# Particular integrals and particular integrability for (co)symplectic and
(co)contact Hamiltonian systems

R. Azuaje and A. M. Escobar-Ruiz
Departamento de Física, Universidad Autónoma Metropolitana Unidad Iztapalapa,
San Rafael Atlixco 186, 09340, Ciudad de México, México

###### Abstract

In this paper we study the notions of particular integral and particular
integrability, in classical mechanics, under a geometrical approach. For
Hamiltonian systems on cosymplectic, contact and cocontact manifolds, it is
demonstrated that the existence of a particular integral implies that some of
the trajectories can be found from a simpler reduced set of Hamilton
equations. In the case of particular integrability, these trajectories can be
obtained by quadratures. For dissipative systems described using contact
geometry, a particular integral can be viewed as a generalization of the
important concept of dissipated quantity as well.

Keywords: particular integral, integrability by quadratures, symmetric
reduction, contact Hamiltonian systems, dissipated quantities.

##  1 Introduction

It is well-known that the existence of symmetries for a classical mechanical
system simplifies the problem of solving the equations of motion. Also, it is
common knowledge that in the symplectic framework the constants of motion for
a Hamiltonian system are related to symmetries [1, 2, 3]. The famous Liouville
theorem on integrability asserts that for a Hamiltonian system with $n$
degrees of freedom, the knowledge of $n$ functionally independent constants of
motion (or integrals of motion) in involution allows us to find the solutions
of the Hamilton equations of motion by quadratures, i.e., by using finitely
many algebraic operations (including taking inverse functions) and calculating
the integrals of known functions [1, 2, 4, 5, 6, 7, 8, 9, 10].

Physically, a constant of motion is viewed as a conserved quantity during time
evolution for any set of initial conditions in the corresponding domain. For
this reason, they can also be called global integrals. The concept of
particular integral introduced in [11] generalizes the notion of a global
integral. Specifically, a particular integral is a conserved quantity for
possibly certain (sub)sets of initial conditions only. One important rationale
to investigate particular integrals is that they allow us to study non-
integrable systems in certain points and regions where the dynamics of the
system satisfies the regularity requirements for integrability which, in turn,
leads to the notion of particular integrability. In [12] the concepts of
particular integral and particular integrability were introduced within the
formalism of symplectic geometry, i.e., they were considered for Hamiltonian
systems defined on symplectic manifolds. Symplectic geometry very often is
regarded as the natural geometric formalism where the Hamiltonian theory of
classical mechanics is developed. Nevertheless, only autonomous conservative
systems can be described under this framework.

Other geometric constructions such as cosymplectic geometry, contact geometry
and cocontact geometry allow alternative formulations of Hamiltonian
mechanics. They have been proven to be instrumental in describing non-
autonomous and dissipative mechanical systems within the Hamiltonian theory.
For completeness, in section 3 of this paper, we present a brief review of
symplectic geometry, cosymplectic geometry, contact geometry, cocontact
geometry and the corresponding geometric formulations of Hamiltonian
Mechanics. Currently, contact and cocontact Hamiltonian mechanics are subjects
of intense active research, see for example [13, 14, 15, 16, 17]. For
instance, in the case of dissipative systems, the so called dissipated
quantities were studied in relation with Noether symmetries [14, 18, 13],
giving as a result dissipation laws analogous to the conservation theorems
occurring in conservative systems. In fact, the notion of particular integral
generalizes the concept of dissipated quantity.

In [12] the concept of particular integrability has been presented as a more
general notion than Liouville integrability. Accordingly, they share important
features. For example, a common key element is the property of functional
independence of the involved functions. It is worth reminding this property
and the corresponding reduction of the relevant manifold: let $M$ be a smooth
manifold, we say that the functions $f_{1},\cdots,f_{k}\in C^{\infty}(M)$ with
$k\leq dim(M)$ are functionally independent if for any $p$ in a dense subset
of $M$ the differential maps $df_{1}|_{p},\cdots,df_{k}|_{p}$ are linearly
independent; this means that $p$ is a regular point of the differentiable
function $F:M\longrightarrow\mathbb{R}^{k}$ defined by
$F=(f_{1},\cdots,f_{k})$. Let us consider the level set $M_{f}=\\{x\in
M:f_{i}(x)=\alpha_{i},\alpha_{i}\in\mathbb{R}\\}$, and suppose that
$f_{1},\ldots,f_{k}$ are functionally independent, from the Regular Level Set
Theorem [19] we have that $M_{f}$ is a smooth submanifold of $M$ of dimension
$dim(M_{f})=dim(M)-k$.

The aim of this paper is to present the notions of particular integral and
particular integrability in Hamiltonian mechanics under the formalisms of
cosymplectic geometry, contact geometry and cocontact geometry in detail. The
associated reduction of the equations of motion, using particular integrals,
is investigated in each formalism.

This paper is organized as follows. In section 2, we present the notion of
particular integral for a classical mechanical system defined by a smooth
vector field on a smooth generic manifold; in this section we emphasize the
similarities and differences between constants of motion (also called
integrals of motion or first Liouville integrals) and particular integrals. In
section 3, in order to set up the notation and language used in this study, a
brief review of symplectic, cosymplectic, contact and cocontact geometry as
well as their corresponding formulations of Hamiltonian mechanics is provided.
Afterwards, in section 4 we describe the notions of particular integral and
particular integrability for Hamiltonian systems defined on a smooth manifold
equipped with different geometric structures. This section is divided into
four parts, in the first one we give a summary of the symplectic case as
presented in [12] whilst in the second, third and fourth subsections we
introduce the notions of particular integral and particular integrability for
Hamiltonian systems within the framework of cosymplectic, contact and
cocontact geometry, respectively.

##  2 Particular integrals for classical mechanical systems

Classical mechanics considers the motion of dynamical systems whose future and
past are uniquely determined by the initial positions and initial velocities
of all points of the system; such systems are called (classical) mechanical
systems. The phase space of a mechanical system is the set whose elements are
the sets of positions and velocities of all points of the given system [20].
Poincaré visualized a mechanical system as a field of vectors on phase space,
in which a trajectory is a smooth curve tangent at each of its points to the
vector based at that point; which lead him to the notion of a smooth manifold
as the phase space in classical mechanics [1].

Let $M$ be a smooth manifold of dimension $n$ and $V$ a smooth vector field on
$M$. $V$ defines a (classical) mechanical system on $M$ whose trajectories are
the integral curves of $V$ [21]. Let us remember that a integral curve of $V$
is a curve $\gamma:I\subset\mathbb{R}\longrightarrow M$ such that
$\frac{d}{dt}\gamma(t)=V_{\gamma(t)}$ $\forall t\in I$. Let
$(x^{1},\cdots,x^{n})$ be local coordinates on $M$, the local expression of
$V$ is $V=V^{1}(x^{1},\cdots,x^{n})\frac{\partial}{\partial
x^{1}}+\cdots+V^{n}(x^{1},\cdots,x^{n})\frac{\partial}{\partial x^{n}}$, and
the integral curves $\gamma(t)=(x^{1}(t),\cdots,x^{n}(t))$ of $V$ satisfy the
system of equations

$\left\\{\begin{array}[]{c}\dot{x}^{1}=V^{1}(x^{1},\cdots,x^{n}),\\\ \vdots\\\
\dot{x}^{n}=V^{n}(x^{1},\cdots,x^{n}).\end{array}\right.$ (1)

Let us consider the mechanical system $(M,V)$, $M$ is the phase space of the
system and $V$ is the dynamical vector field. The system of first order $n$
differential equations (1) is the system of (local) equations of motion (the
motion of a system in classical mechanics is described using ordinary
differential equations [20]). The evolution of a function $f\in C^{\infty}(M)$
(an observable) along the trajectories of the system is given by

$\frac{d}{dt}f(\gamma(t))=df_{\gamma(t)}(\frac{d}{dt}\gamma(t))=(\frac{d}{dt}\gamma(t))(f)=V_{\gamma(t)}(f)=(Vf)(\gamma(t))\
,$ (2)

so the temporal evolution of $f$ is given by

$\dot{f}\ =\ Vf\ .$ (3)

A function $f\in C^{\infty}(M)$ is called a constant of motion (or an integral
of motion) of the mechanical system $(M,V)$ if $f$ is constant along the
trajectories of the system, which is equivalent to $Vf=0$. The existence of a
constant of motion enable us to construct a reduced mechanical system whose
dynamics is contained into the dynamics of the bigger one, i.e., we can
construct a reduced system such that its trajectories are trajectories of the
original system. So the existence of a constant of motion allows us to look
for trajectories of the system that are (locally) solutions of a reduced
system of differential equations of motion. Indeed, let us suppose that $f$ is
a constant of motion of the system $(M,V)$, we have that the level set

$M_{f}\ =\ \\{\,x\in M:\ f(x)\,=\,c,\,c\in\mathbb{R}\\}\ ,$

is a smooth submanifold of $M$ of codimension $1$ for regular values of $f$
[19] and it is closed under the dynamics of $V$, i.e., if
$\gamma:I\longrightarrow M$ is an integral curve of $V$ such that
$\gamma(t_{0})\in M_{f}$ for some $t_{0}\in I$ then $\gamma(t)\in M_{f}$ for
every $t\in I$. We can prove this last statement as follows: let us suppose
that $\gamma:I\longrightarrow M$ is an integral curve of $V$ such that
$\gamma(t_{0})\in M_{f}$ for some $t_{0}\in I$, then $f(\gamma(t_{0}))=c$ and
since $f$ is a constant of motion it is constant along the whole trajectory,
i.e., $f(\gamma(t))=c$ for every $t\in I$, so we conclude that $\gamma(t)\in
M_{f}$ for every $t\in I$. Now we can look for trajectories of the system
$(M,V)$ that live in $M_{f}$. It is important remarking that the set of the
trajectories of the system $(M,V)$ that live in $M_{f}$ is not the whole set
of the trajectories of the system, but the element of this first set can be
found by solving a reduced system of differential equations. Indeed, if $f$ is
a constant of motion of $(M,V)$ then the integral curves that live in $M_{f}$
are (locally) solutions of a system of $n-1$ differential equations: the
integral curves of $V$ that live in $M_{f}$ are integral curves of the
restriction $V|_{M_{f}}$ of the vector field $V$ to $M_{f}$, so if
$(y^{1},\cdots,y^{n-1})$ are local coordinates on $M_{f}$ then such integral
curves are solutions of the vector differential equation
$\dot{Y}=V|_{M_{f}}(Y)$, where $Y=(y^{1},\cdots,y^{n-1})$, which is equivalent
to a system of $n-1$ differential equations.

Now we present the concept of particular integral explaining how it is related
with that one of constant of motion.

######  Definition 1.

We say that $p\in C^{\infty}(M)$ is a particular integral of $(M,V)$ if
$Vp=a\,p$ for some function $a\in C^{\infty}(M)$ such that

$\lim_{x\to x_{0}}a(x)\ =\ a(x_{0})\in\mathbb{R}\ ,\quad\forall\,x_{0}\in
M_{p}=\\{\,x\in M:\ p(x)\,=\,0\,\\}\ .$

If so, we say that $a\in C^{\infty}(M)$ is a function with real values on
$p=0$.

Hence, the special case $a=0$ corresponds to the definition of a constant of
motion. Similarly to the previous case where a constant of motion is involved,
the existence of a particular integral allows us to look for certain
trajectories of the system that are (locally) solutions of a reduced system of
differential equations of motion. We have the following result.

######  Lemma 1.

If $p$ is a particular integral of $(M,V)$ then $M_{p}$ is closed under the
dynamics of $V$.

