M.C. Muñoz-Lecanda and N. Román-Roy,
Geometry of Mechanics.

We study
the geometry underlying mechanics and its application
to describe autonomous and nonautonomous conservative dynamical systems of different types; as well as dissipative dynamical systems.
We use different geometric descriptions to study
the main properties and characteristics of these systems;
such as their Lagrangian, Hamiltonian and unified formalisms,
their symmetries, the variational principles, and others.
The study is done mainly for the regular case,
although some comments and explanations about singular systems are also included.

Symplectic manifolds, cosymplectic manifolds, Hamiltonian systems,  Riemannian manifolds, Newtonian systems, contact manifolds, dissipative systems, Lagrangian formalism, Hamiltonian formalism, symmetries, conservation laws,  actions of Lie groups, fiber bundles, variational principles.

MSC 2020 codes:

Primary: 53B20, 53D05, 53D10, 70H03, 70H05, 70H33.

Secondary: 53B21, 53D20, 53D22, 53Z05, 58A30, 70A05, 70F20, 70F25, 70G10, 70G45, 70G65, 70G75, 70H15, 70H20, 70H25.

§ ACKNOWLEDGMENTS

Our special thanks to Prof. Xavier Gràcia-Sabaté for the many clarifying discussions held on many of the contents of this book,
and to Prof. Xavier Rivas-Guijarro for his careful reading of the entire manuscript and his very helpful comments and suggestions.

We also want to thank all our colleagues from the
Geometry, Dynamics, and Field Theory Network for their enriching collaboration over all these years.

Finally, we would also like to acknowledge the contributions of Dr. J. Andreu Lázaro-Camí in the drafting of Sections <ref>, <ref>, and <ref>.

We acknowledge the financial support from the
Spanish Ministry of Science and Innovation, grants  PID2021-125515NB-C21, and RED2022-134301-T of AEI,
and Ministry of Research and Universities of
the Catalan Government, project 2021 SGR 00603 Geometry of Manifolds and Applications, GEOMVAP.

The authors both dedicate this work to our respective families for their constant support and understanding over the years.

To Arturo Echeverría-Enríquez (in memoriam).

0pt plus 0.1mm

CHAPTER: GLOSSARY OF NOTATION AND TERMINOLOGY

$\Cinfty(M)$     Smooth functions (of class $\Cinfty$) on a manifold $M$.

$\df^k(M)$     Differentiable $k$-forms on a manifold $M$.

$Z^k(M)$     Differentiable closed $k$-forms on a manifold $M$.

$\vf(M)$     Vector fields on a manifold $M$.

${\cal T}^m_k(M)$     Tensor fields of type $(k,m)$ on a manifold $M$.

${\rm Diff}\,(M)$     Diffeomorphisms of the manifold $M$.

$\vf^{{\rm V}(\pi)}(M)$     $\pi$-vertical vector fields on a bundle $\pi\colon M\to N$.

$\vf(M,\pi)$     Vector fields along a map $\pi$ on a manifold $M$.

${\mit\Gamma}(\pi)$     Sections of the projection $\pi$ on a bundle $\pi\colon M\to N$.

$\d$ & Exterior differential. \\
$β(X)≡⟨X|β⟩$ & Natural pairing between a differential $1$-form $β$ and a vector field $X$. \\
$(X)β$ & Contraction of a differential $k$-form $β$ with a vector field $X$. \\
$(X)β$ & Lie derivative of a differential form $β$ by a vector field $X$. \\
$X(f)≡(X)f$ & Action of a vector field $X$ (Lie derivative) on a function $f$. \\
$Q$ , $τ_QQ→Q$ & Tangent bundle of a manifold $Q$ , canonical projection. \\
$^*Q$ , $π_Q^*Q→Q$ & Cotangent bundle of a manifold $Q$ , canonical projection. \\
$(q^i,v^i) ,  (q^i,p_i)$ & Natural coordinates on $Q$ and $^*Q$. \\
$Ω∈^2(M)$ , $(M,Ω)$ & (Pre)symplectic form , (pre)symplectic manifold. \\
$Θ∈^1(M)$ & (Pre)symplectic potential ($Ω=Θ̣$). \\
$♭_ΩM→^*M$ & \\
$♯_Ω=♭_Ω^-1^* M→M$ & Canonical isomorphisms on a symplectic manifold $(M,Ω)$. \\
$(M,Ω,α)$\ , \ $(M,Ω,h)$ & Hamiltonian system on a symplectic manifold $(M,Ω)$. \\
${ f,g}$ & Poisson bracket between the functions $f$ and $g$. \\
$_lh (M)$ , $_H (M)$ & Local/global Hamiltonian vector fields on a symplectic manifold $(M,Ω)$. \\
$X_h∈(M)$ &
Hamiltonian vector field of $f∈(M)$
on a symplectic manifold $(M,Ω)$. \\
$G$ , $g$ & Lie group, Lie algebra. \\
$ξ_X∈(M)$ & Fundamental vector field of $ΦG→Diff(M)$ associated to $X∈g$. \\
$j^*g→(M)$ & Comomentum map associated to an action $ΦG→Diff(M)$.  \\
$JM→g^*$ & Momentum map associated to an action $ΦG→Diff(M)$.  \\
$Δ∈(M)$ & Liouville (dilation) vector field on a vector bundle $πM→N$. \\
$ϕM→N$ & Canonical lift of a map $ϕM→N$ to the tangent bundles.  \\
$Z^V∈(Q)$ & Vertical lift of $Z∈(Q)$ to the tangent bundle $Q$.  \\
$Z^C∈(Q)$ & Complete canonical lift of $Z∈(Q)$ to the tangent bundle $Q$.  \\
$γ→Q$ & Canonical lift of a curve $γ→Q$ to the tangent bundle $Q$.  \\
$^*ϕ^*N→^*M$ & Canonical lift of a map $ϕM→N$ to the cotangent bundles.  \\
& \\
\hline
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{|c|c|}
\hline
& \\
$Z^*\in\vf(\Tan^*Q)$ & Canonical lift of $Z\in\vf(Q)$ to the cotangent bundle $\Tan^*Q$.  \\
${\cal FL}\colon\Tan Q\to\Tan^*Q$ &
Legendre map defined by $\Lag\in\Cinfty(\Tan Q)$. \\
$J\in{\cal T}^1_1(\Tan Q)$ & Canonical endomorphism on $\Tan Q$. \\
$\Lag\in\Cinfty(\Tan Q)$ , $(\Tan Q,\Lag)$ & Lagrangian function, Lagrangian system. \\
%$(\Tan Q,\Lag)$ & Lagrangian system. \\
$\Theta_\Lag\in\df^1(\Tan Q)$ & Lagrangian (Cartan) $1$-form associated with $\Lag\in\Cinfty(\Tan Q)$. \\
$\Omega_\Lag=-\d\Theta_\Lag\in\df^2(\Tan Q)$ &
Lagrangian (Cartan) $2$-form associated with $\Lag\in\Cinfty(\Tan Q)$. \\
$X_\Lag\in\vf(\Tan Q)$ \ , \ $\Gamma_\Lag\in\vf(\Tan Q)$ &
Lagrangian/Euler-Lagrange vector field
of a Lagrangian system $(\Tan Q,\Lag)$. \\
$\Phi\colon Q\times\Real^n\longrightarrow\Tan^*Q$ &
Complete solution to the Hamilton-Jacobi equation. \\
${\cal S}\colon Q\times\Real^n\longrightarrow\Real$ &
Generating function of a complete solution to the H-J eq. \\
${\rm h}\in\Cinfty(\Tan^*Q)$ , $(\Tan^*Q,\Omega,{\rm h})$ & Canonical Hamiltonian function , Canonical Hamiltonian system. \\
${\cal W}=\Tan Q\times_Q\Tan^*Q$ &
Unified (Pontryagin) bundle. \\
$\varrho_1\colon{\cal W}\to\Tan Q$ & \\
$\varrho_2\colon{\cal W}\to\Tan^*Q$ & \\
$\varrho_0\colon{\cal W}\to Q$ &
Canonical projections of the unified  bundle ${\cal W}$. \\
$\Theta_{\cal W}\in\df^1({\cal W})$ &
Canonical $1$-form on the unified bundle ${\cal W}$. \\
$\Omega_{\cal W}=-\d\Theta_{\cal W}\in\df^2({\cal W})$ &
Canonical $2$-form on the unified bundle ${\cal W}$. \\
${\cal C}\colon{\cal W}\to\Real$ &
Coupling function on the unified bundle ${\cal W}$. \\
${\cal H}\colon{\cal W}\to\Real$ &
Hamiltonian function on the unified bundle ${\cal W}$. \\
$X_{\cal H}\in\vf({\cal W})$ &
Dynamical vector field of the dynamical system $({\cal W},\Omega_{\cal W},{\cal H})$. \\
$\jmath_0\colon\mathcal{W}_0\hookrightarrow\mathcal{W}$ &
Compatibility submanifold of the dynamical system $({\cal W},\Omega_{\cal W},{\cal H})$. \\
$(\eta,\omega)$ , $(M,\eta,\omega)$ & (Pre)cosymplectic structure , (pre)cosymplectic manifold. \\
$R\in\vf(M)$ & Reeb vector field on a cosymplectic manifold $(M,\eta,\omega)$. \\
$\flat_{(\eta,\omega)}\colon\Tan M\to\Tan^*M$ & \\
$\sharp_{(\eta,\omega)}=\flat_{(\eta,\omega)}^{-1}\colon\Tan^* M\to\Tan M$ & Canonical isomorphisms on a cosymplectic manifold $(M,\eta,\omega)$. \\
$X_f\in\vf(M)$ & Hamiltonian vector field on a cosymplectic manifold $(M,\eta,\omega)$. \\
${\rm grad}\,f\in\vf(M)$ & Gradient vector field on a cosymplectic manifold $(M,\eta,\omega)$. \\
${\cal E}_f\in\vf(M)$ & Evolution vector field on a cosymplectic manifold $(M,\eta,\omega)$. \\
$\rho\colon\Real\times Q\to\Real$ & Canonical projection. \\
$\tau_1\colon\Real \times\Tan Q \rightarrow \Real$ & \\
$\tau_2\colon\Real \times \Tan Q \rightarrow \Tan Q$ & \\
$\tau_0\colon\Real \times \Tan Q \rightarrow Q$ & \\
$\tau_{1,0}\colon\Real \times \Tan Q \rightarrow\Real\times Q$ &
Canonical projections of the bundle $\Real\times\Tan Q$. \\
$\pi_1\colon\Real \times\Tan^*Q \rightarrow \Real$ & \\
$\pi_2\colon\Real \times \Tan^*Q \rightarrow\Tan^*Q$ & \\
$\pi_0\colon\Real \times \Tan^*Q \rightarrow Q$ & \\
$\pi_{1,0}\colon\Real \times \Tan^*Q \rightarrow\Real\times Q$ &
Canonical projections of the bundle $\Real\times\Tan^*Q$. \\
$\mbox{\boldmath$\gamma$}\colon\Real \to\Real\times Q$ &
Canonical lift of a curve $\gamma\colon\Real \to  Q$ to $\Real\times Q$.  \\
$\widehat{\bf c} \colon\Real  \to\Real\times \Tan Q$ &
Canonical lift of a curve ${\bf c}\colon\Real \to\Real\times Q$ to $\Real\times\Tan Q$.  \\
${\cal J}\in{\cal T}^1_1(\Real\times\Tan Q)$ & Extension of the canonical endomorphism to $\Real\times\Tan Q$. \\
${\rm F}\Lag\colon\Real\times\Tan Q\to\Real\times\Tan^*Q$ &
Legendre map defined by $\Lag\in\Cinfty(\Real\times\Tan Q)$. \\
$\vartheta_\Lag\in\df^1(\Real\times\Tan Q)$ &
Cartan Lagrangian $1$-form associated with $\Lag\in\Cinfty(\Real\times\Tan Q)$. \\
$\omega_\Lag=-\d\vartheta_\Lag\in\df^1(\Real\times\Tan Q)$ &
Cartan Lagrangian $2$-form associated with $\Lag\in\Cinfty(\Real\times\Tan Q)$. \\
$R_\Lag\in\vf(\Real\times\Tan Q)$ &
Reeb vector field for the cosymplectic manifold
$(\Real\times\Tan Q,\d t,\eta_\Lag)$. \\
$\mbox{\boldmath $\Theta$}_{\cal L}\in\df^1(\Real\times\Tan Q)$ &
Poincar\'e--Cartan $1$-form associated with $\Lag\in\Cinfty(\Real\times\Tan Q)$. \\
$\mbox{\boldmath $\Omega$}_{\cal L}=-\d\mbox{\boldmath $\Theta$}_{\cal L}\in\df^1(\Real\times\Tan Q)$ &
Poincar\'e--Cartan $2$-form associated with $\Lag\in\Cinfty(\Real\times\Tan Q)$. \\
& \\
\hline
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{|c|c|}
\hline
& \\
$\mbox{\boldmath $\Theta$}_{\rm h}\in\df^2(\Real\times\Tan Q$ &
Hamilton--Cartan $1$-form associated with ${\rm h}\in\Cinfty(\Real\times\Tan^*Q)$. \\
$\mbox{\boldmath $\Omega$}_{\rm h}=-\d\mbox{\boldmath $\Theta$}_{\rm h}\in\df^2(\Real\times\Tan^*Q)$ &
Hamilton--Cartan $2$-form associated with ${\rm h}\in\Cinfty(\Real\times\Tan^*Q$. \\
$\nu\colon\Tan(\Real \times Q)\rightarrow \Real\times\Tan Q$ & \\
$\varpi\colon\Tan(\Real \times Q)\rightarrow \Real$ &
Canonical projections of the bundle $\Tan(\Real \times Q)$. \\
$pr_1\colon\Tan^*(\Real \times Q)\rightarrow \Tan^*\Real$ & \\
$pr_2\colon\Tan^*(\Real \times Q)\rightarrow\Tan^*Q$ & \\
$\mu\colon\Tan^*(\Real \times Q)\rightarrow\Real\times\Tan Q$ & \\
$u\colon\Tan^*(\Real \times Q)\rightarrow\Real$ &
Canonical projections of the bundle $\Tan^*(\Real\times Q)$. \\
${\mathbf \Theta}_{{\cal L}_{ext}}\in\df^1(\Tan (\Real\times Q))$ &
Lagrangian $1$-form associated with $\Lag_{ext}\in\Cinfty(\Tan(\Real\times Q))$. \\
${\mathbf \Omega}_{{\cal L}_{ext}}=-\d{\mathbf \Theta}_{{\cal L}_{ext}}\in\df^2(\Tan (\Real\times Q))$ &
Lagrangian $2$-form associated with $\Lag_{ext}\in\Cinfty(\Tan(\Real\times Q))$. \\
${\mathbf \Theta}_{ext}\in\df^1(\Tan^*(\Real\times Q))$ &
Canonical $1$-form on $\Tan^*(Q\times\Real)$. \\
$\mbox{\boldmath $\Omega$}_{ext}=-\d\mbox{\boldmath $\Theta$}_{ext}\in\df^2(\Tan^*(\Real\times Q))$  &
Canonical $2$-form on $\Tan^*(Q\times\Real)$. \\
${\cal M}=\Real\times\Tan Q\times_Q\Tan^*Q$ &
Extended unified (Pontryagin) bundle. \\
$\kappa_1\colon{\cal M}\to\Real\times\Tan Q$ & \\
$\kappa_2\colon{\cal M}\to\Real\times\Tan^*Q$ & \\
$\kappa_0\colon{\cal M}\to\Real\times Q$ &
Canonical projections of the extended unified  bundle ${\cal M}$. \\
$\Theta_{\cal M}\in\df^1({\cal M})$ &
Canonical $1$-form on the extended unified bundle ${\cal M}$. \\
$\Omega_{\cal M}=-\d\Theta_{\cal M}\in\df^2({\cal M})$ &
Canonical $2$-form on the extended unified bundle ${\cal M}$. \\
${\rm C}\colon{\cal M}\to\Real$ &
Coupling function on the unified bundle ${\cal M}$. \\
${\rm H}\colon{\cal M}\to\Real$ &
Hamiltonian function on the unified bundle ${\cal M}$. \\
$X_{\rm H}\in\vf({\cal M})$ &
Dynamical vector field of the dynamical system $({\cal M},\Omega_{\cal M},{\rm H})$. \\
$\jmath_0\colon\mathcal{M}_0\hookrightarrow\mathcal{M}$ &
Compatibility submanifold of the system $({\cal M},\Omega_{\cal M},{\rm H})$. \\
${\tt g}\in{\cal T}_2(Q)$ & Riemannian metric on a manifold $Q$. \\
${\rm F}\in\vf (Q)$ & Vector force field. \\
$\omega\in\df^1 (Q)$ & Work differential form. \\
$(Q,{\tt g})$ & Riemannian manifold. \\
$(Q,{\tt g},{\rm F})$ , $(Q,{\tt g},\omega)$ & Newtonian system. \\
$\nabla$ & Connection on a manifold (Levi-Civita). \\
$\nabla_X$ & Covariant derivative by a vector field $X\in\vf(M)$. \\
$\nabla_t$ & Covariant derivative along a curve. \\
${\mathtt T}\in{\mathcal T}_2^1(M)$ & Torsion tensor of a connection. \\
${\mathit R}\in{\mathcal T}_3^1(M)$ & Curvature tensor of a connection. \\
${\rm Rie}\in{\mathcal T}_4(M)$ & Riemann's curvature tensor of a Riemannian manifold. \\
${\rm Ric}\in{\mathcal T}_2(M)$ & Ricci tensor of a Riemannian manifold. \\
${\rm S}\in\Cinfty(M)$ & Scalar curvature of a Riemannian manifold. \\
$\bmeta$\ ,\ $(M,\bmeta)$ & (Pre)contact form, (pre)contact manifold. \\
$\mathcal{D}^{\rm C}$\ ,\ $\mathcal{D}^{\rm R}$ &
Contact and Reeb distributions on a contact manifold $(M,\bmeta)$.\\
$\Reeb\in\vf(M)$ & Reeb vector field on a contact manifold $(M,\bmeta)$. \\
$\flat_{\bmeta}\colon\Tan M\to\Tan^*M$ & \\
$\sharp_{\bmeta}=\flat_{\bmeta}^{-1}\colon\Tan^* M\to\Tan M$ & Canonical isomorphisms on a contact manifold $(M,\bmeta)$. \\
$\X_f\in\vf(M)$ & Hamiltonian vector field on a contact manifold $(M,\bmeta)$. \\
${\bf grad}\,f\in\vf(M)$ & Gradient vector field on a contact manifold $(M,\bmeta)$. \\
$\evo_f\in\vf(M)$ & Evolution vector field on a contact manifold $(M,\bmeta)$. \\
$\tau_1\colon\Tan Q\times\Real\rightarrow\Tan Q$ &
Canonical projection of the bundle $\Tan Q\times\Real$. \\
$\mathfrak{F}\Lag\colon\Tan Q\times\Real\to\Tan^*Q\times\Real$ &
Legendre map defined by $\Lag\in\Cinfty(\Tan Q\times\Real)$. \\
& \\
\hline
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{|c|c|}
\hline
& \\
$\theta_\Lag\in\df^1(\Tan Q\times\Real)$ &
Cartan Lagrangian $1$-form associated with $\Lag\in\Cinfty(\Tan Q\times\Real)$. \\
$\omega_\Lag=-\d\theta_\Lag\in\df^2(\Tan Q\times\Real)$ &
Cartan Lagrangian $2$-form associated with $\Lag\in\Cinfty(\Tan Q\times\Real)$. \\
$\bmeta_\Lag=\d s-\theta_\Lag\in\df^1(\Tan Q\times\Real)$ &
(Pre)contact Lagrangian form associated with $\Lag\in\Cinfty(\Tan Q\times\Real)$. \\
$\Reeb_\Lag\in\vf(\Tan Q\times\Real)$ &
Reeb vector field for the (pre)contact manifold
$(\Tan Q\times\Real,\bmeta_\Lag)$. \\
$\mathfrak{M}=\Tan Q\times_Q\Tan^*Q\times\Real$ &
Extended precontact unified (or Pontryagin) bundle. \\
$\bmeta_{\mathfrak{M}}\in\df^1(\mathfrak{M})$ &
Canonical precontact form on the precontact unified bundle $\mathfrak{M}$.
\\
$\mathfrak{C}\colon\mathfrak{M}\to\Real$ &
Coupling function on the unified bundle $\mathfrak{M}$. \\
${\bf H}\in\Cinfty(\mathfrak{M})$ &
Hamiltonian function on the precontact unified bundle $\mathfrak{M}$.\\
$\X_{\bf H}\in\vf(\mathfrak{M})$ &
Dynamical vector field of the dynamical system $(\mathfrak{M},\Omega_\mathfrak{M},{\bf H})$. \\
$\jmath_0\colon\mathfrak{M}_0\hookrightarrow\mathfrak{M}$ &
Compatibility submanifold of the dynamical system $(\mathfrak{M},\Omega_\mathfrak{M}
,{\bf H})$. \\
& \\
\hline
\end{tabular}
\end{center}

\chapter{Introduction}

\pagenumbering{arabic}
\setcounter{page}{1}

The study of mechanics experienced a substantial advance during
the 18th and 19th centuries with the emergence of what was called
{\sl Analytical} or {\sl Rational Mechanics}.
Many relevant mathematicians contributed to its development,
such as {\it Euler, Hamilton, Jacobi, Lagrange, Legendre, Poisson}, among others.
Their techniques are essentially based in the application of variational methods
to obtain the equations of motion of dynamical systems, and their subsequent development;
and are widely collected in a lot of classical treaties, such as
[156, 160, 174, 215, 223, 226, 349]
or, in a more modern perspective, [8, 142, 212, 235, 307, 321, 352],
among many others.

