[intoc]

# Some contributions to $k$-contact Lagrangian field equations, symmetries and
dissipation laws

Xavier Rivasa, , Modesto Salgadob,c, and Silvia Soutob,c,

aEscuela Superior de Ingeniería y Tecnología,
Universidad Internacional de La Rioja, Logroño, Spain.
bDepartamento de Xeometría e Topoloxía, Universidade de Santiago de
Compostela,
15782 Santiago de Compostela, Spain.
cCentro de Investigación e Tecnoloxía Matemática de Galicia (CITMAga),
15782 Santiago de Compostela, Spain<EMAIL_ADDRESS>ORCID:
<EMAIL_ADDRESS>ORCID:
<EMAIL_ADDRESS>ORCID: 0000-0003-0755-1211

###### Abstract

It is well known that $k$-contact geometry is a suitable framework to deal
with non-conservative field theories. In this paper, we study some relations
between solutions of the $k$-contact Euler–Lagrange equations, symmetries,
dissipation laws and Newtonoid vector fields. We review the $k$-contact
Euler–Lagrange equations written in terms of $k$-vector fields and sections
and provide new results relating the solutions in both approaches. We also
study different kind of symmetries depending on the structures they preserve:
natural (preserving the Lagrangian function), dynamical (preserving the
solutions), and $k$-contact (preserving the underlying geometric structures)
symmetries. For some of these symmetries, we provide Noether-like theorems
relating symmetries and dissipation laws. We also analyse the relation between
$k$-contact symmetries and Newtonoid vector fields. Throughout the paper, we
will use the damped vibrating string as our main illustrative example.

Keywords: dissipation theorem, symmetries, Lagrangian field theories,
dissipation law, $k$-contact structure.

MSC 2020 codes: 70S05, 70S10, 70G45, 53C15, 53D10, 35R01

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 $k$-vector fields and integral sections
    2. 2.2 The tangent bundle of $k^{1}$-velocities
    3. 2.3 Second-order partial differential equations
  3. 3 $k$-contact Euler–Lagrange equations
  4. 4 Geometric $k$-contact Lagrangian equations
  5. 5 Relations between solutions and Lagrangian $k$-vector fields
  6. 6 Dissipation laws
  7. 7 Symmetries
    1. 7.1 Natural symmetries
    2. 7.2 Dynamical symmetries
    3. 7.3 $k$-contact symmetries
    4. 7.4 Dissipation laws given by vector fields which are not symmetries
    5. 7.5 Newtonoid vector fields
  8. 8 Conclusions and further research
  9. 9 Acknowledgments

##  1 Introduction

Since the 1950s, geometric methods have been used to provide descriptions of
mechanical systems and field theories with many applications in mathematics,
physics, engineering, etc. Some of the most frequent geometric structures
involved in geometric mechanics and field theory are symplectic,
$k$-symplectic or multisymplectic manifolds (see [7, 17, 43] and references
therein). In general, all these geometric methods are applied to conservative
systems, that is, without any dissipation or loss of energy, both in the
Lagrangian and Hamiltonian sides.

In the last decade, the interest in the geometrization of systems with
dissipation of energy has risen drastically. Contact geometry [1, 25, 32] is
the suitable geometric framework to describe many types of damping [2, 4, 8,
13, 21, 35]. This formulation has proved to be very useful in thermodynamics
[3, 45], quantum mechanics [10], circuit theory [26], Lie systems [18] and
control theory [40] among others [5, 11, 15, 16, 27, 28]. Also, there have
been several generalizations of contact geometry in order to describe non-
conservative field theory, namely the $k$-contact [20, 22, 29], $k$-cocontact
[41] and multicontact [12] formulations. The field equations obtained by means
of these formalisms, called the $k$-contact Euler–Lagrange equations, coincide
with the ones obtained from the recently devised generalized Herglotz
principle [23].

The study of symmetries of dynamical systems is of great interest because it
can provide new ways of finding conservation (or dissipation) laws. In
addition, reduction procedures can be performed in order to simplify the
description of a system whose group of symmetries is known. Since the seminal
work by E. Noether [38] (see also [33, 37]), the relation between symmetries
and conserved quantities has been a topic of great relevance in mathematical
physics and dynamical systems. Since the dawn of geometric mechanics, many
papers have been devoted to the study of symmetries and conserved quantities
of Hamiltonian and Lagrangian systems [9, 19, 36, 39, 46]. Recently, this
study has been performed for contact and cocontact systems [14, 21, 24], where
the notion of conserved quantity has to be replaced with the notion of
dissipated quantity.

The aim of this paper is to deepen in the study of the symmetries of non-
conservative autonomous field theories using the $k$-contact formalism. There
exist many types of symmetries depending on the structures they preserve. In
this work, we are focused in natural symmetries (symmetries of the Lagrangian
function), dynamical symmetries (those preserving the solutions) and
$k$-contact symmetries (those preserving the underlying geometric structures).
Some of these symmetries allow us to obtain dissipation laws following the
ideas of E. Noether. Throughout the work, several examples are used, in
particular the vibrating string with damping.

The structure of the paper is as follows. Section 2 is devoted to review some
basic concepts on $k$-vector fields, integral sections and sopdes which are
fundamental tools in this paper. Roughly speaking, a sopde is a $k$-vector
field whose integral sections are first prolongations of maps defined on the
base manifold. In addition, some geometric structures in the tangent bundle of
$k^{1}$-velocities of a manifold are introduced using the theory of lifts of
functions and vector fields. These structures are necessary to develop the
geometrical $k$-contact equations. In Section 3 we introduce the $k$-contact
Euler–Lagrange equations, or Herglotz–Euler–Lagrange equations, and several
examples of systems of this form are provided. Section 4 is devoted to
describe how to obtain these equations geometrically via Poincaré–Cartan
forms. We present an example of the geometric $k$-contact Lagrangian
equations. In Section 5 we discuss the relation between solutions of the
$k$-contact Euler–Lagrange equations and the contact Lagrangian $k$-vector
fields, provided by the geometrical $k$-contact equations.

In Section 6 the notion of dissipation law is introduced and some examples are
provided. The characterization of these laws in Lemma 6.5 is a new tool for
the study of dissipation law at the rest of the paper. Section 7 is devoted to
present several types of symmetry, depending on the structure they preserve.
Along this section, several Noether-like theorems relating symmetries and
dissipation laws are provided. Finally, we generalize the notion of Newtonoid
vector field, see [6], from $k$-symplectic geometry to the $k$-contact case
and we give some results relating Newtonoid vector fields to $k$-contact
symmetries.

Throughout the paper, all the manifolds are real, second countable and of
class $\mathscr{C}^{\infty}$, and the mappings are assumed to be smooth.
Einstein’s notation for sums over crossed repeated indices is hereafter
assumed.

##  2 Preliminaries

The notion of $k$-vector field is of great interest in the geometric study of
partial differential equations. In this section, we review the main concepts
on $k$-vector fields and integral sections. In addition, we give some insights
on the natural structures of the bundle of $k^{1}$-velocities: the Liouville
vector field and the canonical $k$-tangent structure. These structures allow
us to define the notion of second-order partial differential equation. For
more information, we refer to [17] and references therein.

###  2.1 $k$-vector fields and integral sections

Let $M$ be a smooth $n$-dimensional manifold. Consider the Whitney sum of $k$
copies of its tangent bundle:
$\oplus^{k}\mathrm{T}M=\mathrm{T}M\oplus\overset{k}{\dotsb}\oplus\mathrm{T}M$.
We have the natural projections

$\tau^{k}_{M}\colon\oplus^{k}\mathrm{T}M\longrightarrow
M\,,\qquad\tau^{k,\alpha}_{M}\colon\oplus^{k}\mathrm{T}M\longrightarrow\mathrm{T}M\,,\qquad\alpha=1,\dotsc,k\,.$

A $k$-vector field on a manifold $M$ is a section ${\bf X}\colon
M\rightarrow\oplus^{k}\mathrm{T}M$ of the natural projection $\tau^{k}_{M}$.
We will denote by $\mathfrak{X}^{k}(M)$ the set of all $k$-vector fields on
$M$. Thus, a $k$-vector field ${\bf X}\in\mathfrak{X}^{k}(M)$ can be
understood as a family of $k$ vector fields
$X_{1},\dotsc,X_{k}\in\mathfrak{X}(M)$, given by
$X_{\alpha}=\tau^{k,\alpha}_{M}\circ{\bf X}$. With this in mind, we can denote
${\bf X}=(X_{1},\dotsc,X_{k})$. A smooth map $F:M\rightarrow N$ induces the
map
$\oplus^{k}\mathrm{T}F:\oplus^{k}\mathrm{T}M\rightarrow\oplus^{k}\mathrm{T}N$
given by

$\oplus^{k}\mathrm{T}_{m}F({v_{1}}_{m},\dotsc,{v_{k}}_{m})=(\mathrm{T}_{m}F({v_{1}}_{m}),\dotsc,\mathrm{T}_{m}F({v_{k}}_{m}))\,,$
(1)

where $m\in M$, ${v_{1}}_{m},\dotsc,{v_{k}}_{m}\in\mathrm{T}_{m}M$, and
$\mathrm{T}_{m}F:\mathrm{T}_{m}M\rightarrow\mathrm{T}_{F(m)}N$ is the tangent
map of $F$.

Given a map $\psi\colon U\subset\mathbb{R}^{k}\rightarrow M$, we define its
first prolongation to $\oplus^{k}\mathrm{T}M$ as the map $\psi^{(1)}\colon
U\subset\mathbb{R}^{k}\rightarrow\oplus^{k}\mathrm{T}M$ given by

$\psi^{(1)}(t)=\oplus^{k}\mathrm{T}_{t}\psi\left(\displaystyle\frac{\partial}{\partial
t^{1}}\bigg{|}_{t},\dotsc,\displaystyle\frac{\partial}{\partial
t^{k}}\bigg{|}_{t}\right)=\left(\mathrm{T}_{t}\psi\left(\displaystyle\frac{\partial}{\partial
t^{1}}\bigg{|}_{t}\right),\dotsc,\mathrm{T}_{t}\psi\left(\displaystyle\frac{\partial}{\partial
t^{k}}\bigg{|}_{t}\right)\right)\,,$ (2)

where $t=(t^{1},\dotsc,t^{k})$ are the canonical coordinates of
$\mathbb{R}^{k}$.

In the same way as one has integral curves of vector fields, the notion of
integral section of a $k$-vector field is defined as follows. Consider a
$k$-vector field ${\bf X}=(X_{1},\dotsc,X_{k})\in\mathfrak{X}^{k}(M)$. An
integral section of ${\bf X}$ is a map $\psi\colon
U\subset\mathbb{R}^{k}\rightarrow M$ such that $\psi^{(1)}={\bf X}\circ\psi$,
namely $\mathrm{T}_{t}\psi\left({\frac{\partial}{\partial
t^{\alpha}}}\big{|}_{t}\right)=(X_{\alpha}\circ\psi)(t)$ for every
$\alpha=1,\dotsc,k$. We say that a $k$-vector field ${\bf
X}\in\mathfrak{X}^{k}(M)$ is integrable if every point of $M$ is in the image
of an integral section of ${\bf X}$.

Consider a $k$-vector field ${\bf X}=(X_{\alpha})$ with local expression
$X_{\alpha}=X_{\alpha}^{i}\frac{\partial{}}{\partial{x^{i}}}\,$ in a
coordinate system $(x^{i})$ on $U\subset M$. Then, $\psi\colon
U\subset\mathbb{R}^{k}\rightarrow M$ is an integral section of ${\bf X}$ if,
and only if, it is a solution of the system of partial differential equations

$\frac{\partial{\psi^{i}}}{\partial{t^{\alpha}}}\Big{|}_{t}=X_{\alpha}^{i}(\psi(t))\,.$

A $k$-vector field ${\bf X}=(X_{1},\dotsc,X_{k})$ on $M$ is integrable if, and
only if, $[X_{\alpha},X_{\beta}]=0$ for every $\alpha,\beta$, which are
precisely the necessary and sufficient conditions for the integrability of the
above system of partial differential equations [34]. Consider a diffeomorphism
$\Phi:M\rightarrow M$ and a $k$-vector field $\mathbf{X}=(X_{1},\dotsc,X_{k})$
on $M$. If $\psi$ is an integral map of $\mathbf{X}$, then $\Phi\circ\psi$ is
an integral section of $\Phi_{*}\mathbf{X}=(\Phi_{*}X_{\alpha})$. In
particular, if $\mathbf{X}$ is integrable then so is $\Phi_{*}\mathbf{X}$.

###  2.2 The tangent bundle of $k^{1}$-velocities

Let $\tau_{Q}:\mathrm{T}Q\rightarrow Q$ be the tangent bundle of a smooth
manifold $Q$. The vector bundle $\oplus^{k}\mathrm{T}Q$ is called the tangent
bundle of $k^{1}$-velocities of $Q$ (see [17]).

If $(q^{i})$ are local coordinates on $U\subseteq Q$, the natural coordinates
$(q^{i},v^{i})$ on $\mathrm{T}U=\tau_{Q}^{-1}(U)$, namely

$q^{i}(v_{q})=q^{i}(q)\,,\qquad v^{i}(v_{q})=v_{q}(q^{i})\,,$

induce natural coordinates $(q^{i},v_{\alpha}^{i})$, with $i=1,\dotsc,n$ and
$\alpha=1,\dotsc,k$, on $(\tau^{k}_{Q})^{-1}(U)$ which are given by

$q^{i}({v_{1}}_{q},\ldots,{v_{k}}_{q})=q^{i}(q)\,,\qquad
v_{\alpha}^{i}({v_{1}}_{q},\ldots,{v_{k}}_{q})={v_{\alpha}}_{q}(q^{i})\,.$

#### Lifts of functions.

If $f$ is a differentiable function on $Q$, the vertical lift $f^{V}$ and the
$\alpha$-lift $f^{(\alpha)}$ of $f$, are the functions on
$\oplus^{k}\mathrm{T}Q$ given by

$f^{V}(v_{q})=(\tau_{Q}^{k})^{*}f(v_{q})=f(q)\,,\qquad
f^{(\alpha)}(v_{q})=v_{\alpha_{q}}(f)\equiv
v^{i}_{\alpha}(v_{q})\displaystyle\frac{\partial{f}}{\partial{q^{i}}}\Big{|}_{q}\,,\quad\text{for
every}\,\,\,\alpha=1,\ldots,k\,.$

Since $(q^{i})^{V}=q^{i}$ and $(q^{i})^{(\alpha)}=v^{i}_{\alpha}$, we deduce
that vector fields on $\oplus^{k}\mathrm{T}Q$ are characterized by its action
on vertical and $\alpha$-lifts of functions.

