Theta^{r}_{k}\to\Lambda^{r}_{k+1}/\Theta^{r}_{k+1}$
[57] (see also [58]):

######  Theorem 4.3.

Let a vector field $Z$ on $J^{r}\boldsymbol{Y}$ be a contact symmetry. Then
the Lie derivative $L_{Z}$ commutes with all morphisms in the variational
sequence. More explicitly, for all $k\geq 1$

$\displaystyle{\cal L}_{Z}{\cal E}_{k}[\rho]={\cal E}_{k}({\cal
L}_{Z}[\rho])={\cal E}_{k}[Z\rfloor d\rho].$ (4.5)

###### Proof.

Since the Lie derivative commutes with the exterior derivative, we have for
any $k$-form $\rho$ on $J^{r}\boldsymbol{Y}$

$\displaystyle{\cal L}_{Z}[d\rho]=[L_{Z}d\rho]=[dL_{Z}\rho]=[dZ\rfloor
d\rho].$

Writing this formula in terms of the morphism ${\cal E}_{k}$ we get (4.5). ∎

From now on, we shall restrict to $\pi_{s}$-projectable contact symmetries
(recall that then we have $Z=J^{s}\Xi$ for a projectable vector field $\Xi$ on
$\boldsymbol{Y}$). From the definition it is clear that the Lie derivative
then preserves even individual contact components. Namely, if $Z$ is a
projectable contact symmetry on $J^{r+1}\boldsymbol{Y}$ then $Z=J^{r+1}\Xi$
and

$\displaystyle L_{J^{r+1}\Xi}p_{k}\rho=p_{k}L_{J^{r}\Xi}\rho,\qquad k\geq 0,$

generalizing (4.2) to variational forms of any degree. Note that we may write

$\displaystyle
L_{J^{r+1}\Xi}\pi_{r+1,r}^{*}\rho=L_{J^{r+1}\Xi}\left(\sum_{k=0}^{q}p_{k}\rho\right)=\sum_{k=0}^{q}p_{k}(L_{J^{r}\Xi}\rho).$

Let us turn back to formula (4.4). It means that the Lie derivative of classes
of forms, i.e., variational Lie derivative, can be correctly defined as the
equivalence class of the standard Lie derivative of forms, and thus
represented by forms. Such a point of view opens a possibility to express the
infinitesimal first variation formula (4.1) in terms of the variational
morphisms, and, in this way, to obtain a generalization of the first variation
formula for any degree of forms (with higher degree analogs of Noether
theorems, indeed).

Some results in this direction have been already achieved, namely explicit
formulae for the quotient Lie derivative operators were provided, as well as
corresponding versions of Noether theorems interpreted in terms of conserved
currents for Lagrangians and Euler–Lagrange morphisms. However, only classes
of forms up to degree $n+2$, were considered (the latter assumed to be exact)
[26]. The representation made use of intrinsic decomposition formulae for
vertical morphisms due to Kolář [42], which express geometrically the
integration by part procedure, however, in a different way compared to that
based on the splitting of the Cartan form [32, 46]. The decomposition formulae
introduce local objects such as momenta which could be globalized by means of
connections, and besides the usual momentum associated with a Lagrangian, a
‘generalized’ momentum is associated with an Euler–Lagrange type morphism. Its
interpretation in the calculus of variations has not yet been exhaustively
exploited; as conjectured in [80], generalized momenta could play a rôle
within the multisymplectic framework for field theories.

In the sequel of this section we propose an approach to the problem of
representation of the Lie derivative of classes of forms which encompasses
previous results and generalizes to any degree of forms. Actually,
representation of (non exact) classes of forms of any degree appears relevant,
especially in degree $n+3$, for the study of nonvariational differential
equations arising from Helmholtz-like forms (related with the problem of the
existence of closed $n+2$ forms) [69, 75]. We shall exploit the relation
between the interior Euler operator and the Cartan formula for the Lie
derivative of differential forms. The representation of the variational Lie
derivative will provide, in a quite simple and immediate way, ‘any degree’
generalizations of (4.1), together with the generalized Noether theorems, as
‘quotient Cartan formulae’.