Although the dynamical equations of Analytical Mechanics had been well-stablished
previously by {\it Lagrange} and others,
the connection between the variational methods and Mechanics was done
by the {\sl minimum action principles}
[29, 30, 154, 167, 223, 235]. Thus, the so-called
{\sl Hamilton principle} leads to the Euler--Lagrange equations
and  the so-called {\sl Lagrangian formalism} of mechanics.
Introducing the {\sl Legendre transformation} and the ``momentum coordinates''
the dynamical equations become the {\sl Hamilton equations},
originating in this way the {\sl canonical} or {\sl Hamiltonian formalism}
of Mechanics. As in the Lagrangian case, these Hamiltonian equations
can also be derived from the so-called {\sl Hamilton--Jacobi variational principle}.

In the second half of the 20th century,
an extensive group of mathematicians and physicists used
the techniques of differential geometry to formalize analytical mechanics
and physics in general,
and studied the properties of dynamical systems intrinsically,
giving rise to what is known today as {\sl geometric mechanics}.
Among the most relevant books and contributions, we can cite
[1, 7, 65, 85, 172, 200, 207, 208, 217, 236, 238, 239, 244, 254, 280, 292, 307, 316, 346],
in addition to many others.

In the geometric formulation,
differentiable manifolds that model the phase spaces of dynamical systems
are endowed with different kinds of geometric structures,
which are used to study the properties of these systems.
Thus, for {\sl conservative systems}
the {\sl symplectic manifolds}, are used to describe
{\sl autonomous} or {\sl time-independent} dynamical systems
[1, 7, 85, 132, 236, 238, 239, 254, 265, 316, 334, 335, 346];
meanwhile for the {\sl nonautonomous} or {\sl time-dependent} case
also other types of manifolds can be considered ,
such as {\sl jet bundles} and {\sl contact} and {\sl cosymplectic manifolds}
[1, 49, 83, 99, 102, 148, 244, 264, 306].
In particular, for variational dynamical systems;
that is, those of {\sl Lagrangian type},
their phase spaces have the geometric structure of a
tangent bundle [98, 101, 159, 172, 217, 325, 326],
or a cotangent bundle in the dual canonical Hamiltonian formalism
[1, 7, 85, 238, 325, 327].
These formalisms can also be described in a single unified description
[311, 312].
Furthermore, for those dynamical systems which are called of {\sl mechanical type}, their configuration spaces are also endowed
with a (semi)Riemannian metric  [1, 7, 42, 85, 173, 280]
which is used to construct their Lagrangian functions in a canonical way.
Finally, symplectic mechanics has also been  used to describe  {\sl nonholonomic systems};
that is, systems subjected to constraints depending on positions and velocities,
[7, 93, 120, 125, 160, 237, 271, 332];
and also {\sl vakonomic systems}
which consists in considering those systems but modifying the variational principle
using only the variations allowed by the constraints
[7, 30, 154, 262].

In addition, there are other more general geometric structures
which are used to model some special types of dynamical systems.
These are {\sl Poisson and Jacobi manifolds} [238, 245] or also
{\sl Lie algebroids} [72, 123, 181, 185, 259, 261, 348].
Finally, {\sl dissipative} or {\sl nonconservative dynamical systems} are modelled using {\sl contact manifolds}
[34, 36, 87, 107, 110, 135, 163, 214, 224].
In all these cases, there are deep relations between the differential equations
describing the dynamics and the underlying geometric structures.

Another very important fact that characterize the behaviour of dynamical systems is
the existence of symmetries.
The interest of studying symmetries arises as a consequence of the well known fact
that their existence is closely related to that of conserved quantities,
and the main result comes from the work of {\it Emmy Noether} [272]
(see [219] for a review of her results).
On turn, the existence of conserved quantities
facilitates the integration of  the dynamical equations,
applying the suitable reduction procedures.
Concerning to this, it is interesting to mention the fundamental {\sl Arnold-Liouville Theorem}
for {\sl integrable systems} [7]
(see also [15, 266]), and also those by
{\it J.E. Marsden} and {\it A. Weinstein} on the problem of the {\sl (symplectic) reduction} by symmetries
[255] (see also [252, 254, 256, 281, 282, 283, 283] and the references therein).

The physical systems for which these geometric approaches were initially developed
had the characteristic of being {\sl regular}
\footnote{
For Lagrangian systems, this means, locally, that the Hessian matrix with respect to the generalized velocities is regular everywhere.}.
The dynamical systems in classical analytical mechanics are of these kind.
Nevertheless, with the advent of Relativity Theory and the
development of field theories in modern physics, in general,
many systems appear for which that property does not hold; that is,
they are the {\sl singular systems\/}.
For autonomous systems, this feature of being regular or not manifests in the fact that
the underlying geometric structure is
{\sl symplectic} or {\sl presymplectic}
[55, 58, 68, 69, 177, 178, 179, 189, 247, 269, 270, 313].

The aim of this book is to do a detailed exposition of
the geometry underlying dynamical systems and how it is applied
to describe the different kinds of them:  conservative autonomous and nonautonomous dynamical systems in general, and Newtonian systems in particular; as well as dissipative systems.
We use different geometric descriptions to study
the main properties and characteristics of these systems;
such as their Lagrangian, Hamiltonian and unified formalisms,
their symmetries, the variational principles, and others.
The study is done mainly for the regular case,
although comments and explanations about singular systems
are also introduced throughout the exposition.

The organization of the book is the following:

The first chapter of the exposition, Chapter \ref{ch2}, is devoted
to present the most general geometric framework for autonomous mechanics;
that is, {\sl symplectic manifolds} and their structures and characteristics
({\sl Poisson brackets, symplectomorphisms,} etc.),
as well as the fundamental notion of {\sl Hamiltonian dynamical system}.
A general description of {\sl symmetries, conserved quantities} and the
{\sl Noether theorem} is also done for this general situation.

In Chapter \ref{sdl}, we continue the exposition about {\sl symplectic mechanics}
studying, in particular, autonomous mechanical systems of variational type;
that is, those which are described by means of time-independent Lagrangian functions.
The phase spaces of these kinds of systems are represented by tangent
and cotangent bundles of manifolds representing
the configuration spaces of the systems; so the canonical geometric structures
of these bundles are introduced first.
Starting from the Lagrangian function and using the canonical structures of the tangent bundle,
we can define a (pre)symplectic form and the {\sl Legendre map}
which allow us to establish the Lagrangian formalism,
the associated canonical Hamiltonian formalism for these systems,
and the equivalence between both formalisms.
We also state the foundations of the {\sl Hamilton--Jacobi theory}
for Hamiltonian systems.
In addition, we also give a description of the so-called {Skinner-Rusk unified formalism},
which combines in a single framework the two previous formalisms.
The study of symmetries, conserved quantities and Noether's theorem is
also performed for Lagrangian systems in both formalisms,
which lead to introduce some new types of symmetries.
As these dynamical systems are of variational type,
a section is devoted to introduce the variational formulation
and derive the dynamical equations from the corresponding variational principle.
Finally, two of the most classic examples in mechanics,
the {\sl harmonic oscillator\/} and the {\sl Kepler problem},
are analyzed.

Chapter \ref{chap:cosym} is devoted to develop one of the most interesting
and generic geometric formulations for describing
{\sl nonautonomous dynamical systems},
both the Lagrangian and Hamiltonian formalisms.
It is based in using {\sl cosymplectic manifolds},
which are presented in the first section of this chapter.
Then, the dynamics and symmetries for these time-dependent systems are studied
and the formalism is used to describe the classical examples introduced in the previous chapter
when external time-dependent forces act on the oscillator
or for variable mass systems in the case of the Kepler problem.
In this chapter we also give a brief presentation
of two other very common formulations of nonautonomous mechanics;
namely, the {\sl contact} and the {\sl extended symplectic formulations},
and showing their equivalence with the cosymplectic picture.

Next, Chapter \ref{sdn}
deals with the study of those dynamical systems whose configuration spaces are
endowed with additional geometric structures, such as a {\sl metric}.
First, we review the foundations of connections in manifolds and Riemannian geometry,
which are the geometric structures needed to develop what is called {\sl Newtonian mechanics}.
In particular, the existence of a
metric allows us to construct what are known as
{\sl mechanical Lagrangians}. We analyze
different types of them; namely, {\sl conservative} and {\sl coupled systems},
systems with {\sl holonomic} and {\sl nonholonomic constraints};
stating their dynamics and variational formulation.
As a particular situation, the case of {\sl (nonautonomous) Newtonian systems} is
displayed along this exposition.

The last chapter, Chapter \ref{chap:contactmech}, is an introduction to the study of {\sl autonomous dissipative systems},
using the {\sl contact geometry}.
After reviewing the foundations on {contact manifolds},
we establish the generic concepts about {\sl contact Hamiltonian systems}.
and develop the Lagrangian, Hamiltonian and unified Lagrangian-Hamiltonian formalisms for these kinds of systems.
Next, we study symmetries and the concept of {\sl dissipated quantities} for contact Hamiltonian and Lagrangian systems;
establishing the so-called ``dissipation theorems'',
and showing how to associate dissipated and conserved quantities to these symmetries.
As in the above chapters, the examples of the damped harmonic oscillator and the Kepler problem with friction
are analyzed in this context.

The exposition ends with an appendix where various contents on other
geometric structures that appear throughout the exposition are collected;
in particular, tangent and cotangent bundles, and Lie groups and Lie algebras.

%Furthermore, a collection of proposal exercises is displayed at the end of each chapter.

This book is intended for readers who have completed at least
the first courses of a degree on mathematics or physics,
and who have a basic training on differential geometry of smooth manifolds and analytical mechanics.

Along the exposition, all the manifolds are supposed to be real, second-countable and $$.
All the maps and the structures are smooth.
The summation criterion for repeated crossed indices is adopted.

\chapter{Symplectic mechanics (I): Autonomous Hamiltonian dynamical systems}
\label{ch2}

The general geometric framework for describing autonomous mechanical systems (with a finite number of degrees of freedom)
uses some types of differentiable manifolds to model the phase spaces of these systems,
they are the {\sl symplectic} and {\sl presymplectic manifolds}.
This formulation is known as {\sl symplectic mechanics}. With these kinds of manifolds as phase spaces,
we have a very general setting to study dynamical systems,
from which othedescriptions such as the Lagrangian and the Hamiltonian formalisms of Lagrangian systems can be analyzed as particular situations
\footnote{
This general formulation allows us to describe also
dynamical systems which are not of Lagrangian type,
for instance, the system of a classical spin particle [316].}.

The symplectic description of (autonomous) Hamiltonian systems has been exposed in many works and books
(see, for instance, [1, 7, 132, 220, 238, 239, 254, 316, 346] and the references quoted therein).

In this chapter, after reviewing the fundamental concepts on symplectic geometry,
we present the symplectic description of the autonomous Hamiltonian systems
and we introduce several types of symmetries and their associated conserved quantities
from a geometric perspective.

\section{Notions on symplectic and presymplectic geometry}
\label{sympgeom}

In this section, we state the fundamental concepts and properties
of symplectic (and presymplectic) manifolds
(see, for instance, [1, 2, 45, 200, 238, 346]).

\subsection{Symplectic and presymplectic vector spaces}

(All the vector spaces are on the field $$).
Let $E$ be a finite dimensional vector space.

\begin{definition}
A \textbf{symplectic inner product}, or a
\textbf{linear symplectic structure}, on $\textbf{E}$ is a non--degenerate skew symmetric bilinear function $\omega$ on $\textbf{E}$.

We say that the pair $(\textbf{E},\omega)$ is a \textbf{symplectic vector space}.
%We suppose that $\omega\neq 0$.
\end{definition}

Non degeneracy means that, if $ω(x,y)=0$, for every $y∈E$, then $𝐱=0$.
Let $𝐞_1,…,𝐞_r$ be a basis of $E$ and $^1,…,^r$ its dual basis. If $ω_ij=ω(𝐞_i,𝐞_j)$, then the expression of $ω$ on these bases is
\omega=\omega_{ij}\,\mbox{\boldmath $\alpha$}^{i}\otimes\mbox{\boldmath $\alpha$}^{j}\ ,
being $(ω)=(ω_ij)$ the matrix of $ω$ in these bases.

The linear mapping $ω^♭:E→E^*$ is defined by
(\omega^{\flat}(\mathbf{x}))(\mathbf{y})=\langle\omega^{\flat}(\mathbf{x}),\mathbf{y}\rangle=\omega(\mathbf{x},\mathbf{y})\ ,
and its matrix relative to these bases is  $(ω_ij)$,
since $ω(𝐞_i)=ω_ij^j$.
By the skew symmetry of the bilinear mapping $ω$, we have that the matrix $(ω_ij)$ is skew symmetric. The non degeneracy gives us that the mapping $ω^♭$ is one--to--one, hence an isomorphism, and $(ω_ij)$ is regular.
As a consequence, the dimension of $E$ is even, that is $r=2n$, because $(ω)=-(ω)^t$ (the transpose matrix),
and $(ω)=(ω)^t=(-1)^r(ω)$.

A basis $𝐚_1,…,𝐚_n, 𝐚_n+1,…,𝐚_2n$ of $E$ is called \textsl{symplectic} if
\begin{matrix}
\omega(\mathbf{a}_{i},\mathbf{a}_{j})=0 \ ;&  i,j=1,\ldots,n \ ; \\
\omega(\mathbf{a}_{i},\mathbf{a}_{j})=0 \ ;&  i,j=n+1,\ldots,2n\\
\omega(\mathbf{a}_{i},\mathbf{a}_{n+j})=\delta_{ij} \ ;& i,j=1,\ldots,n \ .
\end{matrix}\ .

%A basis $\mathbf{a}_{1},\ldots,\mathbf{a}_{n}, \mathbf{b}_{1},\ldots,\mathbf{b}_{n}$ of $\textbf{E}$ is called \textsl{symplectic} if
%\omega(\mathbf{a}_{i},\mathbf{a}_{j})=0,\,\, \omega(\mathbf{b}_{i},\mathbf{b}_{j})=0,\,\, \omega(\mathbf{a}_{i},\mathbf{b}_{j})=\delta_{ij}\,\, ,
%\forall\, i,j=1,\ldots,n\, .

The existence of symplectic basis is given by the following Lemma:
%$\mbox{\boldmath $\alpha$}$
\begin{lem}
Let $({\bf E},\omega)$ be a $2n$ dimensional symplectic vector space.
\begin{enumerate}
\item There exists a symplectic basis $({\bf e}_k)$ ($k=1,\ldots ,2n$) on ${\bf E}$.

\item If $(\mbox{\boldmath $\alpha$}^k)$ is the corresponding dual basis on ${\bf E}^*$,
\(\dst\omega=\sum_{i=1}^n\mbox{\boldmath $\alpha$}^i\wedge\mbox{\boldmath $\alpha$}^{i+n}\) or,
what is the same, the matrix of $\omega$ in this basis is
\left(\begin{matrix} 0_n & I_n \\ -I_n & 0_n \end{matrix}\right)
where $I_n$ denotes the identity matrix of order $n$ and $0_n$ the zero square matrix $n\times n$.
\end{enumerate}
\label{lemaaux}
\end{lem}
\begin{proof}
By induction on the dimension of ${\bf E}$. Let $\mathbf{e}_{1},\mathbf{e}_{n+1}\in{\bf E}$ with  $\omega(\mathbf{e}_{1},\mathbf{e}_{n+1})\neq 0$. We can choose two such vectors unless $\omega=0$. Dividing $\mathbf{e}_{1}$ by a scalar, we have that $\omega(\mathbf{e}_{1},\mathbf{e}_{n+1})=1$. Then, on the plane ${\bf P}_{1}$ spanned by $\mathbf{e}_{1},\mathbf{e}_{n+1}$ the matrix of $\omega$ is
\left(\begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix}\right)\, .
Let ${\bf E}_{1}$ the $\omega$--orthogonal complement of ${\bf P}_{1}$ in ${\bf E}$, that is:
{\bf E}_{1}=\{{\bf a}\in{\bf E}\,|\, \omega({\bf a},{\bf a}_{1})=0, \forall{\bf a}_{1}\in{\bf P}_{1}\}\ .
Observe that ${\bf E}_{1}\cap{\bf P}_{1}=\{0\}$ and ${\bf E}_{1}+{\bf P}_{1}={\bf E}$. If $\mathbf{a}\in{\bf E}$, we have
\mathbf{a}-\omega({\bf a},{\bf e}_{n+1}){\bf e}_{1}+\omega({\bf a},{\bf e}_{1}){\bf e}_{n+1}\in{\bf E}_{1}\ .
Then, ${\bf E}_{1}\oplus{\bf P}_{1}={\bf E}$ and we can repeat the process on ${\bf E}_{1}$, with dimension less that $2n$, and we have finished the proof.

The result on the expression of $\omega$ as \(\dst\omega=\sum_{i=1}^n\mbox{\boldmath $\alpha$}^i\wedge\mbox{\boldmath $\alpha$}^{i+n}\) is immediate.
\\ \qed  \end{proof}

\begin{remark}{\rm
\begin{itemize}
\item
The non degeneracy condition is equivalent to say that $\omega^{n}=\omega\overbrace{\wedge\ldots\wedge}^{n}\omega\equiv\wedge^n\omega$ is a volume form on ${\bf E}$.
\item
If we take out the non degeneracy condition, then the mapping $\omega^{\flat}$ is not one--to--one. In this case, we need to add the basis $\mathbf{u}_{1},\ldots,\mathbf{u}_{h}$ of $\ker \omega^{\flat}$ to the given basis for the decomposition of ${\bf E}$. Then the corresponding matrix is,
\left(\begin{matrix} 0_n & I_n&0_{n\times h} \\ -I_n & 0_n& 0_{n\times h}\\0_{h\times n} & 0_{h\times n}&0_{h}
\end{matrix}\right)
being $2n+h$ the dimension of ${\bf E}$ and denoting by $0_{n\times h}$ the zero matrix of $n$ rows and $h$ columns. We say that $2n$ is the rank of $\omega$.
\end{itemize}

\subsection{Subspaces of a symplectic linear space}

Associated to a linear symplectic structure we have a notion of orthogonality in parallel to the ideas related to the Euclidean scalar product, but the results are very different. In the next lines, we develop these ideas that are necessary to understand our later description of Lagrangian and Hamiltonian systems.

