\label{project}
\end{definition}

Natural coordinates in $\mathfrak{M}$ are $(q^i,v^i,p_i,s)$.

\begin{definition}
A curve $\mbox{\boldmath $\sigma$}\colon\Real\rightarrow\mathfrak{M}$
is a \textbf{holonomic} in $\mathfrak{M}$ if
$\rho_1\circ\mbox{\boldmath $\sigma$}\colon\Real\to\Tan Q\times\Real$ is holonomic curve.
A vector field $\Gamma\in\vf(\mathfrak{M})$
satisfies the \textbf{second-order condition} in $\mathfrak{M}$
(for short: it is a {\sc sode} in $\mathfrak{M}$) if all of its integral curves
are holonomic in $\mathfrak{M}$.
\end{definition}

In coordinates, a holonomic curve and a {\sc sode} in $\mathfrak{M}$ are expressed as
\beann
\mbox{\boldmath $\sigma$}&=&
\left(\sigma_1^i(t),\frac{d\sigma_1^i}{d t}(t),\sigma_{2\,i}(t),\sigma_0(t) \right) \ ,
\\
\Gamma&=&v^i \frac{\partial}{\partial q^i}+F^i \frac{\partial}{\partial v^i}+G_i \frac{\partial}{\partial p_i}+f\,\frac{\partial}{\partial s} \ .
\eeann

\begin{definition}
The bundle $\mathfrak{M}$ is endowed with the following canonical structures:
\ben
\item
The \textbf{coupling function} in $\mathfrak{M}$ is the
map $\mathfrak{C}\colon\mathfrak{M}\to\Real$  defined as follows: for every
$w=({\rm v}_q,{\rm p}_q,s)\in\mathfrak{M}$, where $q\in Q$,
${\rm p}_q\in\Tan^*Q$, and ${\rm v}_q\in\Tan Q$, then
$\mathfrak{C}(w):=\langle{\rm p}_q,{\rm v}_q\rangle$.
\item
The \textbf{canonical precontact $1$-form} on $\mathfrak{M}$
is the $\rho_1$-semibasic form
d s-\pi_1^*\rho_2^*\pi_1^*\Theta\in\df^1(\mathfrak{M})\ ;$$
where $\bmeta$ is the canonical contact form on $\Tan^*Q\times\Real$
(and $\Theta$ is the canonical $1$-form on
\een
\label{coupling}
\end{definition}

In natural coordinates of $\mathfrak{M}$ we have that
$\bmeta_{\mathfrak{M}}=\d s-p_i\,\d q^i$.

\begin{definition}
Given a Lagrangian function $\Lag\in\Cinfty(\Tan Q\times\Real)$, let
The \textbf{Hamiltonian function} is defined as
\beq
{\bf H}:=\mathfrak{C}-\mathfrak{L} =p_i v^i-\Lag (q^j,v^j,s)\in\Cinfty(\mathfrak{M}) \ .
\label{Hamf}
\eeq
\end{definition}

\begin{remark}{\rm
Observe that $(\mathfrak{M},\bmeta_{\mathfrak{M}})$ is a precontact manifold.
As a consequence, the Reeb vector fields are not uniquely defined and
in natural coordinates of $\mathfrak{M}$ the general solution to \eqref{eq-Reeb} are the vector fields
$\displaystyle \Reeb=\derpar{}{z}+F^i\derpar{}{v^i}$
for arbitrary functions $F^i$.
Nevertheless, as we have pointed out,
the dynamics obtained from the formalism is independent of the choice of the Reeb vector fields;
therefore, as $\mathfrak{M}=\Tan Q\times_Q\Tan^*Q\times\Real$
is a trivial bundle over $\Real$,
the canonical vector field $\displaystyle \derpar{}{s}$ on $\Real$
can be lifted canonically to a vector field in $\mathfrak{M}$,
which can be taken as a representative of the family of
these Reeb vector fields.

Thus, we have that $(\mathfrak{M},\eta_\mathfrak{M},{\bf H})$
is a precontact Hamiltonian system and then,
we can pose the dynamic problem for this system which consists of finding
a Hamiltonian vector field which is a solution to the precontact Hamiltonian equations
\begin{equation}
\label{Whamilton-contact-eqs}
\inn(\X_{\bf H})\d\bmeta_{\mathfrak{M}}=\d{\bf H}-({\Reeb}({\bf H}))\bmeta_{\mathfrak{M}}
\quad ,\quad
\inn(\X_{\bf H})\bmeta_{\mathfrak{M}}=-{\bf H}\ .
\end{equation}
Then, the integral curves $\mbox{\boldmath $\sigma$}\colon I\subset\Real\to\mathfrak{M}$ of $X_{\bf H}$,
are the solutions to the equations
\begin{equation}
\label{Whamilton-contactc-curves-eqs}
\inn(\widetilde{\mbox{\boldmath $\sigma$}})(\d\bmeta_{\mathfrak{M}}\circ\mbox{\boldmath $\sigma$})=\left(\d{\bf H}-({\Reeb}({\bf H}))\bmeta_{\mathfrak{M}}\right)\circ\mbox{\boldmath $\sigma$}
\quad ,\quad
\inn(\widetilde{\mbox{\boldmath $\sigma$}})(\bmeta_{\mathfrak{M}}\circ\mbox{\boldmath $\sigma$})=-{\bf H}\circ\mbox{\boldmath $\sigma$} \ .
\end{equation}

As we shall see next,
these equations are not compatible on $\mathfrak{M}$,
and we have to implement the constraint algorithm
in order to find the final constraint submanifold of $\mathfrak{M}$
where there are consistent solutions
to the equations.
In fact, in natural coordinates of $\mathfrak{M}$ we have that
\d{\bf H}=v^i\d p_i+\left(p_i -\derpar{\Lag}{v^i}\right)\d v^i-\derpar{\Lag}{q^i}\,\d q^i-\derpar{\Lag}{s}\,\d s \ ,
and, if the local expression of
a vector field $\X_{\bf H}\in\vf(\mathfrak{M})$ is
\X_{\bf H} = f^i\derpar{}{q^i}+F^i\derpar{}{v^i}+G_i\derpar{}{p_i}+f\derpar{}{s} \ ,
we obtain that
\inn(\X_{\bf H})\bmeta_{\mathfrak{M}}=f-f^ip_i \quad , \quad
\inn(\X_{\bf H})\d\bmeta_{\mathfrak{M}}=f^i\,\d p_i-G_i\,\d q^i
\quad , \quad
({\Reeb}({\bf H}))\bmeta_{\mathfrak{M}}=-\derpar{\Lag}{s}(\d s-p_i\d q^i) \ .
Then, equations \eqref{Whamilton-contact-eqs} give
\bea
f&=&(f^i-v^i)\,p_i+\Lag \ ,
\label{zero} \\
f^i&=&v^i \ , \label{one} \\
p_i&=&\derpar{\Lag}{v^i} \ ,
\label{two} \\
+p_i\derpar{\Lag}{s} \ ,
\label{three}
\eea
The equations (\ref{one}) are the holonomy conditions
(i.e., $\X_{\bf H}$ must be a {\sc sode}),
which arise straightforwardly from the unified formalism.
The algebraic equations (\ref{two}) are compatibility conditions
defining the submanifold $\mathfrak{M}_0\hookrightarrow\mathfrak{M}$,
where the equations are compatible.
This $\mathfrak{M}_0$ is the {\sl first constraint submanifold} of the Hamiltonian precontact system $(\mathfrak{M},\bmeta_{\mathfrak{M}},{\bf H})$, and is the graph of $\mathfrak{F}\Lag$; that is,
\mathfrak{M}_0=\{ ({\rm v}_q,\mathfrak{F}\Lag({\rm v}_q))\in\mathfrak{M} \ ,\ \mbox{\rm for ${\rm v}_q\in\Tan Q$}\} \ .
In this way, as it is usual, the unified formalism includes the definition of
the Legendre map as a consequence of the constraint algorithm.

Thus, vector fields which are solutions to \eqref{Whamilton-contact-eqs} are,
\X_{\bf H}=
\left(\derpar{\Lag}{q^i}
\quad \mbox{\rm (on $\mathfrak{M}_0$)}\ .
The constraint algorithm continues by demanding that
$\X_{\bf H}$ must be tangent to $\mathfrak{M}_0$. As
$\displaystyle \xi^1_j=p_j-\derpar{\Lag}{v^j}\in\Cinfty(\mathfrak{M})$
are the constraints defining $\mathfrak{M}_0$, this condition is
\beq
\X_{\bf H}\left(p_j-\derpar{\Lag}{v^j}\right)=
-\frac{\partial^2\Lag}{\partial q^i\partial v^j}v^i
-\frac{\partial^2\Lag}{\partial v^i\partial  v^j}F^i-
\Lag\frac{\partial^2\Lag}{\partial z\partial v^j}
\quad \mbox{\rm (on $\mathfrak{M}_0$)} \ ,
\label{tangcond}
\eeq
and there are two options:
\bit
\item
If $\Lag$ is a regular Lagrangian, these equations determine all the
functions \(F^i\).
Then the solution is unique and the algorithm ends.
\item
If $\Lag$ is singular, then these equations establish relations
among the arbitrary functions $F^i$;
some of them remain undetermined and the solutions are not unique.
Furthermore, new constraints $\xi^2_\mu\in\Cinfty(\mathfrak{M})$
could appear, then defining a new submanifold
Then, the algorithm continues by demanding the tangency of
$\X_{\bf H}$ to $\mathfrak{M}_1$, and so on,
until we obtain a final constraint submanifold $\mathfrak{M}_f$ (if it exists)
where tangent solutions $X_{\bf H}$ exist.
\eit
If $\mbox{\boldmath $\sigma$}(t)=(q^i(t),v^i(t),p_i(t),s(t))$
is an integral curve of $\X_{\bf H}$, we have that
$\displaystyle f^i=\frac{d q^i}{d t}$,
$\displaystyle F^i=\frac{d v^i}{d t}$,
$\displaystyle G_i=\frac{d p_i}{d t}$,
$\displaystyle f=\frac{d s}{d t}$,
and equations (\ref{zero}), (\ref{one}), (\ref{two}), and (\ref{three})
give to the coordinate expression of the equations
\eqref{Whamilton-contactc-curves-eqs}; in particular:
\bit
\item
From (\ref{one}), we have that $\dst v^i=\frac{d q^i}{d t}$;
that is, the holonomy condition.
\item
Using (\ref{one}) again, the equation (\ref{zero}) gives the equation \eqref{ELeqs2} again.
\item
The equations \eqref{three} read,
\frac{dp_i}{dt}=\derpar{\Lag}{q^i}+p_i\derpar{\Lag}{s}=
-\left(\derpar{{\bf H}}{q^i}+p_i\derpar{{\bf H}}{s}\right) \ ,
which are the second group of Hamilton's equations \eqref{hel-eqs01}.
Then, using \eqref{two},
these equations are (on $\mathfrak{M}_0$),
\frac{d}{dt}\left(\derpar{\Lag}{v^i}\right)=\derpar{\Lag}{q^i}
+\derpar{\Lag}{v^i} \derpar{\Lag}{s} \ ,
which are the Herglotz--Euler-Lagrange equations (\ref{ELeqs3}).
The first group of Hamilton's equations \eqref{hel-eqs01} arises
from the definition of the Hamiltonian function
\eqref{Hamf}, using the holonomy condition.
\item
Using \eqref{two} and \eqref{ELeqs2},
the tangency condition \eqref{tangcond} gives
again the Herglotz--Euler-Lagrange equations \eqref{ELeqs3}.
If $\Lag$ is singular, these equations can be incompatible.
\eit

\subsection{Recovering the Lagrangian and Hamiltonian formalisms and equivalence}
\label{recovering}

Now we  study the equivalence of the unified formalism
with the Lagrangian and Hamiltonian formalisms
for the hyperregular case
(the regular case is the same, at least locally).
See [107] more details about the singular case.