These results, which have an intrinsic importance from a theoretical point of
view, as shown in Section $5$, have also various concrete applications;
besides the ones presented here, it is worth mentioning that applications of
iterated variational Lie derivatives in variational problems on gauge-natural
bundles produced physically relevant results concerning existence and
globality of conserved quantities and on existence of Higgs fields on spinor
bundles (see, e.g., [23, 24, 81, 82, 83, 86]).

Let $\Xi$ be a projectable vector field on $\boldsymbol{Y}$. We define the
interior product of a prolongation of $\Xi$ with the equivalence class of
$\rho$ by the formula

$\displaystyle J^{r}\Xi\rfloor[\rho]=\big{[}J^{s}\Xi\rfloor
R_{q}([\rho])\big{]}$ (4.6)

(the equivalence class of the interior product of the vector field with the
representation of the equivalence class of $\rho$). This definition is well
posed since the representation $R_{q}$ is unique.

Given a $q$-form $\rho$ defined (locally) on $J^{r}\boldsymbol{Y}$, we have a
commutative diagram defining an operator $\hat{R}_{q}$

$\displaystyle\hat{R}_{q}(\mathcal{L}_{J^{r}\Xi}[\rho])\equiv\hat{R}_{q}([L_{J^{r}\Xi}\rho])=L_{J^{s}\Xi}R_{q}([\rho]).$

This operator is uniquely defined and is equal, respectively, to the following
expressions:

$\displaystyle L_{J^{s}\Xi}h\rho,\qquad$ $\displaystyle 0\leq q\leq n,\qquad$
$\displaystyle s=r+1,$ $\displaystyle L_{J^{s}\Xi}\mathcal{I}(\rho),\qquad$
$\displaystyle n+1\leq q\leq P,\qquad$ $\displaystyle s=2r+1,$ $\displaystyle
L_{J^{s}\Xi}\rho,\qquad$ $\displaystyle q>P+1,\qquad$ $\displaystyle s=r,$

where $P$ denotes the corank of the Cartan distribution on
$J^{r}\boldsymbol{Y}$.

This definition enables us to deal with ordinary Lie derivatives of forms on
${\Lambda}^{s}_{q}$, thus we can apply the standard Cartan formula.

We recall a result in [43, Theorem III.11], see also [45].

######  Lemma 4.4.

Let $\Xi$ be a $\pi$-vertical vector field on $\boldsymbol{Y}$ and $\rho$ a
differential $q$-form on $J^{r}\boldsymbol{Y}$. Then the following holds true
for $i=1,\ldots,q$

$\displaystyle J^{r+2}\Xi\rfloor
p_{i}dp_{i}\rho=-p_{i-1}d\big{(}J^{r+1}\Xi\rfloor p_{i}\rho\big{)},$

and

$\displaystyle L_{J^{r+2}\Xi}(\pi_{r+2,r+1})^{*}p_{i}\rho=J^{r+2}\Xi\rfloor
p_{i+1}dp_{i}\rho+p_{i}d\big{(}J^{r+1}\Xi\rfloor p_{i}\rho\big{)}.$

As above, we denote $\omega_{0}=dx^{1}\land dx^{2}\land\cdots\land dx^{n}$.
Next, let us denote

$\displaystyle\omega_{i}=\frac{\partial}{\partial
x^{i}}\rfloor\omega_{0},\qquad\omega_{ij}=\frac{\partial}{\partial
x^{j}}\rfloor\omega_{i}.$

It is well known that in order to obtain a representation of classes of degree
$n+1$ in the variational sequence by dynamical forms the following integration
formula is used [49]

$\displaystyle\omega^{\alpha}_{Ji}\wedge\omega_{0}=-d\omega^{\alpha}_{J}\wedge\omega_{i},$

and the corresponding representation is obtained by taking the $p_{1}$
component obtained by iterated integrations by parts. In order to integrate by
parts $(p+k)$-forms with $p<n$, we need to generalize as

$\displaystyle\gamma_{\alpha}^{J[ij]}\omega^{\alpha}_{J[i}\wedge\omega_{j]}=-\gamma_{\alpha}^{J[ij]}d\omega^{\alpha}_{J}\wedge\omega_{ij}.$

This enables us to generalize results given in [45].