Let $(E,ω)$ be a $2n$-dimensional symplectic vector space and $F⊂E$ a linear subspace. The {\sl \textbf{$\omega$--orthogonal complement}} of $F$ is defined as
{\bf F}^\bot=\{{\bf u}\in{\bf E}\ |\ \omega({\bf u},{\bf u}')=0\, , \, \mathrm{for\,\, every}\,\, {\bf u}'\in{\bf F}\}.
Observe that, in general, $F∩F^≠0$ (for example, if $F=span {u}$, then $F⊂F^$).

\begin{definition}
Let ${\bf F}$ be a subspace of a linear symplectic $2n$--dimensional vector space $({\bf E},\omega)$.
%We say:
\begin{enumerate}
\item $ {\bf F}$ is \textbf{isotropic} if ${\bf F}\subset  {\bf F}^\bot$; that is, $\omega({\bf u},{\bf u}')=0$, for every ${\bf u},{\bf u}'\in{\bf F}$.
\item $ {\bf F}$ is \textbf{coisotropic} if ${\bf F}\supset  {\bf F}^\bot$; that is, if
$\omega({\bf u},{\bf u}')=0$, for every ${\bf u}'\in{\bf F}$, then ${\bf u}\in{\bf F}$.
\item $ {\bf F}$ is \textbf{Lagrangian} if ${\bf F}$ is isotropic and there exists an isotropic complement ${\bf F}'$; that is, an isotropic subspace ${\bf F}'\subset{\bf E}$ such that ${\bf F}\oplus{\bf F}'={\bf E}$
\item $ {\bf F}$ is \textbf{symplectic} if $\omega$ restricted to ${\bf F}$ is non degenerates; that is, $({\bf F},\omega_{\bf F}=\left.\omega\right|_{\bf F})$ is a symplectic vector space.
\end{enumerate}
\end{definition}

These notions have several associated properties that we resume in the following two propositions, with some proofs and indications.

\begin{prop}
\label{propos1}
Let $({\bf E},\omega)$ be a symplectic vector space and
${\bf F},{\bf G}$ subspaces of ${\bf E}$.
\begin{enumerate}
\item If ${\bf F}\subset{\bf G}$, then ${\bf F}^\bot\supset{\bf G}^\bot$\,.
\item  ${\bf F}^\bot\cap{\bf G}^\bot=({\bf F}+{\bf G})^\bot$.
\item $\dim{\bf E}=\dim{\bf F}+\dim {\bf F}^\bot$.
\item ${\bf F}= {\bf F}^{\bot\bot}$.
\item $({\bf F}\cap{\bf G})^\bot={\bf F}^\bot+{\bf G}^\bot$.
\end{enumerate}
\end{prop}
\begin{proof}
\begin{description}
\item[{\rm 3}.]
Consider the linear map $\omega^{\flat}\colon\textbf{E}\to \textbf{E}^{*}$.
Observe that, if ${\bf u}\in{\bf F}$, then $\omega^{\flat}({\bf u})\in\textbf{E}^{*}$
annihilates the subspace ${\bf F}^\bot$. Hence, the restricted linear map
$\omega_{\bf F}^{\flat}\colon\textbf{F}\to \textbf{E}^*$ induces another one
$\hat{\omega}_{\bf F}^{\flat}\colon\textbf{F}\to\left(\textbf{E}/{\bf F}^\bot\right)^{*}$.
This last map is injective, but not necessarily onto. Thus, we have
\dim{\bf F}\leq\dim\left(\textbf{E}/{\bf F}^\bot\right)^{*}=\dim\left(\textbf{E}/{\bf F}^\bot\right)=\dim\textbf{E}-\dim{\bf F} \ .
Conversely, consider the map
${\bf F}\stackrel{\omega^{\flat}}{\longrightarrow}\textbf{E}^{*}\stackrel{j}{\longrightarrow}{\bf F}^{*}$,
where $j$ is the restriction from $\textbf{E}$ to ${\bf F}$.
Let  $\bar{\omega}_{\bf F}^{\flat}=\omega^{\flat}\circ j$ and observe that
$\ker\bar{\omega}_{\bf F}^{\flat}={\bf F}^\bot$. Hence,
\dim{\bf F}=\dim{\bf F}^{*}\geq\dim(\mathrm{ img}\,\bar{\omega}_{\bf F}^{\flat})=
\dim\textbf{E}-\dim{\bf F}^{*}=\dim\textbf{E}-\dim{\bf F} \ .
From both expressions, we obtain the result we wanted.
\item[{\rm 4}.]
It is clear that ${\bf F}\subset{\bf F}^{\bot\bot}$.
If we apply the previous result to ${\bf F}$ and to ${\bf F}^\bot$ we have
\dim{\bf E}=\dim{\bf F}+\dim {\bf F}^\bot=\dim{\bf F}^\bot+\dim {\bf F}^{\bot\bot} \ ;
that is, $\dim{\bf F}=\dim {\bf F}^{\bot\bot}$, hence ${\bf F}={\bf F}^{\bot\bot}$ as we wanted.
\item[{\rm 5}.]
From 2. and 4. we have
({\bf F}\cap{\bf G})^\bot\stackrel{4.}{=}({\bf F}^{\bot\bot}\cap{\bf G}^{\bot\bot})^\bot\stackrel{2.}{=}
({\bf F}^\bot\cap{\bf G}^\bot)^{\bot\bot}\stackrel{4.}{=}{\bf F}^\bot\cap{\bf G}^\bot  \ .
\end{description}
\qed  \end{proof}

And the next proposition gives alternative definitions for a subspace to be Lagrangian.

\begin{prop}
Let $({\bf E},\omega)$ be a symplectic vector space $({\bf E},\omega)$ and ${\bf F}$ a subspace of ${\bf E}$. The following statements are equivalent:
\begin{enumerate}
\item ${\bf F}$ is Lagrangian.
\item ${\bf F}={\bf F}^\bot$.
\item ${\bf F}$ is isotropic and $\dim{\bf F} =\frac{1}{2} \dim{\bf E}$.
\end{enumerate}
\end{prop}
\begin{proof}
(1. $\Longrightarrow$  2).
As ${\bf F}$ is  Lagrangian it is isotropic, hence ${\bf F}\subset{\bf F}^\bot$, and there exists another isotropic subspace ${\bf F}'$ with ${\bf F}\oplus{\bf F}'={\bf E}$.

Suppose now that ${\bf u}\in{\bf F}^\bot$ and put ${\bf u}={\bf a}+{\bf b}$ with ${\bf a}\in{\bf F}$, ${\bf b}\in{\bf F}'$. We prove that ${\bf b}=0$. In fact, ${\bf b}\in{\bf F}'^{\bot}$ since ${\bf F}'$ is isotropic. We also have that ${\bf b}={\bf u}-{\bf a}\in{\bf F}^\bot$ since  ${\bf u},{\bf a}\in{\bf F}$ and ${\bf F}$ is isotropic. Then
{\bf b}\in{\bf F}'^{\bot}\cap{\bf F}^\bot=({\bf F}'+{\bf F})^\bot={\bf E}^\bot=\{0\}\ ,
because $\omega$ is non--degenerate. Then we have that ${\bf u}={\bf a}\in{\bf F}$ and
${\bf F}^\bot\subset{\bf F}$.
%Thus from 1. we obtain 2.

(2. $\Longrightarrow$  3).
The hypothesis and the item 3 of the above proposition give the result.

(3. $\Longrightarrow$  2).
From 3 and the above proposition we obtain that $\dim{\bf F}=\dim{\bf F}^\bot$ and being ${\bf F}$ isotropic, we have that ${\bf F}={\bf F}^\bot$; that is,
the statement 2.

(2. $\Longrightarrow$  1).
We have ${\bf F}={\bf F}^\bot$, hence ${\bf F}$ is isotropic. We need to construct a subspace ${\bf F}'$.
Let ${\bf a}_{1}\notin{\bf F}$ and ${\bf F}_{1}=\mathrm{span}\{{\bf a}_{1}\}$. Then
${\bf F}\cap{\bf F}_{1}=\{0\}$, hence
${\bf F}^\bot+{\bf F}_{1}^\bot={\bf F}\oplus{\bf F}_{1}^\bot={\bf E}$, that is ${\bf F}\oplus{\bf F}_{1}^\bot={\bf E}$ by the above proposition.
Now let ${\bf a}_{2}\notin{\bf F}+{\bf F}_{1}$, ${\bf a}_{2}\in{\bf F}_{1}^\bot$. We have two alternatives:
\begin{enumerate}
\item This vector ${\bf a}_{2}$ does not exist. In this case ${\bf F}_{1}^\bot\subset{\bf F}+{\bf F}_{1}$, hence ${\bf F}+{\bf F}_{1}={\bf E}$ and
{\bf F}_{1}^\bot=(\mathrm{span}\{{\bf a}_{1}\})^\bot\supset\mathrm{span}\{{\bf a}_{1}\}={\bf F}_{1}\ ,
then ${\bf F}_{1}$ is isotropic and we can take ${\bf F}'={\bf F}_{1}$.
\item
There exists such ${\bf a}_{2}$. Let
${\bf F}_{2}={\bf F}_{1}+\mathrm{span}\{{\bf a}_{2}\}$, following the same procedure as above, we have that
${\bf F}\cap{\bf F}_{2}=\{0\}$, hence
${\bf F}^\bot+{\bf F}_{2}^\bot={\bf F}\oplus{\bf F}_{2}^\bot={\bf E}$, that is ${\bf F}\oplus{\bf F}_{2}^\bot={\bf E}$. But
{\bf F}_{2}^\bot=(\mathrm{span}\{{\bf a}_{1},{\bf a}_{2}\})^\bot
\supset\mathrm{span}\{{\bf a}_{1},{\bf a}_{2}\}={\bf F}_{2} \ ,
hence ${\bf F}_{2}$ is isotropic and we can take ${\bf F}'={\bf F}_{2}$.
\end{enumerate}
Inductively we can continue and, as $\dim{\bf E}$ is finite, we arrive to some ${\bf F}_{k}$ such that ${\bf F}'={\bf F}_{k}$ and we have finished the proof.
\\ \qed  \end{proof}

This last proposition says that a Lagrangian subspace is a \textsl{maximal isotropic subspace}.

\subsection{Symplectic and presymplectic manifolds}
\protect\label{vs}

\begin{definition}
\begin{description}
\item[{\rm (a)}]
Let $M$ be a differentiable manifold.
A \textbf{symplectic form}
\footnote{
The word {\sl  symplectic} comes from the Greek word
``$\sigma\iota\mu\pi\lambda\epsilon\kappa\tau\iota\kappa o$''  which means ``what unites''.
It was introduced by H. Weyl, who substituted
the Latin root of the term ``complex'',
to refer to a structure of the group
${\bf Sp}(n,\Complex )$.
Although the structure of the symplectic manifolds
had been implicitly considered before,
it is not until the decade of 1950 when the symplectic geometry appears
as a differentiated branch of Differential Geometry, being
A. Lichnerowicz the first in introducing the term
{\sl  symplectic manifold}.}
in $M$ is a differential $2$-form
$\Omega \in {\mit\Omega}^2(M)$ such that:
\begin{enumerate}
\item
It is closed: $\d\, \Omega = 0$, (we write $\Omega \in Z^2(M)$).
\item
It is non-degenerated at every point of $M$; that is, $\Omega_{{\rm p}}$ is a linear symplectic structure in $\Tan_{\rm p}M$ for every ${\rm p}\in M$.
\end{enumerate}
If the form $\Omega$ is closed but degenerated it is said to be a
\textbf{presymplectic form}, and if $\Omega$ is nondegenerated but not closed
it is an \textbf{almost-symplectic form}.
\item[{\rm (b)}]
A \textbf{symplectic} (resp. \textbf{presymplectic}) \textbf{manifold}
is a pair $(M,\Omega )$ where $M$ is a differentiable manifold
and $\Omega$ is a symplectic (resp. presymplectic) form.

If the symplectic (resp. presymplectic)  form is exact;
that is, there exists
$\Theta \in {\mit\Omega}^1(M)$ such that $\d\Theta = \Omega$,
then $\Omega$ is an \textbf{exact symplectic} (resp. \textbf{exact presymplectic})
\textbf{form}, and $(M,\Omega )$ is an
\textbf{exact symplectic} (resp. \textbf{exact presymplectic}) \textbf{manifold}.
The form $\Theta$ is called a \textbf{symplectic} (resp. \textbf{presymplectic}) \textbf{potential}.
\end{description}
\label{vsm}
\end{definition}

\begin{remark}{\rm
\begin{itemize}
\item
As a consequence of the second condition of the definition, a non-degenerated differential $2$-form
can only be defined in manifolds of even dimensi\'on,
thus we write $\dim M = 2n$.
\item
As, since {\sl Poincar\'e's Lemma}, every closed form is locally exact,
if $\Omega$ is a symplectic or a presymplectic form, for every point
${\rm p}\in M$, there is an open neighbourhood
$U \subset M$, ${\rm p}\in U$, and $\vartheta \in {\mit\Omega}^1(U)$ such that
Every $1$-form $\vartheta$ satisfying this condition is called a
{\sl local symplectic ({\rm or} presymplectic) potential}.

Observe that if $\vartheta \in {\mit\Omega}^1(U)$
and $\vartheta ' \in {\mit\Omega}^1(U')$ are two different symplectic
(or presymplectic) potentials, then $\vartheta ' = \vartheta + \d f$
in $U\cap U'$, for some $f \in \Cinfty(U\cap U')$.
\item
In the case of a presymplectic form, the dimension of $\ker \omega_{\rm p}^{\flat}$, for ${\rm p}\in M$, may depend on the chosen point ${\rm p}\in M$. Usually, we demand that this dimension does not depend on the point and say that the presymplectic form is {\sl regular}.
\end{itemize}

The following theorem describes the local structure of the symplectic manifolds [103].

\begin{teor}
{\rm (Darboux)} \
Let $(M,\Omega )$ be a $2n$-dimensional symplectic manifold.
For every point ${\rm p}\in M$ there exists an open neighbourhood
$U \subset M$, ${\rm p}\in U$,
which is the domain of a local chart $(U;x^i,y_i)_{i=1 \ldots n}$,
such that $\Omega$ has the expression
\Omega \mid_U = \d x^i \wedge \d y_i \ .
These local charts are called {\sl \textbf{symplectic charts}}
and their coordinates are the {\sl \textbf{canonical coordinates}} or
{\sl \textbf{Darboux coordinates}} of the symplectic manifold in this chart.
\end{teor}
\begin{proof}
(This proof is taken from [1]. For other different proofs, see for example
[40, 101, 137]).
The proof is organized in several parts:
\begin{enumerate}
\item As this is a local result, we can suppose that $M$ is $\Real^{2n}$ and ${\rm p}={\bf 0}$.
\item Let $\Omega$ be a symplectic form in $\Real^{2n}$ and $\Omega_{0}=\Omega({\rm p})$ the constant symplectic form in $\Real^{2n}$ equal to  $\Omega$ at ${\rm p}$.

It is enough to prove that there exists a neighbourhood $U$ of ${\rm p}$, and a diffeomorphism $\phi:U\to U$ such that $\phi^{*}\Omega_{0}=\Omega$ on $U$. This is true since, by the above Lemma, we can choose a global coordinate system in $\Real^{2n}$ such that the symplectic form $\Omega_{0}$ take the expression we need.

\item Consider the 2--form $\omega_{t}\equiv\Omega+t(\Omega_{0}-\Omega)$ in $\Real^{2n}$ for $t\in[0,1]$. We have that
\begin{enumerate}
\item $\omega_{0}=\Omega$, $\omega_{1}=\Omega_{0}$, $\d\,\omega_{t}=0$, for every $t\in[0,1]$.
\item $\omega_{t}({\rm p})=\Omega({\rm p})+t(\Omega_{0}({\rm p})-\Omega({\rm p}))=\Omega({\rm p})=\Omega_{0}({\rm p})$ is non--degenerated.
\item As the interval $[0,1]$ is compact, there exists an open set $U_{0}\subset U$  with ${\rm p}\in U_{0}$ such that $\left.\omega_{t}\right|_{U_{0}}$ is non--degenerate for every $t\in[0,1]$.
\item Being $\Omega_{0}-\Omega$ closed,  we can suppose that $U_{0}$ is a ball with center in ${\rm p}$ such that there exists $\alpha\in\Omega^{1}(U_{0})$ with $\Omega_{0}-\Omega=\d\,\alpha$, by Poincar\'e Lemma.
\end{enumerate}
\item Let $X_{t}\in\vf(U_{0})$ be defined by $\inn (X_t)\omega_t=-\alpha$. This time-dependent vector field is well-defined because $\omega_t$ is non--degenerated. Observe that $X_t({\rm p})=0$ since  $\alpha({\rm p})=0$.
\item Let $F_{t,s}$ be the time-dependent flux of $X_{t}$, satisfying $F_{t,s}\circ F_{s,r}=F_{t,r}$ and $F_{t,t}=I$, and we can consider it is defined in $U_{0}$, or reduce it if necessary. Let $F_{t}=F_{t,0}$ the associated diffeomorphic flux with $F_{t+h}=F_{t+h,t}\circ F_{t,0}$. Then, we have:
\begin{eqnarray*}
\frac{\d}{\d\, t}F_{t}^{*}\omega_{t} &=&
\lim_{h\to 0}\frac{F_{t+h}^{*}\omega_{t}  -F_{t}^{*}\omega_{t}}{h}+F_{t}^{*}(\Omega_{0}-\Omega)\\
&=&F_{t}^{*}( \Lie(X_{t})\omega_{t} )+F_{t}^{*}(\Omega_{0}-\Omega)
=F_{t}^{*}(\d\, \inn (X_t)\omega_t +\Omega_{0}-\Omega)
\end{eqnarray*}
that is $F_{t}^{*}\omega_{t}$ is constant, hence $F_{1}^{*}\omega_{1}=F_{0}^{*}\omega_{0}$. Then,
F_{1}^{*}\Omega_{0}=F_{1}^{*}\omega_{1}=F_{0}^{*}\omega_{0}=\Omega\, ,
and $F_{1}$ is the diffeomorphism transforming the constant symplectic form $\Omega_{0}$ into
our given symplectic form
\end{enumerate}
\qed  \end{proof}

\begin{remark}{\rm
For presymplectic manifolds there is a similar result (see, for instance, [101, 137]).