#### Lifts of vector fields.

Given a vector field $X\in\mathfrak{X}(Q)$, the vertical $\alpha$-lifts
$X^{V_{\alpha}}$, and the complete lift $X^{C}$ are the vector fields on
$\oplus^{k}\mathrm{T}Q$ given by

$\begin{array}[]{lcl}X^{V_{\alpha}}(f^{V})=0,&&X^{V_{\alpha}}(f^{(\beta)})=\delta^{\alpha}_{\beta}(X(f))^{V},\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
X^{C}(f^{V})=(X(f))^{V},&&X^{C}(f^{(\alpha)})=(X(f))^{(\alpha)}\,,\end{array}$
(3)

with $\alpha=1,\ldots,k.$

Taking adapted coordinates $(q^{i},v_{\alpha}^{i})$ on
$\oplus^{k}\mathrm{T}Q$, if $X=X^{i}\displaystyle\frac{\partial}{\partial
q^{i}}$, then

$X^{V_{\alpha}}=X^{i}\frac{\partial{}}{\partial{v^{i}_{\alpha}}}\,,\qquad
X^{C}=X^{i}\frac{\partial{}}{\partial{q^{i}}}+v^{j}_{\alpha}\frac{\partial{X^{i}}}{\partial{q^{j}}}\frac{\partial{}}{\partial{v^{i}_{\alpha}}}\,,$
(4)

and we have

$\left(\frac{\partial{}}{\partial{q^{i}}}\right)^{V_{\alpha}}=\frac{\partial{}}{\partial{v^{i}_{\alpha}}}\,,\qquad\left(\frac{\partial{}}{\partial{q^{i}}}\right)^{C}=\frac{\partial{}}{\partial{q^{i}}}\,.$
(5)

As a consequence of (4) and (5) we deduce that the tensor fields of type
$(1,1)$ on $\oplus^{k}\mathrm{T}Q$ are characterized by the action on these
lifts of vector fields.

The Liouville vector field $\Delta\in\mathfrak{X}(\oplus^{k}\mathrm{T}Q)$ is
the vector field defined by $\Delta(f^{V})=0$ and
$\Delta(f^{(\alpha)})=f^{(\alpha)}$. In adapted coordinates, it reads

$\Delta=v^{i}_{\alpha}\frac{\partial{}}{\partial{v^{i}_{\alpha}}}\,.$ (6)

######  Remark 2.1.

  1. (1)

If $X$ is a vector field on $Q$ with local one-parameter group
$\phi_{t}:Q\rightarrow Q$, then $X^{C}$ is the infinitesimal generator of the
flow
$\mathrm{T}\phi_{t}:\oplus^{k}\mathrm{T}Q\rightarrow\oplus^{k}\mathrm{T}Q$.

  2. (2)

$\Delta$ is the infinitesimal generator of the flow
$\phi:\mathbb{R}\times\oplus^{k}\mathrm{T}Q\rightarrow\oplus^{k}\mathrm{T}Q\,,\,\phi(t,(v_{1_{q}},\ldots,v_{k_{q}}))=(e^{t}v_{1_{q}},\ldots,e^{t}v_{k_{q}})$.

The canonical $k$-tangent structure is the family $(J^{1},\ldots,J^{k})$ of
tensor fields of type $(1,1)$ defined by $J^{\alpha}(X^{C})=X^{V_{\alpha}}$
and $J^{\alpha}(X^{V_{\alpha}})=0\,.$ In local adapted coordinates it is
written as

$J^{\alpha}=\displaystyle\frac{\displaystyle\partial}{\displaystyle\partial
v^{i}_{\alpha}}\otimes\mathrm{d}q^{i}\,.$ (7)

###  2.3 Second-order partial differential equations

Since
$\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}\rightarrow\oplus^{k}\mathrm{T}Q$ is
a trivial vector bundle, the canonical structures in $\oplus^{k}\mathrm{T}Q$
(the canonical $k$-tangent structure and the Liouville vector field described
above) can be extended to $\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$ in a
natural way, and are denoted with the same notation ($J^{\alpha}$ and
$\Delta$).

######  Definition 2.2.

Consider a map $\phi_{s}\colon U\subset\mathbb{R}^{k}\rightarrow
Q\times\mathbb{R}^{k}$ with $\phi_{s}(t)=(\phi(t),s^{\alpha}(t))\,,$ where
$\phi\colon U\subset\mathbb{R}^{k}\rightarrow Q$. The first prolongation of
$\phi_{s}$ to $\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$ is the map
$\phi_{s}^{(1)}\colon
U\subset\mathbb{R}^{k}\rightarrow\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$
given by

$\phi_{s}^{(1)}(t)=(\phi^{(1)}(t),s^{\alpha}(t))\,,$

where

$\phi^{(1)}(t)=\left(\mathrm{T}_{t}\phi\left(\displaystyle\frac{\partial}{\partial
t^{1}}\Big{|}_{t}\right),\ldots,\mathrm{T}_{t}\phi\left(\displaystyle\frac{\partial}{\partial
t^{k}}\Big{|}_{t}\right)\right)$

is the first prolongation of $\phi$ to $\oplus^{k}\mathrm{T}Q$ defined in (2),
and $(t^{1},\ldots,t^{k})$ are the canonical coordinates of $\mathbb{R}^{k}$.

In local coordinates, if $\phi(t)=(\phi^{i}(t))$, then we have

$\phi_{s}^{(1)}(t)=\left(\phi^{i}(t),\frac{\partial\phi^{i}}{\partial
t^{\alpha}}\Big{|}_{t},s^{\alpha}(t)\right).$ (8)

######  Definition 2.3.

A $k$-vector field $\mathbf{\Gamma}=(\Gamma_{1},\ldots,\Gamma_{k})$ on
$\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$ is a second-order partial
differential equation (or a sopde) if, and only if,
$J^{\alpha}(\Gamma_{\alpha})=\Delta$.

In local adapted coordinates, a sopde
$\mathbf{\Gamma}=(\Gamma_{1},\ldots,\Gamma_{k})$ is given by

$\Gamma_{\alpha}=v^{i}_{\alpha}\frac{\partial}{\partial
q^{i}}+\Gamma_{\alpha\beta}^{i}\frac{\partial}{\partial
v^{i}_{\beta}}+\Gamma_{\alpha}^{\beta}\frac{\partial}{\partial
s^{\beta}}\,,\qquad\alpha,\beta=1,\dotsc,k\,,$ (9)

where $\Gamma_{\alpha\beta}^{i}$ and $\Gamma_{\alpha}^{\beta}$ are smooth
functions defined on domains of induced charts on
$\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$.

If $\psi\colon
U\subset\mathbb{R}^{k}\rightarrow\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$,
locally given by $\psi(t)=(\psi^{i}(t),\psi_{\alpha}^{i}(t),s^{\alpha}(t))$,
is an integral section of a sopde
$\mathbf{\Gamma}=(\Gamma_{1},\ldots,\Gamma_{k})$, from Definition 2.2 and
equation (9), it follows that

$\frac{\partial\psi^{i}}{\partial
t^{\alpha}}\Big{|}_{t}=\psi^{i}_{\alpha}(t)\,,\qquad\frac{\partial\psi^{i}_{\beta}}{\partial
t^{\alpha}}\Big{|}_{t}=\Gamma_{\alpha\beta}^{i}(\psi(t))\,,\qquad\frac{\partial
s^{\beta}}{\partial
t^{\alpha}}\Big{|}_{t}=\Gamma_{\alpha}^{\beta}(\psi(t))\,.$ (10)

Then, we have

$\psi(t)=(\psi^{i}(t),\psi_{\alpha}^{i}(t),s^{\alpha}(t))=\phi_{s}^{(1)}(t)=(\phi^{(1)}(t),s^{\alpha}(t))\,,\qquad\frac{\partial
s^{\beta}}{\partial
t^{\alpha}}\Big{|}_{t}=\Gamma_{\alpha}^{\beta}(\phi_{s}^{(1)})\,,$

where $\phi=\mathrm{pr}_{Q}\circ\psi\colon
U\subset\mathbb{R}^{k}\overset{\psi}{\longrightarrow}\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}\overset{\mathrm{pr}_{Q}}{\longrightarrow}Q$.

Thus we obtain the following characterization for the integral maps of a
sopde.

######  Proposition 2.4.

Let $\mathbf{\Gamma}=(\Gamma_{1},\ldots,\Gamma_{k})$ be an integrable sopde on
$\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$. If
$\psi:U\subset\mathbb{R}^{k}\rightarrow\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$
is an integral section of $\mathbf{\Gamma}$, then

$\psi(t)=(\phi^{(1)}(t),s^{\alpha}(t))=\phi_{s}^{(1)}(t)\,,$

and $\phi_{s}(t)=(\phi^{i}(t),s^{\alpha}(t))$ is a solution of the following
system of second-order partial differential equations

$\frac{\partial^{2}\phi^{i}}{\partial t^{\alpha}\partial
t^{\beta}}\Big{|}_{t}=\Gamma_{\alpha\beta}^{i}\left(\phi^{j}(t),\frac{\partial\phi^{j}}{\partial
t^{\gamma}}\Big{|}_{t},s^{\gamma}(t)\right)\,,\qquad\frac{\partial
s^{\beta}}{\partial
t^{\alpha}}\Big{|}_{t}=\Gamma_{\alpha}^{\beta}\left(\phi^{j}(t),\frac{\partial\phi^{j}}{\partial
t^{\gamma}}\Big{|}_{t},s^{\gamma}(t)\right)\,.$ (11)

Conversely, if $\phi_{s}\colon U\subset\mathbb{R}^{k}\rightarrow
Q\times\mathbb{R}^{k}$ is a map satisfying the system (11), then
$\phi_{s}^{(1)}$ is an integral section of
$\mathbf{\Gamma}=(\Gamma_{1},\ldots,\Gamma_{k})$.

######  Definition 2.5.

A map $\phi_{s}\colon U\subset\mathbb{R}^{k}\rightarrow Q\times\mathbb{R}^{k}$
satisfying (11) is called a solution of the sopde vector field
$\mathbf{\Gamma}$.

##  3 $k$-contact Euler–Lagrange equations

In this section we introduce the $k$-contact Euler–Lagrange equations, also
called Herglotz–Euler–Lagrange equations, and several examples of systems of
this form are provided.

The $k$-contact Euler–Lagrange equations for a Lagrangian function
$L:\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}\rightarrow\mathbb{R}$ read

$\frac{\partial}{\partial t^{\alpha}}\left(\frac{\partial L}{\partial
v^{i}_{\alpha}}\circ\phi_{s}^{(1)}\right)-\frac{\partial L}{\partial
q^{i}}\circ\phi_{s}^{(1)}=\left(\frac{\partial L}{\partial
s^{\alpha}}\frac{\partial L}{\partial
v^{i}_{\alpha}}\right)\circ\phi_{s}^{(1)}\,,\qquad\frac{\partial
s^{\alpha}}{\partial t^{\alpha}}=L\circ\phi_{s}^{(1)}\,,$ (12)

for a map $\phi_{s}\colon U\subset\mathbb{R}^{k}\rightarrow
Q\times\mathbb{R}^{k},\,\,\phi_{s}(t)=(\phi(t),s^{\alpha}(t))$, where
$\phi\colon U\subset\mathbb{R}^{k}\rightarrow Q$. Equations (12) can be
written as

$\frac{\partial^{2}L}{\partial v_{\alpha}^{i}\partial
v_{\beta}^{j}}\frac{\partial^{2}\phi^{j}}{\partial t^{\alpha}\partial
t^{\beta}}+\frac{\partial^{2}L}{\partial q^{j}\partial
v^{i}_{\alpha}}\frac{\partial\phi^{j}}{\partial
t^{\alpha}}+\frac{\partial^{2}L}{\partial s^{\beta}\partial
v^{i}_{\alpha}}\frac{\partial s^{\beta}}{\partial t^{\alpha}}-\frac{\partial
L}{\partial q^{i}}=\frac{\partial L}{\partial s^{\alpha}}\frac{\partial
L}{\partial v^{i}_{\alpha}}\,,\qquad\frac{\partial s^{\alpha}}{\partial
t^{\alpha}}=L\,,$ (13)

where $\phi_{s}^{(1)}(t)=\left(\phi^{j}(t),\dfrac{\partial\phi^{j}}{\partial
t^{\alpha}}\bigg{|}_{t},s^{\alpha}(t)\right)$, with $t\in\mathbb{R}^{k}$,
which represents a system of second-order partial differential equations on
$Q\times\mathbb{R}^{k}$. It is important to point out that these equations can
be obtained from two different variational principles [23] generalizing the
usual Herglotz variational principle [31] and the variational principle for
the $k$-symplectic Euler–Lagrange equations [17, Sec. 6.2]. Throughout the
paper, we will call solutions of equations (12) to the maps $\phi_{s}$ or
$\phi_{s}^{(1)}$ indistinctly, since we are only considering solutions which
are first prolongations of maps $\phi_{s}:U\subset\mathbb{R}^{k}\rightarrow
Q\times\mathbb{R}^{k}$.