First, observe that, in this case, denoting by
the natural embedding, we have that
(\rho_1\circ\jmath_0)(\mathfrak{M}_0)=\Tan Q\times\Real
\quad , \quad
(\rho_2\circ\jmath_0)(\mathfrak{M}_0)=\Tan^*Q\times\Real \ .
Furthermore, as $\mathfrak{M}_0$ is the graph of the Legendre map
$\mathfrak{F}\Lag$, it is diffeomorphic to $\Tan Q\times\Real$,
being the restricted projection $\rho_1\circ\jmath_0$
this diffeomorphism.
So, we have the diagram
\xymatrix{
\ & \ & \mathfrak{M}<EMAIL_ADDRESS><EMAIL_ADDRESS>& \ & \ \\
\ & \ & \mathfrak{M}_0 \ar[dll] \ar[drr] \ar@{^{(}->}[u]^{\jmath_0} & \ & \ \\
\Tan Q\times\Real \ar[rrrr]^<(0.40){\mathfrak{F}\Lag}
& \ & \ & \ & \Tan^*Q\times\Real
As in the other unified formalisms,
functions, differential forms on $\mathfrak{M}$,
and vector fields on $\mathfrak{M}$ tangent to $\mathfrak{M}_0$
can be restricted to $\mathfrak{M}_0$.
Then, they can be translated to the Lagrangian or the Hamiltonian side
by using that $\mathfrak{M}_0$ is diffeomorphic to $\Tan Q\times\Real$,
or projecting to $\Tan^* Q\times\Real$.
In particular, a simple calculation in coordinates shows that
\rho_1^*E_L={\bf H}=\rho_2^*{\rm h} \ ,
every curve $\mbox{\boldmath $\sigma$}\colon I\subseteq\Real\to\mathfrak{M}$
which takes values in $\mathfrak{M}_0$, can be split as
$\mbox{\boldmath $\sigma$}=(\mbox{\boldmath $\sigma$}_L,\mbox{\boldmath $\sigma$}_H)$, where
$\mbox{\boldmath $\sigma$}_L=\rho_1\circ\mbox{\boldmath $\sigma$}\colon I\subseteq\Real \to\Tan Q\times\Real$
and $\mbox{\boldmath $\sigma$}_H=
\mathfrak{F}\Lag\circ\mbox{\boldmath $\sigma$}_L\colon I\subseteq\Real\to\Tan^*Q\times\Real$.
Therefore, taking all of this into account, the results, and the discussion in the above section
lead to state:

\begin{teor}
Let $\mbox{\boldmath $\sigma$}\colon\Real\to\mathfrak{M}$,
with  ${\rm Im}\,(\mbox{\boldmath $\sigma$})\subset\mathfrak{M}_0$,
be a curve solution to the equations \eqref{Whamilton-contactc-curves-eqs}
on $\mathfrak{M}_0$.
$\mbox{\boldmath $\sigma$}_L$ is the lift of the projected curve
${\bf c}=\rho_0\circ\mbox{\boldmath $\sigma$}\colon\Real\to Q\times\Real$
to $\Tan Q\times\Real$ (that is, $\mbox{\boldmath $\sigma$}_L$ is
a holonomic curve),
and it is a solution to the equations \eqref{hec}.
Moreover, the curve
$\mbox{\boldmath $\sigma$}_H=
\mathfrak{F}\Lag\circ\widetilde{\bf c}$
is a solution to the equations \eqref{hamilton-contactc-curves-eqs} on $\Tan^*Q\times\Real$.

Conversely, for every curve ${\bf c}\colon\Real\to Q\times\Real$ such that
$\widetilde{\bf c}$ is a solution to the equations \eqref{hec}, we have that the curve
$\mbox{\boldmath $\sigma$}=(\widetilde{\bf c},\mathfrak{F}\Lag\circ\widetilde{\bf c})$
is a solution to the equations \eqref{Whamilton-contactc-curves-eqs} on $\mathfrak{M}_0$,
and the curve $\mathfrak{F}\Lag\circ\widetilde{\bf c}$
is a solution to the equations \eqref{hamilton-contactc-curves-eqs} on $\Tan^*Q\times\Real$.
\label{mainteor01}
\end{teor}

The curves $\mbox{\boldmath $\sigma$}\colon\Real\to\mathfrak{M}$
solution to the equations \eqref{Whamilton-contactc-curves-eqs}
are the integral curves of {\sc sode} vector fields $\X_{\bf H}\in\vf(\mathfrak{M})$
solution to \eqref{Whamilton-contact-eqs} on $\mathfrak{M}_0$.
The curves $\mbox{\boldmath $\sigma$}_\Lag\colon\Real\to\Tan Q\times\Real$
are the integral curves of {\sc sode} vector fields $\X_\Lag\in\vf(\Tan Q\times\Real)$
solution to  \eqref{hec}.
Finally, the curves
$\mbox{\boldmath $\sigma$}_{\rm h}\colon\Real\to\Tan^*Q\times\Real$
are the integral curves of the contact Hamiltonian vector field $\X_{\rm h}\in\vf(\Tan^*Q\times\Real)$
solution to \eqref{evolvf11b} on $\Tan^*Q\times\Real$.
Then, an immediate corollary of the above theorem is the following:

\begin{teor}
Let $\X_{\bf H}\in\vf(\mathfrak{M})$ be the contact Hamiltonian vector field
solution to the equations \eqref{Whamilton-contact-eqs}
on $\mathfrak{M}_0$, and tangent to

Then the vector field $\X_\Lag\in\vf(\Tan Q\times\Real)$, defined by
$\X_\Lag\circ\rho_1=\Tan\rho_1\circ\X_{\bf H}$,
is a {\sc sode} vector field which is a
solution to the equations \eqref{eq-E-L-contact1}; that is, the contact Euler-Lagrange vector field,
on $\Tan Q\times\Real$.

Furthermore, the vector field $\X_{\rm h}\in\vf(\Tan^*Q\times\Real)$, defined by
$\X_{\rm h}\circ\rho_2=\Tan\rho_2\circ X_{\bf H}$,
is the contact Hamiltonian vector field solution to the equations \eqref{evolvf11b} on $\Tan^*Q\times\Real$.
\end{teor}

These results are analogous to those obtained for the unified
formalism of  dynamical systems.

\begin{remark}{\rm
For singular almost-regular Lagrangians, the equivalence between the constraint algorithms
in the unified and in the Lagrangian formalism
only holds when the second-order condition is imposed,
since, unlike in the unified formalism, this condition does not hold in the Lagrangian case (see [270, 311]).

\section{Symmetries of contact dynamical systems}
\label{sims}

(See [163] and also [111] for another complementary approach to these topics. See also [166] for a complete classification of symmetries of autonomous and nonautonomous contact systems, and [4] for a reduction scheme).

\subsection{Symmetries of contact Hamiltonian systems. Dissipated and conserved quantities}

Let $(M,\bmeta,h)$ be a contact Hamiltonian system with Reeb vector field $\Reeb$,
and let $\X_h$ be the contact Hamiltonian vector field for this system;
that is, the solution to the Hamilton equations \eqref{evolvf1}.

As, for other kinds of dynamical systems,
a \textsl{dynamical symmetry} of a contact Hamiltonian system  is a diffeomorphism
$\Phi\colon M\longrightarrow M$ such that $\Phi_*\X_h=X_h$.
Similarly, an \textsl{infinitesimal dynamical symmetry} of a contact Hamiltonian system  is a vector field
$Y\in\vf(M)$ whose local flux is a dynamical symmetry; that is,
This means that dynamical symmetries map solutions into solutions.

Nevertheless, we are mainly interested in those symmetries that let
the geometric structures invariant:

\begin{definition}
\label{defsym}
A \textbf{contact Noether symmetry} of a contact Hamiltonian system $(M,\bmeta,h)$ is a diffeomorphism
$\Phi\colon M\rightarrow M$ such that
\Phi^*\bmeta=\bmeta
\quad ,\quad
\Phi^*h=h \ .
An \textbf{infinitesimal contact Noether symmetry} of a contact Hamiltonian system $(M,\bmeta,h)$ is a vector field
$Y\in \vf(M)$ whose local flux is a contact Noether symmetry; that is,
\Lie(Y)\bmeta=0
\quad ,\quad
\Lie(Y)h=0 \ .
\end{definition}

\begin{prop}
\label{symreeb}
Every (infinitesimal) contact Noether symmetry preserves the Reeb vector field; that is, $\Phi_*\Reeb=\Reeb$
(or $[Y,\Reeb]=0$).
\end{prop}
\begin{proof}
We obtain that,
\beann
\inn(\Phi^{-1}_*\Reeb)(\Phi^*\d\bmeta)&=&
\Phi^*(\inn(\Reeb)\d\bmeta)=0 \ , \\
\inn(\Phi^{-1}_*\Reeb)(\Phi^*\bmeta)&=&
\Phi^*(\inn(\Reeb)\bmeta)=1 \ ,
\eeann
and, as $\Phi^*\bmeta=\bmeta$ and the Reeb vector field is unique,
from these equalities we get $\Phi^{-1}_*\Reeb=\Reeb$ or, equivalently,

For the infinitesimal case, if $Y\in\vf(M)$ is an infinitesimal Noether symmetry, we have:
\bea
\label{aux2}
\Lie([Y,\Reeb])\bmeta&=&
\Lie(Y)\Lie(\Reeb)\bmeta-\Lie(\Reeb)\Lie(Y)\bmeta=
\Lie(Y)\inn(\Reeb)\d\bmeta+\d\inn(\Reeb)\eta=0 \ ,
\\
\Lie([Y,\Reeb])\d\bmeta&=& \d\Lie([Y,\Reeb])\bmeta=0 \ , \nonumber
\eea
hence $[Y,\Reeb]=\Lie(Y)\Reeb\in \ker\bmeta\cap\ker\d\bmeta=\{ 0\}$.
\\ \qed\end{proof}

As in the dynamical systems studied in the previous chapters,
we have:

\begin{prop}
\label{csis}
Every (infinitesimal) contact Noether symmetry is a (infinitesimal) dynamical symmetry.
\end{prop}
\begin{proof}
In fact, using that $\Phi$ is a contact Noether symmetry and that, by the above proposition, $\Phi_*\Reeb=\Reeb$, we obtain:
\beann
\inn(\Phi^{-1}_*\X_h)\d\bmeta&=&
\inn(\Phi^{-1}_*\X_h)(\Phi^*\d\bmeta)=
\Phi^*(\inn(\X_h)\d\bmeta)=
\Phi^*(\d h-(\Reeb(h))\,\bmeta)=\d h-(\Reeb(h) )\,\bmeta \ ,
\\
\inn(\Phi^{-1}_*\X_h)\bmeta&=&
\inn(\Phi^{-1}_*\X_h)(\Phi^*\bmeta)=
\Phi^*(\inn(\X_h)\bmeta)=
\Phi^*(-h)=-h \ ,
\eeann
then $\Phi^{-1}_*\X_h$ is a solution to the dynamical equation \eqref{evolvf11b} and,
as the solution is unique, we conclude that $\Phi_*\X_h=\X_h$, and
hence $\Phi$ is a symmetry.