###### Proof.

Let $\gamma:I\longrightarrow M$ be an integral curve of $V$ such that
$\gamma(t_{0})\in M_{p}$ for some $t_{0}\in I$, so we have
$p(\gamma(t_{0}))=0$ and

$\begin{split}\frac{d}{dt}p(\gamma(t))&=(Vp)(\gamma(t))\\\
&=(ap)(\gamma(t))\\\ &=a(\gamma(t))p(\gamma(t)),\end{split}$ (4)

i.e. $\frac{d}{dt}p(\gamma(t))$ is proportional to $p(\gamma(t))$; we can see
that $\frac{d^{k}}{dt^{k}}p(\gamma(t))$ is proportional to $p(\gamma(t))$ for
each $k=1,2,\ldots$, so we have that

$\frac{d^{k}}{dt^{k}}p(\gamma(t))|_{t=t_{0}}\ =\ 0\ ,$ (5)

for every $k=1,2,\ldots$, i.e., $p(\gamma(t))$ and all its derivatives are
zero on $t=t_{0}$, therefore since we suppose that $p(\gamma(t))$ is an
analytic function then $p(\gamma(t))=0$ $\forall\,t\in I$ [22, 23, 24], i.e.,
$\gamma(t)\in M_{p}\ \forall\,t\in I$. ∎

Now given a particular integral we can look for the trajectories of the system
$(M,V)$ that live in $M_{p}$, and like for a constant of motion, if $p$ is a
particular integral of $(M,V)$ then the integral curves that live in $M_{p}$
are (locally) solutions of a reduced system of $n-1$ differential equations.

Up to this point general classical mechanical systems have been considered. We
are interested in Hamiltonian systems, these are mechanical systems on smooth
manifolds where the dynamical vector fields are the Hamiltonian vector fields
for distinguished functions called the Hamiltonian functions. In order to
construct a Hamiltonian system, a smooth manifold equipped with a geometric
structure is required. The corresponding Hamiltonian vector field for a given
function is assigned according to a one-to-one correspondence, between vector
fields and 1-forms on such manifold, defined by means of the geometric
structure. So that the geometric structure plays a fundamental roll in the
construction of the Hamiltonian system. Symplectic geometry is considered the
natural geometric framework where the theory of Hamiltonian mechanics is
developed, nevertheless, only autonomous conservative systems can be described
under this formalism. Other geometric frameworks allow alternative
formulations of Hamiltonian mechanics. They are instrumental to describe non-
autonomous and dissipative mechanical systems as Hamiltonian systems. These
formalisms are cosymplectic geometry, contact geometry and cocontact geometry
which we briefly review in the next section.

##  3 Geometric formalisms for Hamiltonian mechanics

###  3.1 Symplectic geometry

In this subsection, in order to set up the notation and language used in this
paper, we present a brief review of symplectic geometry and the formulation of
symplectic (time-independent conservative) Hamiltonian mechanics (for details
see [1, 6, 25, 19, 26]).

Let $M$ be a smooth manifold. A symplectic structure on $M$ is a closed non-
degenerate 2-form $\omega$ on $M$. Closed means that $d\omega=0$ and non-
degenerate implies that for each 1-form $\alpha$ on $M$ there is one and only
one vector field $X$ which obeys $X\lrcorner\omega=\alpha$. A symplectic
manifold is a pair $(M,\omega)$ where $\omega$ is a symplectic structure on
$M$. By definition, each symplectic manifold is of even dimension. Let
$(M,\omega)$ be a symplectic manifold of dimension $2n$. Around any point
$x\in M$ there exist local coordinates
$(q^{1},\cdots,q^{n},p_{1},\cdots,p_{n})$, called canonical coordinates or
Darboux coordinates, such that

$\omega=dq^{i}\wedge dp_{i}.$ (6)

In this paper we adopt the Einstein summation convention ( i.e., a summation
over repeated indices is assumed).

For each $f\in C^{\infty}(M)$ it is assigned a vector field $X_{f}$ on $M$,
called the Hamiltonian vector field for $f$, according to

$X_{f}\lrcorner\omega\ =\ df\ .$ (7)

In canonical coordinates, $X_{f}$ reads

$X_{f}=\frac{\partial f}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-\frac{\partial f}{\partial q^{i}}\frac{\partial}{\partial p_{i}}\ .$
(8)

The assignment $f\longmapsto X_{f}$ is linear, that is

$X_{f+\alpha g}\ =\ X_{f}+\alpha X_{g}\ ,$ (9)

$\forall f,g\in C^{\infty}(M)$ and $\forall\alpha\in\mathbb{R}$. The
symplectic structure $\omega$ on $M$ defines a Poisson structure on
$C^{\infty}(M)$ as follows: given $f,g\in C^{\infty}(M)$ the Poisson bracket
of $f$ and $g$ is defined by

$\\{f,g\\}=X_{g}f,$ (10)

or equivalently

$\\{f,g\\}=\omega(X_{f},X_{g}).$ (11)

In canonical coordinates we have

$\\{f,g\\}\ =\ \frac{\partial f}{\partial q^{i}}\frac{\partial g}{\partial
p_{i}}\,-\,\frac{\partial f}{\partial p_{i}}\frac{\partial g}{\partial q^{i}}\
.$ (12)

The assignment $f\mapsto X_{f}$ is a Lie algebra antihomomorphism between the
Lie algebras $(C^{\infty}(M),\\{,\\})$ and $(\mathfrak{X}(M),[,])$ [19], i.e.,
$X_{\\{f,g\\}}=-[X_{f},X_{g}]$.

The theory of time-independent conservative Hamiltonian systems is naturally
constructed within the mathematical formalism of symplectic geometry. Given
$H\in C^{\infty}(M)$, the dynamical system defined on $M$ by the Hamiltonian
vector field $X_{H}$ is a Hamiltonian system denoted by $(M,\omega,H)$.
$(M,\omega)$ is called the phase space of the system, $H$ is the Hamiltonian
function and $n=\frac{dim(M)}{2}$ is the number of degrees of freedom of the
system. The trajectories of the system are the integral curves of $X_{H}$, in
canonical coordinates $(q^{1},\cdots,q^{n},p_{1},\cdots,p_{n})$ its expression
$\psi(t)=(q^{1}(t),\cdots,q^{n}(t),p_{1}(t),\cdots,p_{n}(t))$ satisfies the
Hamilton equations of motion

$\dot{q^{i}}=\frac{\partial H}{\partial p_{i}},\hskip
28.45274pt\dot{p_{i}}=-\frac{\partial H}{\partial q^{i}}\qquad;\qquad
i=1,2,3,\ldots,n\ .$ (13)

The evolution (the temporal evolution) of a function $f\in C^{\infty}(M)$ (a
physical observable) along the trajectories of the system is given by

$\dot{f}=X_{H}f=\\{f,H\\}\ ,$ (14)

so that $f$ is a constant of motion if and only if $\\{f,H\\}=0$.

###  3.2 Cosymplectic geometry

In this subsection, a review of the cosymplectic geometry and the associated
formulation of time-dependent Hamiltonian systems is presented.

######  Definition 2.

Let $M$ be a $2n+1$ dimensional smooth manifold. A cosymplectic structure on
$M$ is a couple $(\Omega,\eta)$, where $\Omega$ is a closed 2-form on $M$ and
$\eta$ is a closed 1-form on $M$ such that $\eta\wedge\Omega^{n}\neq 0$. If
$(\Omega,\eta)$ is a cosymplectic structure on $M$ we say that
$(M,\Omega,\eta)$ is a cosymplectic manifold.

Let $(M,\Omega,\eta)$ be a cosymplectic manifold of dimension $2n+1$. Around
any point $x\in M$ there exist local coordinates
$(q^{1},\cdots,q^{n},p_{1},\cdots,p_{n},t)$, called canonical coordinates or
Darboux coordinates, such that

$\Omega=dq^{i}\wedge dp_{i}\hskip 28.45274pt\rm{and}\hskip 28.45274pt\eta=dt.$
(15)

There exists a distinguished vector field $R$ on $M$, called the Reeb vector
field, which obeys

$R\lrcorner\Omega=0\hskip 28.45274pt\rm{and}\hskip
28.45274ptR\lrcorner\eta=1.$ (16)

In canonical coordinates we have $R=\frac{\partial}{\partial t}$.

For each $f\in C^{\infty}(M)$ it is assigned a vector field $X_{f}$ on $M$,
called the Hamiltonian vector field for $f$, according to

$X_{f}\lrcorner\Omega=df-(Rf)\eta\hskip 28.45274pt\rm{and}\hskip
28.45274ptX_{f}\lrcorner\eta=0.$ (17)

Or equivalently, since $\mathcal{X}_{\eta\Omega}:TM\longrightarrow T^{*}M$
defined by $\mathcal{X}_{\eta\Omega}(X)=X\lrcorner\Omega+(X\lrcorner\eta)\eta$
is a bundle isomorphism [27], $X_{f}$ is the only vector field such that

$X_{f}\lrcorner\Omega+(X_{f}\lrcorner\eta)\eta=df-(Rf)\eta.$ (18)

In canonical coordinates we have

$X_{f}=\frac{\partial f}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-\frac{\partial f}{\partial q^{i}}\frac{\partial}{\partial p_{i}}.$ (19)

The assignment $f\longmapsto X_{f}$ is linear, that is

$X_{f+\alpha g}=X_{f}+\alpha X_{g},$ (20)

$\forall f,g\in C^{\infty}(M)$ and $\forall\alpha\in\mathbb{R}$. As in the
symplectic case, we have that the cosymplectic structure $(\Omega,\eta)$ on
$M$ defines a Poisson structure on $C^{\infty}(M)$ as follows: given $f,g\in
C^{\infty}(M)$ the Poisson bracket of $f$ and $g$ is defined by

$\\{f,g\\}=X_{g}f,$ (21)

or equivalently

$\\{f,g\\}=\Omega(X_{f},X_{g}).$ (22)

In canonical coordinates, it reads

$\\{f,g\\}=\frac{\partial f}{\partial q^{i}}\frac{\partial g}{\partial
p_{i}}-\frac{\partial f}{\partial p_{i}}\frac{\partial g}{\partial q^{i}}.$
(23)

As in the symplectic framework, we have that the assignment $f\mapsto X_{f}$
is a Lie algebra antihomomorphism between the Lie algebras
$(C^{\infty}(M),\\{,\\})$ and $(\mathfrak{X}(M),[,])$ [27], i.e.,
$X_{\\{f,g\\}}=-[X_{f},X_{g}]$. In addition $X_{Rf}=-[X_{f},R]$.

The theory of time-dependent conservative Hamiltonian systems is naturally
constructed within the mathematical formalism of cosymplectic geometry (for
details see [6, 27, 28]). Given $H\in C^{\infty}(M)$, the dynamical system
defined on $M$ by the so called evolution vector field $E_{H}=X_{H}+R$ is a
Hamiltonian system denoted by $(M,\Omega,\eta,H)$. $(M,\Omega,\eta)$ is called
the phase space of the system, $H$ is the Hamiltonian function and
$n=\frac{dim(M)-1}{2}$ is the number of degrees of freedom of the system. The
trajectories of the system are the integral curves of $E_{H}$, in canonical
coordinates $(q^{1},\cdots,q^{n},p_{1},\cdots,p_{n},t)$ its expression
$\psi(t)=(q^{1}(t),\cdots,q^{n}(t),p_{1}(t),\cdots,p_{n}(t),t)$ satisfies the
Hamilton equations of motion

$\dot{q^{i}}=\frac{\partial H}{\partial p_{i}},\hskip
28.45274pt\dot{p_{i}}=-\frac{\partial H}{\partial q^{i}}.$ (24)

The evolution (the temporal evolution) of a function $f\in C^{\infty}(M)$ (a
physical observable) along the trajectories of the system is given by

$\dot{f}=E_{H}f=X_{H}f+Rf=\\{f,H\\}+Rf.$ (25)

So that $f$ is a constant of motion if and only if $\\{f,H\\}+Rf=0$.

###  3.3 Contact geometry

In this subsection we move to the formalism of contact Hamiltonian systems
which naturally describe dissipative systems (for details see [29, 28, 30]),
and they also have applications in thermodynamics, statistical mechanics and
others [31].

######  Definition 3.

Let $M$ be a $2n+1$ dimensional smooth manifold. A contact structure on $M$ is
a 1-form $\theta$ on $M$ such that $\theta\wedge d\theta^{n}\neq 0$. If
$\theta$ is a contact structure on $M$ we say that $(M,\theta)$ is a contact
manifold.