######  Lemma 4.5.

Let $\rho\in{\Lambda}^{r}_{p+k}$, $1\leq p\leq n$. Let
$p_{k}\rho=\sum\limits^{r}_{|J|=0}\omega^{\alpha}_{J}\wedge\eta^{J}_{\alpha}$,
with $\eta^{J}_{\alpha}$ $(k-1)$-contact $(p+k-1)$-forms. Then we have the
decomposition

$\displaystyle p_{k}\rho=\mathfrak{I}(\rho)+p_{k}dp_{k}\mathfrak{R}(\rho),$

where $\mathfrak{R}(\rho)$ is a local $k$-contact $(p+k-1)$-form such that

$\displaystyle\mathfrak{I}(\rho)+p_{k}dp_{k}\mathfrak{R}(\rho)=\omega^{\alpha}\wedge\sum^{r}_{|J|=0}(-1)^{|J|}d_{J}\eta^{J}_{\alpha}+\sum^{r}_{|I|=1}d_{I}\big{(}\omega^{\alpha}\wedge\zeta^{I}_{\alpha}\big{)},$

with

$\displaystyle\zeta^{I}_{\alpha}=\sum^{r-|I|}_{|J|=0}(-1)^{J}\binom{|I|+|J|}{|J|}d_{J}\eta^{JI}_{\alpha}.$

Notice that $d_{J}\eta^{J}_{\alpha}$ are also $(k-1)$-contact $p$-horizontal
(i.e., containing the wedge of $p$ of the forms $dx^{i}$) $(p+k-1)$-forms. It
is therefore well defined a local splitting of $p_{k}\rho$,
$\rho\in{\Lambda}^{r}_{p+k}$, $1\leq p\leq n$ and $k>1$.

Of course, $\mathfrak{I}=\mathcal{I}$ and $\mathfrak{R}=\mathcal{R}$ in the
case $p=n$.

######  Example 4.6.

Let $p=n-1$. We can write
$\omega^{\alpha}\wedge\zeta^{I}_{\alpha}=\chi^{Il}\wedge\omega_{l}$, where
$\chi^{Il}$ are some local $k$-contact $k$-forms on $J^{2r}\boldsymbol{Y}$.
Therefore

$\displaystyle\sum^{r}_{|I|=1}d_{I}\big{(}\omega^{\alpha}\wedge\zeta^{I}_{\alpha}\big{)}=\sum^{r}_{|I|=1}d_{I}\chi^{Il}\wedge\omega_{l}=d_{i}\sum^{r-1}_{|I|=0}d_{I}\chi^{Ili}\wedge\omega_{l}=d_{i}\sum^{r-1}_{|I|=0}d_{I}\chi^{Ili}\wedge
dx^{j}\wedge\omega_{lj}$
$\displaystyle=(-1)^{(n-1)+k-1}d_{i}\left[\sum^{r-1}_{|I|=0}(-1)^{k}d_{I}\chi^{{I}{[lj]}}\wedge\omega_{lj}\right]\wedge
dx^{i}=d_{H}\left[\sum^{r-1}_{|I|=0}(-1)^{k}d_{I}\chi^{I[lj]}\wedge\omega_{lj}\right].$

We denote by
$\mathfrak{R}(\rho)=\sum\limits^{r-1}_{|I|=0}(-1)^{k}d_{I}\chi^{I[lj]}\wedge\omega_{lj}$
so that

$\displaystyle\sum^{r}_{|I|=1}d_{I}\big{(}\omega^{\alpha}\wedge\zeta^{I}_{\alpha}\big{)}=p_{k}dp_{k}\mathfrak{R}(\rho).$

In the following we shall use such a splitting in order to split the vertical
differential of a $p$-density and define a corresponding ‘momentum form’.

######  Theorem 4.7.

Let $0\leq q\leq n$, $\Xi$ a $\pi$-projectable vector field on
$\boldsymbol{Y}$ and $\rho$ a differential $q$-form on $J^{r}\boldsymbol{Y}$.
We have

  1. $1.$

For $0\leq q\leq n-1$

$\displaystyle\hat{R}_{q}(\mathcal{L}_{J^{r}\Xi}[\rho])\equiv
L_{J^{r+1}\Xi}h\rho$
$\displaystyle\hphantom{\hat{R}_{q}(\mathcal{L}_{J^{r}\Xi}[\rho])}{}=J^{r+2}\Xi_{H}\rfloor
d_{H}h\rho+J^{r+2}\Xi_{V}\rfloor\mathfrak{I}(d\rho)+d_{H}\big{(}J^{r+1}\Xi_{V}\rfloor\tilde{p}_{d_{V}h\rho}+J^{r+1}\Xi_{H}\rfloor
h\rho\big{)},$

where a ‘generalized momentum’ is defined by

$\displaystyle\tilde{p}_{d_{V}h\rho}=-p_{1}\mathfrak{R}(d\rho).$ (4.7)