Finally, as a straightforward consequence of the definition, we have:

\begin{prop}
\label{Liouville}
Every symplectic manifold is an {\sl oriented manifold}.
\end{prop}
\begin{proof}
In fact, using the symplectic form we can define the volume form
\(\dst\Omega^n:=\bigwedge^n\Omega \in {\mit\Omega}^{2n}(M)\),
which is called the {\sl \textbf{Liouville volume form}} on $M$.
\\ \qed  \end{proof}

\bigskip
\noindent{\bf Some examples of symplectic manifolds}:
\begin{enumerate}
\item
The cotangent bundle $\Tan^*Q$ of a manifold $Q$ is an example of symplectic manifold which, moreover, is the canonical model of these kinds of manifolds
(see Theorem \ref{ficotvs} in the Appendix \ref{sec:cotbun} for all the details). Here we do a short survey:
We have a canonical 1-form $\Theta \in\Omega^1(\Tan^*Q)$ which, in natural coordinates $(q^i,p_i)$ of $\Tan^*Q$, is
$\Theta=p_i\d q^i$. Then $\Omega=-\d\Theta=\d q^i\wedge\d p_i$ is a symplectic form.
Observe that the natural coordinates of $\Tan^*Q$ are Darboux coordinates for $\Omega$.
\item
$\Real^{2n}$ with the usual Cartesian coordinate system, $(x^{1},\ldots,x^{n},y_{1},\ldots,y_{n})$, and the 2-form $\Omega=\d x^{i}\wedge\d y_{i}$ is a symplectic manifold.
\item
The 2-sphere $S^{2}\subset \Real^{3}$ as a Riemannian submanifold with the induced  area 2-form is a symplectic manifold.
\item
Another relevant example which will be used later is given by the following:

\begin{prop}
\label{productsymp}
Let $(M_1,\Omega_1)$, $(M_2,\Omega_2)$ be symplectic manifolds
with $\dim M_1 = \dim M_2$,
and $\pi_j\colon M_1\times M_2\to M_j$, $j=1,2$ be the natural projections.
Then the product manifold $(M_1\times M_2,\Omega_{12}=\pi_1^*\Omega_1-\pi_2^*\Omega_2)$
is a symplectic manifold.
\end{prop}
\end{enumerate}

\subsection{Submanifolds of a symplectic manifold}

As in the case of linear symplectic structures, in a symplectic manifold there exist some interesting kinds of submanifolds. They are associated to the idea of orthogonality in the tangent space at every point with respect to the induced linear symplectic structure at this point. Lagrangian submanifolds play an important role in the study of dynamics of Hamiltonian systems.

\begin{definition}
Let $(M,\Omega)$ be a symplectic manifold and $j:L\to M$ an immersion.
\begin{enumerate}
\item
$L$ is an \textbf{isotropic immersed submanifold} of $(M,\Omega)$ if
$\Tan_{\rm p}j(\Tan_{\rm p}L)\subset\Tan_{j(\rm p)}M$ is an isotropic subspace of
$(\Tan_{j(\rm p)}M,\Omega_{j(\rm p)})$ as a linear symplectic space, for every ${\rm p}\in L$;
\item
$L$ is a \textbf{coisotropic immersed  submanifold} of $(M,\Omega)$ if
$\Tan_{\rm p}j(\Tan_{\rm p}L)\subset\Tan_{j(\rm p)}M$ is a coisotropic subspace of
$(\Tan_{j(\rm p)}M,\Omega_{j(\rm p)})$ as a linear symplectic space, for every ${\rm p}\in L$;
\item
$L$ is a \textbf{symplectic immersed submanifold} of $(M,\Omega)$ if
$\Tan_{\rm p}j(\Tan_{\rm p}L)\subset\Tan_{j(\rm p)}M$ is a symplectic subspace of
$(\Tan_{j(\rm p)}M,\Omega_{j(\rm p)})$ as a linear symplectic space, for every ${\rm p}\in L$;
\end{enumerate}
This same terminology is used for subbundles of $\Tan M$ over submanifolds of $M$.
\end{definition}

\begin{definition}
Let $(M,\Omega)$ be a symplectic manifold and $L\subset M$ a submanifold. We say that $L$ is a \textbf{Lagrangian submanifold}  if it is isotropic and $\dim L=(1/2)\dim M$.\end{definition}

\begin{remark}{\rm
\bit
\item
A submanifold
$j\colon L\to M$ is isotropic if, and only if, $j^{*}\Omega=0$.
\item
From the above study about linear Lagrangian subspaces, we have that, if $L\subset M$ is Lagrangian, then $\dim L=(1/2)\dim M$ and $(\Tan_{\rm p}L)^{\bot}=\Tan_{\rm p}L$.
\eit

\bigskip
\noindent {\bf Examples of Lagrangian submanifolds}:
\begin{enumerate}
\item We know that $\Real^{2n}$ with elements denoted by $(x,y)\in\Real^{2n}$ and coordinates $(x,y)=(x^{i} ,y_{j})$ has a natural symplectic form given by $\omega= \d x^{i}\wedge\d y_{i}$. Examples of Lagrangian submanifolds are the following:
L_{1}=\{(x,y)|x=0\} \quad , \quad
L_{2}=\{(x,y)|y=0\} \quad , \quad
L_{3}=\{(x,y)|x=y\} \ .
\item Taking the cotangent bundle of a manifold $Q$, we have a symplectic manifold, $(\Tan^*Q,\omega)$. Then Lagrangian submanifolds are the fibers of the bundle, that is $q=\mathrm{constant}$, or the section zero, that is the manifold $Q$ as a submanifold of $\Tan^*Q$.
\item In the symplectic manifold $(\Tan^*Q,\omega)$, let $\alpha:Q\to\Tan^*Q$ be a differential form and $N=\{(q,p)\in\Tan^*Q\ | \ p=\alpha(q)\}$ the graph of $\alpha$. Then $N$ is a Lagrangian submanifold of $\Tan^*Q$ if, and only if, $\alpha$ is a closed form. In fact, we have
\d\alpha=\d(\alpha^{*}\theta)=\alpha^{*}\d\theta=-\alpha^{*}\d\omega\, ,
by the properties defining the canonical forms in $\Tan^*Q$. You can see the corresponding section in the coming chapters for a detailed account of all this.
\item As a particular case, given a function $f:Q\to\Real$, the graph of $\d f$ is a Lagrangian submanifold of $\Tan^*Q$.
\end{enumerate}

\subsection{Canonical isomorphism. Hamiltonian vector fields}
\protect\label{icch}

The fact that a symplectic form is non-degenerated
has important consequences. One of the main ones is the following:
every differential form $Ω∈^k(M)$
defines a linear map
\begin{array}{ccccc}
\flat_\Omega & \colon & \Tan M & \longrightarrow &\wedge^{k-1} \Tan^*M
\\
& & ({\rm p},X_{\rm p}) & \mapsto & ({\rm p},\inn(X_{\rm p})\Omega_{\rm p})
\end{array} \ ,
and its natural extension (which we denote with the same notation)
\begin{array}{ccccc}
\flat_\Omega & \colon & \vf (M) & \longrightarrow & {\mit\Omega}^{k-1}(M)
\\
& & X & \mapsto & \inn(X)\Omega
\end{array} \ .
The inverse of this isomorphism is denoted $♯_Ω:=♭_Ω^-1$.

Given a differentiable manifold $M$
and a form $Ω∈^2(M)$;
it is obvious that $Ω$ is non-degenerated (symplectic) if, and only if,
$♭_Ω$ is an isomorphism between $M$ and $^* M$. Then:

\begin{definition}
If $(M,\Omega)$ is a symplectic manifold, the map
$\flat_\Omega$ is called the \textbf{canonical isomorphism} induced by
\end{definition}

Given a symplectic manifold $(M,Ω)$,
every function $f ∈(M)$ has associated a unique
vector field $X_f ∈(M)$ by means of the map
\sharp_\Omega\circ\d\colon \Cinfty (M)\mapping{\d}
\df^1(M)\mapping{\sharp_\Omega}\vf (M) \ ;
that is, defined as $X_f:=♯_Ω(f̣)$
or, what is equivalent, given implicitly by
\beq
\inn(X_f)\Omega:=\d f \ .
\label{ch}
\eeq

\begin{remark}{\rm
Observe that the map $\sharp_\Omega\circ\d$ is not surjective.
This means that, although the canonical isomorphism allows us to
associate to every vector field $X$ a differential $1$-form
it is not always possible to associate a function since,
in order to do this, the form would be exact necessarily;
but that form is not even closed, in general.
Neither is the map $\sharp_\Omega\circ\d$ injective,
since functions differing in an additive constant have
the same vector field associated by this map.

Bearing in mind this comment, we define:

\begin{definition}
Let $(M,\Omega )$ be a symplectic manifold. A vector field $X \in \vf (M)$ is a
\textbf{ (global) Hamiltonian vector field} if
$\inn(X)\Omega$ is an exact form.
In this case, the function $f \in \Cinfty (M)$
such that $\inn(X)\Omega = \d f$
is called a {\sl \textbf{(global) Hamiltonian function}}
of the vector field $X$.

The set of global Hamiltonian vector fields in $M$ is denoted $\vf_H (M)$.
\end{definition}

Observe that, taking into account the comment before Equation
(\ref{ch}), every function $f∈(M)$
is a Hamiltonian function of a global Hamiltonian vector field $X_f$.

Nevertheless, the requirement in this definition is too restrictive
and,  for the physical interest, it is sufficient to demand that:

\begin{definition}
\label{locHvf}
Let $(M,\Omega )$ be a symplectic manifold. A vector field
$X \in \vf (M)$ is a \textbf{local Hamiltonian vector field} if
$\inn(X)\Omega$ is a closed form.

In this case, for every point ${\rm p} \in M$, Poincar\'e's Lemma assures
the existence of a neighbourhood $U \subset M$, ${\rm p}\in U$,  and a function
$f \in \Cinfty (U)$ such that $\inn(X)\Omega = \d f$, in $U$.
This function is called a \textbf{local Hamiltonian function}
of the vector field $X$, in $U$.

The set of local Hamiltonian vector fields in $M$ is denoted $\vf_{lh} (M)$.
\end{definition}

\begin{remark}{\rm
\bit
\item
Obviously $\vf_H (M) \subset \vf_{lh} (M)$.
Thus, all we state for local Hamiltonian vector fields
holds also for global Hamiltonian vector fields.
\item
The above definitions are also valid for presymplectic manifolds.
The difference is that, in this case, the map $\flat_\Omega$
is not an isomorphism because it is not injective, thus not exhaustive, and, hence,
not every function in the manifold is associated to a Hamiltonian vector field.
\eit

Remember that a curve $cI⊆→M$ is an {\sl integral curve}
of a vector field $X∈(M)$ if $ċ(t)=(X∘c)(t)$, for $t∈I$;
where $ċ(t)$ denotes de {\sl derivative} of $c$ at $t$
(i.e, the {\sl tangent vector} of the curve at $c(t)$).
Let $cI⊆→M$ the canonical lift of $c$
to the tangent bundle $M$; that is, $c(t)=(c(t),ċ(t))$,
for $t∈I$ (see Definition \ref{canlifcurv}).
Then, bearing in mind these definitions and equation  \eqref{Hvecf},
it is immediate to prove that:

\begin{teor}
\label{teo:Hcurv}
A vector field  $X\in\vf(M)$ in a symplectic manifold $(M,\Omega)$, is the (local) Hamiltonian vector field corresponding to the function $f \in \Cinfty (M)$, that is $X=X_{f}$, if, and only if,
the integral curves $c\colon I\subset\Real\to M$ of $X$ are the solutions to the equation
\begin{equation}
\label{Hcurv}
\inn( \widetilde c)(\Omega\circ c)=\d f\circ c \ ;
\eeq
Equation \eqref{Hcurv} is the {\sl \textbf{Hamilton equation}} for the
integral curves of $X$.
\end{teor}

Remember that equation \eqref{Hcurv} is a straightforward consequence of equation \eqref{ch},
of the definitions of integral curve of a vector field, and of the contraction
$\inn(\widetilde c)(\Omega\circ c)$
(see Remark \ref{intcurve}).

\noindent {\bf Local expressions}:
If $(U;x^i,y_i)$ is a symplectic chart, we have that
\begin{eqnarray*}
X_{f}\mid_U &=& A^i\derpar{}{x^i}+B_i\derpar{}{y_i} \ ,
\\
\d f\mid_U &=& \derpar{f}{x^i}\,\d x^i + \derpar{f}{y_i}\,\d y_i \ ,
\end{eqnarray*}
then, for $X_{f}$, the solution to equation (\ref{ch}), we have that
0 = (\inn(X_{f})\Omega - \d f)\mid_U
=\left( -B_i - \derpar{f}{x^i}\right)\d x^i +
\left( A^i - \derpar{f}{y_i}\right)\d y_i \ ;
that is,
\beq
A^i = \derpar{f}{y_i} \quad , \quad B_i= -\derpar{f}{x^i} \ ,
\label{heqs1}
\eeq
and then
\beq
\label{eq:locHvf}
X_{f}\mid_U = \derpar{f}{y_i}\derpar{}{x^i}-\derpar{f}{x^i}\derpar{}{y_i} \ ,
\eeq
therefore, the integral curves $c(t)=(x^i(t),y_i(t))$ of $X_{f}$ are the solutions to the system of
first-order differential equations
\beq
\frac{d y_i}{d t} = -\derpar{f}{x^i}(c(t))
\quad , \quad
\frac{d x^i}{d t} = \derpar{f}{y_i}(c(t)) \ .
\label{heqs2}
\eeq
Equations \eqref{heqs1} and  \eqref{heqs2} are the local expression of equations
\eqref{ch} and \eqref{Hcurv} respectively, and are called the {\sl Hamilton equations}
of the (local) Hamiltonian vector field $X_{f}$ and its integral curves, respectively.

An important result which is later used is:

\begin{lem}
Let $(M,\Omega)$ be a symplectic manifold.
For every point ${\rm p}\in M$, there exist vector fields $X_j\in\vf_{lh}(M)$, $j=1,\ldots ,2n$,
such that $\{ X_j({\rm p})\}$ is a basis of $\Tan_{\rm p}M$
\footnote{
That is, local Hamiltonian vector field expand locally the
tangent bundle of $M$.
\label{auxil}
\end{lem}
\begin{proof}
It is immediate using symplectic charts,
since the local coordinate vector fields
are locally Hamiltonian vector fields associated to the Hamiltonian functions
$y_i$ and $-x^i$, respectively.
\\ \qed  \end{proof}

\subsection{Invariant forms}

The properties of Hamiltonian vector field are closely related with the properties
of the symplectic form.
Originally, this relationship was established by studying the so-called {\sl integral invariants} of mechanics [73, 144, 160].
Next, we explore this relation.
First, we introduce the following concept:

\begin{definition}
Let $M$ be a differentiable manifold and $X \in \vf (M)$.
A form $\beta \in \df^p(M)$ is an \textbf{absolute invariant form}
for $X$ if $\Lie (X)\beta = 0$.
\end{definition}

\begin{remark}{\rm
Reminding the interpretation of the Lie derivative,
to be an absolute invariant form for $X$ means that
$\beta$ is invariant along the integral curves of $X$,
and, if $F_t$ denotes the flux of the vector field $X$,
this is equivalent to demand that $F_t^*\beta = \beta$ for every $t$.

Now we can state the following result, which is usually taken
as an alternative definition of Hamiltonian vector field:

\begin{teor}
Let $(M,\Omega )$ be a symplectic (resp. presymplectic) manifold.
The vector field $X \in \vf (M)$ is a
local Hamiltonian vector field if, and only if,
$\Omega$ is an absolute invariant form for $X$.
\label{teoin}
\end{teor}
\begin{proof}
As $\Omega$ is a closed form, we have that
\Lie (X)\Omega = \inn (X)\d \Omega + \d\inn (X)\Omega =
\d\inn (X)\Omega = 0
\ \Longleftrightarrow \  \inn (X)\Omega \in Z^1(M)
\ \Longleftrightarrow \ X \in \vf_{lh}(M) \ .
\qed  \end{proof}

\begin{remark}{\rm
This result relates the Hamiltonian vector fields with the fact that $\Omega$
is closed, although it is less precise than Definition \ref{locHvf},
since it does not allow us to distinguish the global Hamiltonian vector fields inside the set of the local Hamiltonian ones.

From this theorem we deduce:

\begin{teor} {\rm (Liouville)}:
Let $(M,\Omega )$ be a symplectic manifold
and $\Omega^n$ the Liouville volume form in $M$.
Then $\Lie (X)\Omega^n =0$, for every $X \in \vf_{lh}(M)$.
\end{teor}
\begin{proof}
It is immediate since, as the Lie derivative is a derivation,
\vspace{-3mm}
\Lie (X)\Omega^n =n\,(\Lie (X)\Omega)\wedge
\overbrace{\Omega\wedge\ldots\wedge\Omega}^{(n-1\ times)}=0 \ .
\qed  \end{proof}

\begin{prop}
Let $(M,\Omega )$ be a symplectic (resp. presymplectic) manifold.
The set $\vf_{lh}(M)$ is closed for the Lie bracket of vector fields
and it is a  real Lie algebra.
\label{allie}
\end{prop}
\begin{proof}
We have to prove that $[X,Y]\in\vf_{lh}(M)$, for every $X,Y\in\vf_{lh}(M)$.
Taking into account the relation
\inn([X,Y])\Omega =
\Lie (X)\inn(Y)\Omega - \inn(Y)\Lie (X)\Omega
and Theorem \ref{teoin}, we have that, if \(\dst\inn (Y)\Omega\vert_U=\d f\), on an open set $U$, we have
\inn([X,Y])\Omega =\Lie (X)\inn(Y)\Omega\vert_U=\Lie(X)\d f=\d\Lie(X)f \ ,
that is, in $U$, the local Hamiltonian function for $[X,Y]$ is $\Lie(X)f$.

Linearity, skew symmetry and the Jacobi identity of the Lie bracket
complete the proof.
\\ \qed  \end{proof}

So, given a symplectic manifold
$(M,\Omega )$, we have proven that the symplectic and the Liouville volume forms are invariant
by the local Hamiltonian vector fields; that is,
by the groups of local diffeomorphisms generated by their fluxes.
Actually, there is a more general property stating that some geometrical structures,
such as those defined by a volume or a symplectic
form on a differentiable manifold,
are determined by their automorphism groups
(i.e.; the groups of volume-preserving and symplectic diffeomorphisms), as it was shown by {\it A. Banyaga} [12, 13, 14].

Now we can try to characterize all the elements in
${\mit\Omega}^k(M)$ with the above property.
The answer to this problem was established by
{\it Lee Hwa Chung} [232], who studied the uniqueness
of  the {\sl  integral invariant forms} by local transformations generated
by the flux of  the local Hamiltonian vector fields.
Next, we state the geometrical version of this theorem and prove a partial result of it
\footnote{
The original proof of this theorem is local, using Darboux coordinates.}.

\begin{teor} {\rm (Lee Hwa Chung)}.
Let $(M,\Omega )$ be a symplectic manifold
and $\alpha \in {\mit\Omega}^k(M)$ an absolute invariant form for
every $X \in \vf_{lh}(M)$. Then:
\begin{enumerate}
\item
If $k$ is odd, that is $k=2r-1$ with $r\in\Nat$, then
$\alpha = 0$.
\item
If $k$ is even, that is $k=2r$ with $r\in\Nat$, then
$\displaystyle\alpha = c\,\overbrace{\Omega\wedge\ldots\wedge\Omega}^{(r\ times)}
\equiv c \,\bigwedge^r\Omega$,
where $c \in \Real$.
\end{enumerate}
\end{teor}
\begin{proof}
We prove the statement for the case $k\leq 2$, which is the only we need later
(for the proof of the general  case, see  [241]).