There is a wider notion of contact manifolds, some authors define a contact
structure on a manifold as a one-codimensional maximally non-integrable
distribution [13, 32, 33, 34]. Locally a contact structure can be expressed as
the kernel of a one-form $\theta$ satisfying $\theta\wedge d\theta^{n}\neq 0$,
$\theta$ is called a local contact form. However, not every contact structure
admits a global contact form. When a contact structure admits a global contact
form, the contact manifold is called co-oriented. Every contact manifold
$(M,\theta)$ from definition 3 is a co-oriented contact manifold in this wider
sense by taking the distribution ${\rm Ker}(\theta)$. In this paper we
restrict ourselves to co-oriented contact manifold and we refer to them as
contact manifolds.

Let $(M,\theta)$ be a contact manifold of dimension $2n+1$. Around any point
$x\in M$ there exist local coordinates
$(q^{1},\cdots,q^{n},p_{1},\cdots,p_{n},z)$, again called canonical or Darboux
coordinates, such that

$\theta=dz-p_{i}dq^{i}.$ (26)

There exists a distinguished vector field $R$ on $M$, called the Reeb vector
field, which obeys

$R\lrcorner\theta=1\hskip 28.45274pt\rm{and}\hskip 28.45274ptR\lrcorner
d\theta=0.$ (27)

In canonical coordinates we simply have $R=\frac{\partial}{\partial z}$.

For each $f\in C^{\infty}(M)$ it is assigned a vector field $X_{f}$ on $M$,
called the contact Hamiltonian vector field for $f$, according to

$X_{f}\lrcorner\theta=-f\hskip 28.45274pt\rm{and}\hskip
28.45274ptX_{f}\lrcorner d\theta=df-(Rf)\theta.$ (28)

Or equivalently, since $\mathcal{X}_{\theta}:TM\longrightarrow T^{*}M$ defined
by $\mathcal{X}_{\theta}(X)=X\lrcorner d\theta+(X\lrcorner\theta)\theta$ is a
bundle isomorphism [29], $X_{f}$ is the only vector field such that

$X_{f}\lrcorner d\theta+(X_{f}\lrcorner\theta)\theta=df-(Rf+f)\theta.$ (29)

In canonical coordinates we have

$X_{f}=\frac{\partial f}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-\left(\frac{\partial f}{\partial q^{i}}+p_{i}\frac{\partial f}{\partial
z}\right)\frac{\partial}{\partial p_{i}}+\left(p_{i}\frac{\partial f}{\partial
p_{i}}-f\right)\frac{\partial}{\partial z}.$ (30)

It can be checked that the assignment $f\longmapsto X_{f}$ is linear, that is

$X_{f+\alpha g}=X_{f}+\alpha X_{g},$ (31)

$\forall f,g\in C^{\infty}(M)$ and $\forall\alpha\in\mathbb{R}$.

Symplectic and cosymplectic manifolds are Poisson manifolds (symplectic and
cosymplectic structures define Poisson brackets), but a contact manifold is
strictly a Jacobi manifold, i.e., a contact structure on a manifold defines a
Jacobi bracket. Given $f,g\in C^{\infty}(M)$ the Jacobi bracket of $f$ and $g$
is defined by

$\\{f,g\\}=X_{g}f+fRg,$ (32)

or equivalently

$\\{f,g\\}=\theta([X_{f},X_{g}]).$ (33)

In canonical coordinates we have

$\\{f,g\\}=\frac{\partial f}{\partial q^{i}}\frac{\partial g}{\partial
p_{i}}-\frac{\partial f}{\partial p_{i}}\frac{\partial g}{\partial
q^{i}}+\frac{\partial f}{\partial z}\left(p_{i}\frac{\partial g}{\partial
p_{i}}-g\right)-\frac{\partial g}{\partial z}\left(p_{i}\frac{\partial
f}{\partial p_{i}}-f\right).$ (34)

In the contact case the assignment $f\mapsto X_{f}$ also defines a Lie algebra
antihomomorphism between the Lie algebras $(C^{\infty}(M),\\{,\\})$ and
$(\mathfrak{X}(M),[,])$, indeed, for $f,g\in C^{\infty}(M)$ we have

$-[X_{f},X_{g}]=X_{\\{f,g\\}}.$ (35)

For details see [29].

Hamiltonian systems on contact manifolds are called contact Hamiltonian
systems. Given $H\in C^{\infty}(M)$, the dynamics of the dynamical system on
$(M,\theta)$ (the phase space) with Hamiltonian function $H$ is defined by the
Hamiltonian vector field $X_{H}$. $(M,\theta,H)$ is a contact Hamiltonian
system with $n=\frac{dim(M)-1}{2}$ degrees of freedom. In canonical
coordinates we have

$X_{H}=\frac{\partial H}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-\left(\frac{\partial H}{\partial q^{i}}+p_{i}\frac{\partial H}{\partial
z}\right)\frac{\partial}{\partial p_{i}}+\left(p_{i}\frac{\partial H}{\partial
p_{i}}-H\right)\frac{\partial}{\partial z}.$ (36)

The trajectories
$\psi(t)=(q^{1}(t),\cdots,q^{n}(t),p_{1}(t),\cdots,p_{n}(t),z(t))$ of the
system are the integral curves of $X_{H}$, they satisfy the dissipative
Hamilton equations of motion

$\dot{q^{i}}=\frac{\partial H}{\partial p_{i}},\hskip
28.45274pt\dot{p_{i}}=-\left(\frac{\partial H}{\partial
q^{i}}+p_{i}\frac{\partial H}{\partial z}\right),\hskip
28.45274pt\dot{z}=p_{i}\frac{\partial H}{\partial p_{i}}-H.$ (37)

The evolution of a function $f\in C^{\infty}(M)$ along the trajectories of the
system is given by

$\dot{f}=\\{f,H\\}-fRH.$ (38)

So that $f$ is a constant of motion if and only if $\\{f,H\\}-fRH=0$.

###  3.4 Cocontact geometry

This subsection together with the previous ones, completes the notation and
language employed in this paper. Here we briefly review the formalism of time-
dependent contact Hamiltonian systems introduced in [35].

######  Definition 4.

Let $M$ be a $2n+2$ dimensional smooth manifold. A cocontact structure on $M$
is a couple $(\theta,\eta)$ of 1-forms on $M$ such that $\eta$ is closed and
$\eta\wedge\theta\wedge(d\theta)^{n}\neq 0$. If $(\theta,\eta)$ is a cocontact
structure on $M$ we say that $(M,\theta,\eta)$ is a cocontact manifold.

Let $(M,\theta,\eta)$ be a cocontact manifold of dimension $2n+2$. Around any
point $p\in M$ there exist local coordinates
$(t,q^{1},\cdots,q^{n},p_{1},\cdots,p_{n},z)$, called canonical coordinates or
Darboux coordinates, such that

$\theta=dz-p_{i}dq^{i}\hskip 28.45274pt\rm{and}\hskip 28.45274pt\eta=dt.$ (39)

There exist two distinguished vector fields $R_{z}$ and $R_{t}$ on $M$, called
the contact Reeb vector field and the time Reeb vector field respectively,
such that

$\left\\{\begin{array}[]{c}R_{z}\lrcorner\eta=0\\\ R_{z}\lrcorner\theta=1\\\
R_{z}\lrcorner d\theta=0\end{array}\right.$ (40)

and

$\left\\{\begin{array}[]{c}R_{t}\lrcorner\eta=1\\\ R_{t}\lrcorner\theta=0\\\
R_{t}\lrcorner d\theta=0\end{array}\right..$ (41)

In canonical coordinates, we have $R_{z}=\frac{\partial}{\partial z}$ and
$R_{t}=\frac{\partial}{\partial t}$.

For each $f\in C^{\infty}(M)$ it is assigned a vector field $X_{f}$ on $M$,
called again the contact Hamiltonian vector field for $f$, according to

$X_{f}\lrcorner\theta=-f,\hskip 28.45274ptX_{f}\lrcorner
d\theta=df-(R_{z}f)\theta-(R_{t}f)\eta\hskip 28.45274pt\rm{and}\hskip
28.45274ptX_{f}\lrcorner\eta=0.$ (42)

Or equivalently, since $\mathcal{X}_{\theta,\eta}:TM\longrightarrow T^{*}M$
defined by $\mathcal{X}_{\theta,\eta}(X)=X\lrcorner
d\theta+(X\lrcorner\theta)\theta+(X\lrcorner\eta)\eta$ is a bundle
isomorphism, $X_{f}$ is the only one vector field such that

$X_{f}\lrcorner
d\theta+(X_{f}\lrcorner\theta)\theta+(X_{f}\lrcorner\eta)\eta=df-(R_{z}f+f)\theta-
R_{t}f\eta.$ (43)

In canonical coordinates, we have

$X_{f}=\frac{\partial f}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-\left(\frac{\partial f}{\partial q^{i}}+p_{i}\frac{\partial f}{\partial
z}\right)\frac{\partial}{\partial p_{i}}+\left(p_{i}\frac{\partial f}{\partial
p_{i}}-f\right)\frac{\partial}{\partial z}.$ (44)

We can observe that the assignment $f\longmapsto X_{f}$ is linear, that is

$X_{f+\alpha g}=X_{f}+\alpha X_{g},$ (45)

$\forall f,g\in C^{\infty}(M)$ and $\forall\alpha\in\mathbb{R}$. Like contact
manifolds, cocontact manifolds are Jacobi manifolds; given $f,g\in
C^{\infty}(M)$ the Jacobi bracket of $f$ and $g$ is defined by

$\\{f,g\\}=X_{g}f+fR_{z}g.$ (46)

In canonical coordinates, we have

$\\{f,g\\}=\frac{\partial f}{\partial q^{i}}\frac{\partial g}{\partial
p_{i}}-\frac{\partial f}{\partial p_{i}}\frac{\partial g}{\partial
q^{i}}+\frac{\partial f}{\partial z}\left(p_{i}\frac{\partial g}{\partial
p_{i}}-g\right)-\frac{\partial g}{\partial z}\left(p_{i}\frac{\partial
f}{\partial p_{i}}-f\right).$ (47)

Of course the assignment $f\mapsto X_{f}$ also defines a Lie algebra
antihomomorphism between the Lie algebras $(C^{\infty}(M),\\{,\\})$ and
$(\mathfrak{X}(M),[,])$, indeed, for $f,g\in C^{\infty}(M)$ we have

$-[X_{f},X_{g}]=X_{\\{f,g\\}}.$ (48)

Time-dependent contact Hamiltonian systems are defined under the formalism of
cocontact geometry. Given $H\in C^{\infty}(M)$, the dynamics of the
Hamiltonian system on $(M,\theta,\eta)$ (the phase space) with Hamiltonian
function $H$ is defined by the evolution vector field $E_{H}=X_{H}+R_{t}$.
$(M,\theta,\eta,H)$ is a cocontact Hamiltonian system with
$n=\frac{dim(M)-2}{2}$ degrees of freedom. In canonical coordinates

$E_{H}=\frac{\partial H}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-\left(\frac{\partial H}{\partial q^{i}}+p_{i}\frac{\partial H}{\partial
z}\right)\frac{\partial}{\partial p_{i}}+\left(p_{i}\frac{\partial H}{\partial
p_{i}}-H\right)\frac{\partial}{\partial z}+\frac{\partial}{\partial t}.$ (49)

The trajectories
$\psi(s)=(t(s),q^{1}(s),\cdots,q^{n}(s),p_{1}(s),\cdots,p_{n}(s),z(s))$ of the
system are the integral curves of $E_{H}$, they satisfy the equations

$\dot{q^{i}}=\frac{\partial H}{\partial p_{i}},\hskip
28.45274pt\dot{p_{i}}=-\left(\frac{\partial H}{\partial
q^{i}}+p_{i}\frac{\partial H}{\partial z}\right),\hskip
28.45274pt\dot{z}=p_{i}\frac{\partial H}{\partial p_{i}}-H,\hskip
28.45274pt\dot{t}=1.$ (50)

Since $\dot{t}=1$, we can take $t=s$, which implies that the temporal
parameter for the system is $t$, that is, the trajectories of the system are
parametrized by $t$

$\psi(t)=(t,q^{1}(t),\cdots,q^{n}(t),p_{1}(t),\cdots,p_{n}(s),z(t))$ (51)

and they obey the dissipative Hamilton equations of motion

$\dot{q^{i}}=\frac{\partial H}{\partial p_{i}},\hskip
28.45274pt\dot{p_{i}}=-\left(\frac{\partial H}{\partial
q^{i}}+p_{i}\frac{\partial H}{\partial z}\right),\hskip
28.45274pt\dot{z}=p_{i}\frac{\partial H}{\partial p_{i}}-H.$ (52)

The evolution of a function $f\in C^{\infty}(M)$ along the trajectories of the
system is given by

$\dot{f}=E_{H}f=X_{H}f+R_{t}f=\\{f,H\\}-fR_{z}H+R_{t}f.$ (53)

So that $f$ is a constant of motion if and only if
$\\{f,H\\}-fR_{z}H+R_{t}f=0$.