######  Remark 3.1.

In the case $k=1$, equations (12) are Herglotz’s equations (see [31])

$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial
v^{i}}\circ\gamma\right)-\frac{\partial L}{\partial
q^{i}}\circ\gamma=\left(\frac{\partial L}{\partial s}\frac{\partial
L}{\partial
v^{i}}\right)\circ\gamma\,,\qquad\frac{\mathrm{d}s}{\mathrm{d}t}=L\circ\gamma\,,$

for a Lagrangian function
$L:\mathrm{T}Q\times\mathbb{R}\longrightarrow\mathbb{R}$, with solution a
curve $\gamma(t)=(q^{i}(t),\dot{q}^{i}(t),s(t))$ on
$\mathrm{T}Q\times\mathbb{R}$.

######  Remark 3.2.

Note that when the Lagrangian function $L$ does not depend on the variables
$s^{\alpha}$, for all $\alpha=1,\ldots,k$, then the above equations are the
Euler–Lagrange field equations for an autonomous Lagrangian
$L(q^{i},v^{i}_{\alpha})$ defined on $\oplus^{k}\mathrm{T}Q$.

Let us see some examples of systems modelled by second-order partial
differential equations which can be described by means of $k$-contact systems
(see [20]).

######  Example 3.3.

(The damped vibrating string). It is well known that a vibrating string can be
described using the $k$-symplectic Lagrangian formalism. Consider the
coordinates $(t,x)$ for the time and the space. Denote by $\phi$ the
separation of a point in the string from its equilibrium point, and hence
$v_{t}$ and $v_{x}$ will denote the derivative of $\phi$ with respect to the
two independent variables. The Lagrangian function
$L_{\circ}\colon\oplus^{2}\mathrm{T}\mathbb{R}\rightarrow\mathbb{R}$ for this
system is given by

$L_{\circ}(q,v_{t},v_{x})=\frac{1}{2}\rho v_{t}^{2}-\frac{1}{2}\tau
v_{x}^{2}\,,$ (14)

where $\rho$ is the linear mass density of the string and $\tau$ is the
tension of the string. We assume that these quantities are constant. The
Euler–Lagrange equation for this Lagrangian density is

$\frac{\partial^{2}\phi}{\partial t^{2}}=c^{2}\frac{\partial^{2}\phi}{\partial
x^{2}}\,,$

where $c^{2}=\dfrac{\tau}{\rho}$, which is the one-dimensional wave equation.

In order to model a vibrating string with linear damping, we modify the
Lagrangian function (14) so that it becomes a $k$-contact Lagrangian function
[22]. The new Lagrangian function $L$ is defined in the phase bundle
$\oplus^{2}\mathrm{T}\mathbb{R}\times\mathbb{R}^{2}$, endowed with coordinates
$(q,v_{t},v_{x},s^{t},s^{x})$, as

$L(q,v_{t},v_{x},s^{t},s^{x})=L_{\circ}-\gamma s^{t}=\frac{1}{2}\rho
v_{t}^{2}-\frac{1}{2}\tau v_{x}^{2}-\gamma s^{t}\,.$ (15)

Consider a solution
$\phi_{s}\colon\mathbb{R}^{2}\rightarrow\mathbb{R}\times\mathbb{R}^{2}$ with

$\phi_{s}(t,x)=(\phi(t,x),s^{t}(t,x),s^{x}(t,x)),$

where $\phi\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$. The first prolongation
of $\phi_{s}$ is the map
$\phi_{s}^{(1)}\colon\mathbb{R}^{2}\rightarrow\oplus^{2}\mathrm{T}\mathbb{R}\times\mathbb{R}^{2}$
given by

$\phi_{s}^{(1)}(t,x)=\left(\phi(t,x),\frac{\partial\phi}{\partial
t}\Big{|}_{(t,x)},\frac{\partial\phi}{\partial
x}\Big{|}_{(t,x)},s^{t}(t,x),s^{x}(t,x)\right)\,.$ (16)

The $k$-contact Euler–Lagrange equations (13), for the Lagrangian $L$, become

$\frac{\partial^{2}\phi}{\partial t^{2}}-c^{2}\frac{\partial^{2}\phi}{\partial
x^{2}}+\gamma\frac{\partial\phi}{\partial t}=0\,,\qquad\frac{\partial
s^{1}}{\partial t}+\frac{\partial s^{2}}{\partial x}=L\circ\phi_{s}^{(1)}\,.$
(17)

The first equation corresponds to a vibrating string with damping.

######  Example 3.4.

(Two coupled vibrating strings with damping). Consider a system of two coupled
strings with damping. The configuration manifold of the system is
$Q=\mathbb{R}^{2}$ equipped with coordinates $(q^{1},q^{2})$, where each
coordinate represent the displacement of each string. The Lagrangian phase
bundle of this system is
$M=\oplus^{2}\mathrm{T}\mathbb{R}^{2}\times\mathbb{R}^{2}$ endowed with
natural coordinates
$(q^{1},q^{2},v_{1}^{t},v_{2}^{t},v_{1}^{x},v_{2}^{x},s^{t},s^{x})$. Consider
the Lagrangian function
$L\colon\oplus^{2}\mathrm{T}\mathbb{R}^{2}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$
given by

$L(q^{1},q^{2},v_{t}^{1},v_{t}^{2},v_{x}^{1},v_{x}^{2},s^{t},s^{x})=\frac{1}{2}\left((v_{t}^{1})^{2}+(v_{t}^{2})^{2}-(v_{x}^{1})^{2}-(v_{x}^{2})^{2}\right)-C(z)-\gamma
s^{t}\,,$

where $\gamma$ is a friction coefficient and $C$ is a function that represents
a coupling of the two strings that depends only on
$z=\sqrt{(q^{1})^{2}+(q^{2})^{2}}$.

The maps
$\phi^{(1)}_{s}=\left(\phi^{1},\phi^{2},\frac{\partial\phi^{1}}{\partial
t},\frac{\partial\phi^{1}}{\partial x},\frac{\partial\phi^{2}}{\partial
t},\frac{\partial\phi^{2}}{\partial x},s^{t},s^{x}\right)$ solution to the
$k$-contact Euler–Lagrange equations (12) satisfy the system of partial
differential equations

$\displaystyle\frac{\partial^{2}\phi^{1}}{\partial
t^{2}}-\frac{\partial^{2}\phi^{1}}{\partial
x^{2}}+\gamma\frac{\partial\phi^{1}}{\partial
t}+C^{\prime}(z)\frac{\phi^{1}}{z}=0\,,$ (18)
$\displaystyle\frac{\partial^{2}\phi^{2}}{\partial
t^{2}}-\frac{\partial^{2}\phi^{2}}{\partial
x^{2}}+\gamma\frac{\partial\phi^{2}}{\partial
t}+C^{\prime}(z)\frac{\phi^{2}}{z}=0\,,$ (19) $\displaystyle\frac{\partial
s^{t}}{\partial t}+\frac{\partial s^{x}}{\partial x}=L\circ\phi_{s}\,.$ (20)

The first two equations correspond to two damped coupled strings with coupling
function $C$. The Hamiltonian formulation of this system was studied in [22].

######  Example 3.5.

(The telegrapher’s equation). The current and voltage on a uniform electrical
transmission line is described by the so-called telegrapher’s equations [30,
p. 306], [44, p. 653]:

$\begin{dcases}\frac{\partial V}{\partial x}=-L\frac{\partial I}{\partial
t}-RI\,,\\\ \frac{\partial I}{\partial x}=-C\frac{\partial V}{\partial
t}-GV\,,\end{dcases}$

where $V(x,t)$ is the voltage, $I(x,t)$ is the current, $L$ is the inductance,
$R$ is the resistance, $C$ is the capacitance, and $G$ is the conductance.
This system can be uncoupled, obtaining the system of second-order partial
differential equations

$\begin{dcases}\frac{\partial^{2}V}{\partial
x^{2}}=LC\frac{\partial^{2}V}{\partial t^{2}}+(LG+RC)\frac{\partial
V}{\partial t}+RGV\,,\\\ \frac{\partial^{2}I}{\partial
x^{2}}=LC\frac{\partial^{2}I}{\partial t^{2}}+(LG+RC)\frac{\partial
I}{\partial t}+RGI\,.\\\ \end{dcases}$

Note that both equations in the system above are identical, and are also known
as telegrapher’s equation. Both of them can be written as

$\square q+\gamma\frac{\partial q}{\partial t}+m^{2}q=0\,,$ (21)

where

$\gamma=\dfrac{LG+RC}{LC}\,,\qquad
m^{2}=\dfrac{RG}{LC}\,,\qquad\square=\dfrac{\partial^{2}}{\partial
t^{2}}-\dfrac{1}{LC}\dfrac{\partial^{2}}{\partial x^{2}}\,.$

Equation (21) is the $k$-contact Euler–Lagrange equation (see [29]) for the
Lagrangian function
$L:\oplus^{2}\mathrm{T}\mathbb{R}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$
given by

$L(q,v_{t},v_{x},s^{t},s^{x})=\frac{1}{2}v_{t}^{2}-\frac{1}{2LC}v_{x}^{2}-\frac{1}{2}m^{2}q^{2}-\gamma
s^{t}\,.$

######  Example 3.6.

(Laplace’s equation with damping). It is well known that Laplace’s equation is

$\Delta\phi:=\frac{\partial^{2}\phi}{\partial
x_{1}^{2}}+\dotsb+\frac{\partial^{2}\phi}{\partial x_{n}^{2}}=0\,,$

where $\phi:\mathbb{R}^{n}\rightarrow{\mathbb{R}}$, and it can be understood
as the Euler–Lagrange equation for a Lagrangian
$L_{\circ}\colon\oplus^{n}\mathrm{T}\mathbb{R}\rightarrow\mathbb{R}$ given by

$L_{\circ}(q,v_{1},\dotsc,v_{n})=\frac{1}{2}(v_{1}^{2}+\dotsb+v_{n}^{2})\,.$

Consider now the phase bundle
$\oplus^{n}\mathrm{T}\mathbb{R}\times\mathbb{R}^{n}$ with canonical
coordinates $(q,v_{1},\dotsc,v_{n},s^{1},\dotsc,s^{n})$ and the $n$-contact
Lagrangian function
$L\colon\oplus^{n}\mathrm{T}\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$
given by

$L=L_{\circ}-\gamma_{i}s^{i}\,,$

where $\gamma_{i}\in\mathbb{R}$ are constants. The $n$-contact Euler–Lagrange
equations (13) for this Lagrangian $L$ become

$\frac{\partial^{2}\phi}{\partial
x_{1}^{2}}+\dotsb+\frac{\partial^{2}\phi}{\partial
x_{n}^{2}}+\gamma_{i}\frac{\partial\phi}{\partial
x_{i}}=0\,,\quad\frac{\partial s^{1}}{\partial x_{1}}+\dotsb+\frac{\partial
s^{n}}{\partial x_{n}}=L\,,$ (22)

where
$\phi^{(1)}_{s}:\mathbb{R}^{n}\rightarrow\oplus^{n}\mathrm{T}\mathbb{R}\times\mathbb{R}^{n}$,
is the first prolongation of the map $\phi$, given by
$\phi^{(1)}_{s}=\left(\phi,\frac{\partial\phi}{\partial
x_{1}},\ldots,\frac{\partial\phi}{\partial x_{n}},s^{1},\ldots,s^{n}\right)$.

Note that the first equation in (22) is an elliptic partial differential
equation that can be understood as a Laplace’s equation with some additional
first-order terms.

##  4 Geometric $k$-contact Lagrangian equations

In this section we review some of the main aspects of the $k$-contact
Lagrangian formalism for non-conservative field theories, first introduced in
[22].

######  Definition 4.1.

A Lagrangian function is a function
$L\in\mathscr{C}^{\infty}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$.

  * •

The Lagrangian energy associated to $L$ is the function defined by
$E_{L}=\Delta(L)-L\in\mathscr{C}^{\infty}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$.

  * •

The Poincaré–Cartan forms associated to $L$ are

$\theta^{\alpha}_{L}=\mathrm{d}L\circ
J^{\alpha}\in\Omega^{1}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})\,.$

  * •

We define the following one-forms associated to $L$

$\eta^{\alpha}_{L}=\mathrm{d}s^{\alpha}-\theta^{\alpha}_{L}\in\Omega^{1}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})\,,$

called contact Lagrangian one-forms.

Note that the contact Lagrangian forms introduced above are not contact forms.
However, in favourable cases, they define a $k$-contact structure on
$\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$, motivating their name. In natural
coordinates, the local expressions of the objects introduced in Definition 4.1
are

$E_{L}=v^{i}_{\alpha}\frac{\partial L}{\partial v^{i}_{\alpha}}-L\,,$ (23)
$\eta^{\alpha}_{L}=\mathrm{d}s^{\alpha}-\displaystyle\frac{\partial
L}{\partial
v^{i}_{\alpha}}\mathrm{d}q^{i},\quad\quad\mathrm{d}\eta^{\alpha}_{L}=\displaystyle\frac{\partial^{2}L}{\partial
q^{j}\partial
v^{i}_{\alpha}}\mathrm{d}q^{i}\wedge\mathrm{d}q^{j}+\displaystyle\frac{\partial^{2}L}{\partial
v^{j}_{\beta}\partial
v^{i}_{\alpha}}\mathrm{d}q^{i}\wedge\mathrm{d}v^{j}_{\beta}+\displaystyle\frac{\partial^{2}L}{\partial
s^{\beta}\partial v^{i}_{\alpha}}\mathrm{d}q^{i}\wedge\mathrm{d}s^{\beta}.$
(24)

######  Definition 4.2.

A Lagrangian function
$L\colon\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}\rightarrow\mathbb{R}$ is
said to be regular if the Hessian of the Lagrangian function $L$ with respect
to the fibre coordinates, namely

$g^{\alpha\beta}_{ij}=\frac{\partial^{2}L}{\partial v^{i}_{\alpha}\partial
v^{j}_{\beta}}\,,$ (25)

has maximal rank $nk$ on $\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$.
Otherwise, the Lagrangian function is singular.