As $\alpha$ is invariant under the action of  every $X\in\vf_{lh}(M)$, we have that
\beq
0=\Lie (X)\alpha =\d\inn (X)\alpha +\inn (X)\d\alpha
\quad \Longleftrightarrow \quad
\d\inn (X)\alpha =-\inn (X)\d\alpha \ .
\label{uno}
\eeq
Consider now $X,X'\in\vf_{lh}(M)$ and ${\rm p}\in M$; there exist
$U\subset M$, ${\rm p}\in U$, and $f,g\in\Cinfty (U)$ such that
$\inn (X)\Omega\vert_U=\d f$ and $\inn (X')\Omega\vert_U=\d g$
(from now on we write $X\vert_U\equiv X_f$ y
$X'\vert_U\equiv X_g$).
The vector field $X_{h}$ given in $U$ as $X_h\vert_U=fX_g+gX_f$, is locally Hamiltonian and, in $U$, its local Hamiltonian function is $h=fg\in\Cinfty (U)$, since:
\inn (X_h)\Omega \vert_U=
\inn (fX_g+gX_f)\Omega =f\inn (X_g)\Omega +g\inn (X_f)\Omega =
f\d g+g\d f\equiv\d h \ .
Thus we have
\inn (X_h)\alpha\vert_U= f\inn (X_g)\alpha +g\inn (X_f)\alpha \ ,
and taking the exterior differential
\d\inn (X_h)\alpha\vert_U=\d f\wedge\inn (X_g)\alpha+f\d\inn (X_g)\alpha+
\d g\wedge\inn (X_f)\alpha+g\d\inn (X_f)\alpha \ .
But, having in mind \eqref{uno},
\d\inn (X_h)\alpha=-\inn(X_fh)\d\alpha\vert_U=
-f\inn (X_g)\d\alpha -g\inn (X_f)\d\alpha =
f\d\inn (X_g)\alpha+g\d\inn (X_f)\alpha \ ,
and comparing the last two equations, we conclude that
\beq
(\d f\wedge\inn (X_g)\alpha+\d g\wedge\inn (X_f)\alpha)\vert_U=0 \ .
\label{dos}
\eeq
Putting now $X_f=X_g$ in this expression,
that is, $f=g$, we obtain that, for every $f\in\Cinfty (U)$,
\d f\wedge\inn (X_f)\alpha\vert_U=0 \ .
Now, we have two options:
\ben
\item
If $k=1$, then $\inn (X_f)\alpha\in\Cinfty (M)$,
and this last equality leads to $\inn (X_f)\alpha =0$,
for every $X_f\in\vf_{lh}(M)$.
Taking into account that, by Lemma \ref{auxil},
local Hamiltonian vector fields span locally $\Tan M$,
we obtain that $\inn (X)\alpha\vert_U=0$,
for every $X\in\vf (U)$, and this implies necessarily that
$\alpha\vert_U=0$ (for every $U$) and then $\alpha =0$.
\item
If $k=2$, we have to conclude:
\bit
\item
either $\inn (X_f)\alpha =0$,
\item
or $\inn (X_f)\alpha\vert_U=\eta_{X_f}\d f$,
where $\eta_{X_f}\in\Cinfty (U)$.
\eit
In the first case, reasoning as in the above item,
we conclude that $\alpha =0$.
In the second case, going to the expression (\ref{dos}),
we obtain
(\d f\wedge\d g\,\eta_{X_g}+\d g\wedge\d f\,\eta_{X_f})\vert_U=0\ ;
that is
(\d f\wedge\d g)(\eta_{X_g}-\eta_{X_f})\vert_U=0 \ ,
for every $f,g\in\Cinfty (U)$. Then it must be
that is, the function $\eta$ does not depend on the
local Hamiltonian vector field.

Therefore, for every $X_f\in\vf_{lh}(M)$, we have that
$\inn (X_f)\alpha\vert_U=\eta\,\d f$, with $\eta\in\Cinfty (M)$, then
\inn (X_f)\alpha\vert_U=\eta\,\d f=\eta\inn (X_f)\Omega=
\inn (X_f)(\eta\Omega ) \ ;
but, taking into account Lemma \ref{auxil},
this equality leads to
\inn (X)(\alpha -\eta\Omega )\vert_U=0 \ ,
for every $X\in\vf (M)$, and we conclude that
$\alpha =\eta\,\Omega$.
Finally, as $\alpha$ is invariant for every local
Hamiltonian vector field, we have that, for every $Y\in\vf_{lh}(M)$,
0=\Lie (Y)\alpha =\Lie (Y)(\eta\Omega )=
(\Lie (Y)\eta)\Omega +\eta\Lie (Y)\Omega=(\Lie (Y)\eta)\Omega \ ,
then $\Lie (Y)\eta =0$ and, by Lemma \ref{auxil}, this result holds for every $Y\in\vf (M)$;
therefore $\eta =c$ (constant) and thus
$\alpha =c\,\Omega$.
\een
\qed  \end{proof}

\begin{remark}{\rm
\begin{itemize}
\item
So that, the absolute invariant forms for every
local Hamiltonian vector field are multiple
of exterior products of the symplectic form and hence
they are of degree even necessarily. The above proof is for 2-forms only.
\item
A similar result can be proved also for presymplectic manifolds [175]
(See also [146] for another interesting generalization of this theorem).
\end{itemize}

\subsection{Poisson brackets}

The symplectic form allows us to introduce
certain well known operations in Analytical Mechanics in a natural way.

\begin{definition}
Let $(M,\Omega )$ be a symplectic manifold.
The \textbf{Lagrange bracket} of two vector fields $X,Y \in \vf (M)$
is the bilinear map
\begin{array}{ccccc}
(\ ,\ ) & \colon & \vf (M) \times \vf (M) & \longrightarrow & \Cinfty (M)
\\
& & X,Y & \mapsto & (X,Y)
\end{array}
defined by
(X,Y) := \Omega (X,Y) := \inn(Y)\inn(X)\Omega \ .
\end{definition}

\begin{remark}{\rm
\begin{itemize}
\item
This bracket is not an internal operation in  $\vf (M)$, as is the {\sl  Lie bracket},
whose result is another vector field.
\item
From the skew symmetry of $\Omega$ we deduce immediately that the Lagrange bracket is also skew symmetric; that is,
$(X,Y) = -(Y,X)$.
\end{itemize}

Taking into account (\ref{ch}), from this concept we obtain:

\begin{definition}
Let $(M,\Omega )$ be a symplectic manifold.
The \textbf{Poisson bracket} of two functions $f,g \in \Cinfty (M)$
is the Lagrange bracket of their associated Hamiltonian vector fields,
that is, the bilinear map
\begin{array}{ccccc}
\{\ ,\ \} & \colon &
\Cinfty (M) \times \Cinfty (M) & \longrightarrow & \Cinfty (M)
\\
& & f,g & \mapsto & \{ f,g \}
\end{array}
defined by
\{ f,g \} := \Omega (X_f,X_g) := \inn(X_g)\inn(X_f)\Omega \ .
\end{definition}

\noindent {\bf Local expressions}:
If $(U;x^i,y_i)$ is a symplectic chart and
X\mid_U = A^i\derpar{}{x^i}+B_i\derpar{}{y_i} \quad ,
\quad
Y\mid_U = C^i\derpar{}{x^i}+D_i\derpar{}{y_i} \ ,
(X,Y)\mid_U = -B_iC^i+A^iD_i \ .
\{ f,g \}\mid_U =
\derpar{f}{x^i}\derpar{g}{y_i} - \derpar{f}{y_i}\derpar{g}{x^i} \ .
In particular, for  the canonical coordinates $x^i,y_i$ we have that
\{ x^i,x^j \} = 0 \ , \
\{ y_i,y_j \} = 0 \ , \
\{ x^i,y_j \} =  \delta^i_j \ .

The main properties of the Poisson bracket are collected in  the following:

\begin{prop}
Let $(M,\Omega )$ be a symplectic manifold and $\{\,,\,\}$ the associate Poisson bracket. Then
\vspace{-3mm}
\begin{enumerate}
\item
$\{ f,g \}=-\{ g,f \}$ ({\sl skew symmetry}).
\item
$\{ f,\{ g,h \}\} + \{ g,\{ h,f \}\} + \{ h,\{ f,g \}\} = 0$ ({\sl Jacobi identity}).
\item
$\{ f,g \}=\Lie (X_g)f =-\Lie (X_f)g$
\item
$X_{\{ f,g \}} = [X_g,X_f]$.
\end{enumerate}
\end{prop}
\begin{proof}
\begin{enumerate}
\item
Immediate from the definition.
\item
It is a consequence of $\Omega$ being closed.
\item
Bearing in mind the Cartan formula for the Lie derivative:
\{ f,g \} = \inn(X_g)\inn(X_f)\Omega = \inn(X_g)\d f = \Lie (X_g)f \ .
In an analogous way, we have that
$\{ f,g \} = -\Lie (X_f)g$.
\item
The statement is equivalent to $\inn([X_g,X_f])\Omega = \d \{ f,g \}$,
and remembering Proposition \ref{allie} we obtain
\beq
\inn([X_g,X_f])\Omega =\Lie (X_g)\inn(X_f)\Omega = \Lie(X_g)\d f=
\d\Lie(X_g)f=\d \{ f,g \} \ .
\label{expre}
\eeq
\end{enumerate}
\qed  \end{proof}

\begin{remark}{\rm
\begin{itemize}
\item
The first two properties establish that
$\Cinfty (M)$  with the Poisson bracket is a real Lie algebra.
\item
The third property allows us to give a geometric interpretation
of the Poisson bracket between two functions: it measures the variation of one of them along
the integral curves of the Hamiltonian vector field associated to the other.
\item
The fourth property tells us that the map  $\Cinfty (M)\to\vf(M)$ given by $f\mapsto X_{f}$, is a Lie algebra
(anti)--homomorphism between $(\Cinfty (M),\{\,\, ,\,\})$ and $(\vf (M),[\, \, ,\,])$.
\end{itemize}

Using again the canonical isomorphism,
and taking into account (\ref{expre}),
we can establish the following generalization:

\begin{definition}
Let $(M,\Omega )$ be a symplectic manifold.
The {\sl Poisson bracket} of two $1$-forms
$\alpha ,\beta \in {\mit\Omega}^1 (M)$
is the bilinear map
\begin{array}{ccccc}
\{\ ,\ \} & \colon &
{\mit\Omega}^1 (M)\times{\mit\Omega}^1(M)&\longrightarrow&\df^1 (M)
\\
& & \alpha ,\beta & \mapsto & \{ \alpha ,\beta \}
\end{array} \ ,
defined by
\{ \alpha ,\beta \} := \inn([X_\alpha,X_\beta ])\Omega \ ,
where $X_\alpha = \sharp_\Omega (\alpha )$
and $X_\beta = \sharp_\Omega (\beta )$.
\end{definition}

It is evident that:

\begin{prop}
Let $(M,\Omega )$ be a symplectic manifold. Then,
for every $f,g \in  \Cinfty (M)$,
\d \{ f,g \} = -\{ \d f,\d g \} \ .
\end{prop}

The properties of the Poisson bracket of $1$-forms
are obviously analogous to those of the Poisson bracket of functions.

\subsection{Canonical transformations and symplectomorphisms}
\protect\label{tcs}

In the previous section we have seen how the properties
of the symplectic structure allow us to introduce the concept of
local Hamiltonian vector field and how the
integral curves of these fields are obtained
as solutions to the Hamilton equations.
We will see also that this kind of vector fields is suitable
to describe dynamical systems.
This means that there is a deep relation between the dynamics
of physical systems and the geometric properties of
their phase spaces.

In this way, from a dynamical perspective,
it is reasonable to suppose that the more relevant transformations
among dynamical systems are those preserving the dynamical equations
which, in our case, means geometrically
that they transform Hamiltonian vector fields into Hamiltonian vector fields.
Consequently, we define:

\begin{definition}
Let $(M_1,\Omega_1)$ and $(M_2,\Omega_2)$ be
symplectic manifolds and $\Phi:M_{1}\to M_{2}$ a diffeomorphism.
%such that
%$\dim M_1=\dim M_2$, and a diffeomorphism
%$\Phi \in {\rm Diff}\,  (M_1,M_2)$.
We say that
$\Phi$ is a \textbf{canonical transformation}
%between these symplectic manifolds
if it maps local Hamiltonian vector fields into
local Hamiltonian vector fields biunivocally; that is,
$\Phi_*(\vf_{lh} (M_1)) = \vf_{lh}(M_2)$.
\end{definition}

%Observe that this implies that $\Phi_*(\vf_{h} (M_1)) = \vf_{h}(M_2)$.

Concerning the geometrical aspects,
the more interesting transformations between symplectic manifolds
are the following:

\begin{definition}
Let $(M_1,\Omega_1)$ and $(M_2,\Omega_2)$ be
symplectic manifolds and $\Phi:M_{1}\to M_{2}$ a diffeomorphism.
We say that $\Phi$ is a \textbf{symplectomorphism}
(or also a \textbf{symplectic  transformation\/})
if it  preserves their symplectic structures;
that is, $\Phi^*\Omega_2 = \Omega_1$.
\end{definition}

As Hamiltonian vector fields are defined using the symplectic form,
we can expect that there is some relation between both kinds of transformations.
In fact, Lee Hwa Chung's Theorem allows us to prove that
both concepts are essentially the same:

\begin{teor}
Let $(M_1,\Omega_1)$ and $(M_2,\Omega_2)$ be
symplectic manifolds
and $\Phi \in {\rm Diff}\,(M_1,M_2)$.
The necessary and sufficient condition for $\Phi$
to be a canonical transformation is that
$\Phi^*\Omega_2 = c\Omega_1$, with $c \in {\bf R}$.
\label{teosim}
\end{teor}
\begin{proof}  Suppose that $\Phi$ is a canonical transformation. For every
$X_1 \in \vf_{lh} (M_1)$, we have that
$X_{2}=\Phi_*X_1\in \vf_{lh}(M_2)$ and,
as we know from Theorem \ref{teoin},
$\Lie (X_2)\Omega_2 = 0$, hence
0 = \Phi^*(\Lie (X_2)\Omega_2) =
\Lie (\Phi_*^{-1}X_2)(\Phi^*\Omega_2) =
\Lie (X_1)(\Phi^*\Omega_2) \ .
But this means that $\Phi^*\Omega_2$ is invariant by any element of $\vf_{lh} (M_1)$ and, from Lee Hwa Chung's Theorem,
we conclude that
$\Phi^*\Omega_2 = c\Omega_1$, $c \in {\bf R}$ and $\Phi$ is a symplectomorphism.

Conversely, suppose that $\Phi$ is a symplectomorphism, $\Phi^*\Omega_2 = c\Omega_1$. Given $X_1 \in \vf_{lh}(M_1)$ we have that $\Lie (X_1)\Omega_1 = 0$. Then
0 = \Phi^{*^{-1}}(\Lie (X_1)\Omega_1) =
\Lie (\Phi_*X_1)(\Phi^{*^{-1}}\Omega_1) =
\frac{1}{c}\Lie (\Phi_*X_1)\Omega_2 \ ;
hence, by Theorem \ref{teoin}, $\Phi_*X_1 \in \vf_{lh}(M_2)$, that is $\Omega_{2}$ is invariant by every element of  $\vf_{lh}(M_2)$,
and $\Phi$ is a canonical transformation.
\\ \qed  \end{proof}

\begin{remark}{\rm
\begin{itemize}
\item
The constant $c$ that appears in this last theorem
is called  the {\sl valence} of the canonical transformation.
It is usual to consider transformations with $c=1$
(that is, symplectomorphisms),
and they are called {\sl univalent {\rm or} restricted canonical transformations}.
Another terminology is also used, calling canonical transformations
to those with valence $c=1$, and then call
{\sl generalized canonical transformations} to the rest [302].
\item
Observe that a diffeomorphism is an {\sl univalent canonical transformation}
if, and only if, it is a {\sl  symplectomorphism}.
\item
All these definitions and properties are also valid for
presymplectic manifolds. In this case we talk about
{\sl presymplectomorphisms}
(the study of this case is done in [24, 58, 59]).
\end{itemize}

Another fundamental result is:

\begin{prop}
Let $(M,\Omega )$ be a symplectic (resp. presymplectic) manifold.
A vector field $X \in\vf(M)$ is a local Hamiltonian vector field
if, and only if, its flux is a group of
local symplectomorphisms (resp. local presymplectomorphisms).
\end{prop}
\begin{proof} Let $F_t$ be the flux of a vector field $X$. We know that $
\Lie (X)\Omega = 0$ is equivalent to $F_t^*\Omega = \Omega$, hence the result follows directly.
\\ \qed  \end{proof}

Finally, it is easy to prove that:

\begin{prop}
The set of canonical transformations  of a symplectic (resp. presymplectic) manifold
$(M,\Omega)$, with the operation of composition, is a group.
\end{prop}

The group of the symplectomorphisms
(resp. presymplectomorphisms)
of a symplectic (resp. presymplectic) manifold is denoted by ${\bf Sp}(M,\Omega )$,
and it has a crucial relevance in the study of {\sl  symmetries} of dynamical systems.

\subsection{Characterization of canonical transformations}

The last theorem has some important corollaries, which give
alternative characterizations for a transformation to be canonical
(or a symplectomorphism).
The most important of them uses the Poisson bracket of functions.
Previously, we have to specify how the Hamiltonian functions associated to
Hamiltonian vector fields are transformed under
these kinds of transformations.

\begin{prop}
Let $(M_1,\Omega_1)$ and $(M_2,\Omega_2)$ be
symplectic manifolds
and $\Phi \in {\rm Diff}\,  (M_1,M_2)$
a canonical transformation of valence $c$.
If $X_1 \in \vf_{lh}(M_1)$, let $X_2 := \Phi_*X_1 \in \vf_{lh}(M_2)$,
and $h_1 \in \Cinfty (U_1)$ and $h_2 \in \Cinfty (U_2)$
local Hamiltonian functions of $X_1$ and $X_2$
on $U_1 \subset M_1$ and $U_2 :=\Phi (U_1) \subset M_2$,
respectively. Then
ch_1 = \Phi^*h_2 + k \quad , \quad k \in {\bf R} \ .
\end{prop}
\begin{proof}
Following the previous theorem, we have that

\begin{eqnarray*}
\d(\Phi^{*^{-1}}h_{1})&=&\Phi^{*^{-1}}\d h_{1} =\Phi^{*^{-1}}(\inn (X_1)\Omega_1)\mid_{U_1} \\&=&
\inn (\Phi_*X_1)(\Phi^{*^{-1}}\Omega_1)\mid_{\Phi (U_1)} =
\frac{1}{c}\inn (X_2)\Omega_2\mid_{U_2} =
\frac{1}{c}\d h_2 =\d\Big( \frac{1}{c}h_2\Big)  \,  ,
\end{eqnarray*}
that is
$\d(ch_{1})=\d(\Phi^{*} h_2)$, and the result follows.
\\ \qed  \end{proof}

%but, by hypothesis,
%$\inn (X_1)\Omega_1 \mid_{U_1} = \d h_1$, then
%\Phi^{*^{-1}}(\inn (X_1)\Omega_1)\mid_{U_1} =
%\Phi^{*^{-1}} \d h_1 = \d (\Phi^{*^{-1}} h_1) \ ,
%therefore, comparing both expressions we lead to the result.
%\qed  \end{proof}

Bearing this in mind, we state:

\begin{teor}
Let $(M_1,\Omega_1)$ and $(M_2,\Omega_2)$ be
symplectic manifolds. A diffeomorphism
$\Phi :M_1\to M_2$ is a canonical transformation of valence $c$
if, and only if
\Phi^*\{ f_2,g_2 \} = \frac{1}{c}\{ \Phi^*f_2,\Phi^*g_2 \} \ .
for every $f_2,g_2 \in \Cinfty (M_2)$,
\end{teor}
\begin{proof}
Let $f_{2},g_2 \in \Cinfty (M_2)$ and
$X_{f_{2}},X_{g_2} \in \vf_{h}(M_2)$ be the corresponding Hamiltonian vector fields.

If $\Phi$ is a canonical transformation, then $\Phi_*^{-1}X_{g_2} \in \vf_{h}(M_1)$
and, by the above proposition we know that
$\inn (\Phi_*^{-1}X_{g_2})\Omega_1 = \d (\frac{1}{c}\Phi^*g_2)$;
that is, $\Phi_*^{-1}X_{g_2} = X_{\frac{1}{c}\Phi^*g_2}$,
and then
\Phi^*\{ f_2,g_2 \} =
\Phi^*(\Lie (X_{g_2})f_2) =
\Lie (\Phi_*^{-1}X_{g_2})\Phi^*f_2
=\Lie (X_{\frac{1}{c}\Phi^*g_2})\Phi^*f_2 =
\frac{1}{c}\{ \Phi^*f_2,\Phi^*g_2 \} \ .