##  4 Particular integrals and particular integrability for Hamiltonian
systems

###  4.1 Symplectic Hamiltonian systems

Let $(M,\omega,H)$ be a Hamiltonian system with $n$ degrees of freedom (${\rm
dim}(M)=2n$). The dynamical vector field is the Hamiltonian vector field
$X_{H}$, so a particular integral of $(M,\omega,H)$ is a function $f\in
C^{\infty}(M)$ such that $X_{H}f=af$ for some function $a\in C^{\infty}(M)$
with real values on $f=0$; in terms of the Poisson bracket we have that $f$ is
a particular integral if and only if $\\{f,H\\}=af$ (this is the original
notion of particular integral introduced in [11] and presented under the
symplectic framework in [12]). Let us suppose that $f\in C^{\infty}(M)$ is a
particular integral of $(M,\omega,H)$, we know that we can look for the
trajectories of the system that live in $M_{f}=\\{\,x\in M:\ f(x)\,=\,0\,\\}$,
which provided that $0$ is a regular value of $f$ is a smooth submanifold of
$M$ of dimension $2n-1$, and these trajectories are (locally) solutions of a
system of $2n-1$ differential equations. In canonical coordinates
$(q^{1},\cdots,q^{n},p_{1},\cdots,p_{n})$ on $(M,\omega)$, the trajectories of
the system are the solutions of the Hamilton’s equations of motion

$\dot{q^{i}}=\frac{\partial H}{\partial p_{i}},\hskip
28.45274pt\dot{p_{i}}=-\frac{\partial H}{\partial q^{i}}\qquad;\qquad
i=1,2,3,\ldots,n\ .$ (54)

In [12] it is shown that the trajectories of $(M,\omega,H)$ that live in
$M_{f}=\\{\,x\in M:\ f(x)\,=\,0\,\\}$ are solutions of the system of $2n-1$
differential equations

$\left\\{\begin{array}[]{c}\dot{Q}^{1}=\frac{\partial K}{\partial
P_{1}}\big{|}_{{}_{f=0}}\\\ \vdots\\\ \dot{Q}^{n-1}=\frac{\partial K}{\partial
P_{n-1}}\big{|}_{{}_{f=0}}\\\ \dot{Q}^{n}=\frac{\partial K}{\partial
f}\big{|}_{{}_{f=0}}\\\ \dot{P}_{1}=-\frac{\partial K}{\partial
Q^{1}}\big{|}_{{}_{f=0}}\\\ \vdots\\\ \dot{P}_{n-1}=-\frac{\partial
K}{\partial Q^{n-1}}\big{|}_{{}_{f=0}}\\\ \end{array}\right.,$ (55)

where $(Q,P)$ are canonical coordinates on $(M,\omega)$ such that $P_{n}=f$
(such a canonical transformation can be described by a generating function of
the form $F=F_{2}(q^{1},\cdots,q^{n},P_{1},\cdots,P_{n-1},f)-Q^{i}P_{i}$ [4,
36]) and $(Q^{1},\cdots,Q^{n},P_{1},\cdots,P_{n-1})$ are local coordinates on
$M_{f}$. As it is remarked in [12], system (55) is not a system of Hamilton
equations of motion since it is a system of an odd number of differential
equations. Nevertheless, in [12] it is shown that the solutions of the system
(55) can be found by solving a system of $2n-2$ Hamilton equations of motion
and quadratures; we present the most important aspects of this reduction
process, for details see [12]. The system (55) is the system of equations of
motion for the dynamical system defined by $X_{H}$ on $M_{f}$ which is not
Hamiltonian. Some remarks are in order:

  * •

Since we are taking $P_{n}=f=0$, thus, $\dot{P}_{n}=-\frac{\partial
K}{\partial Q^{n}}\big{|}_{{}_{f=0}}=0$, so the function $K\big{|}_{M_{f}}$
does not depend on the coordinate $Q^{n}$. Therefore, the time evolution of
each coordinate $Q^{1},\cdots,Q^{n-1},P_{1},\cdots,P_{n-1}$ on $M_{f}$ is
independent of $Q^{n}$.

  * •

Because $P_{n}=f$ is not a coordinate on $M_{f}$, formally the term
$\frac{\partial K}{\partial f}\big{|}_{{}_{f=0}}$ can not be calculated as a
derivative on $M_{f}$. In fact, it is a function defined on $M$ restricted to
$M_{f}$. Hence, it follows that the time evolution of $Q^{n}$,
$\dot{Q}^{n}=\frac{\partial K}{\partial f}\big{|}_{{}_{f=0}}$, may depend on
$Q^{n}$ as well.

  * •

One can solve the original system (55) by solving first the reduced system

$\left\\{\begin{array}[]{c}\dot{Q}^{1}=\frac{\partial K}{\partial
P_{1}}\big{|}_{{}_{f=0}}\\\ \vdots\\\ \dot{Q}^{n-1}=\frac{\partial K}{\partial
P_{n-1}}\big{|}_{{}_{f=0}}\\\ \dot{P}_{1}=-\frac{\partial K}{\partial
Q^{1}}\big{|}_{{}_{f=0}}\\\ \vdots\\\ \dot{P}_{n-1}=-\frac{\partial
K}{\partial Q^{n-1}}\big{|}_{{}_{f=0}}\\\ \end{array}\right.\ ,$ (56)

and afterwards integrating the equation $\dot{Q}^{n}=\frac{\partial
K}{\partial f}(Q^{1},\cdots,Q^{n-1},Q^{n},P_{1},\cdots,P_{n-1})$.

System (56) is a system of Hamilton equations of motion in the coordinates
$(Q^{1},\cdots,Q^{n-1},P_{1},\cdots,P_{n-1})$, which are coordinates on the
level submanifold $M_{f,Q}=\\{x\in M:\ f(x)=0,\ Q^{n}(x)={\rm constant}\\}$.
Indeed, $(M_{f,Q},\omega\big{|}_{M_{f,Q}})$ is a symplectic manifold of
dimension $2n-2$ and system (56) is the system of Hamilton equations of motion
in the lower-dimensional phase space $(M_{f,Q},\omega\big{|}_{M_{f,Q}})$ with
Hamiltonian function
$K\big{|}_{M_{f}}=K\big{|}_{M_{f}}(Q^{1},\cdots,Q^{n-1},P_{1},\cdots,P_{n-1})$.
The dynamics defined by the Hamiltonian system
$(M_{f,Q},\omega\big{|}_{M_{f,Q}},K\big{|}_{M_{f}})$ is the projection of the
dynamics of $X_{H}\big{|}_{M_{f}}$ into the submanifold $M_{f,Q}$ in the sense
that if $\pi:M_{f}\longrightarrow M_{f,Q}$ is the projection map from $M_{f}$
to the submanifold $M_{f,Q}$ defined in the canonical coordinates $(Q,P)$ by

$\pi(Q^{1},\cdots,Q^{n-1},Q^{n},P_{1},\cdots,P_{n-1})=(Q^{1},\cdots,Q^{n-1},P_{1},\cdots,P_{n-1}),$
(57)

then the trajectories of the system
$(M_{f,Q},\omega\big{|}_{M_{f,Q}},K\big{|}_{M_{f}})$ are images by $\pi$ of
trajectories of the system defined by $X_{H}$ on $M_{f}$, i.e.,
$\gamma(t)=(\,Q^{1}(t),\cdots,Q^{n-1}(t),P_{1}(t),\cdots,P_{n-1}(t)\,)$ is a
trajectory of the system $(M_{f,Q},\omega\big{|}_{M_{f,Q}},K\big{|}_{M_{f}})$
if and only if there is a trajectory
$\overline{\gamma(t)}=(Q^{1}(t),\cdots,Q^{n-1}(t),Q^{n}(t),P_{1}(t),\cdots,P_{n-1}(t))$
such that $\pi(\overline{\gamma(t)})=\gamma(t)$ for every $t$. To lift the
dynamics from $M_{f,Q}$ to $M_{f}$ we consider the trajectories
$\gamma(t)=(Q^{1}(t),\cdots,Q^{n-1}(t),P_{1}(t),\cdots,P_{n-1}(t))$ of the
system $(M_{f,Q},\omega\big{|}_{M_{f,Q}},K\big{|}_{M_{f}})$ and construct the
trajectories
$\overline{\gamma(t)}=(Q^{1}(t),\cdots,Q^{n-1}(t),Q^{n}(t),P_{1}(t),\cdots,P_{n-1}(t))$
on $M_{f}$ by integrating the equation $\dot{Q}^{n}=\frac{\partial K}{\partial
f}(Q^{1},\cdots,Q^{n},P_{1},\cdots,P_{n-1})$. It is worth mentioning that this
reduction process is possible due to the fact that the time evolution of each
coordinate $Q^{1},\cdots,Q^{n-1},P_{1},\cdots,P_{n-1}$ is independent of
$Q^{n}$.

On the other hand, the notion of particular integrability as a more general
notion than Liouville integrability has been presented in [12]. Let us see a
brief review.

######  Definition 5.

We say that the functions $f_{1},\cdots,f_{k}\in C^{\infty}(M)$ are in
particular involution if

$\\{f_{i},f_{j}\\}=a^{ij}_{1}f_{1}+\cdots+a^{ij}_{n}f_{k}$ (58)

for some functions $a^{ij}_{1},\cdots,a^{ij}_{k}\in C^{\infty}(M)$ with real
values on $f_{1}=\cdots=f_{k}=0$.

It is worth remarking that the functions $a^{ij}_{1},\cdots,a^{ij}_{k}\in
C^{\infty}(M)$ in the previous definition are such that

$\lim_{x\to x_{0}}a^{ij}_{s}(x)\ =\
a^{ij}_{s}(x_{0})\in\mathbb{R}\quad,\qquad\forall\,\,x_{0}\in M_{f}=\\{x\in
M:\ f_{1}(x)=\cdots=f_{k}(x)=0\\}\ .$

The following theorem is presented as a (partial) generalization of the
Liouville theorem on integrability by quadratures.

######  Theorem 1.

If $H=f_{1},\cdots,f_{n}\in C^{\infty}(M)$ are functionally independent and in
particular involution functions, then for the Hamiltonian system
$(M,\omega,H)$, the trajectories of the system that live in $M_{f}=\\{x\in M:\
f_{1}(x)=\cdots=f_{n}(x)=0\\}$ can be found by quadratures.

######  Definition 6.

A Hamiltonian system $(M,\omega,H)$ with $n$ degrees of freedom is said to be
particularly integrable if there exist $n$ independent functions
$H=f_{1},\cdots,f_{n}\in C^{\infty}(M)$ in particular involution.