######  Remark 4.3.

A Lagrangian function
$L:\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}\rightarrow\mathbb{R}$ is regular
if, and only if, the family of $k$ differential one-forms
$\eta^{1}_{L},\ldots,\eta^{k}_{L}$ defines a $k$-contact structure on
$\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$. In this case,
$(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k},\eta^{\alpha}_{L})$ becomes a
$k$-contact manifold (see [20]).

Using the forms $\eta^{\alpha}_{L}$ we can write equations (12) as follows

$\begin{dcases}i_{\displaystyle\mathrm{T}\phi_{s}^{(1)}\Big{(}\frac{\partial}{\partial
t^{\alpha}}\Big{)}}\mathrm{d}\eta^{\alpha}_{L}=\left(\mathrm{d}E_{L}+\dfrac{\partial
L}{\partial s^{\alpha}}\eta^{\alpha}_{L}\right)\circ\phi_{s}^{(1)}\,,\\\
i_{\displaystyle\mathrm{T}\phi_{s}^{(1)}\Big{(}\frac{\partial}{\partial
t^{\alpha}}\Big{)}}\eta^{\alpha}_{L}=-E_{L}\circ\phi_{s}^{(1)}\,,\end{dcases}$
(26)

where

$\mathrm{T}_{t}\phi_{s}^{(1)}\left(\frac{\partial}{\partial
t^{\alpha}}\Big{|}_{t}\right)=\frac{\partial\phi^{i}}{\partial
t^{\alpha}}\Big{|}_{t}\,\frac{\partial}{\partial
q^{i}}\Big{|}_{\phi_{s}^{(1)}(t)}+\frac{\partial^{2}\phi^{i}}{\partial
t^{\alpha}\partial t^{\beta}}\Big{|}_{t}\,\frac{\partial}{\partial
v^{i}_{\beta}}\Big{|}_{\phi_{s}^{(1)}(t)}+\frac{\partial s^{\beta}}{\partial
t^{\alpha}}\Big{|}_{t}\,\frac{\partial}{\partial
s^{\beta}}\Big{|}_{\phi_{s}^{(1)}(t)}\,.$

######  Definition 4.4.

The geometric $k$-contact Lagrangian equations for a $k$-vector field
$\mathbf{X}=(X_{1},\dotsc,X_{k})$ on
$\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$ are

$\begin{dcases}i_{X_{\alpha}}\mathrm{d}\eta_{L}^{\alpha}=\mathrm{d}E_{L}+\dfrac{\partial
L}{\partial s^{\alpha}}\eta_{L}^{\alpha}\,,\\\
i_{X_{\alpha}}\eta_{L}^{\alpha}=-E_{L}\,.\end{dcases}$ (27)

We will denote by
$\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ the set of
$k$-contact Lagrangian $k$-vector fields, namely the $k$-vector fields
$\mathbf{X}=(X_{1},\dotsc,X_{k})$ on
$\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$, which are solutions to the
equations (27).

For a $k$-vector field
${\mathbf{X}}=(X_{1},\dotsc,X_{k})\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$
with local expression

$X_{\alpha}=X_{\alpha}^{i}\frac{\partial}{\partial
q^{i}}+X_{\alpha\beta}^{i}\frac{\partial}{\partial
v^{i}_{\beta}}+X_{\alpha}^{\beta}\frac{\partial}{\partial s^{\beta}}\,,$

equations (27) read

$\displaystyle 0$
$\displaystyle=\left(X_{\alpha}^{j}-v^{j}_{\alpha}\right)\frac{\partial^{2}L}{\partial
s^{\beta}\partial v^{j}_{\alpha}}\,,$ (28) $\displaystyle 0$
$\displaystyle=\left(X_{\alpha}^{j}-v^{j}_{\alpha}\right)\frac{\partial^{2}L}{\partial
v^{i}_{\beta}\partial v^{j}_{\alpha}}\,,$ (29) $\displaystyle 0$
$\displaystyle=\left(X_{\alpha}^{j}-v^{j}_{\alpha}\right)\frac{\partial^{2}L}{\partial
q^{i}\partial v^{j}_{\alpha}}+\frac{\partial L}{\partial
q^{i}}-X_{\alpha}^{\beta}\,\frac{\partial^{2}L}{\partial s^{\beta}\partial
v^{i}_{\alpha}}-X_{\alpha}^{j}\,\frac{\partial^{2}L}{\partial q^{j}\partial
v^{i}_{\alpha}}-X_{\alpha\beta}^{j}\,\frac{\partial^{2}L}{\partial
v^{j}_{\beta}\partial v^{i}_{\alpha}}+\frac{\partial L}{\partial
s^{\alpha}}\frac{\partial L}{\partial v^{i}_{\alpha}}\,,$ (30) $\displaystyle
0$ $\displaystyle=L+\left(X_{\alpha}^{j}-v^{j}_{\alpha}\right)\frac{\partial
L}{\partial v^{i}_{\alpha}}-X_{\alpha}^{\alpha}\,.$ (31)

If $L$ is a regular Lagrangian, equations (29) lead to
$X_{\alpha}^{j}=v^{j}_{\alpha}$, which are the sopde condition for the
$k$-vector field ${\mathbf{X}}$. Then, (28) holds identically, and (30) and
(31) give

$X_{\alpha\beta}^{j}\,\frac{\partial^{2}L}{\partial v^{j}_{\beta}\partial
v^{i}_{\alpha}}+v_{\alpha}^{j}\,\frac{\partial^{2}L}{\partial q^{j}\partial
v^{i}_{\alpha}}+X_{\alpha}^{\beta}\,\frac{\partial^{2}L}{\partial
s^{\beta}\partial v^{i}_{\alpha}}-\frac{\partial L}{\partial
q^{i}}=\frac{\partial L}{\partial s^{\alpha}}\frac{\partial L}{\partial
v^{i}_{\alpha}}\,,\qquad X_{\alpha}^{\alpha}=L\,.$ (32)

The following lemma is a direct consequence of equations (29), (30) and (31).

######  Lemma 4.5.

Consider a Lagrangian function
$L\in\mathscr{C}^{\infty}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$.

  1. (1)

If $L$ is a regular Lagrangian, then any $k$-vector field
${\mathbf{X}}=(X_{1},\dotsc,X_{k})\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$
is a sopde, which is locally given by formula (9) and satisfies equations
(32). Moreover, if $\mathbf{X}$ is integrable, its integral sections are
canonical lifts $\phi_{s}^{(1)}$ of solutions of the $k$-contact
Euler–Lagrange equations (12).

  2. (2)

If ${\mathbf{X}}$ is sopde and
${\mathbf{X}}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$
then it is locally given by formula (9) and satisfies equations (32).

######  Remark 4.6.

The results of Lemma 4.5 are the foundations of the $k$-contact Lagrangian
formalism, and equations (27) can be seen as a geometric version of the
$k$-contact Euler–Lagrange field equations (13).

######  Remark 4.7.

Notice that the particular case $k=1$ gives the contact Lagrangian formalism
for mechanical systems with dissipation [21].

In the following example we look for solutions
$(X_{1},X_{2})\in\mathfrak{X}^{2}_{L}(\oplus^{2}\mathrm{T}\mathbb{R}\times\mathbb{R}^{2})$
of the geometric $k$-contact Euler–Lagrange equations (27) for the regular
Lagrangian describing the damped vibrating string, and we give an example of
an integrable sopde which is a solution.

######  Example 4.8.

Let us consider again the vibrating string with damping introduced in Example
3.3. Recall that the Lagrangian function
$L:\oplus^{2}\mathrm{T}\mathbb{R}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ of
this system given by (15) is regular. The Lagrangian energy associated to the
Lagrangian function $L$ is

$E_{L}=\Delta(L)-L=\frac{1}{2}\rho v_{t}^{2}-\frac{1}{2}\tau v_{x}^{2}+\gamma
s^{t}\,,$

and the contact one-forms are

$\eta^{t}_{L}=\mathrm{d}s^{t}-\theta^{t}_{L}=\mathrm{d}s^{t}-\rho
v_{t}\mathrm{d}q\,,\qquad\eta^{x}_{L}=\mathrm{d}s^{x}-\theta^{x}_{L}=\mathrm{d}s^{x}+\tau
v_{x}\mathrm{d}q\,.$

Consider now a two-vector field
$\mathbf{X}=(X_{1},X_{2})\in\mathfrak{X}^{2}(\oplus^{2}\mathrm{T}\mathbb{R}\times\mathbb{R}^{2})$
with local expression

$\displaystyle X_{1}$ $\displaystyle=f_{1}\frac{\partial}{\partial
q}+F_{1t}\frac{\partial}{\partial v_{t}}+F_{1x}\frac{\partial}{\partial
v_{x}}+g_{1}^{t}\frac{\partial}{\partial
s^{t}}+g_{1}^{x}\frac{\partial}{\partial s^{x}}\,,$ $\displaystyle X_{2}$
$\displaystyle=f_{2}\frac{\partial}{\partial q}+F_{2t}\frac{\partial}{\partial
v_{t}}+F_{2x}\frac{\partial}{\partial v_{x}}+g_{2}^{t}\frac{\partial}{\partial
s^{t}}+g_{2}^{x}\frac{\partial}{\partial s^{x}}\,.$

The first equation in (27) reads

$\rho f_{1}\mathrm{d}v_{t}-\rho F_{1t}\mathrm{d}q-\tau
f_{2}\mathrm{d}v_{x}+\tau F_{2x}\mathrm{d}q=\rho v_{t}\mathrm{d}v_{t}-\tau
v_{x}\mathrm{d}v_{x}+\gamma\rho v_{t}\mathrm{d}q\,,$

which yields the conditions

$\displaystyle-\rho\,F_{1t}+\tau F_{2x}$ $\displaystyle=\gamma\rho v_{t}$
$\displaystyle\mbox{(coefficients in $\mathrm{d}u$)}\,,$ (33) $\displaystyle
f_{1}$ $\displaystyle=v_{t}$ $\displaystyle\mbox{(coefficients in
$\mathrm{d}v_{t}$)}\,,$ (34) $\displaystyle f_{2}$ $\displaystyle=v_{x}$
$\displaystyle\mbox{(coefficients in $\mathrm{d}v_{x}$)}\,.$ (35)

Notice that the equations (34) and (35) above are the sopde conditions for the
two-vector field $\mathbf{X}$. On the other hand, the second equation in (27)
gives the condition

$g_{1}^{t}+g_{2}^{x}=\frac{1}{2}\rho v_{t}^{2}-\frac{1}{2}\tau
v_{x}^{2}-\gamma s^{t}=L\,.$

Hence, the two-vector field $\mathbf{X}$ solution has the local expression

$\displaystyle X_{1}$ $\displaystyle=v_{t}\frac{\partial}{\partial
q}+\left(\frac{\tau}{\rho}F_{2x}-\gamma v_{t}\right)\frac{\partial}{\partial
v_{t}}+F_{1x}\frac{\partial}{\partial
v_{x}}+\left(L-g_{2}^{x}\right)\frac{\partial}{\partial
s^{t}}+g_{1}^{x}\frac{\partial}{\partial s^{x}}\,,$ $\displaystyle X_{2}$
$\displaystyle=v_{x}\frac{\partial}{\partial q}+F_{2t}\frac{\partial}{\partial
v_{t}}+F_{2x}\frac{\partial}{\partial v_{x}}+g_{2}^{t}\frac{\partial}{\partial
s^{t}}+g_{2}^{x}\frac{\partial}{\partial s^{x}}\,,$

where the functions $F_{1x},F_{2t},F_{2x},g_{1}^{x},g_{2}^{t},g_{2}^{x}$
remain undetermined.

In order to give an example of an integrable sopde solution to (27), we assume

  * •

the functions $F_{1x}=F_{2t}=g_{1}^{x}=g_{2}^{t}=0$,

  * •

the functions $g_{2}^{x}$ only depend on the variables $v_{x}$,

  * •

$\displaystyle\frac{\partial F_{2}^{x}}{\partial s^{x}}=0$,
$\displaystyle\frac{\partial F_{2x}}{\partial v_{t}}\neq 0$,
$\displaystyle\frac{\partial F_{2x}}{\partial s^{t}}\neq 0$.

In this case, the integrability conditions,
$[\Gamma_{\alpha},\Gamma_{\beta}]=0$ for every $\alpha,\beta$, are

$\displaystyle\frac{\partial F_{2x}}{\partial
v_{x}}=0\,,\qquad\displaystyle\frac{\partial g_{2}^{x}}{\partial v_{x}}=-\tau
v_{x}\,,\qquad\displaystyle\frac{\tau}{\rho}F_{2x}-\gamma
v_{t}+L-g_{2}^{x}=0\,.$

Then, an integrable sopde $(X_{1},X_{2})$ is

$\begin{array}[]{l}X_{1}=v_{t}\,\displaystyle\frac{\partial}{\partial
q}+\left(-\frac{1}{2}\rho v_{t}^{2}+\gamma
s^{t}\right)\,\displaystyle\frac{\partial}{\partial
v_{t}}+\left(\frac{1}{2}\rho v_{t}^{2}-\gamma
s^{t}\right)\displaystyle\frac{\partial}{\partial s^{t}}\,,\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr X_{2}=v_{x}\,\displaystyle\frac{\partial}{\partial
q}+\left(\frac{\gamma\rho}{\tau}v_{t}-\frac{\rho^{2}}{2\tau}v_{t}^{2}+\frac{\gamma\rho}{\tau}s^{t}\right)\,\displaystyle\frac{\partial}{\partial
v_{x}}-\frac{1}{2}\tau v_{x}^{2}\,\displaystyle\frac{\partial}{\partial
s^{x}}\,,\end{array}$

and by Lemma 4.5, any integral section of $(X_{1},X_{2})$ is a canonical lift
of a solution $\phi_{s}$ of the $k$-contact Euler–Lagrange equations (17).