For the infinitesimal case we have that,
\beann
\inn([Y,\X_h])\bmeta&=&
\Lie(Y)\inn(\X_h)\bmeta-\inn(\X_h)\Lie(Y)\bmeta=
-\Lie(Y)h=0 \ ,
\\
\inn([Y,\X_h])\d\bmeta&=&
\d\inn([Y,\X_h])\bmeta=0\ ,
\eeann
then $[Y,\X_h]\in \ker\bmeta\cap\ker\d\bmeta=\{ 0\}$
and it is a dynamical symmetry.
\\ \qed\end{proof}

Now, the result stated in Proposition \ref{disipenerg},
induces us to define:

\begin{definition}\label{definition:diss-quan}
A \textbf{dissipated quantity} of the contact Hamiltonian system $(M,\bmeta,h)$ is a function
$F\in\Cinfty(M)$ satisfying that
\beq\label{eq:dissipation}
\Lie(\X_h)F=-(\Reeb(h))\,F \ .
\eeq
A \textbf{conserved quantity} of the contact Hamiltonian system $(M,\bmeta,h)$ is a function
$G\in\Cinfty(M)$ satisfying that
\Lie(\X_h)G=0 \ .
\end{definition}

For contact Hamiltonian systems,
as we are dealing with dissipative systems,
symmetries are associated with dissipated

\begin{teor}
\label{th:dissipation}
{\rm (Dissipation theorem).}
If $Y\in\vf(M)$ is an infinitesimal dynamical symmetry,
then the function $F=-\inn(Y)\bmeta$
is a dissipated quantity.
\end{teor}
\begin{proof}
In fact, Proposition \ref{csis} says that $[Y,\X_h]=0$; then,
\begin{align*}
\Lie(\X_h) F &=
-\Lie(\X_h) \inn(Y) \bmeta =
- \inn(Y) \Lie(\X_h)\bmeta - \inn(\Lie(\X_h)Y)\, \bmeta =
\\
(\Reeb(h)) \inn(Y)\bmeta + \inn([Y,\X_h]) \bmeta =
-(\Reeb(h))\, F + \inn([Y,\X_h]) \bmeta =
-(\Reeb(h))\, F\ ,
\end{align*}
where we have applied that $\Lie(\X_h)\bmeta=-\Reeb(h))\,\bmeta$ (by \eqref{equivmanner}).
\\ \qed\end{proof}

\begin{remark}{\rm
In particular, the Hamiltonian vector field $\X_h$ is
trivially a symmetry, since $[\X_h,\X_h]=0$.
Then, Proposition \eqref{eq:disipenerg}
establishes that its associated dissipated quantity is the energy,

These are ``non conservation theorems''
which account for the non-conservation of these quantities associated with the symmetries.
In particular, every dissipated quantity changes with the same rate, $-\Reeb(h)$,
which suggests that the quotient of two dissipated quantities should be a conserved quantity.

\begin{prop}
\label{prop:disscon}
\begin{enumerate}
\item
If $F_1$ and $F_2$ are dissipated quantities and $F_2\not=0$, then $\dst\frac{F_1}{F_2}$ is a conserved quantity.
\item
If $F$ is a dissipated quantity and $G$ is a conserved quantity, then $FG$ is a dissipated quantity.
\end{enumerate}
\end{prop}
\begin{proof}
In fact, we have
\begin{align*}
\Lie(\X_h)\left(\frac{F_1}{F_2}\right)&=
\frac{1}{F_2}\Lie(\X_h)F_1-\frac{F_1}{{F_2}^2}\Lie(\X_h){F_2}=-
\frac{1}{F_2}(\Reeb(h))F_1+\frac{F_1}{{F_2}^2}(\Reeb(h))F_2=0\ .
\\
\Lie(\X_h)(FG)&=G\,\Lie(\X_h)F+F\,\Lie(\X_h)G=-(\Reeb(h))\,FG.
\end{align*}
\qed\end{proof}

A straightforward consequence of this proposition is that symmetries can also have associated conserved quantities:

\begin{corol}
For every infinitesimal symmetry $Y\in\vf(M)$,
if $h\not=0$, then $\dst -\frac{\inn(Y)\bmeta}{h}$ is a conserved quantity.
\label{consquanquot}
\end{corol}

Contact symmetries also allow us to generate new dissipated quantities
from another one.

\begin{prop}
If $\Phi\colon M\rightarrow M$ is a contact Noether symmetry and
$F\in\Cinfty(M)$ is a dissipated quantity, then so is $\Phi^*F$.
Similarly, if $Y\in\vf(M)$ is an infinitesimal contact Noether symmetry, then
$\Lie(Y)F$ is a dissipated quantity.
\end{prop}
\begin{proof}
In fact, we have
\Lie(\X_h)(\Phi^*F)=
\Phi^*\Lie(\Phi_*\X_h)F=
\Phi^*\Lie(\X_h)F=
\Phi^*(-\Reeb(h))F=
-(\Reeb(h))(\Phi^*F) \ .
For the infinitesimal case, taking into account that $Y$ is a contact infinitesimal $\Lie(\X_h)F=0$, that $[Y,\X_h]=\X_h$ (Proposition \ref{csis}) and that $[Y,\Reeb]=0$ (Proposition \ref{symreeb}), we have:
\beann
\Lie(\X_h)\Lie(Y)F&=&
\Lie([\X_h,Y])F-\Lie(Y)\Lie(\X_h)F+
\\ &=&
\\ &=&
=-(\Reeb(h))(\Lie(Y)F) \ .
\eeann
\qed\end{proof}

\subsection{Symmetries for contact Lagrangian and canonical Hamiltonian systems}

Next, we particularize the results on symmetries and dissipated quantities
to the case of Lagrangian dissipative systems and their canonical Hamiltonian formalism.

Consider a regular contact Lagrangian system $(\Tan Q\times\Real,\Lag)$,
with Reeb vector field $\Reeb_\Lag$,
and let $\X_\Lag$ be the contact Euler--Lagrange vector field for this system;
that is, the solution to the Lagrangian equations \eqref{eq-E-L-contact1}.

All the definitions and results about symmetries and dissipated quantities stated in the preceding section
hold for the contact system $(\Tan Q\times\Real,\bmeta_\Lag,E_\Lag)$.
In particular, the dissipation theorem states that
$-\inn(Y)\bmeta_\Lag$ is a dissipated quantity, for every infinitesimal dynamical symmetry $Y\in\vf(\Tan Q\times\Real)$,
the energy dissipation theorem states that
\beq
\label{energdis}
\Lie(\X_\Lag)E_\Lag=-(\Reeb_\Lag(E_\Lag))\,E_\Lag\ .
\eeq

if $\varphi\colon Q\to Q$ is a diffeomorphism,
we can construct the {\sl canonical lift} of $\varphi$ to $\Tan Q\times\Real$ as the diffeomorphism
$\Phi:=(\Tan\varphi, {\rm {\rm Id}_{\mathbb R}}) \colon
\Tan Q\times\Real
\longrightarrow
\Tan Q\times\Real$.
These kinds of diffeomorfisms $\Phi$ are usually called {\sl natural transformations} on $\Tan Q\times\Real$.
Similarly, for every vector field $Z\in \vf(Q)$
we can define its {\sl complete lift}
to $\Tan Q\times\Real$ as the vector field
$Y\in\vf(\Tan Q\times\Real)$
whose local flux is the canonical lift of
the local flux of $Z$ to $\Tan Q\times\Real$.
The infinitesimal transformation generated by $Y$ are called {\sl infinitesimal natural transformation}
on $\Tan Q\times\Real$.

Then, taking into account the definitions
of the canonical endomorphism ${\cal J}$
and the Liouville vector field $\Delta$ in $\Tan Q\times\Real$,
it can be proved that canonical lifts of diffeomorphisms and vector fields
from $Q$ to $\Tan Q\times\Real$ preserve these canonical structures.
Furthermore, if these lifts leave the Lagrangian function invariant,
they also preserve the Reeb vector field $\Reeb_\Lag$.
Therefore, as an immediate consequence,
we obtain the following relation:

\begin{prop}
If $\Phi\in{\rm Diff}(\Tan Q\times\Real)$ (resp. $Y\in\vf(\Tan Q\times\Real)$) is a canonical lift
to $\Tan Q\times\Real$ of a diffeomorphism $\varphi\in{\rm Diff}(Q)$
(resp.\ of a vector field $Z\in\vf(Q)$)
that leaves the Lagrangian invariant, then
it is a (infinitesimal) contact Noether symmetry, {\it i.e.},
\Phi^*\bmeta_\Lag=\bmeta_\Lag \ ,\
\Phi^*E_\Lag=E_\Lag \qquad
({\rm resp.}\ \Lie(Y)\bmeta_\Lag=0 \ ,\
\Lie(Y)E_\Lag=0 \:)\ .
As a further consequence, it is a (infinitesimal) dynamical symmetry.
\end{prop}

As a direct consequence, if $\displaystyle\frac{\partial \Lag}{\partial q^i}=0$,
then $\displaystyle\frac{\partial}{\partial q^i}$
is an infinitesimal contact Noether symmetry, and its associated dissipated quantity
is the momentum
$\displaystyle\frac{\partial \Lag}{\partial v^i}$;
that is,
\Lie(\X_\Lag)\left(\frac{\partial\Lag}{\partial v^i}\right)=
-(\Reeb_\Lag(E_\Lag))\frac{\partial\Lag}{\partial v^i}=
\derpar{\Lag}{s}\frac{\partial\Lag}{\partial v^i}\ .
(See also [169]for a similar description).

For the canonical Hamiltonian formalism, consider
the canonical contact Hamiltonian system $(\Tan^*Q\times\Real,\bmeta,{\rm h})$.
As in the Lagrangian formalism, if $\varphi\colon Q\to Q$ is a diffeomorphism,
we can construct the {\sl canonical lift} of $\varphi$ to $\Tan^*Q\times\Real$,
which is the diffeomorphism
$\Phi:=(\Tan^*\varphi, {\rm {\rm Id}_{\mathbb R}}) \colon
\Tan^*Q\times\Real
\longrightarrow
\Tan^*Q\times\Real$.
and is called a {\sl natural transformation} on $\Tan^*Q\times\Real$.
In the same way, for every vector field $Z\in \vf(Q)$
we define its {\sl complete lift}
to $\Tan^*Q\times\Real$ as the vector field
whose local flux is the canonical lift of
the local flux of $Z$ to $\Tan^*Q\times\Real$.
It is called a {\sl natural infinitesimal transformations} on $\Tan^*Q\times\Real$.

The canonical forms $\Theta$ and $\Omega=-\d\Theta$
in $\Tan^*Q$ and their extensions to $\Tan^*Q\times\Real$ are invariant under the action
of canonical lifts of diffeomorphisms and vector fields from $Q$ to $\Tan^*Q$ and $\Tan^*Q\times\Real$.
Then, taking into account the definition
of the contact form $\bmeta$ in $\Tan^*Q\times\Real$, we have:

\begin{prop}
$\Phi\in{\rm Diff}(\Tan^*Q\times\Real)$ (resp. $Y\in\vf(\Tan^*Q\times\Real)$) is a canonical lift to $\Tan^*Q\times\Real$ of a diffeomorphism $\varphi\in{\rm Diff}(Q)$
(resp. of $Z\in\vf(Q)$), then
\begin{enumerate}
\item
$\Phi^*\bmeta=\bmeta$ (resp $\Lie(Y)\bmeta=0$).
\item
If, in addition,
$\Phi^*{\rm h}={\rm h}$ (resp. $\Lie(Y){\rm h}=0$),
then it is a  (infinitesimal) contact Noether symmetry.
\end{enumerate}
\end{prop}

In particular, we have the following:

\begin{teor}{\rm (Momentum dissipation theorem).}
If $\displaystyle\frac{\partial {\rm h}}{\partial q^i}=0$,
then $\displaystyle\frac{\partial}{\partial q^i}$
is an infinitesimal contact Noether symmetry,
and its associated dissipated quantity
is the corresponding momentum $p_i$;
that is,
$\Lie(\X_{\rm h})p_i=-(\Reeb({\rm h}))\,p_i$.
\label{disipe3}
\end{teor}
\begin{proof}
A simple computation in local coordinates shows that
\Lie\Big(\frac{\partial}{\partial q^i}\Big)\bmeta=0$
\Lie\Big(\frac{\partial}{\partial q^i}\Big)=0$.
Therefore, $\dst\frac{\partial}{\partial q^i}$
is a contact Noether symmetry and, hence, a dynamical symmetry.
The other results are a consequence of the dissipation theorem.
\end{proof}

\section{Examples}

\subsection{The damped harmonic oscillator}

Consider a harmonic oscillator in a medium with friction.
As in Section \ref{sho},
the configuration bundle of the system is $Q=\Real$ but, in order to develop the contact formulation, we take the manifold $Q\times\Real$. with coordinates $(t,q)$.

\subsubsection{Lagrangian formalism}

The Lagrangian description of the one-dimensional damped harmonic oscillator is done in the phase space
$\Tan Q\times\Real\simeq=\Real^2\times\Real$.
It is described by the hyperregular contact Lagrangian function,
\Lag=\frac{1}{2}mv^{2}-\frac{1}{2}kq^{2}-\gamma s \ ,
where $\gamma\in\Real^+$ is the dissipation parameter.