  2. $2.$

For $q=n$

$\displaystyle\hat{R}_{n}(\mathcal{L}_{J^{r}\Xi}[\rho])\equiv
L_{J^{r+1}\Xi}h\rho=J^{r+2}\Xi_{V}\rfloor
E_{n}(h\rho)+d_{H}\big{(}J^{r+1}\Xi_{V}\rfloor
p_{d_{V}h\rho}+J^{r+1}\Xi_{H}\rfloor h\rho\big{)},$

where

$\displaystyle p_{d_{V}h\rho}=-p_{1}\mathcal{R}(d\rho).$ (4.8)

Notice, that the latter formula, written as

$\displaystyle L_{J^{r+1}\Xi}\lambda=J^{r+2}\Xi_{V}\rfloor
E_{\lambda}+d_{H}\phi_{\lambda,\Xi},$

where

$\displaystyle\phi_{\lambda,\Xi}=J^{r+1}\Xi_{V}\rfloor
p_{d_{V}h\rho}+J^{r+1}\Xi_{H}\rfloor h\rho,$

is obviously the infinitesimal first variation formula (4.1). We observe that
$p_{d_{V}h\rho}$ are momentum forms, and $\phi_{\lambda,\Xi}$ is the Noether
current related with $\Xi$ and $\lambda$.

###### Proof.

For all cases by the Cartan formula

$\displaystyle L_{J^{r+1}\Xi}h\rho=J^{r+1}\Xi\rfloor
dh\rho+d\big{(}J^{r+1}\Xi\rfloor h\rho\big{)}$

we have

$\displaystyle
L_{J^{r+1}\Xi}h\rho=\big{(}J^{r+2}\Xi_{H}+J^{r+2}\Xi_{V}\big{)}\rfloor(d_{V}h\rho+d_{H}h\rho)+(d_{V}+d_{H})\big{(}\big{(}J^{r+1}\Xi_{H}+J^{r+1}\Xi_{V}\big{)}\rfloor
h\rho\big{)}$
$\displaystyle\hphantom{L_{J^{r+1}\Xi}h\rho}{}=d_{H}\big{(}J^{r+1}\Xi_{H}\rfloor
h\rho\big{)}+J^{r+2}\Xi_{H}\rfloor d_{H}h\rho+J^{r+2}\Xi_{V}\rfloor
d_{V}h\rho.$

This expression can be further characterized in more detail.

Let $0\leq q\leq n-1$. Since
$d_{V}h\rho=p_{1}d\rho=\mathfrak{I}(d\rho)+p_{1}dp_{1}\mathfrak{R}(d\rho)$,
and since by Lemma 4.4,

$\displaystyle J^{r+2}\Xi_{V}\rfloor
p_{1}dp_{1}\mathfrak{R}(d\rho)=-p_{0}d\big{(}J^{r+1}\Xi_{V}\rfloor
p_{1}\mathfrak{R}(d\rho)\big{)}=-d_{H}\big{(}J^{r+1}\Xi_{V}\rfloor
p_{1}\mathfrak{R}(d\rho)\big{)},$

we can define a momentum form associated with a density of degree $<n$ by
(4.7) and obtain

$\displaystyle L_{J^{r+1}\Xi}h\rho=d_{H}\big{(}J^{r+1}\Xi_{H}\rfloor
h\rho\big{)}+J^{r+2}\Xi_{H}\rfloor
d_{H}h\rho+J^{r+2}\Xi_{V}\rfloor(\mathfrak{I}(d\rho)+p_{1}dp_{1}\mathfrak{R}(d\rho))$
$\displaystyle\hphantom{L_{J^{r+1}\Xi}h\rho}{}=J^{r+2}\Xi_{V}\rfloor\mathfrak{I}(d\rho)+J^{r+2}\Xi_{H}\rfloor
d_{H}h\rho+d_{H}(J^{r+1}\Xi_{V}\rfloor\tilde{p}_{d_{V}h\rho}+J^{r+1}\Xi_{H}\rfloor
h\rho).$