##  5 Relations between solutions and Lagrangian $k$-vector fields

Now we characterize, locally and globally, the set of sopdes in
$\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$.

######  Proposition 5.1.

Let $L\in\mathscr{C}^{\infty}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ be a
Lagrangian function and
${\bf\Gamma}\in\mathfrak{X}^{k}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ a
$k$-vector field. Then,

  1. (1)

${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$
if, and only if, it satisfies the conditions

$\mathscr{L}_{\Gamma_{\alpha}}\eta^{\alpha}_{L}=\frac{\partial L}{\partial
s^{\alpha}}\eta^{\alpha}_{L}\,,\qquad
i_{\Gamma_{\alpha}}\eta^{\alpha}_{L}=-E_{L}\,.$ (36)

  2. (2)

Moreover, if ${\bf\Gamma}$ is a sopde on
$\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$ with local expression (9), the
above equations (36) can be written locally as

$\Gamma_{\alpha}\left(\frac{\partial L}{\partial
v^{i}_{\alpha}}\right)-\frac{\partial L}{\partial q^{i}}=\frac{\partial
L}{\partial s^{\alpha}}\frac{\partial L}{\partial
v_{\alpha}^{i}}\,,\qquad\Gamma^{\alpha}_{\alpha}=L\,.$ (37)

###### Proof.

  1. (1)

It is a direct consequence of equations (27) and Cartan calculus.

  2. (2)

Using the local expressions of the sopde ${\bf\Gamma}$, the Lagrangian energy
$E_{L}$, and the one-forms $\eta^{\alpha}_{L}$ ((9), (23) and (24)
respectively), a direct computation on local coordinates shows that

$\displaystyle 0$
$\displaystyle=i_{\Gamma_{\alpha}}\eta^{\alpha}_{L}+E_{L}=\left(\mathrm{d}s^{\alpha}-\frac{\partial
L}{\partial
v^{i}_{\alpha}}\mathrm{d}q^{i}\right)(\Gamma_{\alpha})+v^{i}_{\alpha}\frac{\partial
L}{\partial v^{i}_{\alpha}}-L=\Gamma^{\alpha}_{\alpha}-L$

and

$\displaystyle 0$
$\displaystyle=\mathscr{L}_{\Gamma_{\alpha}}\eta^{\alpha}_{L}-\frac{\partial
L}{\partial
s^{\alpha}}\eta_{L}^{\alpha}=-\mathrm{d}E_{L}+i_{\Gamma_{\alpha}}\mathrm{d}\eta^{\alpha}_{L}-\frac{\partial
L}{\partial s^{\alpha}}\eta_{L}^{\alpha}$
$\displaystyle=-\mathrm{d}\left(v^{i}_{\alpha}\frac{\partial L}{\partial
v^{i}_{\alpha}}-L\right)+i_{\Gamma_{\alpha}}\left(\mathrm{d}q^{i}\wedge\mathrm{d}\left(\frac{\partial
L}{\partial v^{i}_{\alpha}}\right)\right)-\frac{\partial L}{\partial
s^{\alpha}}\left(\mathrm{d}s^{\alpha}-\frac{\partial L}{\partial
v^{i}_{\alpha}}\mathrm{d}q^{i}\right)$ $\displaystyle=-\frac{\partial
L}{\partial v^{i}_{\alpha}}\mathrm{d}v^{i}_{\alpha}+\frac{\partial L}{\partial
q^{i}}\mathrm{d}q^{i}+\frac{\partial L}{\partial
v^{i}_{\alpha}}\mathrm{d}v^{i}_{\alpha}+\frac{\partial L}{\partial
s^{\alpha}}\mathrm{d}s^{\alpha}-\Gamma_{\alpha}\left(\frac{\partial
L}{\partial v^{i}_{\alpha}}\right)\mathrm{d}q^{i}-\frac{\partial L}{\partial
s^{\alpha}}\mathrm{d}s^{\alpha}+\frac{\partial L}{\partial
s^{\alpha}}\frac{\partial L}{\partial v^{i}_{\alpha}}\mathrm{d}q^{i}$
$\displaystyle=\left(\frac{\partial L}{\partial
q^{i}}-\Gamma_{\alpha}\left(\frac{\partial L}{\partial
v^{i}_{\alpha}}\right)+\frac{\partial L}{\partial s^{\alpha}}\frac{\partial
L}{\partial v^{i}_{\alpha}}\right)\mathrm{d}q^{i}\,,$

which proves the result.

∎

We present now a new relation between solutions to the $k$-contact
Euler–Lagrange equations (12) and $k$-contact Lagrangian $k$-vector fields on
$\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$. This relation plays a fundamental
role in this paper.

######  Proposition 5.2.

Let $L\in\mathscr{C}^{\infty}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ be a
Lagrangian function.

  1. (1)

A map $\phi_{s}:U\subset\mathbb{R}^{k}\rightarrow
Q\times\mathbb{R}^{k}\,,\,\phi_{s}(t)=(\phi(t),s^{\alpha}(t))$ is a solution
to the $k$-contact Euler–Lagrange equations (13) if, and only if,

$g^{\alpha\beta}_{ij}\circ\phi_{s}^{(1)}\left(\Gamma^{j}_{\alpha\beta}\circ\phi_{s}^{(1)}-\frac{\partial^{2}\phi^{j}}{\partial
t^{\alpha}\partial t^{\beta}}\right)+\frac{\partial^{2}L}{\partial
s^{\beta}\partial
v^{i}_{\alpha}}\circ\phi_{s}^{(1)}\left(\Gamma_{\alpha}^{\beta}\circ\phi_{s}^{(1)}-\frac{\partial{s^{\beta}}}{\partial{t^{\alpha}}}\right)=0\,,$
(38)
$\Gamma^{\alpha}_{\alpha}\circ\phi_{s}^{(1)}=\frac{\partial{s^{\alpha}}}{\partial{t^{\alpha}}}=L\circ\phi_{s}^{(1)}\,,$
(39)

for any sopde
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$,
with local expression (9).

  2. (2)

If a $k$-vector field
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$
is integrable, and
$\phi_{s}^{(1)}:U\subset\mathbb{R}^{k}\rightarrow\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$
is an integral section, then $\phi_{s}$ is a solution to the $k$-contact
Euler–Lagrange equations (13).

###### Proof.

  1. (1)

Consider a map $\phi_{s}:U\subset\mathbb{R}^{k}\rightarrow
Q\times\mathbb{R}^{k}\,,\,\,\phi_{s}(t)=(\phi(t),s^{\alpha}(t))\,$. If
$\phi_{s}$ is a solution to the $k$-contact Euler–Lagrange equations (13),
then we have

$g^{\alpha\beta}_{ij}\circ\phi_{s}^{(1)}\frac{\partial^{2}\phi^{j}}{\partial
t^{\alpha}\partial t^{\beta}}+\frac{\partial^{2}L}{\partial q^{j}\partial
v^{i}_{\alpha}}\circ\phi_{s}^{(1)}\frac{\partial\phi^{j}}{\partial
t^{\alpha}}+\frac{\partial^{2}L}{\partial s^{\beta}\partial
v^{i}_{\alpha}}\circ\phi_{s}^{(1)}\frac{\partial{s^{\beta}}}{\partial{t^{\alpha}}}-\frac{\partial
L}{\partial q^{i}}\circ\phi_{s}^{(1)}=\left(\displaystyle\frac{\partial
L}{\partial s^{\alpha}}\displaystyle\frac{\partial L}{\partial
v^{i}_{\alpha}}\right)\circ\phi_{s}^{(1)}\,,$ (40)
$\frac{\partial{s^{\alpha}}}{\partial{t^{\alpha}}}=L\circ\phi_{s}^{(1)}\,.$
(41)

Using Proposition 5.1, a sopde ${\bf\Gamma}$ is in
$\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ if, and only
if, it satisfies equations equations

$\Gamma_{\alpha}^{\alpha}=L\,,$ (42) $\frac{\partial^{2}L}{\partial
v^{j}_{\beta}\partial
v^{i}_{\alpha}}\Gamma_{\alpha\beta}^{j}+\frac{\partial^{2}L}{\partial
q^{j}\partial v^{i}_{\alpha}}v_{\alpha}^{j}+\frac{\partial^{2}L}{\partial
s^{\beta}\partial
v^{i}_{\alpha}}\Gamma_{\alpha}^{\beta}\displaystyle-\frac{\partial L}{\partial
q^{i}}=\frac{\partial L}{\partial s^{\alpha}}\frac{\partial L}{\partial
v^{i}_{\alpha}}\,.$ (43)

Now, restricting equations (42) and (43) to the image of
$\phi_{s}^{(1)}=(\phi^{(1)},s^{\alpha})$ one gets

$(g^{\alpha\beta}_{ij}\circ\phi_{s}^{(1)})(\Gamma^{j}_{\alpha\beta}\circ\phi_{s}^{(1)})+\frac{\partial^{2}L}{\partial
q^{j}\partial
v^{i}_{\alpha}}\circ\phi_{s}^{(1)}+\frac{\partial\phi^{j}}{\partial
t^{\alpha}}\frac{\partial^{2}L}{\partial s^{\beta}\partial
v^{i}_{\alpha}}\circ\phi_{s}^{(1)}(\Gamma_{\alpha}^{\beta}\circ\phi_{s}^{(1)})-\frac{\partial
L}{\partial q^{i}}\circ\phi_{s}^{(1)}=\left(\frac{\partial L}{\partial
s^{\alpha}}\frac{\partial L}{\partial
v^{i}_{\alpha}}\right)\circ\phi_{s}^{(1)}\,,$ (44)
$\Gamma_{\alpha}^{\alpha}\circ\phi_{s}^{(1)}=L\circ\phi_{s}^{(1)}.$ (45)

Using equations (44) and (45), we have that $\phi_{s}$ satisfies (38) and (39)
if, and only if, it satisfies (40) and (41), that are equivalent to
$k$-contact Euler–Lagrange equations (13).

  2. (2)

Since
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$
it follows that ${\bf\Gamma}$ satisfies equations (28), (29), (30) and (31).
If $\phi^{(1)}_{s}$ is an integral map of ${\bf\Gamma}$, we know that
$\Gamma_{\alpha}^{i}(\phi^{(1)}_{s})=v^{i}_{\alpha}(\phi^{(1)}_{s}),$ and if
we restrict equations (30) to $\phi^{(1)}_{s}$, we obtain that $\phi_{s}$
satisfies the $k$-contact Euler–Lagrange equations (13).

∎

######  Remark 5.3.

Equations (38) and (39) do not require any relationship between the $k$-vector
field
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$
and the solution $\phi_{s}$ to the $k$-contact Euler–Lagrange equations (13).
In other words, one might have a solution $\phi_{s}$ to the $k$-contact
Euler–Lagrange equations (13) which may not be a solution for any
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$.

######  Remark 5.4.

Propositions 5.1 and 5.2 play an important role in subsequent sections of this
paper. When the Lagrangian $L$ does not depend on the variables $s^{\alpha}$,
for $\alpha=1,\ldots,k$, the function $L$ may be defined on the bundle
$\oplus^{k}\mathrm{T}Q$, and the previous results can be formulated in terms
of the $k$-symplectic Lagrangian formalism (see Propositions 2.11 and 2.12 in
[6]).

##  6 Dissipation laws

In this section, we discuss dissipation laws for Lagrangian functions defined
on $\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$ and we give certain relations
between them and integrable sopdes on
$\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$. From now
on, unless otherwise stated, $L$ will denote a (regular or singular)
Lagrangian function on the phase bundle
$\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$ and $\eta_{L}^{\alpha}$ will
denote the associated contact Lagrangian one-forms.

######  Definition 6.1.

A map
$F=(F^{1},\ldots,F^{k})\colon\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$
is called a dissipation law if the divergence of

$F\circ\phi_{s}^{(1)}=(F^{1}\circ\phi_{s}^{(1)},\ldots,F^{k}\circ\phi_{s}^{(1)})\colon
U\subset\mathbb{R}^{k}\longrightarrow\mathbb{R}^{k}$

satisfies

$\operatorname{Div}(F\circ\phi_{s}^{(1)})=\frac{\partial{(F^{\alpha}\circ\phi_{s}^{(1)})}}{\partial{t^{\alpha}}}=\left(\displaystyle\frac{\partial{L}}{\partial{s^{\alpha}}}F^{\alpha}\right)\circ\phi_{s}^{(1)}\,,$
(46)

for every solution $\phi_{s}\colon U\subset\mathbb{R}^{k}\rightarrow
Q\times\mathbb{R}^{k}$ of the $k$-contact Euler–Lagrange equations (12).

Then, we have

$\displaystyle\frac{\partial(F^{\alpha}\circ\phi^{(1)}_{s})}{\partial
t^{\alpha}}\Big{|}_{t}$ $\displaystyle=\frac{\partial F^{\alpha}}{\partial
q^{i}}\Big{|}_{\phi_{s}^{(1)}(t)}\frac{\partial\phi^{i}}{\partial
t^{\alpha}}\Big{|}_{t}+\frac{\partial F^{\alpha}}{\partial
v^{i}_{\beta}}\Big{|}_{\phi_{s}^{(1)}(t)}\frac{\partial^{2}\phi^{i}}{\partial
t^{\alpha}\partial t^{\beta}}\Big{|}_{t}+\frac{\partial F^{\alpha}}{\partial
s^{\beta}}\Big{|}_{\phi_{s}^{(1)}(t)}\frac{\partial{s^{\beta}}}{\partial{t^{\alpha}}}\Big{|}_{t}$
$\displaystyle=\left(\displaystyle\frac{\partial{L}}{\partial{s^{\alpha}}}F^{\alpha}\right)\circ\phi_{s}^{(1)}(t)\,,$
(47)

where $\phi_{s}^{(1)}(t)=\left(\phi^{i}(t),\dfrac{\partial\phi^{i}}{\partial
t^{\alpha}}\Big{|}_{t},s^{\alpha}(t)\right)$.