The proof of theorem (1) is based on the fact that the Hamiltonian vector
fields $X_{f_{1}}|_{M_{f}},\cdots,X_{f_{n}}|_{M_{f}}$, where $M_{f}=\\{x\in
M:\ f_{1}(x)=\cdots=f_{n}(x)=0\\}$ is a smooth submanifold of $M$ of dimension
$dim(M)-n=2n-n=n$, are symmetries of $X_{H}|_{M_{f}}$ and generate an Abelian
(therefore solvable) Lie algebra with the Lie bracket of vector fields; then
by Lie’s theorem on integrability, the solutions of the equations of motion of
the dynamical system defined by $X_{H}|_{M_{f}}$ can be found by quadratures.
The concept of solvable Lie algebra is more general than that of Abelian Lie
algebra, indeed, every Abelian Lie algebra is trivially a solvable Lie algebra
[37, 38]. Let $v$ be a smooth vector field on $\mathbb{R}^{n}$, the Lie’s
theorem on integrability [2, 39, 40, 41, 17] establishes sufficient conditions
for the dynamical system defined by the vector differential equation
$\dot{x}=v(x)$ to be integrable by quadratures, it reads as follows:

######  Theorem 2.

Let $u_{1},\ldots u_{n}$ be linearly independent possible time-dependent
smooth vector fields on $\mathbb{R}^{n}$. If $u_{1},\ldots,u_{n}$ are
symmetries of the possible time-dependent smooth vector field $v$ (i.e.,
$[u,v]=0$) and they generate a solvable Lie algebra with the Lie bracket $[,]$
of vector fields, then the solutions of the vector differential equation
$\dot{x}=v(x,t)$ can be found by quadratures.

Finally, in [12] is is shown that, as the involution condition for Liouville
integrability, the condition for particular integrability is also a maximal
condition in the following sense.

######  Proposition 1.

For a given symplectic Hamiltonian system we have that the maximum number of
functionally independent functions in particular involution (satisfying
condition (58)) is equal to the number of degrees of freedom.

###  4.2 Cosymplectic Hamiltonian systems

Let $(M,\Omega,\eta,H)$ be a cosymplectic Hamiltonian system with $n$ degrees
of freedom (${\rm dim}(M)=2n+1$). In order to introduce the notion of
particular integrability in the cosymplectic formalism, we follow the spirit
of the original notion, i.e., a particular integral must be characterized by
the condition $\\{f,H\\}=af$ for some function $a\in C^{\infty}(M)$ with real
values on $f=0$. So we consider a particular integral of $(M,\Omega,\eta,H)$
as a function $f\in C^{\infty}(M)$ such that $X_{H}f=af$ for some function
$a\in C^{\infty}(M)$ with real values on $f=0$; in terms of the Poisson
bracket we have that $f$ is a particular integral if and only if
$\\{f,H\\}=af$. It is important to remark that in the cosymplectic framework,
the dynamical vector field of the Hamiltonian system $(M,\Omega,\eta,H)$ is
the evolution vector field $E_{H}=X_{H}+R$ (for details see section 3.2), so
in order to be able to look for the trajectories of the system that live in
$M_{f}=\\{\,x\in M:\ f(x)\,=\,0\,\\}$ we must have $E_{H}f|_{f=0}=0$, which is
fulfilled by requiring that $Rf=0$ ($f$ is time-independent). In conclusion,
we propose the following definition.

######  Definition 7.

A particular integral of $(M,\Omega,\eta,H)$ is a function $f\in
C^{\infty}(M)$ such that $Rf=0$ and $X_{H}f=af$ for some function $a\in
C^{\infty}(M)$ with real values on $f=0$.

Let us suppose that $f\in C^{\infty}(M)$ is a particular integral of
$(M,\Omega,\eta,H)$, we know that we can look for the trajectories of the
system that live in $M_{f}=\\{\,x\in M:\ f(x)\,=\,0\,\\}$, which provided that
$0$ is a regular value of $f$ is a smooth submanifold of dimension
$2n+1-1=2n$. In canonical coordinates
$(q^{1},\cdots,q^{n},p_{1},\cdots,p_{n},t)$ on $(M,\Omega,\eta)$, the
trajectories of the system are the solutions of the Hamilton equations of
motion

$\dot{q^{i}}=\frac{\partial H}{\partial p_{i}},\hskip
28.45274pt\dot{p_{i}}=-\frac{\partial H}{\partial q^{i}}\qquad;\qquad
i=1,2,3,\ldots,n\ .$ (59)

Let us suppose we can find a canonical transformation $(q,p,t)\mapsto(Q,P,t)$
such that $P_{n}=f$, then the Hamilton equations of motion in the new
coordinates $(Q,P,t)$ are

$\left\\{\begin{array}[]{c}\dot{Q}^{1}=\frac{\partial K}{\partial P_{1}}\\\
\vdots\\\ \dot{Q}^{n-1}=\frac{\partial K}{\partial P_{n-1}}\\\
\dot{Q}^{n}=\frac{\partial K}{\partial f}\\\ \dot{P}_{1}=-\frac{\partial
K}{\partial Q^{1}}\\\ \vdots\\\ \dot{P}_{n-1}=-\frac{\partial K}{\partial
Q^{n-1}}\\\ \dot{P}_{n}=\dot{f}=-\frac{\partial K}{\partial
Q^{n}}\end{array}\right.\ ,$ (60)

with $K=K(Q,P,t)$ the new Hamiltonian function. Such a canonical
transformation can be described by a generating function of the form
$F=F_{2}(q^{1},\cdots,q^{n},P_{1},\cdots,P_{n-1},f,t)-Q^{i}P_{i}$ [4, 36].
Since we are looking for the trajectories that live in $M_{f}$ then we set up
$f=0$. Therefore, at $f=0$ the system (60) reduces to $P_{n}=f=0$ and system
(55) (formally the reduced system is not exactly system (55) since in this
case the function $K$ is a possible time-dependent function, nevertheless it
has exactly the same form, so we use the same form system). In this case, the
system (55) is the non-autonomous system of equations of motion for the
dynamical system defined by $X_{H}|_{M_{f}}$, i.e., the mechanical system
defined by $X_{H}$ on $M_{f}$ which is not Hamiltonian. The same remarks in
the symplectic case are valid in the cosymplectic framework: the function
$K\big{|}_{M_{f}}$ does not depend on the coordinate $Q^{n}$, therefore, the
time evolution of each coordinate $Q^{1},\cdots,Q^{n-1},P_{1},\cdots,P_{n-1}$
on $M_{f}$ is independent of $Q^{n}$; the time evolution of $Q^{n}$,
$\dot{Q}^{n}=\frac{\partial K}{\partial f}\big{|}_{{}_{f=0}}$, may depend on
$Q^{n}$ as well. So one can solve the original system (55) by solving first
the reduced system (56) and afterwards integrating the equation
$\dot{Q}^{n}=\frac{\partial K}{\partial
f}(Q^{1},\cdots,Q^{n-1},Q^{n},P_{1},\cdots,P_{n-1},t)$.
$(Q^{1},\cdots,Q^{n-1},P_{1},\cdots,P_{n-1},t)$ are coordinates on the level
submanifold $M_{f,Q}=\\{x\in M:\ f(x)=0,\ Q^{n}(x)={\rm constant}\\}$, the
system (56) is the system of Hamilton equations of motion for the Hamiltonian
system with phase space
$(M_{f,Q},\Omega\big{|}_{M_{f,Q}},\eta\big{|}_{M_{f,Q}})$ and Hamiltonian
function
$K\big{|}_{M_{f}}=K\big{|}_{M_{f}}(Q^{1},\cdots,Q^{n-1},P_{1},\cdots,P_{n-1},t)$.
As in the symplectic framework, the dynamics defined by the cosymplectic
Hamiltonian system
$(M_{f,Q},\Omega\big{|}_{M_{f,Q}},\eta\big{|}_{M_{f,Q}},K\big{|}_{M_{f}})$
with $n$ degrees of freedom is the projection of the dynamics of
$X_{H}\big{|}_{M_{f}}$ into the submanifold $M_{f,Q}$.

For cosymplectic Hamiltonian systems the notion of Liouville integrability is
completely analogous to the one in the symplectic case, namely, for a
cosymplectic Hamiltonian system with $n$ degrees of freedom, the existence of
$n$ functionally independent constants of motion in involution allows to find
the solutions of the Hamilton equations of motion by quadratures [42, 43, 44].
Now we introduce the notion of particular integrability in the framework of
cosymplectic geometry. Observe that the concept of particular involution only
involve the Poisson bracket, so in the cosymplectic case we have the same
notion, i.e., the functions $f_{1},\cdots,f_{k}\in C^{\infty}(M)$ are in
particular involution if they satisfy condition (58) for some functions
$a^{ij}_{1},\cdots,a^{ij}_{k}\in C^{\infty}(M)$ with real values on
$f_{1}=\cdots=f_{k}=0$. Eventually, we arrive to the following theorem
analogous to theorem 1.

######  Theorem 3.

Let $f_{1},\cdots,f_{n}\in C^{\infty}(M)$ be functionally independent and in
particular involution functions such that $Rf_{i}=0$. If for each
$i\in\\{1,\cdots,n\\}$ $\\{f_{i},H\\}=c^{i1}f_{1}+\cdots+c^{in}f_{n}$ for some
$c^{ij}\in C^{\infty}(M)$ with real values on $f_{1}=\cdots=f_{n}=0$, then the
trajectories of the Hamiltonian system $(M,\Omega,\eta,H)$ that live in
$M_{f}=\\{x\in M:\ f_{1}(x)=\cdots=f_{n}(x)=0\\}$ can be found by quadratures.

Observe that $f_{1},\cdots,f_{n}$ in theorem 3 may be particular integrals or
even constants of motion in particular involution, of course we can see that
the condition $\\{f_{i},H\\}=c^{i1}f_{1}+\cdots+c^{in}f_{n}$ is more general
than the condition for a particular integral. We can also observe that we have
not taken $H$ as one of the functions $f_{1},\cdots,f_{n}$, it is because in
the cosymplectic framework $H$ is not necessarily a particular integral (we
may have $RH\neq 0$).

Before giving the proof of theorem 3 we present a brief review of the Lie
integrability by quadratures for a special kind of time-dependent smooth
vector fields. For our purposes we need to consider smooth vector fields $v$
on $\mathbb{R}^{n}\times\mathbb{R}$ of the form
$v(x,t)=v^{1}(x,t)\frac{\partial}{\partial
x^{1}}+\cdots+v^{n}(x,t)\frac{\partial}{\partial
x^{n}}+\frac{\partial}{\partial t}$ (observe that this is the local form of
the evolution vector field $E_{H}$), with $t$ the coordinate in $\mathbb{R}$.
If we consider the dynamical system on $\mathbb{R}^{n}\times\mathbb{R}$
defined by the vector field $v$, then the equations of motion are

$\left\\{\begin{array}[]{c}\dot{x}^{1}=v^{1}(x^{1},\ldots,x^{n},t),\\\
\vdots\\\ \dot{x}^{n}=v^{n}(x^{1},\ldots,x^{n},t),\\\
\dot{t}=1\end{array}\right.$ (61)

By integrating the last equation we have the solution for the variable $t$,
then we only have to solve a system of $n$ differential equations

$\left\\{\begin{array}[]{c}\dot{x}^{1}=v^{1}(x^{1},\ldots,x^{n},t),\\\
\vdots\\\ \dot{x}^{n}=v^{n}(x^{1},\ldots,x^{n},t),\end{array}\right.,$ (62)

which can be solved by quadratures provided the existence of $n$ linearly
independent vector fields of the form $u^{1}(x,t)\frac{\partial}{\partial
x^{1}}+\cdots+u^{n}(x,t)\frac{\partial}{\partial x^{n}}$ that form a solvable
Lie algebra of symmetries of the vector field
$\overline{v}=v^{1}(x,t)\frac{\partial}{\partial
x^{1}}+\cdots+v^{n}(x,t)\frac{\partial}{\partial x^{n}}$.

The following result well known in the symplectic framework is fundamental for
the proof of theorem 3.

######  Lemma 2.

If $f_{1},\cdots,f_{k}\in C^{\infty}(M)$ are functionally independent
functions such that $Rf_{i}=0$ for $i=1,\cdots,k$ then the Hamiltonian vector
fields $X_{f_{1}},\ldots,X_{f_{k}}$ are linearly independent.

###### Proof.

Let us suppose that $c_{1}X_{f_{1}}+\cdots+c_{k}X_{f_{k}}=0$ for some
$c_{1},\cdots,c_{k}\in\mathbb{R}$. We have that
$\left(c_{1}X_{f_{1}}+\cdots+c_{k}X_{f_{k}}\right)\lrcorner\Omega=c_{1}df_{1}+\cdots+c_{k}df_{k}$
then

$c_{1}df_{1}+\cdots+c_{k}df_{k}=0,$ (63)

which means that for regular points (points where the differential
$df_{1},\cdots,df_{k}$ are linearly independent) the vector fields
$X_{f_{1}},\ldots,X_{f_{k}}$ are linearly independent. Of course in the
symplectic framework this result is obtained from the fact that
$X_{f}\lrcorner\omega=df$ for each $f\in C^{\infty}(M)$. ∎

Now we present the proof of theorem 3.