Let $q=n$. Then $d_{H}h\rho=0$, hence $J^{r+2}\Xi_{H}\rfloor d_{H}h\rho=0$.
Next, again since
$d_{V}h\rho=p_{1}d\rho=\mathcal{I}(d\rho)+p_{1}dp_{1}\mathcal{R}(d\rho)$ and
since by Lemma 4.4, $J^{r+2}\Xi_{V}\rfloor
p_{1}dp_{1}\mathcal{R}(d\rho)=-d_{H}(J^{r+1}\Xi_{V}\rfloor
p_{1}\mathcal{R}(d\rho))$, we define a local momentum form by (4.8), and
obtain

$\displaystyle
L_{J^{r+1}\Xi}h\rho=J^{r+2}\Xi_{V}\rfloor(\mathcal{I}(d\rho)-d_{H}\big{(}J^{r+1}\Xi_{V}\rfloor
p_{1}\mathcal{R}(d\rho)\big{)}+d_{H}(J^{r+1}\Xi_{H}\rfloor h\rho)$
$\displaystyle\hphantom{L_{J^{r+1}\Xi}h\rho}{}=J^{r+2}\Xi_{V}\rfloor\mathcal{I}(d\rho)+d_{H}\big{(}J^{r+1}\Xi_{V}\rfloor
p_{d_{V}h\rho}+J^{r+1}\Xi_{H}\rfloor h\rho\big{)}$
$\displaystyle\hphantom{L_{J^{r+1}\Xi}h\rho}{}=J^{r+2}\Xi_{V}\rfloor\mathcal{I}(d\rho)+d_{H}\phi_{\lambda,\Xi}.$

However, $\phi_{\lambda,\Xi}$ is a horizontal form, and in Takens
representation, $\mathcal{I}(d\rho)=E_{n}(h\rho)$ and
$d_{H}\phi_{\lambda,\Xi}=hd\phi_{\lambda,\Xi}=E_{n-1}(\phi_{\lambda,\Xi})$,
proving the theorem. ∎

######  Theorem 4.8.

Let $q=n+k$, $k\geq 1$. Let $\Xi$ be a $\pi$-vertical vector field on
$\boldsymbol{Y}$. We have

$\displaystyle\hat{R}_{n+k}(\mathcal{L}_{J^{r}\Xi}[\rho])=J^{s+1}\Xi\rfloor
p_{k+1}d(\mathcal{I}(\rho))+p_{k}d\big{(}J^{s}\Xi\rfloor\mathcal{I}(\rho)\big{)}.$

###### Proof.

For simplicity of notation, in what follows we omit the projections (so we
write $p_{k}\rho$ instead of $\pi_{s,r+1}^{*}p_{k}\rho$, etc.).

Since $R_{n+k}([\rho])=\mathcal{I}(\rho)=p_{k}\rho-
p_{k}dp_{k}\mathcal{R}(\rho)$, we have

$\displaystyle\hat{R}_{n+k}(\mathcal{L}_{J^{r}\Xi}[\rho])=L_{J^{s}\Xi}\mathcal{I}(\rho)=L_{J^{s}\Xi}p_{k}\rho-
L_{J^{s}\Xi}p_{k}dp_{k}\mathcal{R}(\rho).$

By applying Lemma 4.4 to $dp_{k}\mathcal{R}(\rho)$ we have

$\displaystyle L_{J^{s+1}\Xi}p_{k}dp_{k}\mathcal{R}(\rho)=J^{s+1}\Xi\rfloor
p_{k+1}dp_{k}dp_{k}\mathcal{R}(\rho)+p_{k}d\big{(}J^{s}\Xi\rfloor
p_{k}dp_{k}\mathcal{R}(\rho)\big{)}.$

Simple manipulations show that

$\displaystyle p_{k}d(J^{s}\Xi\rfloor
p_{k}dp_{k}\mathcal{R}(\rho))=-J^{s+1}\Xi\rfloor
p_{k+1}dp_{k}dp_{k}\mathcal{R}(\rho),$

so that

$\displaystyle L_{J^{s+1}\Xi}p_{k}dp_{k}\mathcal{R}(\rho)=0.$

On the other hand

$\displaystyle L_{J^{s}\Xi}p_{k}\rho=J^{s+1}\Xi\rfloor
p_{k+1}dp_{k}\rho+p_{k}d\big{(}J^{s}\Xi\rfloor p_{k}\rho\big{)},$

but, since $p_{k}\rho=\mathcal{I}(\rho)+p_{k}dp_{k}\mathcal{R}(\rho)$, by
substituting we get