Conversely, if the condici\'on holds, first we have that,
given $f_2,g_2 \in \Cinfty (M_2)$,
\Phi^*\{ f_2,g_2 \} =
\Lie (\Phi_*^{-1}X_{g_2})\Phi^*f_2 \ ,
and, on the other hand,
\frac{1}{c}\{ \Phi^*f_2,\Phi^*g_2 \} =
\Lie (X_{\frac{1}{c}\Phi^*g_2})\Phi^*f_2 \ ;
hence, combining these two last equalities,
\Phi_*^{-1}X_{g_2} = X_{\frac{1}{c}\Phi^*g_2} \in \vf_{lh}(M_1)
\ ; \ \forall X_{g_2} \in \vf_{lh}(M_2) \ .
Therefore, $\Phi$ is a canonical transformation
with valence $c$ as a consequence of Lee Hwa Chung's Theorem.
\\ \qed  \end{proof}

\begin{remark}{\rm
This result states that a transformation is canonical
if, and only if,
the Poisson bracket is invariant, up to a multiplicative constant, by its action.
Really, this is an alternative way to say that
the symplectic structure is invariant by the transformation.

From the local point of view, if $\Phi:M_{1}\to M_{2}$ is a canonical transformation with valence $c=1$, and we have Darboux coordinates
$(x^i,y_i)$ in $M_1$ and $(\tilde  x^i,\tilde  y_i)$ in $M_2$, we have

\hspace{40mm}\(\begin{array}{ccccccc}
\{\Phi^*\tilde  x^i,\Phi^*\tilde  x^j\}&=&\Phi^*\{\tilde  x^i,\tilde  x^j\}  \ , \\
\{\Phi^*\tilde  y_i,\Phi^*\tilde  y_j\}&=&\Phi^*\{\tilde  y_i,\tilde  y_j\}  \ ,\\
\{\Phi^*\tilde  y_i,\Phi^*\tilde  y_j\} &=&\Phi^*\{\tilde  y_i,\tilde  y_j\}   \,,
\end{array}\)

\noindent that is: $(\Phi^*\tilde  x^i,\Phi^*\tilde  y_i)$ is a symplectic coordinate system in $M_{1}$ if, and only if, $(\tilde  x^i,\tilde  y_i)$ is a symplectic coordinate system in $M_{2}$.

\subsection{Generating functions of canonical transformations}

Another characterization of canonical transformations
is by means of the so called  {\sl generating functions}.

\begin{prop}
Let $(M_1,\Omega_1)$ and $(M_2,\Omega_2)$ be
symplectic manifolds and $\Phi \in {\rm Diff}\,  (M_1,M_2)$.
Let $U_1 \subset M_1$ and $U_2 :=\Phi (U_1) \subset M_2$, and
$\Theta_i \in {\mit\Omega}^1(U_i)$ such that
$\Omega_i\mid_{U_i} = \d \Theta_i$, ($i=1,2$).
Then $\Phi$ is
a canonical transformation, with valence $c$, if, and only if,
there exists a function $F_1 \in \Cinfty (U_1)$ such that
(\Phi^*\Theta_2 - c\Theta_1 - \d F_1)\mid_{U_1} = 0 \ ,
or, equivalently, that there exists a function $F_2 \in \Cinfty (U_2)$
such that
(\Phi^{*^{-1}}\Theta_1 - \frac{1}{c}\Theta_2 - \d F_2)\mid_{U_2} = 0 \ .
These functions $F_1,F_2$ are called {\sl \textbf{(Poincar\'e) generating functions}}
of the canonical transformation, and the relation between them is
$F_1 = c\Phi^*F_2 + k$, $k \in {\bf R}$.
\end{prop}
\begin{proof}
From Theorem \ref{teosim},
$\Phi$ is a canonical transformation if, and only if,
0 = (\Phi^*\Omega_2 - c\Omega_1)\mid_{U_1} = \d (\Phi^*\Theta_2 - \Theta_1) \ ,
then Poincar\'e's Lemma leads to the result.

The statement concerning $F_2$ is obtained in an analogous way,
and comparing both results we arrive to the relation between these functions.
\\ \qed  \end{proof}

In classical texts of Mechanics (see, for example, [160, 226, 174]) a more general concept
of generating function, which include the above ones, is studied.
In the geometrical context, they are introduced as follows
[1]
(without loss of generality, we will restrict ourselves to the case $c=1$;
that is, symplectomorphisms or univalent canonical transformations).

First, taking into account Proposition \ref{productsymp}, we have that:

\begin{prop}
Let $(M_1,\Omega_1)$, $(M_2,\Omega_2)$ be symplectic manifolds
with $\dim M_1 = \dim M_2$,
$\Phi\colon M_1\to M_2$ a diffeomorphism, and
${\jmath}\colon graph\,\Phi\hookrightarrow M_1\times M_2$ the natural embedding.

The map $\Phi$ is a symplectomorphism if, and only if,
$graph\,\Phi$ is a Lagrangian submanifold of the symplectic manifold
$(M_1\times M_2,\Omega_{12}:=\pi_1^*\Omega_1-\pi_2^*\Omega_2)$.
\end{prop}
\begin{proof}
Remember that {\sl Lagrangian submanifolds} of a symplectic manifold
$({\cal M},\Omega)$ are {\sl maximal isotropic submanifolds}
${\jmath}\colon S\hookrightarrow{\cal M}$ or, equivalently,
such that $S$ verify that ${\rm dim}\,S=\frac{1}{2}{\rm dim}\,{\cal M}$
and ${\jmath}^*\Omega=0$.
Obviously ${\rm dim}(graph\,\Phi)=\frac{1}{2}{\rm dim}\,(M_1\times M_2)$ ,
and as $\Phi^*\Omega_2=\Omega_1$,
the other condition holds trivially.
\\ \qed  \end{proof}

If $\Omega_j=-\d\theta_j$, $j=1,2$,
and $\Theta_j$ are local symplectic potentials for $\Omega_j$;
being $graph\,\Phi$ a Lagrangian submanifold we have
\beq
\eeq
which, given a point in  $graph\,\Phi$, is locally equivalent to
\beq
\jmath^*(\pi_2^*\Theta_2-\pi_1^*\Theta_1)\vert_W=-\d{\cal S} \ .
\label{Wlocal}
\eeq
where ${\cal S}$ is a function defined in
an open neighbourhood of the given point, $W\subset graph\,\Phi$.
This function depends on the choice of $\Theta_1$ and $\Theta_2$.

\begin{definition}
${\cal S}$ is called a
\textbf{Weinstein generating function} of the Lagrangian submanifold $graph\,\Phi$
and hence of the symplectomorphism $\Phi$.
\end{definition}

If  $(U_1; x^i,y_i)$, $(U_2; \tilde x^i,\tilde y_i)$ are Darboux charts such that
$W\subset U_1\times U_2$, local coordinates in $W$  can be chosen in several ways.
This leads to six different possible choices for ${\cal S}$.
Thus, for instance,  if $(W; x^i,\tilde x^i)$ is a chart,
then \eqref{Wlocal} gives the symplectomorphism  explicitly as
\tilde y_i\d\tilde x^i-y_i\d x^i=-\d {\cal S}(x,\tilde x)
\quad \Longleftrightarrow\ \quad
\tilde y_i=-\derpar{{\cal S}}{\tilde x^i}(x,\tilde x)\ , \ y_i=\derpar{{\cal S}}{x^i}(x,\tilde x)\ .
Of course, the {\sl Poincar\'e generating functions} are two particular choices
of {\sl Weinstein generating functions}.

\section{Hamiltonian dynamical systems}

Once the foundations of symplectic geometry are established,
we are ready to use it for describing the autonomous Hamiltonian dynamical systems.
As we will see in the next chapter, this geometric formulation includes, as particular cases,
the Lagrangian and the canonical Hamiltonian formalisms of variational dynamical systems;
that is, those which are described by Lagrangian functions.

\subsection{Hamiltonian systems. Hamilton equations}
\protect\label{shreh}

Some of the authors who developed geometric mechanics chose an axiomatic manner in their exposition
(see, for instance, [316]).
Following their ideas, first, we state the postulates
for the geometric study of autonomous Hamiltonian dynamical systems.

The first postulate concerns the physical states:
\begin{pos}
{\rm (First Postulate of Hamiltonian mechanics\/)}:
The state space, or phase space, of a dynamical system
is a differentiable manifold $M$ endowed with a closed form
$\Omega\in Z^2(M)$ such that:
\bit
\item
If $\Omega$ is nondegenerate; that is {\sl symplectic},
the system is {\sl regular} and the dimension of $M$
is twice the number of degrees of freedom of the system.
In this case, every point of this manifold represents a physical state of the system.
\item
If $\Omega$ is degenerate, that is {\sl presymplectic},
the system is {\sl singular} (and the manifold $M$
is not even-dimensional necessarily).
\eit
\label{ax1}
\end{pos}

The second Postulate refers to the {\sl  observables},
that is,  the {\sl physical magnitudes}:

\begin{pos}
{\rm (Second Postulate of Hamiltonian mechanics\/)}:
The observables or physical magnitudes of a  dynamical  system
are functions of $\Cinfty (M)$.

The result of the measure of an observable
is the value that the function which represents it takes
at a point in the phase space $M$
(that is, in a given state, according to the first postulate).
\label{ax2}
\end{pos}

If, according to the Postulate \ref{ax1},
a (symplectic o presymplectic) manifold $(M,\Omega )$
constitutes the phase space of a dynamical system,
there is a very natural way of introducing the dynamics.
Thus, we state:

\begin{pos}
{\rm (Third Postulate of Hamiltonian mechanics\/)}:
The dynamics of a dynamical  system
is given by a closed $1$-form $\alpha \in Z^1(M)$
which is called the {\sl \textbf{Hamiltonian $1$-form}} of the system
\footnote{
In some cases, the 2-form $\Omega$ can also contain dynamical information
as it happens, for instance, in the Lagrangian formalism of  the Lagrangian systems
(see Chapter \ref{sdl}).}.
\end{pos}

And, finally, the dynamical equations are stated in:

\begin{pos}
{\rm (Fourth Postulate of Hamiltonian mechanics\/)}:
The dynamical trajectories of the system are the integral curves
of a vector field $X_{\alpha} \in \vf(M)$, if it exists, associated with the form $\alpha$
by the map $\flat_\Omega$; that is,
of the vector field solution to the equation
\beq
\label{Hvecf}
\inn(X_{\alpha})\Omega = \alpha \ .
\eeq
Then, the integral curves $c\colon I\subseteq\Real\to M$ of $X_\alpha$ are solutions to the equation
\beq
\label{Hvecf2}
\inn(\widetilde c)(\Omega\circ c)=\alpha\circ c \ .
\eeq
The vector field $X_\alpha$ is called a (local or global) {\sl \textbf{Hamiltonian vector field}}
of the Hamiltonian system, and equations \eqref{Hvecf} and \eqref{Hvecf2} are the {\sl \textbf{Hamilton equation}} for $X_\alpha$
and its integral curves.

These equations can be obtained from a variational principle called
the {\sl  minimal action Principle of Hamilton--Jacobi}
\footnote{
As we will see in Section \ref{hamvariational}.}.
\label{ax3}
\end{pos}

Then, we define:

\begin{definition}
\begin{enumerate}
\item
A \textbf{regular} or \textbf{symplectic Hamiltonian dynamical system}
is a triple $(M,\Omega ,\alpha )$,
where $(M,\Omega )$ is a symplectic manifold
and $\alpha \in Z^1 (M)$ is the Hamiltonian $1$-form of the system.
If $(M,\Omega )$ is a presymplectic manifold, then $(M,\Omega ,\alpha )$
is said to be a \textbf{singular} or \textbf{presymplectic Hamiltonian dynamical system}.
\item
By Poincar\'e's Lemma, for every $m \in M$,
there exists $U \subset M$, with $m\in U$,
and $h \in \Cinfty (U)$, such that $\alpha\mid_U = \d h$,
which is called a \textbf{local Hamiltonian function} of the system,
and the abovementioned triple is said to be a
\textbf{local Hamiltonian system}.

If $\alpha$ is an exact form, then there exist
$h \in \Cinfty (M)$ such that $\alpha = \d h$,
which is called a \textbf{global Hamiltonian function} of the system,
and the triple $(M,\Omega ,h)$ sis said to be a
\textbf{global Hamiltonian system}.
\end{enumerate}
\label{sdhr}
\end{definition}

Given a Hamiltonian dynamical system $(M,\Omega,\alpha)$,
the {\sl \textbf{Hamiltonian problem}} posed by  the system
consists in finding a vector field $X_\alpha \in \vf (M)$
verifying  equations (\ref{Hvecf}).

For the case of regular systems, we have

\begin{prop}
\label{teo-hameqs}
If $(M,\Omega,\alpha)$ is a regular Hamiltonian system,
then there exists a unique vector field $X_\alpha\in\vf(M)$ which is a solution to equation  \eqref{Hvecf}
\end{prop}
\begin{proof}
If the system is regular, $\flat_\Omega$ is the canonical  isomorphism of
the symplectic manifold $(M,\Omega)$ and
the existence and  uniqueness of $X_\alpha$ is assured.
\\ \qed  \end{proof}

\bigskip
\begin{remark}{\rm
If $(M,\Omega,\alpha)$ is a presymplectic Hamiltonian system,
the Hamilton equations are not necessarily compatible everywhere on $M$ and we need to look for the maximal subset of $M$ where there exists solution.
This subset is obtained by a procedure called the {\sl constraint algorithm} which, in the most favourable cases,
leads to find  a {\sl final constraint submanifold} $P_f\hookrightarrow M$, where there are Hamiltonian vector fields $X_\alpha\in\vf(M)$,
tangent to $P_f$, which are
solutions to the Hamilton equations on $P_f$; although
these vector fields solution are not necessarily unique.
Functions vanishing on the final constraint submanifold are called {\sl constraints}.
Singular (presymplectic) Hamiltonian systems and the corresponding constraint algorithms
have been widely studied in the literature. The interested reader can
see, for instance, [55, 58, 143, 179, 189, 202, 209, 247, 269, 313]
and others papers cited therein. Apart from some examples in the problems,
in this book we only work with regular Hamiltonian systems.

\bigskip
\noindent {\bf Local expressions}:
If $(M,\Omega, \alpha)$ is a regular Hamiltonian system, in a symplectic chart $(U;x^i,y_i)$ of $M$ with $\alpha=\d h$,
if $\displaystyle X\vert_U=f^i\derpar{}{x^i}+g_i\derpar{}{y_i}\in\vf (M)$
then, according to \eqref{eq:locHvf}, we have that
X_{\alpha}\mid_U = \derpar{h}{y_i}\derpar{}{x^i}-\derpar{h}{x^i}\derpar{}{y_i} \ ,
and the integral curves $c(t)=(x^i(t),y_i(t))$ of $X_{\alpha}$ are the solution to the Hamilton equations \eqref{Hcurv} whose local expression is
\frac{d x^i}{d t} = \derpar{h}{y_i}(c(t))
\quad , \quad
\frac{d y_i}{d t} = -\derpar{h}{x^i}(c(t)) \ .

\begin{remark}{\rm
\begin{itemize}
\item
In a Hamiltonian dynamical  system,
the physical observable represented by the Hamiltonian function
is associated with the energy of the system.
\item
This presentation of dynamical systems
is called {\sl Hamiltonian formalism of Mechanics}
because the dynamical trajectories
are given by  the integral curves of a Hamiltonian vector field.
\end{itemize}

\section{Symmetries of regular Hamiltonian systems}
\label{secsym}

In this section we introduce the ideas of {\sl constants of motion}, or {\sl conserved quantities}, and the {\sl symmetries} of a dynamical system and the relationship between both notions. Noether's theorem will be proved and actions of Lie group on Hamiltonian systems together with reduction theory will be introduced and developed in detail.

The adequate way of doing this study
is by means of the theory of {\sl actions of Lie groups} on
symplectic and presymplectic manifolds
(see, for instance, [1, 101, 150, 171, 199, 238, 249, 279]).
Within this theory, the associated concept of {\sl momentum map} plays a crucial role
in order to introduce conserved quantities and the subsequent
{\sl Marsden--Weinstein reduction theorem}.
Although this theorem was initially stated for regular autonomous Hamiltonian systems,
the Marsden--Weinstein technique has been applied and generalized to many different situations.
For instance,  for time-dependent regular Hamiltonian systems
(with regular values of the momentum map)
in the framework of cosymplectic manifolds [3] and
for autonomous and nonautonomous Hamiltonian systems
with singular values of the momentum map [6, 134, 309],
also for singular systems in the autonomous (presymplectic)
case [46, 150] and in the nonautonomous case [130, 210].
Further generalizations are reduction on Poisson and Jacobi manifolds [56, 186, 242, 253, 273, 274],
on Lie algebroids [70, 71],
on cotangent bundles of Lie groups [257],
Lagrangian reduction [80, 258], Euler-Poincar\'e reduction [74, 79],
Routh reduction for regular and singular Lagrangians [76, 228],
reduction of nonholonomic systems [23, 32, 50, 246],
reduction of optimal control systems [31, 260, 322, 329],
and in the context of Dirac structures and implicit Hamiltonian systems
[27, 263].
Finally, although symmetries and conservation laws in field theories
have been studied in several geometric frameworks (see, for instance,
[127, 176, 198, 250, 301]
and the references quoted therein),
the problem of reduction by symmetries of classical field theories
has been solved only for particular situations such as Lagrangian and Poisson
reduction [77], Euler-Poincar\'e reduction in principal fiber bundles
[75, 78],
reduction in multisymplectic and poly(co)symplectic manifolds [138, 140, 153, 251],
and for discrete field theories [331].
Of course, this list of references is far from being complete.

To develop our study we follow approximately the historical order:
we begin by introducing the basic notions about conserved quantities
and symmetries, paying particular attention to the case of
{\sl Noether symmetries} and {\sl Noether's theorem}. The primitive ideas were the constants of motion, the linear momentum and the angular momentum, and their relation with symmetries as global transformation of the space of positions of a system, and after that the infinitesimal symmetries and the Noether theorem. The modern approach comes from the action of Lie groups on the space of states of the system and the reduction theory, hence we review this theory, the notions of comomentum and momentum maps and the subsequent {\sl Marsden--Weinstein reduction theorem}.
We consider only the case of regular dynamical systems
(although some results can be generalized to the singular case).

Throughout this section $(M,\Omega,\alpha)$ will be
a regular Hamiltonian dynamical system and $\alpha=\ d h$, where the Hamiltonian function $h$ is locally or globally defined.
Usually we will write $(M,\Omega,h)$.
Then, $X_h\in\vf_{lh}(M)$ denotes the dynamical vector field
solution to the system.

\subsection{Conserved quantities (constants of motion)}
\label{conquan}

Let $(M,\Omega,h)$ a regular Hamiltonian system and $X_h\in\vf_{lh}(M)$ the dynamical vector field.

The dynamical evolution of an observable which is represented
by a function $f \in \Cinfty (M)$,
is the variation of this function along the integral curves
$c(t)=(q^i(t),y_i(t))$ of the vector field $X_h$; that is, is given by
\frac{d (f\circ c)(t)}{d t} = \left(\left(\Lie (X_h)f\right)\circ c\right)(t)=\left(X_h(f)\circ c\right)(t) \ ,
and, bearing in mind the definition of Poisson bracket, it can be written as
\frac{d (f\circ c)}{d t} = \{ f,h \}\circ c \ .
%\noindent {\bf Local expression}:
Or, in a symplectic chart $(U;x^i,y_i)$ of $M$,
%the evolution of an observable $f$ is
\Lie (X_h)f)\mid_U=X_h(f)\mid_U = \derpar{h}{y_i}\derpar{f}{x^i}-\derpar{h}{x^i}\derpar{f}{y_i} \ .