###### Proof.

Since the functions $f_{1},\cdots,f_{n}$ are functionally independent on $M$
then $M_{f}=\\{x\in M:\ f_{1}(x)=\cdots=f_{n}(x)=0\\}$ is a smooth submanifold
of $M$ of dimension $dim(M)-n=2n+1-n=n+1$, and since $Rf_{1}=\cdots=Rf_{n}=0$,
i.e., the functions $f_{1},\cdots,f_{n}$ are time-independent, then around any
point in $M_{f}$ we can find local coordinates $(y^{1},\cdots,y^{n},t)$.

On the other hand we have that the evolution vector field is tangent to
$M_{f}$, indeed,
$E_{H}f_{i}=\\{f_{i},H\\}+Rf_{i}=c^{i1}f_{1}+\cdots+c^{in}f_{n}$ which is zero
on $M_{f}$; so the trajectories of the Hamiltonian system $(M,\Omega,\eta,H)$
that live in $M_{f}$ are integral curves of $E_{H}|_{M_{f}}$ which in local
coordinates $(y^{1},\cdots,y^{n},t)$ on $M_{f}$ has the form

$E_{H}(y,t)=v^{1}(y,t)\frac{\partial}{\partial
y^{1}}+\cdots+v^{n}(y,t)\frac{\partial}{\partial
y^{n}}+\frac{\partial}{\partial t},$ (64)

i.e., the trajectories of the Hamiltonian system $(M,\Omega,\eta,H)$ that live
in $M_{f}$ have the local form $\gamma(t)=(y^{1}(t),\cdots,y^{n}(t),t)$ where
the functions $y^{1}(t),\cdots,y^{n}(t)$ are solutions of a system of
differential equations of the form (62), namely,

$\left\\{\begin{array}[]{c}\dot{y}^{1}=v^{1}(y^{1},\ldots,y^{n},t),\\\
\vdots\\\ \dot{y}^{n}=v^{n}(y^{1},\ldots,y^{n},t),\end{array}\right.,$ (65)

which can be solved by quadratures provided the existence of $n$ linearly
independent vector fields of the form $u^{1}(y,t)\frac{\partial}{\partial
y^{1}}+\cdots+u^{n}(y,t)\frac{\partial}{\partial y^{n}}$ that form a solvable
Lie algebra of symmetries of the vector field
$\overline{E}_{H}|_{M_{f}}=v^{1}(y,t)\frac{\partial}{\partial
y^{1}}+\cdots+v^{n}(y,t)\frac{\partial}{\partial y^{n}}$. We have that the
vector fields $X_{f_{1}},\ldots,X_{f_{n}}$ are tangent to the submanifold
$M_{f}$ because $X_{f_{i}}f_{j}=\\{f_{j},f_{i}\\}$ which is zero on $M_{f}$,
so we can consider the vector fields
$X_{f_{1}}|_{M_{f}},\cdots,X_{f_{n}}|_{M_{f}}$ on $M_{f}$, which have the
local form $u_{i}^{1}(y,t)\frac{\partial}{\partial
y^{1}}+\cdots+u_{i}^{n}(y,t)\frac{\partial}{\partial y^{n}}$ and since
$X_{\\{f,g\\}}=-[X_{f},X_{g}]$, we have that they generate an Abelian
(therefore solvable) Lie algebra with the Lie bracket of vector fields. In
addition we have that $X_{f_{1}}|_{M_{f}},\cdots,X_{f_{n}}|_{M_{f}}$ are
symmetries of $\overline{E}_{H}|_{M_{f}}$, indeed, we can observe that
$\overline{E}_{H}|_{M_{f}}=X_{H}|_{M_{f}}$ so

$[\overline{E}_{H}|_{M_{f}},X_{f_{i}}|_{M_{f}}]=[X_{H}|_{M_{f}},X_{f_{i}}|_{M_{f}}]=X_{\\{f_{i},H\\}}|_{M_{f}}=0.$
(66)

We conclude that the vector fields
$X_{f_{1}}|_{M_{f}},\cdots,X_{f_{n}}|_{M_{f}}$ on $M_{f}$, which have the
local form $u_{i}^{1}(y,t)\frac{\partial}{\partial
y^{1}}+\cdots+u_{i}^{n}(y,t)\frac{\partial}{\partial y^{n}}$, form a solvable
Lie algebra of symmetries of the vector field $\overline{E}_{H}|_{M_{f}}$,
therefore the trajectories of the Hamiltonian system $(M,\Omega,\eta,H)$ that
live in $M_{f}$ can be found by quadratures. ∎

######  Definition 8.

A cosymplectic Hamiltonian system $(M,\Omega,\eta,H)$ with $n$ degrees of
freedom is said to be particularly integrable if there exist $n$ functionally
independent functions $f_{1},\cdots,f_{n}\in C^{\infty}(M)$ in particular
involution such that $Rf_{i}=0$ and for each $i\in\\{1,\cdots,n\\}$
$\\{f_{i},H\\}=c^{i1}f_{1}+\cdots+c^{in}f_{n}$ for some $c^{ij}\in
C^{\infty}(M)$ with real values on $f_{1}=\cdots=f_{n}=0$.

We finish this subsection by showing that as in the symplectic framework, the
condition for particular integrability is also a maximal condition.

######  Proposition 2.

For a given cosymplectic Hamiltonian system we have that the maximum number of
functionally independent functions $f_{i}$ in particular involution with
$Rf_{i}=0$ is equal to the number of degrees of freedom of the system.

###### Proof.

The proof is completely analogous to the one given in [12] for the symplectic
case. Let us suppose that for a cosymplectic Hamiltonian system with $n$
degrees of freedom there exist more than $n$ functionally independent
functions $f_{1},\cdots,f_{s}$ with $s>n$ and $Rf_{i}=0$ that satisfy
condition (58). Since the functions $f_{1},\cdots,f_{s}$ are functionally
independent on $M$ then $M_{f}=\\{x\in M:\ f_{1}(x)=\cdots=f_{s}(x)=0\\}$ is a
smooth submanifold of $M$ of dimension $dim(M)-s=2n+1-s<s$. The vector fields
$X_{f_{1}},\ldots,X_{f_{s}}$ are tangent to the submanifold $M_{f}$ because
$X_{f_{i}}f_{j}=\\{f_{j},f_{i}\\}$ which is zero on $M_{f}$, so we can
consider the vector fields $X_{f_{1}}|_{M_{f}},\ldots,X_{f_{s}}|_{M_{f}}$ on
$M_{f}$, we have that they are linearly independent at any regular point $p\in
M_{f}$, so they generate a vector subspace of $T_{p}M_{f}$ of dimension $s$
but the dimension of $T_{p}M_{f}$ is $2n-s$ which is less than $s$, so we have
a contradiction, we conclude that the maximum number of functionally
independent functions in particular involution is $n$ (the number of degrees
of freedom). ∎

###  4.3 Contact Hamiltonian systems

Let $(M,\theta,H)$ be a contact Hamiltonian system with $n$ degrees of freedom
(${\rm dim}(M)=2n+1$). We introduce the notion of particular integral as
follows.

######  Definition 9.

A particular integral of $(M,\theta,H)$ is a function $f\in C^{\infty}(M)$
such that $X_{H}f=af$ for some function $a\in C^{\infty}(M)$ with real values
on $f=0$.

We can observe that with this definition we still have the original
characterization $\\{f,H\\}=af$ (where in this case $\\{,\\}$ defines a Jacobi
bracket, see section 3.3 for details). Indeed, we know that
$X_{H}f=\\{f,H\\}-fRH$, so $f\in C^{\infty}(M)$ is a particular integral if
and only if $af=\\{f,H\\}-fRH$ for some function $a\in C^{\infty}(M)$, if and
only if $\\{f,H\\}=af+fRH=bf$ with $b=a+RH\in C^{\infty}(M)$.

In [14, 18, 13] the concept of constant of motion (conserved quantity) for the
case of dissipative systems is generalized to the concept of dissipated
quantity as follows.

######  Definition 10.

A dissipated quantity of the contact Hamiltonian system $(M,\theta,H)$ is a
function $f\in C^{\infty}(M)$ that is dissipated at the same rate as the
contact Hamiltonian $H$, i.e., $X_{H}f=-fRH$.

The present notion of particular integral generalizes both concepts of
constant of motion and dissipated quantity. Indeed, it is clear that a
particular integral $f$ such that $X_{H}f=af$ with $a=0$ is a constant of
motion and with $a=-RH$ is a dissipated quantity.

Let us suppose that $f\in C^{\infty}(M)$ is a particular integral of
$(M,\theta,H)$, we know that we can look for the trajectories of the system
that live in $M_{f}=\\{\,x\in M:\ f(x)\,=\,0\,\\}$, which provided that $0$ is
a regular value of $f$, is a smooth submanifold of dimension $2n+1-1=2n$. In
canonical coordinates $(q^{1},\cdots,q^{n},p_{1},\cdots,p_{n},z)$ on
$(M,\theta)$, the trajectories of the system are the solutions of the
dissipative Hamilton equations of motion

$\dot{q^{i}}=\frac{\partial H}{\partial p_{i}},\hskip
28.45274pt\dot{p_{i}}=-\left(\frac{\partial H}{\partial
q^{i}}+p_{i}\frac{\partial H}{\partial z}\right),\hskip
28.45274pt\dot{z}=p_{i}\frac{\partial H}{\partial p_{i}}-H.$ (67)

Canonical transformations in contact mechanics are defined as transformations
leaving the contact form invariant [16], they are a particular case of the so
called contact transformations, which are transformation that leaves the
contact form invariant up to multiplication by a conformal factor [45, 30]. A
canonical transformation of the form $(q,p,z)\mapsto(Q,P,Z)$ such that
$Q_{n}=f$ can be obtained by a generating function of the form
$Z=z-F_{1}(q^{1},\cdots,q^{n},Q^{1},\cdots,Q^{n-1},f)$, where
$F_{1}(q^{1},\cdots,q^{n},Q^{1},\cdots,Q^{n-1},f)$ is the generating function
of a symplectic canonical transformation [30]. Let us suppose that we can find
a canonical transformation $(q,p,z)\mapsto(Q,P,Z)$ such that $P_{n}=f$, then
the Hamilton equations of motion in the new coordinates $(Q,P,Z)$ are

$\left\\{\begin{array}[]{c}\dot{Q}^{1}=\frac{\partial K}{\partial P_{1}}\\\
\vdots\\\ \dot{Q}^{n-1}=\frac{\partial K}{\partial P_{n-1}}\\\
\dot{Q}^{n}=\frac{\partial K}{\partial f}\\\ \dot{P}_{1}=-\left(\frac{\partial
K}{\partial Q^{1}}+P_{1}\frac{\partial K}{\partial Z}\right)\\\ \vdots\\\
\dot{P}_{n-1}=-\left(\frac{\partial K}{\partial Q^{n-1}}+P_{n-1}\frac{\partial
K}{\partial Z}\right)\\\ \dot{P}_{n}=\dot{f}=-\left(\frac{\partial K}{\partial
Q^{n}}+f\frac{\partial K}{\partial Z}\right)\\\ \dot{Z}=P_{i}\frac{\partial
K}{\partial P_{i}}-K\end{array}\right.\ ,$ (68)

with $K=K(Q,P,Z)$ the new Hamiltonian function. Since we are looking for the
solutions of the Hamilton equations of motion that live in $M_{f}$ then we set
up $f=0$. Therefore, at $f=0$ the system (68) reduces to $P_{n}=f=0$ and
system

$\left\\{\begin{array}[]{c}\dot{Q}^{1}=\frac{\partial K}{\partial
P_{1}}\big{|}_{{}_{f=0}}\\\ \vdots\\\ \dot{Q}^{n-1}=\frac{\partial K}{\partial
P_{n-1}}\big{|}_{{}_{f=0}}\\\ \dot{Q}^{n}=\frac{\partial K}{\partial
f}\big{|}_{{}_{f=0}}\\\ \dot{P}_{1}=-\left(\frac{\partial K}{\partial
Q^{1}}+P_{1}\frac{\partial K}{\partial Z}\right)\big{|}_{{}_{f=0}}\\\
\vdots\\\ \dot{P}_{n-1}=-\left(\frac{\partial K}{\partial
Q^{n-1}}+P_{n-1}\frac{\partial K}{\partial Z}\right)\big{|}_{{}_{f=0}}\\\
\dot{Z}=\left(P_{i}\frac{\partial K}{\partial
P_{i}}-K\right)\big{|}_{{}_{f=0}}.\end{array}\right..$ (69)

So as in the symplectic and cosymplectic frameworks, by means of a particular
integral we can find solutions of the Hamilton equations of motion that are
solutions of a reduced systems of differential equations, i.e., we can find
trajectories of a contact Hamiltonian system that are trajectories of a
reduced dynamical system. Analogous remarks as in the symplectic framework are
in order:

  * •

Since we are taking $P_{n}=f=0$, thus, $\dot{P}_{n}=-\frac{\partial
K}{\partial Q^{n}}\big{|}_{{}_{f=0}}=0$, so the function $K\big{|}_{M_{f}}$
does not depend on the coordinate $Q^{n}$. Therefore, the time evolution of
each coordinate $Q^{1},\cdots,Q^{n-1},P_{1},\cdots,P_{n-1},Z$ on $M_{f}$ is
independent of $Q^{n}$.