$\displaystyle J^{s+1}\Xi\rfloor p_{k+1}dp_{k}\rho=J^{s+1}\Xi\rfloor
p_{k+1}d(\mathcal{I}(\rho)+p_{k}dp_{k}\mathcal{R}(\rho))$
$\displaystyle\hphantom{J^{s+1}\Xi\rfloor
p_{k+1}dp_{k}\rho}{}=J^{s+1}\Xi\rfloor
p_{k+1}d(\mathcal{I}(\rho))+J^{s+1}\Xi\rfloor
p_{k+1}d(p_{k}dp_{k}\mathcal{R}(\rho))$
$\displaystyle\hphantom{J^{s+1}\Xi\rfloor
p_{k+1}dp_{k}\rho}{}=J^{s+1}\Xi\rfloor
p_{k+1}d(\mathcal{I}(\rho))-p_{k}d\big{(}J^{s}\Xi\rfloor
p_{k}dp_{k}\mathcal{R}(\rho)\big{)}.$

Again by the same substitution in $p_{k}d(J^{s}\Xi\rfloor p_{k}\rho)$, we get

$\displaystyle
p_{k}d\big{(}J^{s}\Xi\rfloor\mathcal{I}(\rho)\big{)}+p_{k}d\big{(}J^{s}\Xi\rfloor
p_{k}dp_{k}\mathcal{R}(\rho)\big{)}.$

Summing up we finally obtain

$\displaystyle\hat{R}_{n+k}(\mathcal{L}_{J^{r}\Xi}[\rho])=L_{J^{s}\Xi}p_{k}\rho=J^{s+1}\Xi\rfloor
p_{k+1}d(\mathcal{I}(\rho))+p_{k}d(J^{s}\Xi\rfloor\mathcal{I}(\rho)).$ ∎

By the above theorem we get a generalization of the famous Noether equation
and Noether–Bessel-Hagen equation (i.e., equation for symmetries of a
Lagrangian and of the Euler–Lagrange form) to a source form of any degree.
Indeed, a projectable vector field $\Xi$ on $\boldsymbol{Y}$ is a symmetry of
$\mathcal{I}(\rho)$, i.e., $L_{J^{s}\Xi}\mathcal{I}(\rho)=0$ if and only if
$L_{J^{s+1}\Xi_{V}}\mathcal{I}(\rho)=0$, that is if and only if

$\displaystyle J^{s+1}\Xi_{V}\rfloor
p_{k+1}d(\mathcal{I}(\rho))+p_{k}d\big{(}J^{s}\Xi_{V}\rfloor\mathcal{I}(\rho)\big{)}=0.$

Theorems 4.7 and 4.8 are the desired generalization of the infinitesimal first
variation formula (4.1) to forms of any degree. The ‘boundary term’ on the
right-hand side in Theorem 4.7 contains ‘generalized momenta’, and provides a
lower-degree analog of Noether theorem in the same way as in the classical
case $q=n$ $(k=0)$.

######  Lemma 4.9.

Let $q=n+k$, $k\geq 1$, and let $\Xi$ be a $\pi$-projectable vector field on
$\boldsymbol{Y}$. We have for every class $[\rho]$ of degree $n+k$

$\displaystyle\mathcal{L}_{J^{r+1}\Xi_{H}}[\rho]=0.$

###### Proof.

In fact, $J^{s+1}\Xi_{H}\rfloor d_{H}\mathcal{I}(\rho)=0$ because
$d_{H}\mathcal{I}(\rho)$ is the horizontal differential of an $n$-horizontal
form. On the other hand since
$d_{V}\mathcal{I}(\rho)=p_{k+1}d(\mathcal{I}(\rho))$ and $p_{k+1}d\circ
p_{k+1}d=0$, we have $J^{s+1}\Xi_{H}\rfloor d_{V}\mathcal{I}(\rho)$ $=$
$J^{s+1}\Xi_{H}\rfloor p_{k+1}d(\mathcal{I}(\rho))$ (a $(k+1)$-contact and
$(n-1)$-horizontal $(n+k)$-form); therefore
$\hat{R}_{n+k}(\mathcal{L}_{J^{r+1}\Xi_{H}}[\rho])$ $=$ $J^{s+1}\Xi_{H}\rfloor
p_{k+1}d(\mathcal{I}(\rho)))+p_{k}d(J^{s+1}\Xi_{H}\rfloor\mathcal{I}(\rho))\in{\Theta}_{n+k}^{s+1}$.
∎