Then, we define:

\begin{definition}
A function $f \in \Cinfty (M)$ is a \textbf{conserved quantity}
or a \textbf{constant of motion} if
\Lie (X_h) f = 0 \ ;
that is, it is invariant by the dynamical vector field.
\end{definition}

To say that $f\in\Cinfty (M)$ is a conserved quantity of the
system $(M,\Omega,h)$ means the following:
if ${\rm p} \in M$ and $X_h$ is the Hamiltonian vector field of the system,
let $c\colon(-\epsilon ,\epsilon )\subset\Real\to M$ be the integral curve of $X_h$
with initial condition $c(0)={\rm p}$.
Then, if $f$ is a conserved quantity, we have that
$f(c(t))=f({\rm p})$, for every $t\in(-\epsilon ,\epsilon )$;
that is, the image of $c$ by $f$ is contained in $S_{\rm p}=\{ {\rm q}\in M \ \mid \ f({\rm q})=f({\rm p})\}$,
the level surface of $f$ passing through ${\rm p}$.

One of the fundamental properties of the autonomous Hamiltonian dynamical systems
is the {\sl  conservation of the Hamiltonian}. Remember that for the physical systems, the Hamiltonian is the energy of the system.
Now, we can state this result in a geometric way:

\begin{prop}
{\rm (Conservation of energy)}:
Let $(M,\Omega,h)$ be a Hamiltonian system.
The (local or global) Hamiltonian function $h$ is a conserved quantity.
\end{prop}
\begin{proof}
It is immediate since, as $\Omega$ is skew symmetric, we have
\Lie (X_h) h = \inn (X_h) \d h = \Omega (X_h,X_h) = 0 \ .
\qed  \end{proof}

\begin{remark}{\rm
We have seen how the fundamental properties
of the symplectic form are essential to describe
geometrically physical systems:
the {\sl  non degeneracy} allows us to assure the existence (and uniqueness)
of the dynamical Hamiltonian vector field, and hence, to determine the dynamical evolution
of the system, and from skew symmetry  and the fact to be closed
we obtain the conservation of the energy.

\subsection{Dynamical symmetries}
\label{dynsym}

Let $(M,\Omega,h)$ be a regular Hamiltonian system and $X_h\in\vf_{h}(M)$ the dynamical vector field.

When talking about symmetries, it is usual to refer to the idea that
``a symmetry of a dynamical system lets invariant the solutions
to the differential equations describing the dynamics of the system''.
In this way we define:

\begin{definition}
A \textbf{dynamical symmetry} of the Hamiltonian dynamical system $(M,\Omega,h)$
is a diffeomorphism $\Phi\colon M \to M$ satisfying that,
\Phi_*X_h=X_h \ .
\label{gsdef}
\end{definition}
\vspace{-10mm}
That is, the dynamical vector field $X_h$ is invariant by $\Phi$. Observe that if $\Phi$ is a dynamical symmetry, then so is $\Phi^{-1}$.

Let $c\colon(-\epsilon ,\epsilon )\subset\Real\to M$ be an integral curve of $X_h$, that is $\dot{c}=X_h\circ c$, then  $\dot{c}=X_h\circ c=\Phi_*X_h\circ c$. If $X_h$ is invariant by $\Phi$ we have:
X_h\circ(\Phi\circ c)=\Phi_*X_h\circ\Phi\circ c=\Tan\Phi\circ X_h\circ\Phi^{-1}\circ\Phi\circ c=\Tan\Phi\circ X_h\circ\circ c=\Tan\Phi\circ\dot{c}=\dot{\overline{\Phi\circ c}}
hence $\Phi$ transforms integral curves of $X_h$ into integral curves of $X_h$. The converse is also true; that is: if $\Phi$ transforms integral curves of $X_h$ into integral curves of $X_h$ then $\Phi$ lets invariant the vector field $X_h$. This is another equivalent version of a dynamical symmetry.

The diffeomorphism $\Phi$ can be considered as
locally generated by a vector field, by means of
the local group of diffeomorphisms generated by its flux.
In this way, the above definition leads to the infinitesimal version of symmetries:

\begin{definition}
\ An \textbf{infinitesimal dynamical symmetry} of a Hamiltonian system
$(M,\Omega,h)$ is a vector field $Y\in\vf(M)$ such that
the local diffeomorphisms  generated by its flux
are dynamical symmetries of the system; that is,
\beq
\Lie(Y)X_h=[Y,X_h]=0 \ .
\label{si}
\eeq
\label{gsdefi}
\end{definition}

Recall that this definition is equivalent to say that the flux of $Y$ is made of dynamical symmetries of $X_h$. Thus, if $\Phi_t$ is the flux of $Y$ and $c$ is an integral curve of $X_h$, the $\Phi_t\circ c$ is also an integral curve of $X_h$.

\begin{remark}{\rm
In the above definition, sometimes the condition (\ref{si}) is relaxed by setting
\beq
[Y,X_h]=gX_h \ , \ g\in\Cinfty(M) \ ,
\label{si2}
\eeq
(and the original condition is recovered taking $g=0$).
This allows us to consider also as symmetries
the reparametrizations of the integral curves of the
dynamical vector field.

The infinitesimal dynamical symmetries of $(M,\Omega,h)$ have a natural structure of real Lie algebra. Clearly they are closed by real linear combinations and we have the following proposition:

\begin{prop}
\label{prop:comsym}
If $Y_1,Y_2\in\vf (M)$ are infinitesimal dynamical symmetries,
then $[Y_1,Y_2]$ is an infinitesimal dynamical symmetry.
\end{prop}
\begin{proof}
Using the Jacobi identity, we have
[[Y_1,Y_2],X_h]= [Y_2,[X_h,Y_1]]+[Y_1,[Y_2,X_h]]=0 \ .
\qed  \end{proof}

A first result relating symmetries with conserved quantities is:

\begin{prop}
\label{prop:consym}
\ben
\item
Let $\Phi\colon M\to M$ be a dynamical symmetry of $(M,\Omega,h)$. If $f\in\Cinfty(M)$
is a constant of motion of the system,
then $\Phi^*f$ is also a conserved quantity.
\item
Let $Y\in\vf(M)$ be an infinitesimal dynamical symmetry of $(M,\Omega,h)$. If $f\in\Cinfty(M)$ is a constant of motion of the system,
then $\Lie(Y)f$ is also a conserved quantity.
\een
\label{generador}
\end{prop}
\begin{proof}
\ben
\item
Directly we have
\Lie(X_h)(\Phi^*f)=\Phi^*(\Lie(\Phi_*X_h)f)=0 \ .
\item
The same proof holds taking the flux $\Phi_t$ of $Y$ as a dynamical symmetry.
\beann
\Lie(X_h)\Lie(Y)f&=&
\inn(X_h)\d\Lie(Y)f=\inn(X_h)\Lie(Y)\d f
\\ &=&\Lie(Y)\inn(X_h)\d f-\inn([Y,X_h])\d f
=\Lie(Y)\Lie(X_h)f=0 \ .
\eeann
\een
\qed  \end{proof}

\subsection{Noether symmetries. Noether Theorem}

In the above study of symmetries of the Hamiltonian system $(M,\Omega,\alpha=\d h)$ we have not used the specific structure of the system. In fact, the only element we have considered is the dynamical vector field $X_h\in\vf(M)$. No mention have been made to the symplectic structure of $M$,
the Hamiltonian function $h$, or the Hamiltonian form $\alpha$. We have studied symmetries of the dynamical vector field independently of its origin.

If we consider the other elements defining the system, we arrive to different types of dynamical symmetries  depending on whether they leave invariant the geometric structure;
that is, the symplectic form or the dynamical elements
(i.e., the Hamiltonian function).
For a more complete analysis of all these kinds of symmetries
and how they generate conserved quantities, see, for instance, [81, 82, 299, 303, 304].

The following kind of symmetries has a special relevance as
generators of constants of motion:

\begin{definition}
A diffeomorphism $\Phi\in{\rm Diff}\, (M)$ is a  \textbf{Noether symmetry}
of the Hamiltonian system $(M,\Omega,h)$ if:
\ben
\item
$\Phi$ is a symplectomorphism in $M$: that is, $\Phi^*\Omega =\Omega$.
\item
$\Phi$ lets the dynamics invariant ; that is, $\Phi^*h =h$.
\een
\label{sime1}
\end{definition}

\begin{remark}{\rm
The second condition can also be expressed in the form
$\Phi^*h =h+c$ (with $c\in\Real$).
This means that it transforms a Hamiltonian function into another
Hamiltonian function of the same dynamical vector field $X_h$.
More generically, if $\alpha=\d h$ is the Hamiltonian $1$-form, this is equivalent to say that $\Phi^*\alpha=\alpha$.

\begin{definition}
A vector field $Y\in\vf (M)$ is an \textbf{infinitesimal Noether symmetry}
of the system if  the local diffeomorphisms  generated by the flux of $Y$
are dynamical symmetries of the system; that is,
\ben
\item
$\Lie (Y)\Omega =0$; that is, $Y\in\vf_{lh}(M)$.
\item
$\Lie (Y)h =0$.
\een
\label{simeinf1}
\end{definition}

\begin{remark}{\rm
In the infinitesimal case, as every infinitesimal Noether symmetry
$Y\in\vf(M)$ is a locally Hamiltonian vector field, by the first item of the definition,
we have that, for every ${\rm p}\in M$, there exists an open set $U_{\rm p}\ni p$
and $f_Y\in\Cinfty (U_{\rm p})$ such that $\inn(Y)\Omega=\d f_Y$, in $U_{\rm p}$. The local Hamiltonian $f$ is unique, up to the sum of constant functions.

A first relevant result is:

\begin{prop}
\ben
\item
Every Noether symmetry of $(M,\Omega,h)$ is a dynamical symmetry.
%If $\Phi\in{\rm Diff}\, (M)$ is a Noether symmetry then it is a
%dynamical symmetry.
\item
Every infinitesimal Noether symmetry of $(M,\Omega,h)$ is an infinitesimal dynamical symmetry.
\een
\label{simdin}
\end{prop}
\begin{proof}
\ben
\item
Let $\Phi\in{\rm Diff}\, (M)$ be a Noether symmetry. If $X_h\in\vf (M)$ is a solution to the dynamical equations, then
$0=\inn(X_h)\Omega-\d h$, and as
$\Phi^*\Omega=\Omega$ and $\Phi^*h=h$, we have
0=\Phi^*(\inn(X_h)\Omega-\d h)=\inn(\Phi_*^{-1}X_h)\Phi^*\Omega-\Phi^*\d h=
\inn(\Phi_*^{-1}X_h)\Omega-\d h=\inn(\Phi_*^{-1}X_h)\Omega-\d h \ ;
but as the system regular, the vector field solution is unique,
then $\Phi_*^{-1}X_h=X_h$, and the result holds.
\item
If $Y\in\vf(M)$ is an infinitesimal Noether symmetry, then
$Y\in\vf_{lh}(M)$ and $\Lie(Y)h~=~0$, hence
\inn([Y,X_h])\Omega=\Lie(Y)\inn(X_h)\Omega-\inn(X_h)
\Lie(Y)\Omega=\Lie(Y)\d h=\d\Lie(Y)h=0 \ ,
but being $\Omega$ nondegenerated, this implies that
$[Y,X_h]=0$, and therefore $Y$ is an infinitesimal dynamical symmetry.
\een
\qed  \end{proof}

Clearly, the set of infinitesimal Noether symmetries is a real vector space. Moreover, it is easy to prove that it is real Lie algebra:

\begin{prop}
If $Y_1,Y_2\in\vf (M)$ are  infinitesimal Noether symmetries,
then $[Y_1,Y_2]$ is also an infinitesimal Noether symmetry.
\end{prop}
\begin{proof}
First we have that
$\Lie([Y_1,Y_2])\Omega=0$, since $[Y_1,Y_2]\in\vf_{lh}(M)$, because $Y_1,Y_2\in\vf_{lh}(M)$.
\beann
\Lie([Y_1,Y_2])h &=& \inn([Y_1,Y_2])\d h=
\Lie(Y_1)\inn(Y_2)\d h-\inn(Y_2)\Lie(Y_1)\d h \\ &=&
\Lie(Y_1)\Lie(Y_2)h-\inn(Y_2)\d\Lie(Y_1) h =0 \ .
\eeann
\qed  \end{proof}

In addition, we get:

\begin{prop}
Let $Y\in\vf(M)$ be an infinitesimal Noether symmetry of $(M,\Omega,h)$.
For every ${\rm p}\in M$, there exists an open set $U_{\rm p}\ni {\rm p}$ such that,
if $\vartheta\in\df^1(U_{\rm p})$ is a symplectic potential of $\Omega$
in $U_{\rm p}$, that is $\d\vartheta=\Omega$, then:
\ben
\item
There exists $\zeta_Y\in\Cinfty(U_{\rm p})$ verifying that
\item
If $f_Y\in\Cinfty(U_{\rm p})$ is a local Hamiltonian function of $Y$, then
in $U_{\rm p}$, we have that,
\beq
\qquad \mbox{\it (up to the sum of a constant function)} \ .
\label{fdos}
\eeq
\een
\label{structure}
\end{prop}
\begin{proof}
\ben
\item
Observe that $\Lie(Y)\vartheta$ is a closed form in $U_{\rm p}$ since
\d(\Lie(Y)\vartheta)=\Lie(Y)\d\vartheta=\Lie(Y)\Omega=0 \ ,
Then, by Poincar\'e' s Lemma,
there exists $\zeta_Y\in\Cinfty(U_{\rm p})$
such that $\Lie(Y)\vartheta=\d\zeta_Y$, in an open subset of $U_{\rm p}$ and, reducing the initial domain if necessary, we can suppose it is in $U_{\rm p}$.
\item
If $\inn(Y)\Omega=\d f_Y$ in $U_{\rm p}$, we obtain that
\d\zeta_Y=\Lie(Y)\vartheta=\d\inn(Y)\vartheta+\inn(Y)\d\vartheta=
\d\inn(Y)\vartheta+\inn(Y)\Omega=\d \{\inn(Y)\vartheta+f_Y\} \ ,
and the result holds.
\een
\qed  \end{proof}

Finally, the fundamental result related with Noether-type symmetries
is the classical {\sl Noether's Theorem}, whose geometric Hamiltonian version is the following:

\begin{teor}
{\rm (Noether)}.
If $Y\in\vf (M)$ is an infinitesimal Noether symmetry of a Hamiltonian system $(M,\Omega,h)$, then
its Hamiltonian function $f_Y$
is a conserved quantity; that is, $$\Lie (X_h)f_Y=0 \ .$$
\label{Nth}
\end{teor}
\begin{proof}
In fact, as $\inn(Y)\Omega=\d f_Y$ we have
\Lie(X_h)f_Y =
\inn(X_h)\d f_Y = \inn(X_h)\inn(Y)\Omega=
-\inn(Y)\inn(X_h)\Omega = -\inn(Y)\d h= -\Lie(Y) h=0 \ .
\qed  \end{proof}

Noethers's Theorem is very relevant, since it gives a way to associate
a constant of motion to every Noether symmetry.
The converse statement of Noether's theorem also holds, and it
allows associating a (Noether) symmetry to every conserved quantity:

\begin{teor} {\rm (Inverse Noether)}:
\label{invNo}
For every conserved quantity $f\in\Cinfty(M)$, its Hamiltonian vector field
$Y_f\in\vf_{lh}(M)$ is an infinitesimal Noether symmetry.
\end{teor}
\begin{proof}
As $Y_f\in\vf_{lh}(M)$, then $\Lie(Y_f)\Omega=0$. Furthermore, as $f$
is a conserved quantity, then $\Lie(X_h)f=0$ and therefore
\Lie(Y_f) h=\inn(Y_f)\d h=\inn(Y_f)\inn(X_h)\Omega=
-\inn(X_h)\inn(Y_f)\Omega=-\inn(X_h)\d f=-\Lie(X_h)f=0 \ .
\qed  \end{proof}

In general, for symmetries which are not of Noether-type,
there is not a so direct way of obtaining constants of motion,
except in some particular cases like in the following:

\begin{teor}
\label{teo:biham}
If $Y\in\vf (M)$ is an infinitesimal dynamical symmetry such that
$\Lie(Y)h\not=0$, then the function
$f=\Lie(Y)h$ is a conserved quantity.
\end{teor}
\begin{proof}
We have that
\Lie(X_h)f=\Lie(X_h)\Lie(Y)h=\Lie([X_h,Y])h+\Lie(Y)\Lie(X_h)h=0 \ .
\qed  \end{proof}

For other kinds of symmetries, the way to obtain conserved quantities is, in general, more complicated (see, for instance, [97, 299] and the references quoted therein).

\subsection{Actions of Lie groups on symplectic manifolds}
\label{cami1}

In this section, we analyze the transformations
(i.e., diffeomorphisms) of a dynamical system
which are generated by Lie groups.
In fact, a family of diffeomorphisms in a differentiable manifold
endowed with the operation of composition
has the structure of group and when the
variations of these transformations are considered, the set of
these variations has a smooth differentiable structure.
In this case, the corresponding group of transformations is a Lie group.
(For more information about actions of Lie groups, see,
for instance, [1, 7, 85, 101, 171, 199, 238, 249, 279, 316, 345].
See also the appendix \ref{Liega} for a review about Lie groups).

\begin{definition}
Let $G$ be a Lie group and $M$ a differentiable manifold.
A \textbf{left-action} of $G$ on $M$ is a map
$\phi \colon G \times M \to M$
verifying the following properties:
\begin{description}
\item[{\rm (i)}] \
$\phi(g_1g_2,{\rm p}) = \phi (g_1,\phi (g_2,{\rm p}))$;
for $g_1,g_2\in G$ and ${\rm p}\in M$.
\item[{\rm (ii)}]
If $e\in G$ denotes the neutral element, then
$\phi (e,{\rm p}) = {\rm p}$.
\end{description}
A \textbf{right action} is defined in an analogous way,
changing the operation law in $G$ by the opposite.
In any case, $M$ is said to be a
\textbf{left $G$-manifold} (resp., a \textbf{right $G$-manifold}).