We have the contact Lagrangian form,
\bmeta_\Lag=\d s-v\d q \ ,
and the energy Lagrangian function is
E_\Lag = \frac{1}{2}mv^2 + \frac{1}{2}kq^2 + \gamma s \ .
For $\displaystyle X_\Lag=f\derpar{}{q}+F\derpar{}{v}+g\derpar{}{s}\in\vf(\Tan Q\times\Real)$,
the contact Lagrangian equations \eqref{ELeqs0}
and \eqref{ELeqs1} are
g=\frac{1}{2}mv^{2}-\frac{1}{2}kq^{2}-\gamma s \quad , \quad f=v \quad , \quad mF=-kq-\gamma mv \ .
whose solution is the {\sc sode} vector field,
\X_\Lag=v\frac{\partial}{\partial q}-
\left(\frac{k}{m}\,q+\gamma v\right) \frac{\partial}{\partial v}+\left(\frac{1}{2}mv^{2}-\frac{1}{2}kq^{2}-\gamma s\right)\frac{\partial}{\partial s}\ .
Its integral curves are the solutions to equations \eqref{ELeqs2} and \eqref{ELeqs3},
which for this system give the Herglotz--Euler--Lagrange equations
\beq
\label{HELdho}
\frac{ds}{dt}=\frac{1}{2}m\left(\frac{dq}{dt}\right)^2-\frac{1}{2}kq^{2}-\gamma s \quad , \quad
\frac{d^2q}{dt^2}=-\frac{k}{m}\,q-\gamma\,\frac{dq}{dt} \ ,
\eeq
and the last equation correspond to the well-known dynamical equation of the damped harmonic oscillator.

The dissipation of the energy is given by
\eqref{eq:disipenerg}, which in the Lagrangian formalism reads
\Lie(\X_\Lag)E_\Lag=-\gamma\left(\frac{1}{2}mv^2 + \frac{1}{2}kq^2 + \gamma s\right)\ .

\subsubsection{Hamiltonian formalism}

For the canonical Hamiltonian formalism of the system,
and the Legendre map is
\mathfrak{F}\Lag^*q=q  \quad , \quad \mathfrak{F}\Lag^*p=mv  \quad , \quad \mathfrak{F}\Lag^*s=s  \ .
The canonical contact form is $\bmeta=\d s-p\,\d q$, and
the canonical Hamiltonian function reads
{\rm h}=\frac{p^2}{2m^2} + \frac{1}{2}kq^2 + \gamma s \ .
For $\displaystyle X_{\rm h}=f\derpar{}{q}+F\derpar{}{p}+g\derpar{}{s}$,
the contact Hamiltonian equations \eqref{evolvf11b} give
f=\frac{p}{m} \quad , \quad F=-kq-\gamma p \quad , \quad g=\frac{p^2}{2m^2}-\frac{1}{2}kq^2-\gamma s  \ ,
and its solution is the Hamiltonian vector field,
\X_{\rm h}=\frac{p}{m}\frac{\partial}{\partial q}-
\left(kq+\gamma p\right) \frac{\partial}{\partial p}+\left(\frac{p^2}{2m^2}-\frac{1}{2}kq^2-\gamma s\right)\frac{\partial}{\partial s}\ ,
whose integral curves are the solutions to equations \eqref{hel-eqs01b},
which for this system give the Hamilton equations,
\beq
\label{Hdho}
\dst\frac{dq}{dt}=\frac{p}{m}\quad ,\quad
\dst\frac{dp}{dt}=-kq-\gamma p\quad , \quad
\dst\frac{ds}{dt}=\frac{p^2}{2m^2}-\frac{1}{2}kq^2-\gamma s \ .
\eeq

Finally, the dissipation of the energy is given by \eqref{energdis},
\Lie(\X_{\rm h}){\rm h}=-\gamma\left(\frac{p^2}{2m^2} + \frac{1}{2}kq^2 + \gamma s\right)\ .

\subsubsection{Unified Lagrangian-Hamiltonian formalism}

The extended unified bundle is $\mathfrak{M}=\Tan Q\times_Q\Tan^*Q\times\Real\simeq\Real^4$
with coordinates $(q,v,p,s)$.
Then we have the precontact Hamiltonian system
$(\mathfrak{M},\bmeta_{\mathfrak{M}},{\bf H})$,
where the canonical contact form is
\bmeta_{\mathfrak{M}}=\d s-p\,\d q \ ,
the Hamiltonian function is
{\bf H}=pv-\frac{1}{2}mv^{2}+\frac{1}{2}kq^{2}+\gamma s \ .
and we can take $\dst \Reeb=\derpar{}{s}$ as Reeb vector field.
For this system, the compatibility condition for equations \eqref{Whamilton-contact-eqs} leads to define the submanifold $\mathfrak{M}_0\hookrightarrow\mathfrak{M}$
defined by
\mathfrak{M}_0=\{ (q,v,p,s)\in\mathfrak{M}\,\mid\,  p-mv=0 \} \ ,
and the Hamiltonian vector field solution to \eqref{Whamilton-contact-eqs} on $\mathfrak{M}_0$ is
\X_{\bf H}\vert_{\mathfrak{M}_0}=
\left(kq+\gamma p\right)\derpar{}{p}
+\left(\frac{p^2}{2m^2}-\frac{1}{2}kq^2-\gamma s\right)\derpar{}{s}\ .
The tangency condition for $\X_{\bf H}$ on $\mathfrak{M}_0$ gives
$\dst F=-\frac{k}{m}q-\gamma\frac{p}{m}$,
and thus finally,
\X_{\bf H}\vert_{\mathfrak{M}_0}=
\left(kq+\gamma p\right)\derpar{}{p}
+\left(\frac{p^2}{2m^2}-\frac{1}{2}kq^2-\gamma s\right)\derpar{}{s}\ .
Therefore, the integral curves of $\X_{\bf H}$ are the solutions to
\frac{dq}{dt}=v \quad , \quad
m\frac{dv}{dt}=-mkq-\gamma p \quad , \quad
\frac{dp}{dt}=-kq-\gamma p \quad ,
\quad
\frac{ds}{dt}=\frac{p^2}{2m^2}-\frac{1}{2}kq^2-\gamma s \ .
Te first two equations of this system are equivalent to the
second Herglotz--Euler--Lagrange equation
\eqref{HELdho}
and, using the constraint $p=mv$ (the Legendre map),
the first and third equations are the first pair of the Hamilton equations \eqref{Hdho}.

\subsection{The Kepler problem with friction}

Consider the motion of a massive particle (of mass $m$)
under the action of Newtonian central force
in a stellar media with friction.
As in the previous cases, the motion is on a plane; hence
$Q=\Real^2$, and we take $(r,\phi,s)$ as coordinates in $Q\times\Real$,
(with the origin of $r$ at the center of the force).

\subsubsection{Lagrangian formalism}

The Lagrangian formalism takes place in $\Tan Q\times\Real\simeq\Real^5$, with local coordinates
The contact Lagrangian function that describes the dynamics is
\Lag=\frac{1}{2}m(v_r^2+r^2v_\phi^2)-\frac{K}{r}-\gamma s \quad , \quad K\not=0 \ ;
which is regular as in the above situations.
The energy Lagrangian function and the contact Lagrangian form are
\beann
E_\Lag&=&\frac{1}{2}m(v_r^2+r^2v_\phi^2)+\frac{K}{r}+\gamma s \ , \\
\bmeta_\Lag&=&\d s-m(v_r\,\d r+r^2v_\phi\,\d\phi) \ ,
\eeann
and the Lagrangian is regular.
For $\displaystyle \X_\Lag=f_r\derpar{}{r}+f_\phi\derpar{}{\phi}+
g_r\derpar{}{v_r}+g_\phi\derpar{}{v_\phi}+g\derpar{}{s}\in\vf(\Tan^*Q\times \Real)$,
the contact Lagrangian equations \eqref{ELeqs0}
and \eqref{ELeqs1} give
\beann
g=\frac{1}{2}m(v_r^2+r^2v_\phi^2)-\frac{K}{r}-\gamma s \quad , \quad
f_r=v_r \quad , \quad f_\phi=v_\phi \ , \\
mg_r=2mrv_\phi f_\phi-mrv_\phi^2+\frac{K}{r^2}-\gamma mv_r \quad , \quad
g_\phi=-\frac{2v_\phi f_r}{r}-\gamma r^2v_\phi   \ ,
\eeann
and the contact Euler--Lagrange vector field is
\beann
\X_\Lag&=&v_r\derpar{}{r}+v_\phi\derpar{}{\phi}+\left(rv_\phi^2+\dst\frac{K}{mr^2}-\gamma v_r\right)\derpar{}{v_r}
+\left(-\frac{2v_\phi v_r}{r}-\gamma v_\phi\right)\derpar{}{v_\phi} \\
& & +
\left(\frac{1}{2}m(v_r^2+r^2v_\phi^2)-\frac{K}{r}-\gamma s\right)\derpar{}{s}\ .
\eeann
Then, its integral curves are the solutions to equations \eqref{ELeqs2} and \eqref{ELeqs3} which read as
\bea
\frac{ds}{dt}=\frac{1}{2}m\left(\Big(\frac{dr}{dt}\Big)^2+r^2\Big(\frac{d\phi}{dt}\Big)^2\right)-\frac{K}{r}-\gamma s \quad , \quad
\nonumber \\
m\frac{d^2r}{dt^2}=mr\Big(\frac{d\phi}{dt}\Big)^2+\frac{K}{r^2}-m\gamma v_r
\quad , \quad  \frac{d^2\phi}{dt^2}=
-\frac{2}{r}\,\frac{d\phi}{dt}\,\frac{dr}{dt}-\gamma \frac{d\phi}{dt} \ ,
\label{eqEL-M}
\eea
and are the Herglotz--Euler--Lagrange equations for this system.

Similarly to the above example,
the dissipation of the energy reads
\Lie(\X_\Lag)E_\Lag=-\gamma \left(\frac{1}{2}m(v_r^2+r^2v_\phi^2)+\frac{K}{r}+\gamma s\right)\ .
Furthermore, there exists a Lagrangian contact Noether symmetry
which is generated by the vector field $\dst Y=\derpar{}{\phi}$; in fact,
bearing in mind \eqref{NsymKep1} and \eqref{NsymKep2}, we obtain
\beann
\Lie(Y)\bmeta_\Lag&=&
\Lie\left(\derpar{}{\phi}\right)\left(\d s-m(v_r\,\d r+r^2v_\phi\,\d\phi)\right) \\ &=&
\Lie\left(\derpar{}{\phi}\right)\d s-
\Lie\left(\derpar{}{\phi}\right)\left(m(v_r\,\d r+r^2v_\phi\,\d\phi)\right)=0 \ ,
\\
\Lie(Y)E_\Lag&=&
\Lie\left(\derpar{}{\phi}\right)\left(\frac{1}{2}m(v_r^2+r^2v_\phi^2)+\frac{K}{r}+\gamma s\right)=0 \ ,
\eeann
and the corresponding dissipated quantity is the
angular momentum map
F=-\inn\left(\derpar{}{\phi}\right)\bmeta_\Lag=mr^2v_\phi \ .
Finally, using Corollary \ref{consquanquot},
we obtain that $\dst\frac{F}{E_\Lag}$ is a conserved quantity associated with $Y$.