In the following example, we present two dissipation laws for the equation of
the damped vibrating string given by the Lagrangian function (15).

######  Example 6.2.

The two maps
$F_{1}=(F_{1}^{t},F_{1}^{x}),F_{2}=(F_{2}^{t},F_{2}^{x})\colon\oplus^{2}\mathrm{T}\mathbb{R}\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$,
where

  * a)

$F_{1}^{t}=\rho v_{t},\quad F_{1}^{x}=-\tau v_{x}$,

  * b)

$F_{2}^{t}=\displaystyle\frac{1}{\rho}s^{t}-\displaystyle\frac{1}{2}qv_{t},\quad
F_{2}^{x}=\displaystyle\frac{1}{\rho}s^{x}+\displaystyle\frac{\tau}{2\rho}qv_{x}$,

give dissipation laws for the $k$-contact Euler–Lagrange equations (17). Thus,
if $\phi_{s}$ is a solution to (17), we deduce that

$\frac{\partial(F_{1}^{t}\circ\phi^{(1)}_{s})}{\partial
t}+\frac{\partial(F_{1}^{x}\circ\phi^{(1)}_{s})}{\partial
x}=\rho\frac{\partial^{2}\phi}{\partial
t^{2}}-\tau\frac{\partial^{2}\phi}{\partial
x^{2}}=-\gamma\rho\frac{\partial\phi}{\partial t}=\left(\frac{\partial
L}{\partial s^{t}}F_{1}^{t}\right)\circ\phi_{s}^{(1)},$

where $\phi^{(1)}_{s}=\left(\phi,\dfrac{\partial\phi}{\partial
t},\dfrac{\partial\phi}{\partial x},s^{t},s^{x}\right)$.

One also verifies that $(F_{2}^{t},F_{2}^{x})$ satisfies (47) by a
straightforward computation.

######  Remark 6.3.

Note that if the Lagrangian $L$ does not depend on the variables $s^{\alpha}$,
for $\alpha=1,\ldots,k$, we can consider $L$ and the functions $F^{\alpha}$
defined on $\oplus^{k}\mathrm{T}Q$, and hence the above Definition 6.1 becomes
the definition of conservation law for $k$-symplectic Lagrangian systems (see
[42]).

The following result gives us a first relationship between dissipation laws
and integrable sopdes on
$\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$.

######  Lemma 6.4.

Let
$F=(F^{1},\ldots,F^{k})\colon\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$
be a dissipation law. Then, every integrable sopde
${\bf\Gamma}=(\Gamma_{1},\dots,\Gamma_{k})\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$
satisfies

$\Gamma_{\alpha}(F^{\alpha})=\frac{\partial L}{\partial
s^{\alpha}}F^{\alpha}\,.$ (48)

###### Proof.

Since ${\bf\Gamma}$ is an integrable sopde, for every point
$x\in\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$ there exists an integral
section
$\phi_{s}^{(1)}:U\subset\mathbb{R}^{k}\rightarrow\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$
such that

  1. (1)

$\phi_{s}$ is a solution to the $k$-contact Euler–Lagrange equations, because
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$,

  2. (2)

$\phi_{s}$ satisfies

$\phi_{s}^{(1)}(0)=x\,,\qquad(\phi_{s}^{(1)})_{*}(t)\left(\frac{\partial}{\partial
t^{\alpha}}\Big{|}_{t}\right)=\Gamma_{\alpha}(\phi_{s}^{(1)}(t))\,,$

for every $t\in U$ and $\alpha=1,\ldots,k$.

Condition (2) above means that

$v^{i}_{\alpha}(\phi_{s}^{(1)}(t))=\frac{\partial\phi^{i}}{\partial
t^{\alpha}}\Big{|}_{t}\,,\qquad\Gamma^{i}_{\alpha\beta}(\phi_{s}^{(1)}(t))=\frac{\partial^{2}\phi^{i}}{\partial
t^{\alpha}\partial
t^{\beta}}\Big{|}_{t}\,,\qquad\Gamma_{\alpha}^{\beta}(\psi(t))=\frac{\partial
s^{\alpha}}{\partial t^{\beta}}\Big{|}_{t}\,.$ (49)

Since $F=(F^{1},\ldots,F^{k})$ is a dissipation law, using equation (47) at
$t=0$, and equation (49), we have

$\displaystyle\left(\displaystyle\frac{\partial{L}}{\partial{s^{\alpha}}}F^{\alpha}\right)\circ\phi_{s}^{(1)}(0)$
$\displaystyle=\displaystyle\frac{\partial(F^{\alpha}\circ\phi_{s}^{(1)})}{\partial
t^{\alpha}}\Big{|}_{0}$ (50) $\displaystyle=\displaystyle\frac{\partial
F^{\alpha}}{\partial
q^{i}}\Big{|}_{\phi_{s}^{(1)}(0)}\frac{\partial\phi^{i}}{\partial
t^{\alpha}}\Big{|}_{0}+\frac{\partial F^{\alpha}}{\partial
v^{i}_{\beta}}\Big{|}_{\phi_{s}^{(1)}(0)}\frac{\partial^{2}\phi^{i}}{\partial
t^{\alpha}\partial t^{\beta}}\Big{|}_{0}+\frac{\partial F^{\alpha}}{\partial
s^{\beta}}\Big{|}_{\phi_{s}^{(1)}(0)}\frac{\partial{s^{\beta}}}{\partial{t^{\alpha}}}\Big{|}_{0}$
(51) $\displaystyle=\displaystyle\frac{\partial F^{\alpha}}{\partial
q^{i}}\Big{|}_{x}v^{i}_{\alpha}(x)+\displaystyle\frac{\partial
F^{\alpha}}{\partial
v^{i}_{\beta}}\Big{|}_{x}\Gamma^{i}_{\alpha\beta}(x)+\displaystyle\frac{\partial
F^{\alpha}}{\partial
s^{\beta}}\Big{|}_{x}\Gamma_{\alpha}^{\beta}(x)=\Gamma_{\alpha}(x)(F^{\alpha})\,.$
(52)

∎

The converse of Lemma 6.4 is not true, and the reason is that, as we can see
from equations (38) and (39), we might have solutions $\phi_{s}$ of the
$k$-contact Euler–Lagrange equations (13) which are not solutions to some
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$.
However, we show in the following lemma that, under some assumption on the
functions $F^{\alpha}$, this converse is true.

######  Lemma 6.5.

Let $L\in\mathscr{C}^{\infty}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ be a
Lagrangian and assume that there exists a vector field
$X\in\mathfrak{X}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ such that

$i_{X}\mathrm{d}\eta_{L}^{\alpha}=\mathrm{d}F^{\alpha}\,,\quad\text{for
every}\,\,\,\alpha=1,\ldots,k\,,$ (53)

for some functions
$F^{\alpha}:\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}\rightarrow\mathbb{R}$.
Then, $F=(F^{\alpha})$ is a dissipation law for the $k$-contact Euler–Lagrange
equations (12) if, and only if,

$\Gamma_{\alpha}(F^{\alpha})=\frac{\partial L}{\partial s^{\alpha}}F^{\alpha}$

for every integrable sopde
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$.

###### Proof.

The direct implication is given by Lemma 6.4. For the converse implication,
let

$X=X^{i}\dfrac{\partial}{\partial
q^{i}}+X^{i}_{\alpha}\dfrac{\partial}{\partial
v^{i}_{\alpha}}+X^{\alpha}\dfrac{\partial}{\partial s^{\alpha}}$

be a vector field on $\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$ satisfying
(53). In view of equations (24), we can write both sides of the equations (53)
as

$\displaystyle i_{X}\mathrm{d}\eta_{L}^{\alpha}$
$\displaystyle=\left[\left(\frac{\partial^{2}L}{\partial q^{i}\partial
v^{j}_{\alpha}}-\frac{\partial^{2}L}{\partial q^{j}\partial
v^{i}_{\alpha}}\right)X^{j}-g^{\alpha\beta}_{ij}X^{j}_{\beta}-\frac{\partial^{2}L}{\partial
s^{\beta}\partial
v^{j}_{\alpha}}X^{\beta}\right]\mathrm{d}q^{i}+g^{\alpha\beta}_{ij}X^{i}\mathrm{d}v^{j}_{\beta}+\frac{\partial^{2}L}{\partial
s^{\beta}\partial v^{j}_{\alpha}}X^{j}\mathrm{d}s^{\beta}\,,$ (54)
$\displaystyle\mathrm{d}F^{\alpha}$ $\displaystyle=\frac{\partial
F^{\alpha}}{\partial q^{i}}\mathrm{d}q^{i}+\frac{\partial F^{\alpha}}{\partial
v^{j}_{\beta}}\mathrm{d}v^{j}_{\beta}+\frac{\partial F^{\alpha}}{\partial
s^{\beta}}\mathrm{d}s^{\beta}\,,$ (55)

and necessarily we have

$\frac{\partial F^{\alpha}}{\partial
v^{j}_{\beta}}=g^{\alpha\beta}_{ij}X^{i}\,,\qquad\frac{\partial
F^{\alpha}}{\partial s^{\beta}}=\frac{\partial^{2}L}{\partial
s^{\beta}\partial v^{i}_{\alpha}}X^{i}\,.$ (56)

Consider now $\phi_{s}$ to be any solution to the $k$-contact Euler–Lagrange
equations (13) (which may not be a solution of any ${\bf\Gamma}$). It follows
that $\phi_{s}$ satisfies equations (38) and (39), since ${\bf\Gamma}$ is
assumed to be an integrable sopde.

Contracting equations (39) with $X^{i}\circ\phi_{s}^{(1)}$, we obtain

$(X^{i}\circ\phi_{s}^{(1)})(g^{\alpha\beta}_{ij}\circ\phi_{s}^{(1)})\left(\Gamma^{j}_{\alpha\beta}\circ\phi_{s}^{(1)}-\frac{\partial^{2}\phi^{j}}{\partial
t^{\alpha}\partial
t^{\beta}}\right)+(X^{i}\circ\phi_{s}^{(1)})\left(\frac{\partial^{2}L}{\partial
s^{\beta}\partial
v^{i}_{\alpha}}\circ\phi_{s}^{(1)}\right)\left(\Gamma_{\alpha}^{\beta}\circ\phi_{s}^{(1)}-\frac{\partial{s^{\beta}}}{\partial{t^{\alpha}}}\right)=0.$
(57)

If we replace formulas (56) in equation (57), we have

$\displaystyle 0$ $\displaystyle=\displaystyle\frac{\partial
F^{\alpha}}{\partial
v^{j}_{\beta}}\circ\phi_{s}^{(1)}\left(\Gamma^{j}_{\alpha\beta}\circ\phi_{s}^{(1)}-\displaystyle\frac{\partial^{2}\phi^{j}}{\partial
t^{\alpha}\partial t^{\beta}}\right)+\displaystyle\frac{\partial
F^{\alpha}}{\partial
s^{\beta}}\circ\phi_{s}^{(1)}\left(\Gamma_{\alpha}^{\beta}\circ\phi_{s}^{(1)}-\displaystyle\frac{\partial
s^{\beta}}{\partial t^{\alpha}}\right)$ (58)
$\displaystyle=-\displaystyle\frac{\partial(F^{\alpha}\circ\phi_{s}^{(1)})}{\partial
t^{\alpha}}+\Gamma_{\alpha}(F^{\alpha})\circ\phi_{s}^{(1)}\,,$ (59)

therefore we conclude

$\frac{\partial(F^{\alpha}\circ\phi_{s}^{(1)})}{\partial
t^{\alpha}}=\Gamma_{\alpha}(F^{\alpha})\circ\phi_{s}^{(1)}=\left(\frac{\partial
L}{\partial s^{\alpha}}F^{\alpha}\right)\circ\phi_{s}^{(1)}\,,$

and the result follows. ∎

######  Remark 6.6.

Notice that the above Lemma 6.5 is analogous to Lemma 3.4 in [6] for the
$k$-symplectic Lagrangian formalism.

##  7 Symmetries

In this section we study several different notions of symmetry for $k$-contact
Lagrangian field theories depending on the structure they preserve: natural
symmetries (preserving the Lagrangian), dynamical symmetries (preserving the
solutions) and $k$-contact symmetries (preserving the underlying geometric
structures). We investigate the relations between these symmetries and prove
several Noether-like theorems relating the different types of symmetries to
dissipation laws.

###  7.1 Natural symmetries

We will now try to understand the symmetries of a $k$-contact Lagrangian
system which are lifts of vector fields on the configuration space. One can
see in [14] some interesting results related with this kind of symmetries in
the case $k=1$.

For any vector field $Z\in\mathfrak{X}(Q)$, we denote by
$Z^{C},Z^{V_{\alpha}}\in\mathfrak{X}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$
its complete and vertical $\alpha$-lifts, defined in Section 3 and extended to
$\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$ in a natural way. Their local
expressions are the same as in (4).

######  Definition 7.1.

A vector field $Z\in\mathfrak{X}(Q)$ is said to be an infinitesimal natural
symmetry of a Lagrangian function
$L\in\mathscr{C}^{\infty}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ if
$Z^{C}(L)=0$.

Then, we have the following result relating natural symmetries and dissipation
laws.

######  Theorem 7.2.

Let $Z\in\mathfrak{X}(Q)$ be an infinitesimal natural symmetry of a Lagrangian
function $L$. Then, the functions $F^{\alpha}=Z^{V_{\alpha}}(L)$ give a
dissipation law.

###### Proof.

Consider the local expressions of $Z^{V_{\alpha}}$, $Z^{C}$ and ${\bf\Gamma}$
in (4) and (9). From (37), and taking into account that the functions $Z^{i}$
only depend on the variables $q^{i}$, we get

$\Gamma_{\alpha}(Z^{V_{\alpha}}(L))=\left(Z^{i}\frac{\partial}{\partial
q^{i}}+v^{j}_{\alpha}\displaystyle\frac{\partial Z^{i}}{\partial
q^{j}}\frac{\partial}{\partial
v^{i}_{\alpha}}\right)(L)+\displaystyle\frac{\partial L}{\partial
s^{\alpha}}\left(Z^{i}\frac{\partial}{\partial
v^{i}_{\alpha}}\right)(L)=Z^{C}(L)+\displaystyle\frac{\partial L}{\partial
s^{\alpha}}Z^{V_{\alpha}}(L)\,,$ (60)

for all integrable sopde
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$.