Since, for every $g \in G$ and $p\in M$, the map \
$\Phi_g \colon {\rm p}\mapsto \phi (g,{\rm p})$ \
is a diffeomorphism in $M$, we have that a left-action of $G$ in $M$
is also a homomorphism
\begin{array}{ccccc}
\Phi&\colon&G&\to&\Diff (M)
\\
& &g&\mapsto&\Phi_g
\end{array} \ .
\end{definition}

From now on, we only consider left-actions.

\begin{definition}
let $G$ be a Lie group, $M$ a differentiable manifold,
$\Phi$ an action of $G$ on $M$, and ${\rm p}\in M$.
\begin{enumerate}
\item
The \textbf{isotropy group} of $p$  (with respect to $\Phi$) is the set (subgroup of $G$)
G_{\rm p}:= \{ g \in G \ | \ \Phi_g({\rm p})={\rm p} \} \ .
\item
The \textbf{orbit} of ${\rm p}$ (with respect to $\Phi$) is the set
{\cal O}_{\rm p}:= \{{\rm p}' \in M \ | \ {\rm p}' = \Phi_g({\rm p}) \ , \ \mbox{\rm for every $g \in G$} \}
\item
The action is \textbf{effective} or \textbf{faithful}
if $\dst\bigcap_{{\rm p} \in M}G_{\rm p}=\{ e \}$
or, what is equivalent, if
$\Phi_g = {\rm Id}_M \ \Longleftrightarrow \ g=e$
(that is, $\Phi$ is injective).
\item
The action is \textbf{free} if the following map, defined for a fixed ${\rm p}\in M$,
\begin{array}{ccc}
\\
g&\mapsto&\Phi_g({\rm p})
\end{array}
is injective, for every ${\rm p}\in M$
(this means that there are no invariant points in $M$ under the action).
\item
The action is  \textbf{transitive} if, for every ${\rm p}_1,{\rm p}_2\in M$,
there exists $g \in G$ such that \ $\Phi_g({\rm p}_1)={\rm p}_2$ \
or, what is the equivalent, if \
${\cal O}_{\rm p}=M$, for every ${\rm p}\in M$
(that is, $\Phi$ has only one orbit).
In this case, $M$ is said to be a \textbf{homogeneous $G$-space}.
\item
The action is \textbf{proper} if the anti-image of every  compact set by the map
$(g,{\rm p}) \mapsto (\Phi_g({\rm p}),{\rm p})$, for every $g\in G$ and ${\rm p}\in M$,
is also a compact set.
\end{enumerate}
\end{definition}

\begin{definition}
Let $M_1,M_2$ be differentiable manifolds,
$G$ a Lie group and
$\Phi_1 \colon G\times M_1 \to M_1$, $\Phi_2 \colon G\times M_2 \to M_2$
actions of $G$ on $M_1$ and $M_2$.
A map $F \colon M_1 \to M_2$ is \textbf{equivariant}
with respect to these actions if, for every $g\in G$, the following diagram commutes
\begin{array}{ccccc}
& M_1 & \mapping{\Phi_{1_g}} & M_1 &
\\
F & \Big\downarrow & & \Big\downarrow & F
\\
& M_2 & \mapping{\Phi_{2_g}} & M_2 &
\end{array}
\end{definition}

Remember that the {\sl Lie algebra} of  a Lie group $G$ is
${\bf g}:=\Tan_eG$ or, equivalently, the set of
{\sl left-invariant vector fields} of $\vf(G)$
(that is, the set of vector fields which are invariant
by the action induced on $\vf(G)$ by the left-action $\Phi$ of $G$ on $G$).
Then, every action of $G$ on $M$ induces a Lie algebra-homomorphism
\begin{array}{ccccc}
\xi&\colon&{\bf g}&\to&\vf(M)
\\
& &X&\mapsto&\xi_X
\end{array}
defined as follows: the uniparametric subgroup of $G$
generated by $X$ determines another
uniparametric subgroup of transformations in $M$
\begin{array}{ccccc}
\sigma_t&\colon&M&\to&M
\\
& & {\rm p}&\mapsto&\Phi_{\alpha(-t)}({\rm p}) := \Phi_{{\rm exp}(-tX_e)}({\rm p})
\end{array} \ ,
where ${\rm exp}\colon\Tan_eG\to G$ denotes the
{\sl exponential map} of the group. Then, the map
\begin{array}{ccccc}
\sigma&\colon&\Real \times M&\to&M
\\
& &(t,{\rm p})&\mapsto&\sigma_t({\rm p})
\end{array}
is the flux of some vector field $\xi_X \in\vf(M)$;
that is, for ${\rm p}\in M$ and $t \in \Real$,
$\sigma (t,{\rm p})$ is the integral curve of $\xi_X$
passing through ${\rm p}$. This vector field is given by
\xi_X({\rm p}) = \frac{\d }{\d t}\sigma (t,{\rm p}) =
\frac{\d }{\d t}\sigma_t({\rm p}) =
\frac{\d }{\d t}\Phi_{{\rm exp}(-tX_e)}({\rm p})\Big\vert_{t=0} \ .
Hence, for every $f \in\Cinfty(M)$, we have that
(\xi_X(f))({\rm p}) = \frac{\d }{\d t}f(\Phi_{{\rm exp}(-tX_e)}({\rm p}))\Big\vert_{t=0}\ .

\begin{definition}
For $X \in {\bf g}$,  the vector field $\xi_X\in\vf(M)$ is the
\textbf{infinitesimal generator} or \textbf{fundamental vector field}
of the action $\Phi$ associated to $X$.
\end{definition}

\begin{definition}
Let $\Phi$ be an action of $G$ on $M$.
\begin{enumerate}
\item
A $k$-form $\vartheta \in\df^k(M)$ is \textbf{$G$-invariant} by $\Phi$
if $\Phi_g^*\vartheta =\vartheta$, for every $g \in G$;
or what is equivalent, if
$\Lie(\xi_Z)\vartheta = 0$, for every $Z \in {\bf g}$.
\item
A vector field $Z\in\vf(M)$ is \textbf{$G$-invariant} by $\Phi$
if $\Phi_{g_*}Z =Z$, for every $g \in G$;
or what is equivalent, if
$\Lie(\xi_X)Z= 0$, for every $X\in{\bf g}$.
\end{enumerate}
\end{definition}

\begin{prop}
Let $\psi \colon G_1 \to G_2$ be a Lie group homomorphism.
Then the following diagram commutes:
\begin{array}{ccccc}
& G_1 & \mapping{\psi} & G_2 &
\\
{\rm exp} & \Big\uparrow & & \Big\uparrow & {\rm exp}
\\
& \Tan_{e_1}G_1 \simeq {\bf g}_1 & \mapping{\psi_*}
& \Tan_{e_2}G_2 \simeq {\bf g}_2 &
\end{array}
that is,
$\psi({\rm exp}(tX_{e_1})) = {\rm exp}(t(\psi_*X)_{e_2}$.
\label{conmutexp}
\end{prop}
\begin{proof}
Immediate observing that  \
$t \mapsto {\rm exp}(t(\psi_*X)_{e_2})$ \
is the only uniparametric subgroup of $G_2$
whose tangent vector field at $t=0$ is $(\psi_*X)_{e_2}$.
\\ \qed  \end{proof}

A very relevant type of group actions are the following:

\begin{definition}
Let $G$ be a Lie group, $(M,\Omega)$ a symplectic manifold, and
$\Phi \colon G\times M \to M$ an action of $G$ on $M$.
We say that $\Phi$ is a \textbf{symplectic action} of $G$ on $M$
(also that $G$  \textbf{acts symplectically} on $M$ by $\Phi$)
if $\Phi_g$ is a symplectomorphism, for every $g \in G$;
that is, $\Phi_g^*\Omega = \Omega$.
Then $M$ is said to be a \textbf{symplectic $G$-space}.
\end{definition}

According to this definition, if $G$ acts symplectically on $M$ by $\Phi$,
then the fundamental vector field $\xi_X$ associated to $X$ by $\Phi$
is a locally Hamiltonian vector field and, conversely,
if  $\xi_X\in\vf_{lh}(M)$, for every $X \in{\bf g}$, then
$\Phi$ is a symplectic action of $G$ on $M$. Therefore:

\begin{prop}
Let $\xi\colon{\bf g}\to\vf(M)$ be the map such that
$\xi(X):=\xi_X$, for $X\in {\bf g}$.
Then, $\Phi$ is a symplectic action of $G$ on $M$ if, and only if,
${\rm Im}\,\xi\subseteq\vf_{lh}(M)$.

This means that
$\Lie(\xi_X)\Omega = 0$, for every $X \in {\bf g}$,
(that is, $\inn(\xi_X)\Omega$ is a closed form).
\end{prop}

\begin{definition}
Let $G$ be a Lie group, $(M,\Omega )$ a symplectic manifold and
$\Phi \colon G \to M$ a symplectic action of $G$ on $M$.
We say that $\Phi$ is a \textbf{strongly symplectic action} of $G$ on $M$
if $\xi_X\in\vf_{h}(M)$, for every $X \in{\bf g}$;
or, what is equivalent, $\inn(\xi_X)\Omega$ is an exact form.

In this case $M$ is said to be a
\textbf{strongly symplectic $G$-space} and
$\Phi$ is a \textbf{Hamiltonian action} of $G$ on $M$.
Otherwise, it is a \textbf{locally Hamiltonian action} of $G$ on $M$.
\end{definition}

In order to discuss the obstruction for a symplectic action to be strongly symplectic,
let ${\bf g}_h:=\{ X\in{\bf g}\ \mid\ \xi(X)=\xi_X\in\vf_H(M)\}$
and consider the following sequences of Lie algebras:
\beq
\begin{array}{ccccccccc}
0&\to&{\bf g}_h&\to&
{\bf g}&\to&{\bf g}/{\bf g}_h&\to&0
\\
& &\Big\downarrow \ \xi& &\Big\downarrow \ \xi& &\Big\downarrow \ \widetilde\xi& &
\\
0&\to&{\cal X}_h(M)&\to&
{\cal X}_{lh}(M)&\to&H^1(M)&\to&0
\end{array}\ ,
\label{diag}
\eeq
where $H^1(M)$ denotes the {\sl first de Rham's cohomology group} of $M$,
and $\widetilde \xi$ is a Lie algebra homomorphism which makes the diagram commutative.
Then, the image of {\bf g} by $\xi$ is in ${\cal X}_h(M)$
(that is, the action is strongly symplectic) if, and only if,
$\widetilde \xi = 0$.

There are two specially relevant cases
for which every symplectic action is strongly symplectic:
when $M$ is simply connected
(then every closed form is exact and,
therefore, ${\cal X}_{lh}(M) ={\cal X}_h(M)$) and
if $G$ is semisimple (then ${\bf g} = [{\bf g},{\bf g}]$
and  $[{\cal X}_{lh}(M),{\cal X}_{lh}(M)] \subset {\cal X}_h(M)$).

As a particular case, we have:

\begin{definition}
Let $G$ be a Lie group, $(M,\Omega)$ an exact symplectic manifold
(that is, $\Omega=\d\Theta$ for some $\Theta\in\df^1(M)$)
and $\Phi \colon G \to M$ an action of $G$ to $M$.
$\Phi$ is said to be an
\textbf{exact action} of $G$ on $M$
if, $\Phi_g^*\Theta = \Theta$, for every $g \in G$.
\end{definition}

Of course,  every exact action is strongly symplectic.

\subsection{Comomentum and momentum maps}
\label{cami2}

Next, we introduce new geometric elements
which are very relevant in the theory of symplectic group actions.
First, following [316] we define:

\begin{definition}
Let $G$ a Lie group which acts symplectically on a symplectic manifold $(M,\Omega )$.
A \textbf{comomentum map} associated to this action
is every Lie algebra linear application (if it exists)
\begin{array}{ccccc}
{\rm j}^*&\colon&{\bf g}&\to&\Cinfty (M)
\\
& &X&\mapsto&f_X:={\rm j}^*(X)
\end{array} \ ,
such that the following diagram commutes
\xymatrix{
& & \ar[dl]_<(0.45){{\rm j}^*} {\bf g} \ar[d]^<(0.45){\xi}&
\\
0 \to \Real
\ar[r]&\Cinfty (M)\ar[r]_<(0.25){\sharp_\Omega\circ\d}&\vf_{lh}(M)\ar[r]&H^1(M) \to 0
} \ ,
or, what is equivalent,
\inn(\xi_X)\Omega = \d f_X \ .
\label{com}
\end{definition}

The obstruction to the existence of comomentum maps
is given in the following:

\begin{prop}
Let $G$ be a Lie group which acts symplectically on $(M,\Omega)$.
A comomentum map associated to this action exists
if, and only if, the map \
$\widetilde\xi \colon {\bf g}/{\bf g}_h \to H^1(M)$ \
in \eqref{diag} reduces to be $\widetilde \xi = 0$.
\end{prop}
\begin{proof}
By definition, if a comomentum map exists, then \
$\sharp_\Omega~\circ~\d~\circ~{\rm j}^* = \xi$.
But, as ${\rm Im}\ \xi \subset {\rm Im}\ \sharp_\Omega$,
this implies that $\widetilde\xi = 0$.

Conversely, if $\widetilde \xi = 0$, then \
${\rm Im}\ \xi := \xi ({\bf g}) \subset {\cal X}_h(M)$ \
and, for all $X \in {\bf g}$, there exists
$f_X \in \Cinfty (M)$ such that $\inn(\xi_X)\Omega = \d f_X$.
\\ \qed  \end{proof}

Observe that this is also the obstruction
for a symplectic action to be strongly symplectic.

\begin{prop}
Let $G$ be a Lie group which acts symplectically on $(M,\Omega)$.
A comomentum map associated to this action exists
if, and only if, the action is strongly symplectic.
\end{prop}

\begin{remark}{\rm
If a comomentum map exists, it is not unique.
In fact, if $f_C \colon {\bf g} \to \Real$
is a continuous linear function, then $f_C \in {\bf g}^*$
and then $f_C(X) = ctn.$, for every $X \in {\bf g}$.
Therefore, if ${\rm j}^*$ is a comomentum map,
so is ${\rm j}^{'*}={\rm j}^*+f_C$.

Furthermore, if a comomentum map exists, it is not a
Lie algebra homomorphism necessarily since,
for $X,Y \in {\bf g}$, we have that
\sharp_\Omega \circ \d \{ f_X,f_Y \} =
\sharp_\Omega\inn([\xi_X,\xi_Y])\Omega
= -\sharp_\Omega\inn(\xi_{[X,Y]})\Omega
= -\sharp_\Omega \d f_{[X,Y]}
$\d \{ f_X,f_Y \} = -\d f_{[X,Y]}$ and then
$\{ f_X,f_Y \} = -f_{[X,Y]} + \sigma (X,Y)$
where $\sigma \colon {\bf g}\times{\bf g} \to \Real$
is a skew symmetric bilinear function
whose existence measures the obstruction for the
comomentum map to be a Lie algebra homomorphism.
This leads to state:

\begin{definition}
Let $G$ be a Lie group which acts symplectically on $(M,\Omega)$.
The action is said to be a \textbf{Poissonian action}
or also a \textbf{strongly Hamiltonian action} if
\begin{description}
\item[{\rm (i)}] \
There exists a comomentum map for this action.
\item[{\rm (ii)}]
The comomentum map is a Lie algebra (anti)homomorphism.
\end{description}
Then the triple $(M,\Omega,{\rm j}^*)$ is called a
\textbf{Hamiltonian $G$-space}.
\end{definition}

As a particular case, we have that:

\begin{prop}
If $(M,\Omega )$ is an exact symplectic manifold
and the action of $G$ on $M$ is exact, then it is Poissonian.
\label{exact}
\end{prop}
\begin{proof}
Let $\Omega =\d \Theta$.
Defining $f_X := -\Theta (\xi_X)$, then
${\rm j}^* \colon X \mapsto f_X$ is a comomentum map and
f_{[X,Y]}=-\Theta (\xi_{[X,Y]}) =
\Theta ([\xi_X,\xi_Y]) = \inn([\xi_X,\xi_Y])\Theta
=\Lie(\xi_X)\inn(\xi_Y)\Theta =
-\Lie(\xi_X)f_Y = -\{ f_X,f_Y \} \ .
\qed  \end{proof}

\begin{definition}
Let $G$ be a Lie group which acts
in a strongly symplectic way on a symplectic manifold $(M,\Omega)$.
A \textbf{momentum map} associated to this action
is the dual map of a comomentum map; that is, a map
\begin{array}{ccccc}
{\rm J}&\colon&M&\to&{\bf g}^*
\\
& &{\rm p}&\mapsto&\mu
\end{array}
such that, for every $X \in {\bf g}$ and for ${\rm p}\in M$,
\langle X,{\rm J}({\rm p})\rangle:={\rm j}^*(X)({\rm p})=f_X({\rm p}) \ .
(Observe that the map
$\mu:= {\rm J}({\rm p})\colon{\bf g}\to\Real$ such that
$X\mapsto f_X({\rm p})$, for $X\in{\bf g}$,
is, in fact, an element of ${\bf g}^*$).
\end{definition}

Obviously, the obstruction to the existence of momentum maps is
the same that for the comomentum maps.
In particular, as an immediate corollary of Proposition
\ref{exact} we have:

\begin{prop}
If $(M,\Omega )$ is an exact symplectic manifold
and the action of $G$ on $M$ is exact,
then a momentum map exists. It is given by
$\langle X,{\rm J}({\rm p})\rangle:= -\Theta (\xi_X)({\rm p})$,
for every $X \in {\bf g}$, for ${\rm p}\in M$.
\end{prop}

\begin{remark}{\rm
As a more general result,
if $G$ is a connected Lie group and $\Phi$ a strongly symplectic action
of $G$ on $(M,\Omega)$, then the action is Poissonian
if, and only if,
the momentum maps associated to this action
are $Ad^*$-equivariant, that is,
for every $g \in G$, we have that $Ad^*\circ {\rm J}=\Phi_g\circ{\rm J}$;
where $Ad^*\colon{\bf g}^*\to{\bf g}^*$ is the {\sl coadjoint action} of $G$.
(See [316] for the proof).

\subsection{Reduction by symmetries}
\label{cami3}

Finally, we introduce the concept of
{\sl group of symmetries} of a Hamiltonian system,
its relation with the momentum map,
and give the main insights about the theory of reduction by symmetries.

\begin{definition}
Let $G$ be a Lie group, $(M,\Omega,h)$ a Hamiltonian system,
and $\Phi \colon G \to M$ an action of $G$ on $M$.
We say that $G$ is a \textbf{symmetry group} of the Hamiltonian system if
\begin{description}
\item[{\rm (i)}] \
$\Phi$ is a symplectic action.
\item[{\rm (ii)}]
For every $g \in G$, we have $\Phi_g^*h = h$.
\end{description}
The diffeomorphism $\Phi_g$, for every $g \in G$, is called a
\textbf{symmetry} of the Hamiltonian system.
\end{definition}

From this definition, it is obvious that
$G$ is a symmetry group of a Hamiltonian system $(M,\Omega,h)$
if, and only if,  $\Lie(\xi_X)h = 0$, for every $X \in {\bf g}$.

In this context, the Noether Theorem is stated as follows::

\begin{teor}
({\sl \textbf{Noether's theorem}}).
Let $G$ be a symmetry group which acts in a strongly symplectic way
on the Hamiltonian system $(M,\Omega,h)$.
Then, for every $X \in {\bf g}$, the functions $f_X$
are constants of motion; that is, $\Lie(X_h)f_X = 0$.
\end{teor}
\begin{proof}
We have that $\inn(X_h)\Omega=\d h$, then
\Lie(X_h)f_X=\inn(X_h)\d f_X
= \inn(X_h)\inn(\xi_X)\Omega
=-\inn(\xi_X)\inn(X_h)\Omega = -\inn(\xi_X)\d h
=-\Lie(\xi_X)h = 0 \ .
%(A similar result can also be stated for locally Hamiltonian systems).
\\ \qed  \end{proof}

One of the most important features on the study of
Hamiltonian systems with symmetry is the so-called
{\sl reduction theory}.
Now we are going to state the main results on this topic
(see, for instance, [100, 252, 254, 255, 256, 281, 282, 283, 346] for deeper explanations and details).

First, we have to introduce the following concept:

\begin{definition}
Let $G$ be a Lie group and a strongly symplectic action
of $G$ on $(M,\Omega)$.
Let ${\rm J}$ be a momentum map associated to this action. Then,
$\mu \in {\bf g}^*$ is a
\textbf{weakly regular value} of ${\rm J}$ if:
\begin{description}
\item[{\rm (i)}] \
${\rm J}^{-1}(\mu )$ is a submanifold of $M$.
\item[{\rm (ii)}]
For every ${\rm p}\in {\rm J}^{-1}(\mu )$,
$\Tan_{\rm p}({\rm J}^{-1}(\mu)) = \ker \Tan_{\rm p}{\rm J}$.
\end{description}
If $\Tan_{\rm p}{\rm J}$ is surjective, then $\mu$ is said to be a \textbf{regular value}.
(Of course,  every regular value is weakly regular).
\end{definition}

Then we have:

\begin{teor}
Let $G$ be a Lie group acting on a symplectic manifold
$(M,\Omega )$ such that the action is
Poissonian, free and proper.
Let $\mu$ be a weakly regular value of the
momentum map associated to this action,
and let $G_{\mu}$ be the isotropy group of
${\rm J}^{-1}(\mu )$ for the action of $G$ on $M$.