\subsubsection{Hamiltonian formalism}

For the Hamiltonian formalism, $\Tan^*Q\times\Real\simeq\Real^5$,
with local coordinates $(r,\phi,p_r,p_\phi,s)$. The Legendre transformation is,
\mathfrak{F}\Lag^*r=r  \quad , \quad \mathfrak{F}\Lag^*\phi=\phi \quad , \quad
\mathfrak{F}\Lag^*p_r=mv_r  \quad , \quad \mathfrak{F}\Lag^*p_\phi=mr^2v_\phi \quad , \quad \mathfrak{F}\Lag^*s=s  \ ,
which is a diffeomorphism, and the Lagrangian is hyperregular.
The canonical Hamiltonian function and the canonical contact form are
\beann
{\rm h}&=&\frac{p_r^2}{2m}+\frac{p_\phi^2}{2mr^2}+\frac{K}{r}+\gamma s \\
\bmeta&=&\d s-p_r\,\d r-p_\phi\,\d\phi \ .
\eeann
For $\displaystyle \X_{\rm h}=F_r\derpar{}{r}+F_\phi\derpar{}{\phi}+
the contact Hamiltonian equations \eqref{evolvf11b} lead to
F_r=\frac{p_r}{m} \ , \ F_\phi=\frac{p_\phi}{mr^2} \ , \
G_r=\frac{p_\phi^2}{mr^3}+\frac{K}{r^2}-\gamma p_r \ , \ G_\phi=-\gamma p_\phi  \ , \ g=\frac{p_r^2}{2m}+\frac{p_\phi^2}{2mr^2}-\frac{K}{r}-\gamma s \ .
and then, the contact Hamiltonian vector field is
\X_{\rm h}=\frac{p_r}{m}\derpar{}{r}+\frac{p_\phi}{mr^2}\derpar{}{\phi}+\left(\frac{p_\phi^2}{mr^3}+\frac{K}{r^2}-\gamma p_r\right)\derpar{}{p_r}-\gamma p_\phi\derpar{}{p_\phi}+\left(\frac{p_r^2}{2m}+\frac{p_\phi^2}{2mr^2}-\frac{K}{r}-\gamma s\right)\derpar{}{s} \ .
Hence, its integral curves are the solutions to the equations \eqref{hel-eqs01b} which are
\bea
m\frac{dr}{dt} =p_r \quad , \quad mr^2\frac{d\phi}{dt} =p_\phi \quad , \quad
\frac{dp_r}{dt}=\frac{p_\phi^2}{mr^3}+\frac{K}{r^2}-\gamma p_r & , &
\frac{dp_\phi}{dt}=-\gamma p_\phi \quad ,
\label{dislawKeppler} \\
\frac{ds}{dt}=\frac{p_r^2}{2m}+\frac{p_\phi^2}{2mr^2}-\frac{K}{r}-\gamma s & , & \nonumber
\eea
and are the Hamiltonian equations for this system.

As it is usual, using the Legendre map
one can easily check that $\mathfrak{F}\Lag_*X_\Lag=X_{\rm h}$.

Similarly to the above example,
the dissipation of the energy reads
\Lie(\X_{\rm h}){\rm h}=-\gamma \left(\frac{p_r^2}{2m}+\frac{p_\phi^2}{2mr^2}+\frac{K}{r}+\gamma s\right)\ .
In addition, we have a Hamiltonian contact Noether symmetry which is again the
vector field $\dst Y=\derpar{}{\phi}$, since
\beann
\Lie(Y)\bmeta&=&
\Lie\left(\derpar{}{\phi}\right)\left(\d s-p_r\,\d r-p_\phi\,\d\phi\right) =0 \ ,
\\
\Lie(Y){\rm h}&=&
\Lie\left(\derpar{}{\phi}\right)\left(\frac{p_r^2}{2m}+\frac{p_\phi^2}{2mr^2}+\frac{K}{r}+\gamma s\right)=0 \ .
\eeann
The associated dissipated quantity is again the angular momentum,
F=-\inn\left(\derpar{}{\phi}\right)\bmeta=p_\phi \ .
and the corresponding dissipation law is given directly by the last Hamilton equation in \eqref{dislawKeppler}.
As above, $\dst\frac{F}{{\rm h}}$ is a conserved quantity associated with $Y$.

\subsubsection{Unified Lagrangian-Hamiltonian formalism}

In the precontact extended unified bundle
$\mathfrak{M}=\Tan Q\times_Q\Tan^*Q\times\Real\simeq\Real^7$,
with coordinates $(r,\phi,v_r,v_\phi,p_r,p_\phi,s)$,
the canonical precontact form and the Hamiltonian function are
\beann
\bmeta_{\mathfrak{M}}&=&\d s-p_r\,\d r-p_\phi\,\d\phi \ , \\
{\bf H}&=&p_rv_r+p_\phi v_\phi-\frac{1}{2}m(v_r^2+r^2v_\phi^2)+\frac{K}{r}+\gamma s \ ,
\eeann
and we can take $\dst \Reeb=\derpar{}{s}$ as Reeb vector field.

For $\displaystyle \X_{\bf H}=f_r\derpar{}{r}+
equations \eqref{Whamilton-contact-eqs} lead to
\bea
f_r=v_r \quad , \quad f_\phi=v_\phi \quad , \quad
G_r=\frac{K}{r^2}+mrv_\phi^2-\gamma p_r \quad , \quad G_\phi=-\gamma p_\phi & , &
\nonumber \\
p_r=mv_r \quad , \quad p_\phi=mr^2v_\phi \quad , \quad
g=\frac{p_r^2}{2m}+\frac{p_\phi^2}{2mr^2}-\frac{K}{r}-\gamma s & . & \label{eqsM}
\eea
The first two equations in \eqref{eqsM} are constraints defining the submanifold
$\mathfrak{M}_0\hookrightarrow\mathfrak{M}$ which give the Legendre map.
Thus, the Hamiltonian vector field is
\beann
\X_{\bf H}\vert_{\mathfrak{M}_0}&=&
+\left(\frac{K}{r^2}+mrv_\phi^2-\gamma p_ r\right)\derpar{}{p_r}-\gamma p_\phi\derpar{}{p_\phi} \\ & &
+\left(\frac{p_r^2}{2m}+\frac{p_\phi^2}{2mr^2}-\frac{K}{r}-\gamma s\right)\derpar{}{s} \ ,
\eeann
and the tangency condition of $\X_{\bf H}$ on $\mathfrak{M}_0$ leads to obtain that, finally,
\beann
\X_{\rm H}\vert_{\mathfrak{M}_0}&=&
\left(rv_\phi^2+\dst\frac{K}{mr^2}-\gamma\frac{p_r}{m}\right)\derpar{}{v_r}-\left(\frac{2v_rv_\phi}{r}+\gamma\frac{p_\phi}{m}\right)\derpar{}{v_\phi}
\\ & &
+\left(\frac{K}{r^2}+mrv_\phi^2-\gamma p_ r\right)\derpar{}{p_r}-\gamma p_\phi\derpar{}{p_\phi}+\left(\frac{p_r^2}{2m}+\frac{p_\phi^2}{2mr^2}-\frac{K}{r}-\gamma s\right)\derpar{}{s} \ .
\eeann
Therefore, the integral curves of $\X_{\bf H}$,
on $\mathfrak{M}_0$, are the solutions to
\bea
\frac{dr}{dt}=v_r \quad , \quad
\frac{d\phi}{dt}=v_\phi \quad , \quad
\frac{dv_r}{dt}=\frac{K}{mr^2}+rv_\phi^2-\gamma\frac{p_r}{m} \quad , \quad
\frac{dv_\phi}{dt}=-\frac{2v_rv_\phi}{r}-\gamma\frac{p_\phi}{m} & , &
\label{unieqs1}
\\
\frac{dp_r}{dt}=\frac{K}{r^2}+mrv_\phi^2-\gamma p_r \quad , \quad
\frac{dp_\phi}{dt}=-\gamma p_\phi \quad , \quad
\frac{ds}{dt}=\frac{p_r^2}{2m}+\frac{p_\phi^2}{2mr^2}-\frac{K}{r}-\gamma s & . &
\label{unieqs2}
\eea
Equations \eqref{unieqs1} are equivalent to the Herglotz--Euler--Lagrange equation \eqref{eqEL-M},
and using the constraints $p_r=mv_r$
and $p_\phi=mr^2v_\phi$; that is, the Legendre map,
the first and second equations in \eqref{unieqs1} and \eqref{unieqs2} are
the Hamiltonian equations \eqref{dislawKeppler} of the system.

\begin{appendix}
\chapter{Additional contents}

\section{Tangent and cotangent bundles}
\label{sec:tangentb}

(For a detailed account of these subjects and the proof of the results, see for example
[91, 218, 230, 284, 319]).

\subsection{The tangent bundle of a manifold. Canonical lifts}
\label{canliftq}

Let $Q$ be a differentiable manifold and  $q\in Q$.
Every differentiable curve
$\gamma\colon (-\epsilon ,\epsilon )\subset \Real \to Q$
passing through $q$; that is,
$\gamma (0)=q$ ,
induces a derivation ${\rm D}_\gamma$ in $\Cinfty (Q)$,
in the following way
\begin{array}{cccc}
{\rm D}_\gamma \colon & \Cinfty (Q) & \to & \Real
\\
& f & \mapsto &\displaystyle \lim_{t \mapsto 0}\frac{(f\circ\gamma )(t)-(f\circ\gamma )(0)}{t}
\end{array} \quad .
In the set of such differentiable curves we can define the following equivalence relation
\gamma_1 \sim \gamma_2 \quad \Longleftrightarrow
\quad
{\rm D}_{\gamma_1}={\rm D}_{\gamma_2} \ .
\begin{definition}
\ben
\item
A \textbf{tangent vector} to $Q$ at $q$
is every equivalence class defined  by this relation.
\item
The \textbf{tangent space} of $Q$ at $q$,
denoted by $\Tan_qQ$, is the vector space of all the tangent vectors
to $Q$ at $q$.
\item
The \textbf{tangent bundle} of $Q$ is defined as
\(\dst \Tan Q := \bigcup_{q \in Q}\Tan_qQ\).
We denote $\tau_Q \colon \Tan Q \to Q$ its natural projection.
\een
\end{definition}

Every point ${\rm p}\in\Tan Q$ is a couple
$(q,v)\equiv v_q$, where $q=\tau_Q({\rm p})\in Q$
and $v\in\Tan_qQ$ (it is a tangent vector).

\begin{prop}
The tangent bundle $\Tan Q$ is a $2n$-dimensional differentiable manifold whose differentiable structure is inherited from $Q$.
In addition, the natural projection $\tau_Q$ is a submersion.
\end{prop}
\begin{proof}
If ${\cal A}=\{ (U_\alpha;\phi_\alpha)\}$, with $\phi_\alpha\equiv (\coor{x}{1}{n})$,
is an atlas of local charts on $Q$,
then the induced atlas on $\Tan^*Q$ is
$\Tan{\cal A}=\{ (\tau_Q^{-1}U_\alpha;\psi_\alpha) \}$,
with the coordinate functions $\psi_\alpha$ defined as follows:
\begin{array}{ccccc}
\psi_\alpha&\colon&\tau^{-1}_Q(U_\alpha)&\longrightarrow&
\phi_\alpha(U_\alpha)\times\Real^n\subset\Real^{2n}\\
& & {\rm p}=(q,v) & \mapsto & (q^i(p),v^i({\rm p}))
\end{array} \quad ;
where $i=1,\ldots,n$, and:
\ben
\item
$q^i({\rm p})=(x^i\circ\tau_Q)({\rm p})$
\footnote{
It is usual to commit an abuse of notation denoting by $q^i$
both the coordinates in the base manifold $Q$
and in the tangent bundle $\Tan Q$, and we will do in the sequel.}.
\item
If \(\dst v=\lambda^j\derpar{}{x^j}\Big\vert_q\), then
that is, $v^i(p)$ are the components of the tangent vector
$v\in\Tan_qQ$ in the basis \(\dst\left\{ \derpar{}{x^i}\Big\vert_q\right\}\).
\een
It is obvious that \(\dst\Tan Q=\bigcup_\alpha\tau^{-1}_Q(U_\alpha)\)
and that $\psi_\alpha$ are diffeomorphisms.
Then, it is immediate to prove that $\Tan{\cal A}$ endows $\Tan Q$
with the structure of a differentiable manifold.
If $(U;q^i)$ and $(\bar U;\bar q^i)$ are two local charts
in $Q$ such that
$U\cap\bar U\not=\buit$, and $\bar q^j =\varphi^j (q^i)$ in $U\cap\bar U$,
then for the induced charts $(\tau^{-1}_Q(U);q^i,v^i)$ and
$(\tau^{-1}_Q\bar U);\bar q^i,\bar v^i)$
in $\Tan Q$ we have that, on $\Tan U\cap\Tan\bar U$,
the relation between the coordinates $v^i$ and $\bar v^i$ is
\(\dst\bar v^j=\derpar{\varphi^j}{q^i}v^i\), since
v=v^i\derpar{}{q^i}\Big\vert_q=v^i\derpar{\varphi^j}{q^i}\derpar{}{\bar q^j}\Big\vert_q=
\bar v^j\derpar{}{\bar q^j}\Big\vert_q \ ,
and hence \(\dst\bar v^j=\derpar{\varphi^j}{q^i}v^i\).