Now, since $Z^{C}(L)=0$ we show that
$\Gamma_{\alpha}(Z^{V_{\alpha}}(L))=\dfrac{\partial L}{\partial
s^{\alpha}}Z^{V_{\alpha}}(L)$. Finally, in view of Lemma 6.5, we just need to
check that

$i_{Z^{C}}\mathrm{d}\eta^{\alpha}_{L}=\mathrm{d}\left(Z^{V_{\alpha}}(L)\right)$
(61)

to conclude that the functions $Z^{V_{\alpha}}(L)$ give a dissipation law.

If one applies $\dfrac{\partial}{\partial v^{k}_{\beta}}$ to the identity
$Z^{C}(L)=Z^{i}\dfrac{\partial L}{\partial
q^{i}}+v^{j}_{\alpha}\dfrac{\partial Z^{i}}{\partial q^{j}}\dfrac{\partial
L}{\partial v^{i}_{\alpha}}=0$, we obtain the relation (61) in local
coordinates. ∎

######  Remark 7.3.

When the Lagrangian $L$ does not depend on variables $s^{\alpha}$, for
$\alpha=1,\dotsc,k$, the function $L$ may be defined on the bundle
$\oplus^{k}\mathrm{T}Q$, and the above result is similar to Proposition 3.15
in [42], if we consider the function $g$ to be identically zero. In this case,
the functions $F^{\alpha}$ will give a conservation law, according to the
nature of the system.

The next symmetries we are going to study are those transformations preserving
some structure, and we will distinguish between those that preserve the
solutions of the system (dynamical symmetries) and those that preserve the
geometric structures ($k$-contact symmetries). These notions were introduced
in [21] for the case $k=1$, and in [22] for the general case $k\geq 1$.

###  7.2 Dynamical symmetries

We will begin by introducing the transformations preserving the solutions of
the system.

######  Definition 7.4.

A Lagrangian dynamical symmetry is a diffeomorphism
$\Phi:\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}\rightarrow\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$
such that, for every solution $\phi_{s}^{(1)}$ to the $k$-contact
Euler–Lagrange equations (12), $\Phi\circ\phi_{s}^{(1)}$ is also a solution.

An infinitesimal Lagrangian dynamical symmetry is a vector field
$X\in\mathfrak{X}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ whose local flow
is made of dynamical symmetries.

The following result can be found in [22].

######  Lemma 7.5.

Let $X\in\mathfrak{X}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ be an
infinitesimal dynamical symmetry. Then, for every
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$,
we have $i_{[X,\Gamma_{\alpha}]}\eta^{\alpha}_{L}=0.$

For the case $k=1$ it is known that dynamical symmetries induce dissipated
quantities, and these results are known as dissipation theorems (see [21]).

In the general case $k>1$, in Theorem 3 in [22], it is proved that if $X$ is
an infinitesimal dynamical symmetry, the functions
$F^{\alpha}=-i_{X}\eta^{\alpha}_{L}$ satisfy (48), and give a dissipation law
for integral sections of
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$.

The next result states, again with an extra condition, that these functions
$F^{\alpha}$ are a dissipation law for every solution of the $k$-contact
Euler–Lagrange equations (12).

######  Theorem 7.6 (Dissipation theorem).

Let $L\in\mathscr{C}^{\infty}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ be a
Lagrangian function and consider an infinitesimal dynamical symmetry
$X\in\mathfrak{X}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ satisfying the
conditions

$i_{X}\mathrm{d}\eta^{\alpha}_{L}=\mathrm{d}(-i_{X}\eta^{\alpha}_{L})\,,$ (62)

or, equivalently, $\mathscr{L}_{X}\eta^{\alpha}_{L}=0$. Then, the functions

$F^{\alpha}=-i_{X}\eta^{\alpha}_{L}$ (63)

provide a dissipation law for the $k$-contact Euler–Lagrange equations.

###### Proof.

Let
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$
be an integrable sopde. From Lemma 7.5, we obtain

$\Gamma_{\alpha}(-i_{X}\eta^{\alpha}_{L})=-\mathscr{L}_{\Gamma_{\alpha}}i_{X}\eta^{\alpha}_{L}=-i_{X}\mathscr{L}_{\Gamma_{\alpha}}\eta^{\alpha}_{L}-i_{[\Gamma_{\alpha},X]}\eta^{\alpha}_{L}=\displaystyle\frac{\partial
L}{\partial s^{\alpha}}(-i_{X}\eta^{\alpha}_{L}).$

Now, from (62) and Lemma 6.5, we deduce that the functions
$F^{\alpha}=-i_{X}\eta^{\alpha}_{L}$ give a dissipation law. ∎

###  7.3 $k$-contact symmetries

Among the most relevant symmetries are those that leave the geometric
structures invariant. Let us recall the definition of $k$-contact symmetries
for a Lagrangian function
$L\in\mathscr{C}^{\infty}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ (see
[42]).

######  Definition 7.7.

A diffeomorphism
$\Phi:\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}\rightarrow\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$
is called a Lagrangian $k$-contact symmetry if

$\Phi^{*}\eta^{\alpha}_{L}=\eta^{\alpha}_{L}\,,\qquad\Phi^{*}E_{L}=E_{L}\,.$

A vector field $X\in\mathfrak{X}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$
is called an infinitesimal Lagrangian $k$-contact symmetry if

$\mathscr{L}_{X}\eta^{\alpha}_{L}=0\,,\qquad\mathscr{L}_{X}E_{L}=0\,,$ (64)

that is, its local flow is made of Lagrangian $k$-contact symmetries.

One can obtain the relation between these symmetries that preserve geometric
structures and those that preserve the solutions of the system as follows.

######  Proposition 7.8.

(Infinitesimal) Lagrangian $k$-contact symmetries are (infinitesimal)
Lagrangian dynamical symmetries.

###### Proof.

Let $\psi=\phi_{s}^{(1)}$ be a solution of (26), and $\Phi$ a Lagrangian
$k$-contact symmetry

$\Phi^{*}\eta^{\alpha}_{L}=\eta^{\alpha}_{L}\,,\qquad\Phi^{*}E_{L}=E_{L}\,.$
(65)

We need to prove that

$i_{(\Phi\circ\psi)^{\prime}_{\alpha}}\mathrm{d}\eta^{\alpha}_{L}=\left(\mathrm{d}E_{L}+\dfrac{\partial
L}{\partial s^{\alpha}}\eta^{\alpha}_{L}\right)\circ(\Phi\circ\psi)\,,\qquad
i_{(\Phi\circ\psi)^{\prime}_{\alpha}}\eta^{\alpha}_{L}=-E_{L}\circ(\Phi\circ\psi)\,,$
(66)

where
$(\Phi\circ\psi)^{\prime}_{\alpha}=(\Phi\circ\psi)_{*}(t)(\frac{\partial}{\partial
t^{\alpha}})\,,$ which is a consequence of (26) and (65).

If $X$ is an infinitesimal $k$-contact symmetry, its flow $\phi_{t}$ is made
of local $k$-contact symmetries, namely
$\phi_{t}^{*}\eta^{\alpha}_{L}=\eta^{\alpha}_{L}$ and
$\phi_{t}^{*}E_{L}=E_{L}$. Thus $\phi_{t}$ satisfies (66), and then it
transforms solutions to the $k$-contact Euler–Lagrange equations on solutions
to the $k$-contact Euler–Lagrange equations. ∎

From Dissipation Theorem 7.6 and Proposition 7.8 we deduce that

######  Corollary 7.9.

If $X$ is an infinitesimal Lagrangian $k$-contact symmetry, the functions
$F^{\alpha}=-i_{X}\eta^{\alpha}_{L}$ give a dissipation law for the
$k$-contact Euler–Lagrange equations (12).

######  Example 7.10.

Consider again Example 3.3 (the damped vibrating string). The vector field
$X=\dfrac{\partial}{\partial q}$ is an infinitesimal Lagrangian $k$-contact
symmetry, and therefore, the map given by

$F=\left(-i_{\frac{\partial}{\partial
q}}\eta^{t}_{L},-i_{\frac{\partial}{\partial q}}\eta^{x}_{L}\right)=\left(\rho
v_{t},-\tau v_{x}\right)$

is a dissipation law.

###  7.4 Dissipation laws given by vector fields which are not symmetries

Finally, let us consider any vector field
$Z\in\mathfrak{X}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ satisfying the
conditions

$\mathscr{L}_{Z}\eta^{\alpha}_{L}=\mathrm{d}g^{\alpha}\,,\quad\quad\mathscr{L}_{Z}E_{L}=-g^{\alpha}\displaystyle\frac{\partial
L}{\partial s^{\alpha}}\,,$ (67)

for some functions
$g^{\alpha}\in\mathscr{C}^{\infty}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$.
In the following result we show how to associate dissipation laws to these
particular vector fields.

######  Theorem 7.11.

Let $L:\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}\rightarrow\mathbb{R}$ be a
Lagrangian function and consider a vector field
$Z\in\mathfrak{X}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ satisfying
conditions (67). Then, the functions

$F^{\alpha}=g^{\alpha}-i_{Z}\eta^{\alpha}_{L}$

give a dissipation law for the $k$-contact Euler–Lagrange equations.

###### Proof.

Consider an integrable sopde
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus\mathrm{T}Q\times\mathbb{R}^{k})$.
Then, by Proposition 5.1, the sopde ${\bf\Gamma}$ satisfies equations (36), so
we compute

$\displaystyle\begin{split}\Gamma_{\alpha}(F^{\alpha})&=\Gamma_{\alpha}(g^{\alpha})-\mathscr{L}_{\Gamma_{\alpha}}i_{Z}\eta^{\alpha}_{L}=\Gamma_{\alpha}(g^{\alpha})-i_{Z}\mathscr{L}_{\Gamma_{\alpha}}\eta^{\alpha}_{L}-i_{[\Gamma_{\alpha},Z]}\eta^{\alpha}_{L}\\\
&=\Gamma_{\alpha}(g^{\alpha})-\displaystyle\frac{\partial L}{\partial
s^{\alpha}}i_{Z}\eta^{\alpha}_{L}-i_{[\Gamma_{\alpha},Z]}\eta^{\alpha}_{L}\,.\end{split}$
(68)

Contracting $\mathscr{L}_{Z}\eta^{\alpha}_{L}$ with $\Gamma_{\alpha}$ and
summing for $\alpha$, we have

$i_{\Gamma_{\alpha}}\mathscr{L}_{Z}\eta^{\alpha}_{L}=i_{\Gamma_{\alpha}}(\mathrm{d}g^{\alpha})=\Gamma_{\alpha}(g^{\alpha})\,,$
(69)

and on the other side

$i_{\Gamma_{\alpha}}\mathscr{L}_{Z}\eta^{\alpha}_{L}=\mathscr{L}_{Z}i_{\Gamma_{\alpha}}\eta^{\alpha}_{L}-i_{[Z,\Gamma_{\alpha}]}\eta^{\alpha}_{L}=-\mathscr{L}_{Z}E_{L}+i_{[\Gamma_{\alpha},Z]}\eta^{\alpha}_{L}=\displaystyle\frac{\partial
L}{\partial s^{\alpha}}g^{\alpha}+i_{[\Gamma_{\alpha},Z]}\eta^{\alpha}_{L}\,.$
(70)

Therefore, from (69) and (70) we deduce that

$\Gamma_{\alpha}(g^{\alpha})=\displaystyle\frac{\partial L}{\partial
s^{\alpha}}g^{\alpha}+i_{[\Gamma_{\alpha},Z]}\eta^{\alpha}_{L}\,.$ (71)

Finally, substituting the above expression in equation (68) we obtain

$\Gamma_{\alpha}(g^{\alpha}-i_{Z}\eta^{\alpha}_{L})=\displaystyle\frac{\partial
L}{\partial s^{\alpha}}(g^{\alpha}-i_{Z}\eta^{\alpha}_{L}),$

and by Lemma 6.5 the functions $F^{\alpha}=g^{\alpha}-i_{Z}\eta^{\alpha}_{L}$
give a dissipation law. ∎

######  Remark 7.12.

When the functions $g^{\alpha}=0$, for every $\alpha=1,\ldots,k$, equations
(67) reduce to equations (64) of infinitesimal Lagrangian $k$-contact
symmetries, and the above theorem becomes Corollary 7.9.

To illustrate the above theorem we will give an example of a vector field
satisfying the conditions (67).

######  Example 7.13.

Let us consider again the equations modelling a damped vibrating string given
by the Lagrangian function (15). The vector field
$Z=\displaystyle\frac{\partial}{\partial
q}+\displaystyle\frac{\partial}{\partial
s^{t}}+g^{x}\displaystyle\frac{\partial}{\partial s^{x}}$ satisfies

$\mathscr{L}_{Z}\eta^{t}_{L}=0\,,\qquad\mathscr{L}_{Z}\eta^{x}_{L}=\mathrm{d}g^{x}\,,\qquad\mathscr{L}_{Z}E_{L}=-\displaystyle\frac{\partial
L}{\partial s^{t}}g^{t}-\displaystyle\frac{\partial L}{\partial
s^{x}}g^{x}\,,$

where the function $g^{t}$ is constant, $g^{t}=c$, and $g^{x}$ is an arbitrary
function.

By Theorem 7.11, the corresponding dissipation law is the map given by

$F=\left(k-i_{Z}\eta^{t}_{L},g^{x}-i_{Z}\eta^{x}_{L}\right)=(\rho v_{t},-\tau
v_{x})\,,$

which coincides with the dissipation law given by the vector field
$\displaystyle\frac{\partial}{\partial q}$ in the Example 7.10.