  * •

The time evolution of $Q^{n}$, $\dot{Q}^{n}=\frac{\partial K}{\partial
f}\big{|}_{{}_{f=0}}$, may depend on $Q^{n}$ as well.

  * •

One can solve the original system (69) by solving first the reduced system

$\left\\{\begin{array}[]{c}\dot{Q}^{1}=\frac{\partial K}{\partial
P_{1}}\big{|}_{{}_{f=0}}\\\ \vdots\\\ \dot{Q}^{n-1}=\frac{\partial K}{\partial
P_{n-1}}\big{|}_{{}_{f=0}}\\\ \dot{P}_{1}=-\left(\frac{\partial K}{\partial
Q^{1}}+P_{1}\frac{\partial K}{\partial Z}\right)\big{|}_{{}_{f=0}}\\\
\vdots\\\ \dot{P}_{n-1}=-\left(\frac{\partial K}{\partial
Q^{n-1}}+P_{n-1}\frac{\partial K}{\partial Z}\right)\big{|}_{{}_{f=0}}\\\
\dot{Z}=\left(P_{i}\frac{\partial K}{\partial
P_{i}}-K\right)\big{|}_{{}_{f=0}}\,(i=1,\cdots n-1)\end{array}\right.\ ,$ (70)

and afterwards integrating the equation $\dot{Q}^{n}=\frac{\partial
K}{\partial f}(Q^{1},\cdots,Q^{n-1},Q^{n},P_{1},\cdots,P_{n-1},Z)$.

System (70) is the system of Hamilton equations of motion for the Hamiltonian
function
$K\big{|}_{M_{f}}=K\big{|}_{M_{f}}(Q^{1},\cdots,Q^{n-1},P_{1},\cdots,P_{n-1},Z)$
in the coordinates $(Q^{1},\cdots,Q^{n-1},P_{1},\cdots,P_{n-1},Z)$, which are
coordinates on the level submanifold $M_{f,Q}=\\{x\in M:\ f(x)=0,\
Q^{n}(x)={\rm constant}\\}$. The dynamics defined by the Hamiltonian system
$(M_{f,Q},\omega\big{|}_{M_{f,Q}},K\big{|}_{M_{f}})$ is the projection of the
dynamics of $X_{H}\big{|}_{M_{f}}$ into the submanifold $M_{f,Q}$ in the same
sense as in the symplectic case.

######  Example 1.

Let us consider the motion of a particle in a vertical plane under the action
of constant gravity with air friction. In canonical coordinates
$(x,y,p_{x},p_{y},z)$ the Hamiltonian function is
$H=\frac{1}{2m}(p_{x}^{2}+p_{y}^{2})+mgy+\gamma z$ [18]. We have that
$f=p_{x}$ is a dissipated quantity, therefore a particular integral, indeed
$\\{p_{x},H\\}=0$. By taking $p_{x}=0$ we have that the system of Hamilton
equations of motion reduces to

$\left\\{\begin{array}[]{c}\dot{y}=\frac{\partial K}{\partial p_{y}}\\\
\dot{p}_{y}=-\frac{\partial K}{\partial y}+p_{y}\frac{\partial K}{\partial
z}\\\ \dot{z}=p_{y}\frac{\partial K}{\partial p_{y}}-K\end{array}\right.\ ,$
(71)

and afterwards integrating the equation $\dot{x}=\frac{\partial H}{\partial
p_{x}}$, where $K=H\big{|}_{p_{x}=0}=\frac{1}{2m}p_{y}^{2}+mgy+\gamma z$.

Now we address the problem of introducing the notion of particular
integrability in the framework of contact geometry. The notion of Liouville
integrability in contact Hamiltonian mechanics is a bit different from the one
in the symplectic or cosymplectic framework; we know that there are important
geometric differences between the phase spaces of contact Hamiltonian systems
and symplectic or cosymplectic Hamiltonian systems, namely, symplectic and
cosymplectic manifolds are Poisson manifolds, but a contact manifold is
strictly a Jacobi manifold. In [46, 47] the following definition is presented:

######  Definition 11.

A contact Hamiltonian system $(M,\theta,H)$ with $n$ degrees of freedom is
said to be completely integrable (in the Liouville sense) if there exist $n+1$
constants of motion $H,f_{1},\cdots,f_{n}$ that are independent and in
involution.

In this definition the functions $f_{1},\cdots,f_{k}$ are said to be
independent if the corresponding Hamiltonian vector fields
$X_{f_{1}},\cdots,X_{f_{k}}$ are linearly independent. This is not equivalent
to the condition that the differentials $df_{1},\cdots,df_{k}$ are linearly
independent, since the latter does not hold when one of the Hamiltonian vector
fields is $\mathbb{R}$-proportional to the Reeb field $R$ whose Hamiltonian is
the function $1$ [46], i.e., under this context, the independence of functions
is not equivalent to our notion of functional independence. Contact
Hamiltonian systems with Hamiltonian function $H$ constant are called of Reeb
type, in these cases the Hamiltonian vector field $X_{H}$ is
$\mathbb{R}$-proportional to the Reeb field $R$ and the Hamilton equations of
motion are trivial, so we exclude this case in the notion on particular
integrability. We have the following fundamental result relating functionally
independent functions with linearly independent vector fields.

######  Lemma 3.

If $f_{1},\cdots,f_{k}\in C^{\infty}(M)$ are functionally independent
functions then the Hamiltonian vector fields $X_{f_{1}},\ldots,X_{f_{k}}$ are
linearly independent.

###### Proof.

Let us suppose that $c_{1}X_{f_{1}}+\cdots+c_{k}X_{f_{k}}=0$ for some
$c_{1},\cdots,c_{k}\in\mathbb{R}$. We know that

$\left(c_{1}X_{f_{1}}+\cdots+c_{k}X_{f_{k}}\right)\lrcorner
d\theta=d(c_{1}f_{1}+\cdots+c_{k}f_{k})-R(c_{1}f_{1}+\cdots+c_{k}f_{k})\theta,$
(72)

then

$d(c_{1}f_{1}+\cdots+c_{k}f_{k})-R(c_{1}f_{1}+\cdots+c_{k}f_{k})\theta=0,$
(73)

or equivalently

$d(c_{1}f_{1}+\cdots+c_{k}f_{k})=R(c_{1}f_{1}+\cdots+c_{k}f_{k})\theta.$ (74)

On the other hand, for any $g\in C^{\infty}(M)$ we have $g\theta\wedge
d(g\theta)=g^{2}\theta\wedge d\theta$, then

$g\theta\wedge d(g\theta)^{n}=g^{n}\theta\wedge d\theta^{n}.$ (75)

So if $g\neq 0$ in an open subset of $M$ then $g\theta$ is a contact form on
than open submanifold. Now if we suppose that
$R(c_{1}f_{1}+\cdots+c_{k}f_{k})\neq 0$ in an open subset of $M$ then
$R(c_{1}f_{1}+\cdots+c_{k}f_{k})\theta$ is a contact form on than open
submanifold, but it cannot be since
$d(R(c_{1}f_{1}+\cdots+c_{k}f_{k})\theta)=d(d(c_{1}f_{1}+\cdots+c_{k}f_{k}))=0$;
so we must have that $R(c_{1}f_{1}+\cdots+c_{k}f_{k})$ is zero except possibly
in a zero measure set, therefore $d(c_{1}f_{1}+\cdots+c_{k}f_{k})=0$ which
implies that $c_{1}=\cdots=c_{k}=0$, i.e., for regular points (points where
the differential $df_{1},\cdots,df_{k}$ are linearly independent) the vector
fields $X_{f_{1}},\ldots,X_{f_{k}}$ are linearly independent. ∎

We can observe that in the definition of completely integrable contact
Hamiltonian systems (in the Liouville sense) it is required the existence of
$n+1$ constants of motion and, in addition, the Hamiltonian function is one of
them. In general, the Hamiltonian function is not a constant of motion. In
fact, contact Hamiltonian systems where the Hamiltonian function is a constant
of motion (the Hamiltonian function is invariant under the flow of the Reeb
vector field) are called good contact Hamiltonian systems; and completely
integrable contact Hamiltonian system where the constants of motion are also
invariant under the flow of the Reeb vector field are called completely good
[46]. Even for the notion of non-commutative integrability in the contact
case, it is required that the Hamiltonian function is a constant of motion
[48, 49]. The solutions of the Hamilton equations of motion of a completely
good contact Hamiltonian system can be found by quadratures provided that the
constants of motion form a solvable Lie algebra under the Jacobi bracket [17].

We have the following theorem on integrability involving particular involution
in the contact framework.

######  Theorem 4.

Let $(M,\theta,H)$ be a good contact Hamiltonian system. If
$H=f_{1},\cdots,f_{n}\in C^{\infty}(M)$ are functionally independent and in
particular involution functions such that $Rf_{i}=0$ then the trajectories of
the system $(M,\theta,H)$ that live in $M_{f}=\\{x\in M:\
f_{1}(x)=\cdots=f_{n}(x)=0\\}$ can be found by quadratures.

###### Proof.

Since the functions $f_{1},\cdots,f_{n}$ are functionally independent on $M$
then $M_{f}=\\{x\in M:\ f_{1}(x)=\cdots=f_{n}(x)=0\\}$ is a smooth submanifold
of $M$ of dimension $dim(M)-n=2n+1-n=n+1$, and since $Rf_{1}=\cdots=Rf_{n}=0$,
i.e., the functions $f_{1},\cdots,f_{n}$ are $z$-independent, then around any
point in $M_{f}$ we can find local coordinates $(y^{1},\cdots,y^{n},z)$.

On the other hand we have that the vector fields $X_{f_{1}},\cdots,X_{f_{n}}$
are tangent to $M_{f}$, indeed, $X_{f_{j}}f_{i}=\\{f_{i},f_{j}\\}-f_{i}Rf_{j}$
which is zero on $M_{f}$; so the trajectories of the Hamiltonian system
$(M,\theta,H)$ that live in $M_{f}$ are integral curves of
$X_{H=f_{1}}|_{M_{f}}$, i.e., the trajectories of the Hamiltonian system
$(M,\theta,H)$ that live in $M_{f}$ have the local form
$\gamma(t)=(y^{1}(t),\cdots,y^{n}(t),z(t))$ and are solutions of the vector
differential equation $\dot{x}=X_{H}(x)$ with $x=(y^{1},\ldots,y^{n},z)$ local
coordinates on $M_{f}$, which provided the existence of $n+1$ linearly
independent vector fields on $M_{f}$ that form a solvable Lie algebra of
symmetries of the vector field $X_{H}(x)$, can be solved by quadratures.