Bearing in mind this local description of the tangent bundle,
it is evident that $\dim\,\Tan Q=2n$.

Finally, it is immediate to prove that $\tau_Q$ is an surjective submersion,
since the canonical projection
$\tau_Q \colon \Tan Q \to Q$,
is a surjective map given, in natural coordinates, by
$\tau_Q (q^i,v^i)=q^i$, and hence its tangent map
$\Tan\tau_Q\colon\Tan\Tan Q\to\Tan Q$ is defined as follows:
if $(q,v) \in \Tan Q$ and $X \in \Tan_{(q,v)}(\Tan Q)$
we have that
\(\dst X=\lambda^i\derpar{}{q^i}\Big\vert_{(q,v)}
+\mu^i\derpar{}{v^i}\Big\vert_{(q,v)}\), and
[\Tan_{(q,v)}\tau_Q (X)](q^j)=X(q^j\circ\tau_Q )=X(q^j)=\lambda^j\ ,
(\Tan\tau_Q )((q,v),X) =
(\Tan\tau_Q )\left( (q,v),\lambda^i\derpar{}{q^i}\Big\vert_{(q,v)}+
\mu^i\derpar{}{v^i}\Big\vert_{(q,v)} \right)
= \left(q,\lambda^i\derpar{}{q^i}\Big\vert_q \right) \ ,
with associated matrix is
\beq
\Tan\tau_Q\vert_{(q,v)} = (I_{n\times n},0_{n\times n}) =
\left(\begin{matrix}
1 & \ldots & 0 & 0 & \ldots & 0 \\
\vdots &  & \vdots & \vdots & & \vdots \\
0 & \ldots & 1 & 0 & \ldots & 0\end{matrix}\right) \ .
\label{matrix1}
\eeq
In this way, we conclude that $\tau_Q$ is a surjective submersion.
\\ \qed \end{proof}

\begin{definition}
The above charts of the tangent bundle
are called \textbf{natural charts} and their coordinates \textbf{ natural coordinates}
($q^i$ are the \textbf{base coordinates}
and $v^i$ the \textbf{fiber coordinates}).
\end{definition}

Observe that the coordinate change in $\Tan Q$,
from $(q^i,v^i)$ to $(\bar q^i,\bar v^i)$ has the Jacobian matrix$$
\left(\begin{matrix}
\displaystyle\left(\derpar{\varphi^j}{q^i}\right) & 0 \\
\displaystyle\left(\frac{\partial^2\varphi^j}{\partial q^k\partial q^i}\right)v^i &
\displaystyle\left(\derpar{\varphi^j}{q^i}\right)
\end{matrix}\right) \ ,
and, taking into account that its determinant is positive at every point, we conclude that:

\begin{corol}
$\Tan Q$ is an orientable manifold.
\end{corol}

The tangent bundle of a manifold is an example of a  {\sl vector bundle}.
This structure is defined as follows:

\begin{definition}
A \textbf{vector bundle} is a triple $(E,B,\pi)$,
where $E,B$ are differentiable manifolds
(with $\dim\, B=m$, $\dim\, E=m+n$) and $\pi\colon E\to B$
is a surjective submersion such that, for every $p\in B$,
there exists a local chart $(U,\phi)$, $p\in U$, and a diffeomorphism
$\psi\colon\pi^{-1}(U)\to \phi(U)\times\Real^n$
satisfying that:
\ben
\item
If $\pi_1\colon\phi(U)\times\Real^n\to\phi(U)$
is the natural projection, then $\pi_1\circ\psi=\pi$.
\item
$E_p=\pi^{-1}(p)$ is a vector space and,
if $\pi_2\colon\phi(U)\times\Real^n\to\Real^n$
is the natural projection, then the maps
\begin{array}{ccccc}
\psi_p & \colon & E_p & \longrightarrow & \Real^n \\
& & v &\mapsto & (\pi_2\circ\psi)(p,v)
\end{array}
are vector space morphisms.
\een
$E$ is called the \textbf{total manifold} of the vector bundle,
$B$ is the \textbf{base manifold}, $\pi$ the \textbf{projection of the bundle},
$\Real^n$ is the \textbf{typical fiber},
$n$ is the \textbf{rank of the bundle} and, for every $ p\in B$,
$E_p$ is the \textbf{fiber} over $p$.
The pair $(U,\phi)$ is said to be a \textbf{trivializing open set}
and $\psi$ is the associated \textbf{coordinate map}.
\end{definition}

Observe that the family$(\pi^{-1}(U),\psi)$ is a differentiable atlas of $E$ which is said to be {\sl \textbf{adapted}} to the projection $\pi:E\to B$.

There is a natural way to lift diffeomorphisms, curves and vector fields
on a manifold to its tangent bundle.

\begin{definition}
Let $Q$ be a differentiable manifold and a diffeomorphism
\begin{array}{ccccc}
\varphi&\colon&Q&\longrightarrow&Q
\\
& & q & \mapsto & \varphi(q)
\end{array} \ .
The \textbf{canonical lift} of $\varphi$ to $\Tan Q$
is the diffeomorphism
\begin{array}{ccccc}
\Tan\varphi&\colon&\Tan Q&\longrightarrow&\Tan Q
\\
& &(q,v ) & \mapsto & (q,\Tan_q\varphi (v))
\end{array} \ .
\end{definition}

This definition also holds for every map $\varphi \colon Q\longrightarrow Q$.

The following properties are immediate from the definition:

\begin{prop}
For every $\varphi,\phi\in{\rm Diff}\,Q$,
\ben
\item
$\varphi^{-1}\circ\tau_Q\circ\Tan \varphi =\tau_Q$.
\item
$\Tan (\varphi\circ\phi )=\Tan\varphi\circ\Tan\phi$.
\een
\label{exerc1}
\end{prop}

Using the definition of canonical lift of diffeomorphisms, we set:

\begin{definition}
Let $Z\in\vf(Q)$ be a vector field.
The  \textbf{total}, \textbf{complete}  or \textbf{canonical lift} of $Z$ to $\Tan Q$
is the vector field $Z^C\in\vf (\Tan Q)$
whose local uniparametric groups of diffeomorphisms are the
canonical lifts $\{\Tan F_t\}$ of the
local uniparametric groups of diffeomorphisms
$\{F_t\}$ of $Z$.
\end{definition}

As a direct consequence of the definition and of the theorem of existence and unicity of local uniparametric groups of diffeomorphisms,
we obtain the following result:

\begin{prop}
Let $Z\in\vf (Q)$.
Then $Z^C$ is $\tau_Q$-projectable and $\tau_{Q*}Z^C=Z$;
that is,
$\Tan\tau_Q(Z^C_{(q,v)})=Z_x$, for every $(q,v)\in\Tan Q$.
\label{prolevcan0}
\end{prop}

\noindent{\bf Local expression}:
In a chart of coordinates $(U;q^i)$ of $Q$, if
Z\vert_U=f^i(q^j)\derpar{}{q^i} \ ,
then we have that, in the induced chart of natural coordinates $(\tau_Q^{-1}(U);q^i,v^i)$ of $\Tan Q$,
Z^C\vert_{\tau_Q^{-1}(U)}=f^i(q^j)\derpar{}{q^i}+v^k\derpar{f^i}{q^k}(q^j)\derpar{}{v^i} \ .
In order to prove it, remember that, if $(q,u)\in\tau_Q^{-1}(U)$, then
Z^C(q,u)=\frac{d}{d t}\Big\vert_{t=0}\Tan F_t(q,u) \ ,
where \(\dst\Tan F_t(q^i,v^i)=\left(F_t(q^i),\derpar{F_t}{q^i}v^i\right)\) and,
as a consequence,
\frac{d}{d t}\Big\vert_{t=0}\Tan F_t(q,u)=
\left(\frac{d}{d t}\Big\vert_{t=0}F_t(q),
\frac{d}{d t}\Big\vert_{t=0}\left(\derpar{F_t}{q^i}v^i\right)(q,u)\right)=
\left(f^i(q),v^j\derpar{f^i}{q^j}(q^k)\right) \ ,
and the result follows.

Let $\gamma\colon (a,b)\subseteq\Real\to Q$ be a curve.
If $t_0 \in (a,b)$ and $x=\gamma (t_0)$,
then $\dot\gamma (t_0)$ is the tangent vector to the curve
at the point $\gamma (t_0)$; that is,
$\dot\gamma (t_0)\in\Tan_{\gamma (t_0)}Q$.
As $\dot\gamma (t)$ is well-defined for every $t\in (a,b)$,
we can define:

\begin{definition}
\label{canlifcurv}
The \textbf{canonical lift} of a curve $\gamma$ to $\Tan Q$ is the curve
\begin{array}{cccc}
\widetilde  \gamma \colon &(a,b)\subset\Real&\to&\Tan Q
\\
&t&\mapsto&(\gamma (t),\dot\gamma(t))
\end{array}  \ ,
which is defined as
(\gamma (t_0),\dot\gamma (t_0))=\Tan_{t_0}\gamma\frac{d}{d t} \ ;
that is, for every $f\colon Q \to \Real$, then
(\gamma (t_0),\dot\gamma (t_0))f=
(\Tan_{t_0}\gamma\frac{d}{d t})f=
\frac{d}{d t}\Big\vert_{t_0}(f\circ\gamma )
=\lim_{h\mapsto 0}\frac{f(\gamma (t_0+h))-f(\gamma (t_0))}{h} \ .
Observe that $\tau_Q\circ\widetilde  \gamma=\gamma$.
\end{definition}

\noindent{\bf Local expression}:
If $(q^i)$ are local coordinates of $Q$ in a neighbourhood of
$\gamma (t_0)$, then $\gamma =(\gamma^1,\ldots ,\gamma^n)$,
with $\gamma^i=q^i\circ\gamma$.
If $(q^i,v^i)$ are the natural induced coordinates in $\Tan Q$,
then $\widetilde  \gamma$ is given by
$\widetilde  \gamma =(\gamma^1,\ldots ,\gamma^n,\dot\gamma^1,\ldots ,\dot\gamma^n)$,
where \(\dst\dot\gamma^i=\frac{d\gamma^i}{d t}\).

The tangent vector to $\widetilde  \gamma$ at $\widetilde  \gamma (t_0)$
is \(\dst \Tan_{t_0}\widetilde  \gamma\frac{d}{d t}\)
and, if $f\colon\Tan Q\to \Real$, we have that
\left( \Tan_{t_0}\widetilde  \gamma\frac{d}{d t}\right) f=
\frac{d}{d t}\Big\vert_{t_0}f\circ\widetilde  \gamma \ ,
and hence
\begin{eqnarray*}
\left(\Tan_{t_0}\widetilde  \gamma\frac{d}{d t}\right) q^i&=&
\frac{d}{d t}\Big\vert_{t_0}q^i\circ\widetilde  \gamma=\dot\gamma^i(t_0) \ ,
\\
\left(\Tan_{t_0}\widetilde  \gamma\frac{d}{d t}\right) v^i&=&
\frac{d}{d t}\Big\vert_{t_0}v^i\circ\widetilde  \gamma=
\frac{d}{d t}\Big\vert_{t_0}\dot\gamma^i=\ddot\gamma^i(t_0) \ ;
\end{eqnarray*}
that is
\Tan_{t_0}\widetilde\gamma\Big(\frac{d}{d t}\Big)=
\dot\gamma^i(t_0)\derpar{}{q^i}\Big\vert_{\widetilde  \gamma (t_0)}
+\ddot\gamma^i(t_0)\derpar{}{v^i}\Big\vert_{\widetilde  \gamma (t_0)} \ .

\begin{remark}{\rm
\label{intcurve}
Remember that, given a vector field $X\in\vf(Q)$,
a curve $\gamma\colon (a,b)\subseteq\Real\to Q$
is an {\sl integral curve} of $X$ at $\gamma(t)\in Q$, for $t\in (a,b)$, if
X(\gamma(t))=\dot\gamma(t)\ .
Taking this into account, if $\beta\in\df^k(Q)$,
the contraction of $\beta$ with $X$, denoted by $\inn(X)\beta$, allows us to define a new contraction
$\big(\inn(\widetilde \gamma)(\beta\circ\gamma)$  in a natural way, as follows:
\beq
\inn(\widetilde \gamma(t))\big(\beta(\gamma(t))\big):=
\inn\big(\dot\gamma(t)\big)(\beta\vert_{\gamma(t)}) \quad, \quad \mbox{\rm for $t\in (a,b)$} \ .
\label{contr2}
\eeq

\subsection{The cotangent bundle of a manifold. Canonical lifts}
\label{sec:cotbun}

\begin{definition}
Let $Q$ be a differentiable manifold.
The \textbf{cotangent bundle} of $Q$ is the dual bundle
$\Tan ^*Q$ of the tangent bundle $\Tan Q$;
that is,  \(\dst \Tan^*Q := \bigcup_{q \in Q}\Tan^*_qQ\)~.