Since
$p_{k+1}d(\mathcal{I}(\rho))=\mathcal{I}(d(\mathcal{I}(\rho)))=R_{q+1}(\mathcal{E}_{q}([\rho]))$,
taking into account Theorem 3.15, we finally obtain a ‘quotient’ Cartan
formula for the variational Lie derivative of $n+k$ classes, with $k\geq 1$.

######  Theorem 4.10.

Let $k\geq 1$. Let $\Xi$ be a $\pi$-projectable vector field on
$\boldsymbol{Y}$. For every class
$[\rho]\in\Lambda^{r}_{n+k}/\Theta^{r}_{n+k}$ we have $($up to pull-backs$)$

$\displaystyle\mathcal{L}_{J^{r}\Xi}[\rho]=J^{r+1}\Xi_{V}\rfloor\mathcal{E}_{n+k}([\rho])+\mathcal{E}_{n+k-1}\big{(}J^{r+1}\Xi_{V}\rfloor[\rho]\big{)}.$

###### Proof.

Since $\Xi$ is assumed to be a $\pi$-projectable vector field on
$\boldsymbol{Y}$, up to projections,

$\displaystyle\hat{R}_{n+k}(\mathcal{L}_{J^{r}\Xi}[\rho])=\hat{R}_{n+k}(\mathcal{L}_{J^{r+1}\Xi_{V}}[\rho])+\hat{R}_{n+k}(\mathcal{L}_{J^{r+1}\Xi_{H}}[\rho])$

holds true and the result follows from the above lemma and theorem (the latter
applied for the vertical part of $J^{r}\Xi$). ∎

##  5 Some applications

###  5.1 Helmholtz forms in mechanics

As mentioned above, for $k\geq 2$, a class $[\rho]\in\mathcal{V}^{r}_{n+k}$ in
the variational sequence can be represented by different source forms. We
shall illustrate this feature on the image
$\mathcal{E}_{n+1}([\varepsilon])=[d\varepsilon]\in\mathcal{V}^{2}_{n+2}$,
where $\varepsilon$ is a second-order dynamical form, in the case when $n=1$
(mechanics).

Let $n=\dim X=1$; as above, denote local fibred coordinates on
$\boldsymbol{Y}$ by $(t,q^{\sigma})$ and the associated coordinates on
$J^{3}\boldsymbol{Y}$ by
$(t,q^{\sigma},\dot{q}^{\sigma},\ddot{q}^{\sigma},\dddot{q}^{\sigma})$, and
let

$\displaystyle\omega^{\sigma}=dq^{\sigma}-\dot{q}^{\sigma}dt,\qquad\dot{\omega}^{\sigma}=d\dot{q}^{\sigma}-\ddot{q}^{\sigma}dt,\qquad\ddot{\omega}^{\sigma}=d\ddot{q}^{\sigma}-\dddot{q}^{\sigma}dt.$

Consider a dynamical form $\varepsilon\in\Lambda^{2}_{2}$,
$\varepsilon=E_{\sigma}\omega^{\sigma}\land dt$. Then $[d\varepsilon]$ is
represented by the following canonical source form (called canonical Helmholtz
form), obtained by a straightforward calculation:

$\displaystyle
H_{\varepsilon}=\mathcal{I}(d\varepsilon)=\frac{1}{2}\left(\left(\frac{\partial
E_{\sigma}}{\partial q^{\nu}}-\frac{\partial E_{\nu}}{\partial
q^{\sigma}}-\frac{1}{2}\frac{d}{dt}\left(\frac{\partial
E_{\sigma}}{\partial\dot{q}^{\nu}}-\frac{\partial
E_{\nu}}{\partial\dot{q}^{\sigma}}\right)+\frac{1}{2}\frac{d^{2}}{dt^{2}}\left(\frac{\partial
E_{\sigma}}{\partial\ddot{q}^{\nu}}-\frac{\partial
E_{\nu}}{\partial\ddot{q}^{\sigma}}\right)\right)\omega^{\nu}\right.$
$\displaystyle\left.\hphantom{H_{\varepsilon}=}{}+\left(\frac{\partial
E_{\sigma}}{\partial\dot{q}^{\nu}}+\frac{\partial
E_{\nu}}{\partial\dot{q}^{\sigma}}-2\frac{d}{dt}\frac{\partial
E_{\nu}}{\partial\ddot{q}^{\sigma}}\right)\dot{\omega}^{\nu}+\left(\frac{\partial
E_{\sigma}}{\partial\ddot{q}^{\nu}}-\frac{\partial
E_{\nu}}{\partial\ddot{q}^{\sigma}}\right)\ddot{\omega}^{\nu}\right)\land\omega^{\sigma}\land
dt.$

By construction $H_{\varepsilon}\in\Lambda^{5}_{3}$, however, it is
projectable to $J^{4}\boldsymbol{Y}$.

From the exactness of the representation sequence we get that the condition
$E_{2}(\varepsilon)=H_{\varepsilon}=0$ is necessary and sufficient for
existence of a local Lagrangian $\lambda$ such that
$\varepsilon=E_{1}(\lambda)=E_{\lambda}$. Expressed in terms of the components
of $H_{\varepsilon}$ the condition $H_{\varepsilon}=0$ gives the celebrated
Helmholtz conditions of local variationality for dynamical forms, in the
context of the variational sequence called also source constraints.

The canonical Helmholtz form is not the unique source form providing Helmholtz
conditions. Another distinguished Helmholtz form is [53]

$\displaystyle\bar{H}_{\varepsilon}=\frac{1}{2}\left(\left(\frac{\partial
E_{\sigma}}{\partial q^{\nu}}-\frac{\partial E_{\nu}}{\partial
q^{\sigma}}-\frac{1}{2}\frac{d}{dt}\left(\frac{\partial
E_{\sigma}}{\partial\dot{q}^{\nu}}-\frac{\partial
E_{\nu}}{\partial\dot{q}^{\sigma}}\right)\right)\omega^{\nu}\right.$
$\displaystyle\left.\hphantom{\bar{H}_{\varepsilon}=}{}+\left(\frac{\partial
E_{\sigma}}{\partial\dot{q}^{\nu}}+\frac{\partial
E_{\nu}}{\partial\dot{q}^{\sigma}}-\frac{d}{dt}\left(\frac{\partial
E_{\sigma}}{\partial\ddot{q}^{\nu}}+\frac{\partial
E_{\nu}}{\partial\ddot{q}^{\sigma}}\right)\right)\dot{\omega}^{\nu}+\left(\frac{\partial
E_{\sigma}}{\partial\ddot{q}^{\nu}}-\frac{\partial
E_{\nu}}{\partial\ddot{q}^{\sigma}}\right)\ddot{\omega}^{\nu}\right)\land\omega^{\sigma}\land
dt.$

$\bar{H}_{\varepsilon}$ is of order $3$, and is, indeed, equivalent with
$H_{\varepsilon}$. Note that it provides Helmholtz conditions in an equivalent
and more simple form. The relationship between the canonical and the “reduced”
Helmholtz form is as follows [70]:

$\displaystyle\bar{H}_{\varepsilon}=H_{\varepsilon}+p_{2}d\eta,$

where

$\displaystyle\eta=-\frac{1}{4}\frac{d}{dt}\left(\frac{\partial
E_{\sigma}}{\partial\ddot{q}^{\nu}}-\frac{\partial
E_{\nu}}{\partial\ddot{q}^{\sigma}}\right)\omega^{\nu}\land\omega^{\sigma}.$

It is worth note that in the most frequent case when the components of
$\varepsilon$ are affine in the second derivatives with symmetric
coefficients, i.e.,

$\displaystyle E_{\sigma}=A_{\sigma}+B_{\sigma\nu}\ddot{q}^{\nu},\qquad
B=B^{T},$

the canonical Helmholtz form $H_{\varepsilon}$ is projectable to
$J^{3}\boldsymbol{Y}$ and coincides with the “reduced” form
$\bar{H}_{\varepsilon}$.

Asking about Helmholtz forms of variational order $1$, that is, coming from
the variational sequence of order $1$, we get the following result: The