######  Remark 7.14.

When the Lagrangian $L$ does not depend on the variables $s^{\alpha}$, for
$\alpha=1,\dotsc,k$, the function $L$ is defined on the bundle
$\oplus^{k}\mathrm{T}Q$ and, if we think of $Z$ as vector field on
$\oplus^{k}\mathrm{T}Q$, the conditions (67) are now

$\mathscr{L}_{Z}\mathrm{d}\eta^{\alpha}_{L}=0\,,\qquad\mathscr{L}_{Z}E_{L}=0\,,$

which is the definition of a Cartan symmetry for the $k$-symplectic system
$(\oplus^{k}\mathrm{T}Q,\mathrm{d}\eta^{\alpha}_{L},E_{L})$, see [42].
Moreover, the above Theorem 7.11 becomes the Noether’s Theorem 3.13 in [42].

###  7.5 Newtonoid vector fields

In [6] the set of Newtonoid vector fields were introduced in the framework of
the $k$-symplectic formulation of autonomous first-order field theories,
extending the work of G. Marmo and N. Mukunda in [36] for the case $k=1$. In
this section, we introduce this kind of vector fields in the $k$-contact
framework.

In Proposition 7.16 we prove that, for a regular Lagrangian function $L$,
infinitesimal $k$-contact symmetries are Newtonoid vector fields for every
corresponding sopde
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$.
Finally, we observe that a particular kind of Newtonoid vector fields that
leave the Lagrangian function invariant are also infinitesimal $k$-contact
symmetries and hence they provide dissipation laws.

######  Definition 7.15.

Consider a fixed sopde ${\bf\Gamma}$. A vector field
$X\in\mathfrak{X}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ is Newtonoid
with respect to ${\bf\Gamma}$ if
$J^{\alpha}\left(\left[\Gamma_{\alpha},X\right]\right)=0$. We denote by
$\mathfrak{X}_{\Gamma}$ the set of Newtonoid vector fields associated to a
sopde ${\bf\Gamma}$.

Since

$\displaystyle\left[\Gamma_{\alpha},X\right]$
$\displaystyle=\left[v_{\alpha}^{i}\frac{\partial}{\partial
q^{i}}+\Gamma^{i}_{\alpha\beta}\frac{\partial}{\partial
v^{i}_{\beta}}+\Gamma_{\alpha}^{\beta}\frac{\partial}{\partial
s^{\beta}},X^{i}\frac{\partial}{\partial
q^{i}}+X^{i}_{\gamma}\frac{\partial}{\partial
v^{i}_{\gamma}}+X^{\gamma}\frac{\partial}{\partial s^{\gamma}}\right]$
$\displaystyle=\left(\Gamma_{\alpha}(X^{i})-X^{i}_{\alpha}\right)\frac{\partial}{\partial
q^{i}}+\left(\Gamma_{\alpha}(X^{i}_{\beta})-X(\Gamma^{i}_{\alpha\beta})\right)\frac{\partial}{\partial
v^{i}_{\beta}}+\left(\Gamma_{\alpha}(X^{\beta})-X(\Gamma^{\beta}_{\alpha})\right)\frac{\partial}{\partial
s^{\beta}}\,,$ (72)

we deduce that $J^{\alpha}\left(\left[\Gamma_{\alpha},X\right]\right)=0$ if,
and only if, $\Gamma_{\alpha}(X^{i})=X^{i}_{\alpha}$. Therefore, a Newtonoid
vector field $X$ can be written locally as follows

$X=X^{i}\frac{\partial}{\partial
q^{i}}+\Gamma_{\alpha}(X^{i})\frac{\partial}{\partial
v^{i}_{\alpha}}+X^{\gamma}\frac{\partial}{\partial s^{\gamma}}\,.$ (73)

From the local expressions (4) and (73) it follows that the vector fields of
the form

$Z^{C}+X^{\alpha}\displaystyle\frac{\partial}{\partial s^{\alpha}}\,,$

where $Z^{C}\in\mathfrak{X}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ is the
complete lift of a vector field $Z\in\mathfrak{X}(Q)$ and $X^{\alpha}$ are
arbitrary functions on $\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$, are
Newtonoid vector fields for an arbitrary sopde ${\bf\Gamma}$.

In the next proposition we see that the set of Newtonoid vector fields
contains also infinitesimal $k$-contact symmetries.

######  Proposition 7.16.

Let $X\in\mathfrak{X}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ be an
infinitesimal $k$-contact symmetry of a system described by a regular
Lagrangian function
$L:\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}\rightarrow\mathbb{R}$. Then $X$
is a Newtonoid vector field for every
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$.

###### Proof.

Since
${\bf\Gamma}\in\mathfrak{X}^{k}_{L}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k}$)
and $L$ is regular, from Lemma 4.5 it follows that ${\bf\Gamma}$ is a sopde.
Moreover, ${\bf\Gamma}$ is a solution to the equations

$i_{\Gamma_{\alpha}}\mathrm{d}\eta_{L}^{\alpha}=\mathrm{d}E_{L}+\displaystyle\frac{\partial{L}}{\partial{s^{\alpha}}}\eta_{L}^{\alpha}\,,\quad
i_{\Gamma_{\alpha}}\eta_{L}^{\alpha}=-E_{L}\,.$ (74)

Since $\mathscr{L}_{X}\eta^{\alpha}_{L}=0$ and $\mathscr{L}_{X}E_{L}=0$, and
using the second equation in (74) it follows that

$0=\mathscr{L}_{X}E_{L}=-\mathscr{L}_{X}i_{\Gamma_{\alpha}}\eta^{\alpha}_{L}=-i_{\Gamma_{\alpha}}\mathscr{L}_{X}\eta^{\alpha}_{L}-i_{[X,\Gamma_{\alpha}]}\eta^{\alpha}_{L}=-i_{[X,\Gamma_{\alpha}]}\eta^{\alpha}_{L}\,.$

Now, if we apply $\mathscr{L}_{X}$ to both sides of the first equation in (74)
and use the commutation rules, we obtain

$\mathscr{L}_{X}i_{\Gamma_{\alpha}}\mathrm{d}\eta_{L}^{\alpha}=\mathscr{L}_{X}\mathrm{d}E_{L}+\mathscr{L}_{X}\left(\frac{\partial
L}{\partial
s^{\alpha}}\eta_{L}^{\alpha}\right)=\mathscr{L}_{X}\left(\frac{\partial
L}{\partial s^{\alpha}}\right)\eta_{L}^{\alpha}+\frac{\partial L}{\partial
s^{\alpha}}\mathscr{L}_{X}\eta_{L}^{\alpha}=\mathscr{L}_{X}\left(\frac{\partial
L}{\partial s^{\alpha}}\right)\eta_{L}^{\alpha}$ (75)

and

$\mathscr{L}_{X}i_{\Gamma_{\alpha}}\mathrm{d}\eta_{L}^{\alpha}=i_{\Gamma_{\alpha}}\mathscr{L}_{X}\mathrm{d}\eta^{\alpha}_{L}-i_{[\Gamma_{\alpha},X]}\mathrm{d}\eta^{\alpha}_{L}=-i_{[\Gamma_{\alpha},X]}\mathrm{d}\eta^{\alpha}_{L}\,.$
(76)

Then, from (75) and (76) we have

$i_{[\Gamma_{\alpha},X]}\mathrm{d}\eta^{\alpha}_{L}=-\mathscr{L}_{X}\left(\frac{\partial
L}{\partial s^{\alpha}}\right)\eta_{L}^{\alpha}\,.$ (77)

We will prove now that equation (77) implies that
$J^{\alpha}([\Gamma_{\alpha},X])=0$ and hence $X$ is a Newtonoid vector field
for ${\bf\Gamma}$. Using formula (72), we have

$[\Gamma_{\alpha},X]=V_{\alpha}^{i}\frac{\partial}{\partial
q^{i}}+V^{i}_{\alpha\beta}\frac{\partial}{\partial
v^{i}_{\beta}}+V_{\alpha}^{\beta}\frac{\partial}{\partial s^{\beta}}\,,$ (78)

where

$V^{i}_{\alpha}=\Gamma_{\alpha}(X^{i})-X^{i}_{\alpha}\,,\quad
V^{i}_{\alpha\beta}=\Gamma_{\alpha}(X^{i}_{\beta})-X(\Gamma^{i}_{\alpha\beta})\,,\quad
V^{\beta}_{\alpha}=\Gamma_{\alpha}(X^{\beta})-X(\Gamma^{\beta}_{\alpha})\,.$

Using formula (24), it follows that the differential of the contact one-forms
$\eta^{\alpha}_{L}$ can be written as follows

$\mathrm{d}\eta^{\alpha}_{L}=a^{\alpha}_{ij}\mathrm{d}q^{i}\wedge\mathrm{d}q^{j}+g^{\alpha\beta}_{ij}\mathrm{d}q^{i}\wedge\mathrm{d}v^{j}_{\beta}+h^{\alpha}_{i\beta}\mathrm{d}q^{i}\wedge\mathrm{d}s^{\beta}\,,$
(79)

where

$a^{\alpha}_{ij}=\frac{1}{2}\left(\frac{\partial^{2}L}{\partial q^{j}\partial
v^{i}_{\alpha}}-\frac{\partial^{2}L}{\partial q^{i}\partial
v^{j}_{\alpha}}\right)\,,\quad
g^{\alpha\beta}_{ij}=\frac{\partial^{2}L}{\partial v^{i}_{\alpha}\partial
v^{j}_{\beta}}\,,\quad h^{\alpha}_{i\beta}=\frac{\partial^{2}L}{\partial
s^{\beta}\partial v^{i}_{\alpha}}\,.$

If we replace now formulae (78) and (79) in equation (77) we obtain

$\left(2a^{\alpha}_{ij}V^{j}_{\alpha}-g^{\alpha\beta}_{ij}V^{j}_{\alpha\beta}-h^{\alpha}_{i\beta}V_{\alpha}^{\beta}\right)\mathrm{d}q^{i}+g^{\alpha\beta}_{ij}V^{i}_{\alpha}\mathrm{d}v^{j}_{\beta}+h^{\alpha}_{i\beta}V^{i}_{\alpha}\mathrm{d}s^{\beta}=-\mathscr{L}_{X}\left(\displaystyle\frac{\partial{L}}{\partial{s^{\alpha}}}\right)\left(\mathrm{d}s^{\alpha}-\displaystyle\frac{\partial
L}{\partial v^{i}_{\alpha}}\mathrm{d}q^{i}\right)\,,$

which implies that $g^{\alpha\beta}_{ij}V^{i}_{\alpha}=0$. Using the fact that
the Lagrangian $L$ is regular it follows that $g^{\alpha\beta}_{ij}$ has
maximal rank and hence
$V^{i}_{\alpha}=\Gamma_{\alpha}(X^{i})-X^{i}_{\alpha}=0$, which shows that $X$
is a Newtonoid vector field for the $k$-vector fields ${\bf\Gamma}$. ∎

######  Corollary 7.17.

Let $L\in\mathscr{C}^{\infty}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ be a
regular Lagrangian function.

  1. 1.

Consider a Newtonoid vector field
$X\in\mathfrak{X}(\oplus^{k}\mathrm{T}Q\times\mathbb{R}^{k})$ of the form

$X=Z^{C}+K^{\alpha}\frac{\partial}{\partial s^{\alpha}}\,,\quad\text{with
}Z\in\mathfrak{X}(Q)\,,$ (80)

and such that $X(L)=0$, and $K^{\alpha}$ are constants, for
$\alpha=1,\ldots,k$. Then $X$ is an infinitesimal $k$-contact symmetry.

  2. 2.

The functions
$F^{\alpha}=-i_{X}\eta^{\alpha}_{L}=Z^{V_{\alpha}}(L)-K^{\alpha}$ give a
dissipation law.

###### Proof.

By a straightforward computation in local coordinates, it can be proved that
$X$ is an infinitesimal $k$-contact symmetry, that is, it satisfies equations
(64). Then, by Corollary 7.9 the functions $F^{\alpha}$ give a dissipation law
for the $k$-contact symmetry $X$. ∎

##  8 Conclusions and further research

In this paper we have presented several types of symmetries of non-
conservative Lagrangian field theories using the $k$-contact framework. In
particular, we have introduced natural symmetries (symmetries of the
Lagrangian function), dynamical symmetries (preserving the solutions of the
field equations) and $k$-contact symmetries (preserving the geometric
structures). We have studied the relations among these symmetries and how to
obtain dissipation laws from them. This theory has been illustrated with
several physical examples.

The study of the symmetries of $k$-contact systems in the Lagrangian setting
presented in this paper is another step towards a deeper study of the
symmetries of non-conservative field theories. In future works, this should be
complemented with the analysis of the symmetries and Newtonoid vector fields
in the Hamiltonian counterpart of the $k$-contact formalism. In particular, it
would be interesting to compare both sets of symmetries in the case of
singular Lagrangians, when the Legendre map is not a diffeomorphism. In
addition, it would be very interesting to study the symmetries and Newtonoid
vector fields of these systems and relate them to the symmetries in the
$k$-contact setting presented in this work.

Recently, a more general geometric framework for non-conservative field
theories (generalizing the $k$-contact [20] and $k$-cocontact [41]
formulations) has been introduced: the so-called _multicontact formalism_
[12]. We propose to analyze the relation between the $k$-contact and the
$k$-cocontact formulations with the multicontact setting, both in the
Lagrangian and Hamiltonian formalisms, following the ideas in [43].

##  9 Acknowledgments

X. Rivas, M. Salgado and S. Souto acknowledge financial support of the
Ministerio de Ciencia, Innovación y Universidades (Spain), projects
PGC2018-098265-B-C33 and D2021-125515NB-21.

X. Rivas acknowledges financial support of the Novee Idee 2B-POB II project
PSP: 501-D111-20-2004310 funded by the “Inicjatywa Doskonałości - Uczelnia
Badawcza” (IDUB) program.

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