We denote $\pi_Q \colon \Tan^*Q \to Q$ the natural projection.
\end{definition}

Thus, every point ${\rm p} \in \Tan^*Q$ is a pair
$(q,\xi)\equiv\xi_q$, where $q=\pi_Q ({\rm p}) \in Q$
and $\xi \in \Tan^*_qQ$ (it is a linear form on $\Tan_qQ$).

\begin{prop}
\label{prop:cotbun}
Let $Q$ be a $n$-dimensional differentiable manifold.
The cotangent bundle $\Tan^* Q$ is a $2n$-dimensional differentiable manifold,
whose differentiable structure is induced by the one of $Q$ in a natural way.
Moreover, the natural projection $\pi_Q$ is a submersion.
\end{prop}
\begin{proof}
If ${\cal A}=\{ (U_\alpha;\phi_\alpha=(x^i) \}$ is an atlas of local charts in $Q$,
then the induced atlas
$\Tan^*{\cal A}=\{ (\Tan^*U_\alpha;\eta_\alpha=(q^i,p_i) \}$
is obtained as follows:

Given one of its elements $(U;x^i)$, we take
\begin{enumerate}
\item
$\Tan^* U := \pi_Q^{-1}(U)$.
\item
For every ${\rm p} \in \Tan^*Q$, let $q^i({\rm p}) := x^i \circ \pi_Q ({\rm p})$\footnote
{We commit an abuse of notation, denoting
by $q^i$ both the coordinates $q^i$ and $x^i$.}.
\item
Let ${\rm p} \in \Tan^*Q$ with ${\rm p}=(q,\xi)$. Then \
$\displaystyle p_i({\rm p}) := \Big\langle\xi ,\derpar{}{q^i}\Big\vert_q\Big\rangle$;\
that is, $p_i({\rm p})$ are the components of the linear form
$\xi$ in the natural basis $\d q^i\mid_q$ of $\Tan^*_qM$:
$\xi = p_i({\rm p})\,\d q^i\mid_q$.
\end{enumerate}
If $(q^i)$ and $(\bar q^i)$ are two local systems of coordinates
in $U\subset Q$, and $\bar q^j =\phi^j (q^i)$, then
the relation between $(p_i)$ and $(\bar p_i)$ is the following,
p_i(q,\xi )=\Big\langle\derpar{}{q^i},\xi\Big\rangle=
\Big\langle\derpar{\phi^j}{q^i}\Big\vert_q \derpar{}{\bar q^j},\xi\Big\rangle
=\derpar{\phi^j}{q^i}\Big\vert_q\Big\langle\derpar{}{\bar q^j},\xi\Big\rangle
=\derpar{\phi^j}{q^i}\Big\vert_q \bar p_j(q,\xi ) \ ;
that is, \(\dst p_i=\derpar{\phi^j}{q^i}\bar p_j\).
Then, the changes of coordinates in $\Tan^*Q$,
from $(q^i,p_i)$ to $(\bar q^i,\bar p_i)$, have as Jacobian matrix
\left(\begin{matrix}\displaystyle\left(\derpar{\phi^j}{q^i}\right) & 0 \\
\displaystyle\left(\frac{\partial^2\phi^j}{\partial q^k\partial q^i}\right)^{-1}p_i &
\displaystyle\left(\derpar{\phi^j}{q^i}\right)^{-1}\end{matrix}\right) \ .

From this local description of the cotangent bundle
it is evident that its dimension is $2n$ as we know previously being the dual bundle of $\Tan Q$.
\\ \qed  \end{proof}

\begin{definition}
The above coordinate charts are called
\textbf{natural charts} of the cotangent bundle
and their elements \textbf{natural coordinates}
($q^i$ are the \textbf{ base coordinates}
and $p_i$ the \textbf{fiber coordinates}).
\end{definition}

Observe that the determinant of the above last matrix is the unity and this is the same for any change of natural coordinates, hence the manifold $\Tan^*Q$ is also an orientable manifold.

In an analogous way as for the tangent bundle, there are
natural (canonical) ways to lift
diffeomorphisms and vector fields from a differentiable manifold $Q$
to its cotangent bundle $\Tan^*Q$.

\begin{definition}
Consider a diffeomorphism
\begin{array}{ccccc}
\varphi&\colon&Q&\longrightarrow&Q
\\
& & x & \mapsto & \varphi(x)
\end{array} \ .
The \textbf{canonical lift} of $\varphi$ to $\Tan^*Q$ is the diffeomorphism
\begin{array}{ccccc}
\Tan^*\varphi&\colon&\Tan^*Q&\longrightarrow&\Tan^*Q
\\
& &(\varphi(x),\xi ) & \mapsto & (x,\Tan^*\varphi (\xi))
\end{array}
where $\Tan^*\varphi (\xi)$ is defined by duality as follows: if
\begin{array}{ccccc}
\Tan\varphi&\colon&\Tan Q&\longrightarrow&\Tan Q
\\
& &(q,v) & \mapsto & (x,\Tan\varphi (v))
\end{array}
then, for every $v\in\Tan_qQ$,
(\Tan^*\varphi(\xi))(v) :=\xi (\Tan\varphi(v)) \ .
\end{definition}

The following properties follow straightforwardly from the definition:

\begin{prop}
\label{exerc2}
If $\varphi,\phi\colon Q\to Q$ are diffeomorphisms, then:
\ben
\item
$\varphi\circ\pi_Q\circ\Tan^*\varphi =\pi_Q$.
\item
$\Tan^*(\varphi\circ\phi )=\Tan^*\phi\circ\Tan^*\varphi$,
\een
\end{prop}

A relevant property is:

\begin{prop}
If $\varphi\colon Q\to Q$ is a diffeomorphism,
then $(\Tan^*\varphi)^*\Theta =\Theta$ and,
as a consequence,
$(\Tan^*\varphi)^*\Omega =~\Omega$~\footnote{
With the terminology introduced in Section \ref{tcs},
we say that $\Tan^*\varphi$ is a {\sl simplectomorphism}}.
\label{levdif}
\end{prop}
\begin{proof}
For every $(q,\xi)\in\Tan^*Q$ and
using the definition of $\Theta$ and the first property of the above proposition, we obtain
\beann
\big((\Tan^*\varphi)^*\Theta\big)_{(\varphi(q),\xi)}(V) &=&
\Theta_{(q,\Tan^*\varphi(\xi))}\big(\Tan_{(\varphi(q),\xi)}(\Tan^*\varphi) (V)\big)
\\ &=&
(\varphi_q^*\xi)\Big(\Tan_{(q,\Tan^*\varphi(\xi))}\pi_Q \big(\Tan_{(\varphi(q),\xi)}(\Tan^*\varphi)(V)\big)\Big)
\\ &=&
\xi\Big(\Tan_{\varphi(q)}\varphi\big(\Tan(\pi_Q\circ\Tan^*\varphi)(V)\big)\Big)
\\ &=&
\xi\Big(\Tan_{(\varphi(q),\xi)} \big(\varphi\circ\pi_Q\circ\Tan^*\varphi)(V)\big)\Big)=
\xi\big(\Tan_{(\varphi(q),\xi)}\pi_Q(V)\big)=
\Theta_{(\varphi(q),\xi)}(V) \, .
\eeann
The result for $\Omega$ is immediate from here.
\\ \qed  \end{proof}

\begin{definition}
Consider a vector field $Z\in\vf (Q)$.
The \textbf{canonical lift} of $Z$ to $\Tan^*Q$
is the vector field $Z^*\in\vf (\Tan^*Q)$
whose local uniparametric groups of diffeomorphisms
are the canonical lifts $\Tan^* F_t$
of the local uniparametric groups of diffeomorphisms
$ F_t$ of $Z$.
\end{definition}

As a straightforward consequence of the definition and of the
theorem of existence and unicity of uniparametric groups
of diffeomorphisms, we have that:

\begin{prop}
If $Z\in\vf (Q)$, then $Z^*$ is $\pi_Q$-projectable and,
for every ${\rm p}\equiv (q,\xi)\in\Tan^*Q$,
we have that $\Tan\pi_Q (Z^*_{\rm p})=Z_q$.
\label{prolevcan}
\end{prop}

The coordinate expression of the canonical lift of a vector field to $\Tan^*Q$ is given in \eqref{canlifcot}.

\section{Lie groups and Lie algebras}
\label{Liega}

(For more information about definitions and properties of Lie groups and Lie algebras, see,
for instance, [7, 101, 279, 324, 345]).

\begin{definition}
A (finite-dimensional) \textbf{Lie group}
is a (finite-dimensional) differentiable manifold $G$ such that
{\rm (i)}
$G$ is a group and
{\rm (ii)}
the following group operations are smooth,
\begin{array}{ccccccc}
G \times G &\to& G & \quad , \quad & G &\to& G
\\
(g_1,g_2) &\mapsto& g_1g_2 & \quad , \quad & g &\mapsto& g^{-1}
\end{array} \ .
\end{definition}

\begin{definition}
Let $G$ be a Lie group. A subset $H \subset G$.
is a \textbf{Lie subgroup} of $G$ if
{\rm (i)} $H$ is a subgroup of $G$ and
{\rm (ii)} $H$ has a differentiable structure with respect
to which it is a submanifold of $G$ and a Lie group.
\end{definition}

Given a Lie group $G$ and an element $g \in G$,
we can define the diffeomorphisms
\begin{array}{ccccccccc}
L_g \colon & G & \to & G & \ , \ & R_g \colon & G & \to & G
\\
& g' & \mapsto & gg' & \ , \ & &  g' & \mapsto & g'g
\end{array} \ ,
which are the so-called {\sl \textbf{left}} and {\sl \textbf{right translations}}
respectively, and satisfy the following properties:
\begin{enumerate}
\item
For all $g,g' \in G$;\quad
$L_g \circ L_{g'} = L_gL_{g'}$ , \
$R_g \circ R_{g'} = R_gR_{g'}$.
\item
For all $g \in G$;\quad
$L_g^{-1} = L_{g^{-1}}$ , \
$R_g^{-1} = R_{g^{-1}}$.
\item
For all $g,g' \in G$;
$L_g \circ R_{g'} = R_{g'} \circ L_g$.
\end{enumerate}
These translations also induce translations on the set of
vector fields $\vf(G)$, which are denoted by
$L_{g_*}$ and $R_{g_*}$, respectively. Then,

\begin{definition}
A vector field $X \in\vf(G)$ is
\textbf{left} (resp. \textbf{right}) \textbf{invariant}
if, for every $g \in G$,
L_{g_*}X = X \quad  (resp. \ R_{g_*}X = X) \ .
The sets of these vector fields are denoted
$\vf_L(G)$ and $\vf_R(G)$ respectively.
\end{definition}

\begin{prop}
The sets $\vf_L(G)$ and $\vf_R(G)$ are Lie subalgebras of $\vf(G)$.
\end{prop}
\begin{proof}
The sum of invariant vector fields is also invariant,
and the same thing holds for the Lie bracket,
since, for every $g \in G$ and $X_1,X_2 \in\vf_L(G)$ (or $X_1,X_2 \in\vf_R(G)$),
L_{g_*}[X_1,X_2] = [L_{g_*}X_1,L_{g_*}X_2]=[X_1,X_2] \ .
\qed  \end{proof}

\begin{definition}
The set $\vf_L(G)$ is the
\textbf{Lie algebra} of $G$ and is denoted by ${\bf g}$.
\end{definition}

\begin{prop}
If $e\in G$ is the neutral element of $G$, then there is a
canonical vector space isomorphism between
$\vf_L(G)$ and $\Tan_eG$.
\end{prop}
\begin{proof}
To every $X\in\vf_L(G)$ corresponds a vector of
$\Tan_eG$ by means of the isomorphism
\begin{array}{cccc}
\rho_1 \colon &\vf_L(G)&\to&\Tan_eG
\\
\end{array} \ .
Conversely, from every $X_e \in \Tan_eG$,
we can obtain a unique vector field $X \in\vf_L(G)$
using the map
\begin{array}{ccccc}
X&\colon&G&\to&\Tan G
\\
& &g&\mapsto&(g,\Tan L_gX_e)
\end{array} \ .
which is a vector field since, for every $f \in \Cinfty(G)$,
the map $X(f) \colon G \to \Real$
is smooth and, hence, so is $X$.
Moreover, $X$ is left-invariant, as you can easily see
taking into account its construction from $X_e$.
The map giving $X$ from $X_e$ is denoted $\rho_2 \colon \Tan_eG \to\vf_L(G)$.
\\ \qed  \end{proof}

\begin{itemize}
\item
Thus, it is usual to identify ${\bf g}$ with $\Tan_eG$, and then
the Lie bracket between two vectors of $\Tan_eG$ is defined as
[X_e,X_e'] :=[\rho_2X_e,\rho_2X_e'](e) \ .
Then, if $\{ v_i \}$ is a basis of $\Tan_eG$,
there exist $c_{ij}^k\in\Real$ such that
[v_i,v_j] = c_{ij}^kv_k \ ,
which are called {\sl \textbf{structure constants}}
of the group $G$ (related to the chosen basis of ${\bf g}$).
It can be proved that
$c_{ij}^k = 0$ (for all $i,j,k$)
if, and only if, $G$ is locally isomorphic to
$\Real^n$, that is, $G$ is an {\sl Abelian Lie group}.
\item
As it is obvious, we can take $\vf_R(G)$ instead of $\vf_L(G)$
in order to define ${\bf g}$.
Then, the identification with $\Tan_eG$
would be established in an analogous way,
although the Lie algebra structure so obtained in $\Tan_eG$
is the opposite one to the above.
Nevertheless, both structures are isomorphic
since the map $-{\rm Id}_{\Tan_eG}$ exchanges them.
\end{itemize}

\begin{definition}
Let ${\bf g}$ be a Lie algebra.
Then, we have the following sequence of ideals:
{\bf g} \supseteq {\bf g'} \supseteq {\bf g''} \supseteq \ldots
\supseteq {\bf g^k} \supseteq \ldots
{\bf g'} := [{\bf g},{\bf g}] \ , \
{\bf g''} := [{\bf g'},{\bf g'}] \ , \
\ldots \ , \
{\bf g^k} := [{\bf g^{k-1}},{\bf g^{k-1}}] \ .
Then, ${\bf g^k}$ is called the
{\rm $k$th-derived algebra} of ${\bf g}$.
\end{definition}

Observe that, if  ${\bf g^k} = 0$, then ${\bf g^{k+1}} = 0$.
Then we define:

\begin{definition}
Let ${\bf g}$ be a Lie algebra.
\begin{enumerate}
\item
${\bf g}$ is \textbf{Abelian} if \
${\bf g'} = 0$.
\item
${\bf g}$ is \textbf{solvable} if \
${\bf g^k} = 0$, for some $k>0$.
\item
Consider now the sequence \
${\bf g} = {\bf g_1} \supseteq {\bf g_2} \supseteq \ldots
\supseteq {\bf g_k} \supseteq \ldots$
{\bf g_1} := {\bf g} \ , \
{\bf g_2} := [{\bf g},{\bf g_1}] \ , \
\ldots \ , \
{\bf g_k} := [{\bf g},{\bf g_{k-1}}]
${\bf g}$ is \textbf{nilpotent} if \
${\bf g_k} = \{ 0 \}$, for some $k>0$.
\item
${\bf g}$ is \textbf{simple} if \
it has no ideals other than $\{ 0 \}$ and ${\bf g}$.
\item
${\bf g}$ is \textbf{semisimple} if \
its largest solvable ideal is $\{ 0 \}$;
that is, if ${\bf g^k}={\bf g}$, for every $k$.
\end{enumerate}
\end{definition}

It is easy to prove that
{\bf g} \ {\rm abelian} \ \Rightarrow
{\bf g} \ {\rm nilpotent} \ \Rightarrow
{\bf g} \ {\rm solvable} \ ,
and that every non-Abelian simple Lie algebra
is semisimple.

\begin{definition}
Let ${\bf g}$ be a Lie algebra.
The \textbf{center} of ${\bf g}$ is the set
\{ X \in {\bf g} \ | \ [X,Y]=0 \ ,  Y \in {\bf g} \} \ ,
which is a commutative ideal of ${\bf g}$.
\end{definition}

\begin{definition}
A $p$-form $\beta \in {\mit\Omega}^p(G)$ is
\textbf{left} (resp. \textbf{right}) \textbf{invariant}
if, for every $g \in G$,
L_g^*\beta = \beta \quad (resp. \ R_g^*\beta = \beta ) \ .
The set of the left-invariant $1$-forms
is the dual vector space of $\vf_L(G) := {\bf g}$
and is denoted by ${\bf g}^*$.
\end{definition}

\begin{prop}
${\bf g}^*$ is isomorphic to $\Tan_e^*G$.
\end{prop}
\begin{proof}
It is obvious from the definition.
\\ \qed  \end{proof}

As a consequence, from a covector  $\beta_e \in \Tan_e^*G$,
we can obtain another one at every point $g \in G$
doing $L_{g^{-1}}^*\beta_e := \beta_g$.
Then, the map
\begin{array}{ccccc}
{\rm B}&\colon&G&\to&\Tan^*G
\\
& &g&\mapsto&(g,\beta_g )
\end{array}
is a left-invariant $1$-form.
Therefore, it is easy to prove the following properties:

\begin{prop}
\label{A30}
\begin{enumerate}
\item
For every $g \in G$, $\beta \in {\bf g}^*$,
and $X,Y \in {\bf g}$, we have that:
\begin{enumerate}
\item
$\d \beta$ is left-invariant.
\item
$\langle X_g,\beta_g\rangle = \langle X_e,\beta_e\rangle$;
that is,
$\langle X,\beta\rangle = ctn.$
\item
$\d \beta (X,Y) = -\langle[X,Y],\beta\rangle$.
\end{enumerate}
\item
If $\{ v_i \}$ is a basis of $\Tan_eG$ and
$\{ \gamma^i \}$ is the dual basis in $\Tan_e^*G$, then
\begin{enumerate}
\item
$\d \gamma^k (v_i,v_j) = -c_{ij}^l<v_l,\gamma^k> = - c_{ij}^k$
\item
The following equations holds
\d \gamma^k = -1/2 c^k_{ij}\gamma^i \wedge \gamma^j
The numbers $c^k_{ij}\in\Real$ are called the
\textbf{Maurer--Cartan structure equations}
of the group $G$ (related to the chosen basis of ${\bf g}^*$).
\end{enumerate}
\end{enumerate}
\end{prop}

\begin{definition}
The \textbf{canonical} or  \textbf{Maurer--Cartan form}
of the Lie group $G$ is the $1$-form with values on ${\bf g}$,
$\omega \in {\mit\Omega}^1(G,{\bf g})$,
defined as follows:
for every $g \in G$ and $X \in\vf(G)$,
\omega_g(X_g) := \Tan L_g^{-1}X_g
\end{definition}

Obviously, $\omega$ is left-invariant.

Observe that, in the definition, $X \in {\cal X}(G)$.
In particular, if $X \in {\bf g}$,
(that is, $X$ is left-invariant) then we have that
$\Tan L_g^{-1}X_g = X_e$.
Then an alternative definition of $\omega$ is the following:

\begin{definition}
The \textbf{Maurer--Cartan form}
of the Lie group $G$ is the only left-invariant $1$-form
with values on ${\bf g}$,
$\omega \in {\mit\Omega}^1(G,{\bf g})$,
such that $\omega(X) = X$, for every $X \in {\bf g}$;
that is, it gives the identity one n ${\bf g}$.
\end{definition}

\begin{definition}
The \textbf{uniparametric subgroups} of $G$
are the integral curves passing through $e$
of the left (resp. right) invariant vector fields,
that is, the maps
\begin{array}{ccccc}
\alpha &\colon&\Real&\to&G
\\
& &t&\mapsto&g_t:=\alpha(t)
\end{array}
such that, if $X \in \vf_L(G)$, then
{\rm (i)} \
$\displaystyle X_{g_{t_0}} = \frac{\d}{\d t} \alpha (t)\Big\vert_{t_0}$ and \
{\rm (ii)}
$\displaystyle X_e = \frac{\d}{\d t} \alpha (t)\Big\vert_0$.
\end{definition}

These uniparametric subgroups are complete and,
there exists a bijective correspondence between
${\bf g}$ and the set of uniparametric subgroups of $G$.
The map which implements this correspondence is defined as follows:

\begin{definition}
The \textbf{exponential map}
is defined as,
\begin{array}{ccccc}
{\rm exp}&\colon&\Tan_eG&\to&G
\\
& &tX_e&\mapsto&g_t
\end{array}
for $t \in \Real$, and such that
${\rm exp}(0)=e$; that is, if $\alpha$ is the uniparametric subgroup
corresponding to $\rho_2X_e$, then \
{\rm (i)} \
${\rm exp}(X_e) := \alpha(1)$ and \
{\rm (ii)}\
${\rm exp}(0) := e$.
\end{definition}

\begin{itemize}
\item
Notice that the exponential map maps
a point of the line $tX_e$ on $\Tan_eG$ into
a point of the integral curve of the vector field
$\rho_2X_e$ on $G$.
\item
Taking into account that $\Real$ is a Lie group
with respect to the sum and that
$\alpha \colon \Real \to G$
is a Lie group homomorphism, we have that
\beann
{\rm exp}(t_1X_e){\rm exp}(t_2X_e)
&=& g_{t_1}g_{t_2} = \alpha(t_1)\alpha(t_2) = \alpha(t_1+t_2)
\\
&=& g_{(t_1+t_2)} = {\rm exp}((t_1+t_2)X_e) \ .
\eeann
\frac{\d}{\d t} {\rm exp}(tX_e)\Big\vert_{t_0} =
\frac{\d}{\d t}\alpha(t)\Big\vert_{t_0} = X_{g_{t_0}} \ ,
and, in particular,
\frac{\d}{\d t} {\rm exp}(tX_e)\Big\vert_{t=0} = X_e \ .
All these properties justify the name given to this map
\footnote{
Another justification is that,
if $V$ is a vector space, and we consider
$G={\rm Aut}(V)$, then
${\rm exp} \colon {\rm End}(V) \to {\rm Aut}(V)$
and, for all $a \in {\rm End}(V)$, we obtain
{\rm exp}(a) =e^a := {\bf 1} + a + a^2/2! + a^3/3! + \ldots + a^n/n! + \ldots
where ${\bf 1}$ is the identity and
$a^n := a \circ \ldots \circ a$ ($n$ times).}.
Observe that the last two equalities are natural
from the definition, since
$\alpha(t) = {\rm exp}(tX_e)$
is the integral curve of $X$.
Hence, the flux of $X$ is
\begin{array}{ccccc}
\tau&\colon&\Real \times G&\to&G
\\
& &(t,g)&\mapsto&g\,{\rm exp}(tX_e) \ ;
\end{array}
that is,
$\tau_t(g) = g\,{\rm exp}(tX_e) = R_{{\rm exp}(tX_e)}g$.
\item
The exponential map allows constructing locally the group $G$
from the algebra ${\bf g}$.
\end{itemize}

\end{appendix}

\bibliographystyle{abbrv}
\small
\addcontentsline{toc}{chapter}{Bibliography}
\label{Chap:Bibliography}
\bibliographystyle{AMS_Mod}

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