In the contact case the assignment $f\mapsto X_{f}$ defines a Lie algebra
antihomomorphism between the Lie algebras $(C^{\infty}(M),\\{,\\})$ and
$(\mathfrak{X}(M),[,])$ [29], indeed, for $f,g\in C^{\infty}(M)$ we have

$-[X_{f},X_{g}]=X_{\\{f,g\\}}.$ (76)

So we have that $X_{f_{1}}|_{M_{f}},\cdots,X_{f_{n}}|_{M_{f}}$ generate an
Abelian (therefore solvable) Lie algebra of symmetries of
$X_{H=f_{1}}|_{M_{f}}$ with the Lie bracket of vector fields, in addition the
vector field $R$ is also tangent to $M_{f}$ and is a symmetry of
$X_{H=f_{1}}|_{M_{f}}$, indeed, we know that $[R,X_{H}]=X_{\\{1,H\\}}=0$. We
conclude that the vector fields
$R,X_{f_{1}}|_{M_{f}},\cdots,X_{f_{n}}|_{M_{f}}$ are linearly independent and
generate an Abelian (therefore solvable) Lie algebra of symmetries of
$X_{H=f_{1}}|_{M_{f}}$ with the Lie bracket of vector fields, therefore the
trajectories of the Hamiltonian system $(M,\theta,H)$ that live in $M_{f}$ can
be found by quadratures. ∎

######  Definition 12.

A good contact Hamiltonian system $(M,\theta,H)$ with $n$ degrees of freedom
is said to be particularly integrable if there exist $n$ functionally
independent functions $H=f_{1},\cdots,f_{n}\in C^{\infty}(M)$ in particular
involution such that $Rf_{i}=0$.

The condition for particular integrability is also a maximal condition.

######  Proposition 3.

For a given good contact Hamiltonian system we have that the maximum number of
functionally independent functions $f_{i}$ in particular involution with
$Rf_{i}=0$ is equal to the number of degrees of freedom of the system.

The proof is completely analogous to the one given in the previous section for
the cosymplectic case.

###  4.4 Cocontact Hamiltonian systems

We finish this section by introducing the notions of particular integral and
particular integrability in the cocontact framework. The development of this
subsection follows the guide of the previous ones. Let $(M,\theta,\eta,H)$ be
a cocontact Hamiltonian system with $n$ degrees of freedom (${\rm
dim}(M)=2n+2$). We introduce the notion of particular integral as follows.

######  Definition 13.

A particular integral of $(M,\theta,\eta,H)$ is a function $f\in
C^{\infty}(M)$ such that $R_{t}f=0$ and $X_{H}f=af$ for some function $a\in
C^{\infty}(M)$ with real values on $f=0$.

We can observe that, as in the previous subsection, in terms of the Jacobi
bracket $f$ is a particular integral if and only if $\\{f,H\\}=af$ for some
function $a\in C^{\infty}(M)$. As in the cosymplectic case we require
$R_{t}f=0$ in order to be able to look for the trajectories of the system that
live in $M_{f}=\\{\,x\in M:\ f(x)\,=\,0\,\\}$, indeed, we must have
$E_{H}f|_{f=0}=0$, where $E_{H}=X_{H}+R_{t}$ is the dynamical vector field
(for details see section 3.4).

In [15] dissipated quantities are studied for cocontact Hamiltonian systems.

######  Definition 14.

A dissipated quantity of the cocontact Hamiltonian system $(M,\theta,\eta,H)$
is a function $f\in C^{\infty}(M)$ such that $E_{H}f=-fR_{z}H$.

As it is pointed out in [14, 15], the most general case is
$E_{H}H=-HR_{z}H+R_{t}H$. Thus, whenever $H$ depends explicitly on $t$ it is
not a dissipated quantity itself. A dissipated quantity $f$ such that
$R_{t}f=0$ is a particular integral.

Let us suppose that $f\in C^{\infty}(M)$ is a particular integral of
$(M,\theta,\eta,H)$. We can look for the trajectories of the system that live
in $M_{f}=\\{\,x\in M:\ f(x)\,=\,0\,\\}$, which provided that $0$ is a regular
value of $f$ is a smooth submanifold of dimension $2n+2-1=2n+1$. Using a
similar procedure as in the contact framework, by means of a particular
integral, we can find solutions of the Hamilton equations of motion that are
solutions of a reduced system of differential equations, i.e., we can find
trajectories of a cocontact Hamiltonian system that are trajectories of a
reduced dynamical system.

As in the contact case, the notion of Liouville integrability for cocontact
Hamiltonian systems is restricted to good Hamiltonian systems. A cocontact
Hamiltonian system is good if the Hamiltonian function is invariant under the
flow of the contact Reeb vector field; in this case the Hamiltonian function
is not necessarily a constant of motion, indeed, let $(M,\theta,\eta,H)$ be a
good cocontact Hamiltonian system, i.e., $R_{z}H=0$, then $E_{H}H=R_{t}H$
which is not necessarily zero. We say that a good cocontact Hamiltonian with
$n$ degrees of freedom is a completely good cocontact Hamiltonian system if
there exist $n$ functionally independent constants of motion
$f_{1},\cdots,f_{n}$ with $R_{z}f_{i}=0$ that are in involution. The solutions
of the Hamilton equations of motion of a completely good cocontact Hamiltonian
system can be found by quadratures provided that the constants of motion form
a solvable Lie algebra under the Jacobi bracket [17].

We have the following theorem analogous to theorems 1, 3 and 4.

######  Theorem 5.

Let $(M,\theta,\eta,H)$ be a good cocontact Hamiltonian system. Let
$f_{1},\cdots,f_{n}\in C^{\infty}(M)$ be functionally independent and in
particular involution functions such that $R_{z}f_{i}=R_{t}f_{i}=0$. If for
each $i\in\\{1,\cdots,n\\}$ $\\{f_{i},H\\}=c^{i1}f_{1}+\cdots+c^{in}f_{n}$ for
some $c^{ij}\in C^{\infty}(M)$ with real values on $f_{1}=\cdots=f_{n}=0$,
then the trajectories of the Hamiltonian system $(M,\theta,\eta,H)$ that live
in $M_{f}=\\{x\in M:\ f_{1}(x)=\cdots=f_{n}(x)=0\\}$ can be found by
quadratures.

As in the cosymplectic case, we have not taken $H$ as one of the functions
$f_{1},\cdots,f_{n}$ because $R_{t}H$ is not necessarily zero. We have the
following result analogous to lemmas 2 and 3.

######  Lemma 4.

If $f_{1},\cdots,f_{k}\in C^{\infty}(M)$ are functionally independent
functions such that $R_{t}f_{i}=0$ for $i=1,\cdots,k$ then the Hamiltonian
vector fields $X_{f_{1}},\ldots,X_{f_{k}}$ are linearly independent.

The proof is also analogous to the previous lemmas. Finally we present the
proof of theorem 5.

###### Proof.

Since the functions $f_{1},\cdots,f_{n}$ are functionally independent on $M$
then $M_{f}=\\{x\in M:\ f_{1}(x)=\cdots=f_{n}(x)=0\\}$ is a smooth submanifold
of $M$ of dimension $dim(M)-n=2n+2-n=n+2$, and since
$R_{z}f_{1}=\cdots=R_{z}f_{n}=R_{t}f_{1}=\cdots=R_{t}f_{n}=0$, i.e., the
functions $f_{1},\cdots,f_{n}$ are $z$-independent and time-independent, then
around any point in $M_{f}$ we can find local coordinates
$(t,y^{1},\cdots,y^{n},z)$.

On the other hand we have that the evolution vector field is tangent to
$M_{f}$, indeed,
$E_{H}f_{i}=\\{f_{i},H\\}+R_{t}f_{i}=c^{i1}f_{1}+\cdots+c^{in}f_{n}$ which is
zero on $M_{f}$; so the trajectories of the Hamiltonian system
$(M,\theta,\eta,H)$ that live in $M_{f}$ are integral curves of
$E_{H}|_{M_{f}}$ which in local coordinates $(t,y^{1},\cdots,y^{n},z)$ on
$M_{f}$ has the form

$E_{H}(t,y,z)=v^{1}(t,y,z)\frac{\partial}{\partial
y^{1}}+\cdots+v^{n}(t,y,z)\frac{\partial}{\partial
y^{n}}+v^{z}(t,y,z)\frac{\partial}{\partial z}+\frac{\partial}{\partial t},$
(77)

i.e., the trajectories of the Hamiltonian system $(M,\theta,\eta,H)$ that live
in $M_{f}$ have the local form $\gamma(t)=(t,y^{1}(t),\cdots,y^{n}(t),z(t))$
where the functions $y^{1}(t),\cdots,y^{n}(t),z(t)$ are solutions of the
system of differential equations

$\left\\{\begin{array}[]{c}\dot{y}^{1}=v^{1}(t,y^{1},\ldots,y^{n},z),\\\
\vdots\\\ \dot{y}^{n}=v^{n}(t,y^{1},\ldots,y^{n},z),\\\
\dot{z}=v^{z}(t,y^{1},\ldots,y^{n},z),\end{array}\right.,$ (78)

which can be solved by quadratures provided the existence of $n+1$ linearly
independent vector fields of the form $u^{1}(t,y,z)\frac{\partial}{\partial
y^{1}}+\cdots+u^{n}(t,y,z)\frac{\partial}{\partial
y^{n}}++u^{z}(t,y,z)\frac{\partial}{\partial z}$ that form a solvable Lie
algebra of symmetries of the vector field
$\overline{E}_{H}|_{M_{f}}=v^{1}(t,y,z)\frac{\partial}{\partial
y^{1}}+\cdots+v^{n}(t,y,z)\frac{\partial}{\partial
y^{n}}+v^{z}(t,y,z)\frac{\partial}{\partial z}$. As in the contact case we
have that $R_{z},X_{f_{1}}|_{M_{f}},\cdots,X_{f_{n}}|_{M_{f}}$ are linearly
independent and generate an Abelian (therefore solvable) Lie algebra of
symmetries of $\overline{E}_{H}|_{M_{f}}=X_{H}|_{M_{f}}$ with the Lie bracket
of vector fields. Therefore, the trajectories of the Hamiltonian system
$(M,\theta,\eta,H)$ that live in $M_{f}$ can be found by quadratures. ∎

######  Definition 15.

A good cocontact Hamiltonian system $(M,\theta,\eta,H)$ with $n$ degrees of
freedom is said to be particularly integrable if there exist $n$ functionally
independent functions $f_{1},\cdots,f_{n}\in C^{\infty}(M)$ in particular
involution such that for each $i\in\\{1,\cdots,n\\}$,
$R_{z}f_{i}=R_{t}f_{i}=0$ and $\\{f_{i},H\\}=c^{i1}f_{1}+\cdots+c^{in}f_{n}$
for some $c^{ij}\in C^{\infty}(M)$ with real values on $f_{1}=\cdots=f_{n}=0$.

Of course, like in the contact case, the condition for particular
integrability in the cocontact framework is a maximal condition.

##  5 Conclusions

We have introduced the notions of particular integral and particular
integrability for classical Hamiltonian systems on cosymplectic, contact and
cocontact manifolds. These results extend those presented in [12] for the
symplectic case. A constant of motion (a first Liouville integral) can be
regarded as a special case of the concept of particular integral. Especially,
in the contact framework the present study generalizes the notion of a
dissipated quantity. For a non necessarily integrable (in the Liouville sense)
cosymplectic, contact or cocontact Hamiltonian system we have shown that,
provided the existence of a sufficient number of functionally independent
particular integrals, it is possible to find some of the trajectories by
quadratures. If we do not have enough number of particular integrals required
to achieve particular integrability, we are able to reduce the Hamilton
equations of motion to a simpler form at least.

For future work, we plan to investigate the existence of infinitesimal
symmetries related to particular integrals in the above mentioned geometric
frameworks.

## Acknowledgments

The author R. Azuaje wishes to thank CONAHCYT (México) for the financial
support through a postdoctoral fellowship in the program Estancias
Posdoctorales por México 2022. In addition, R. Azuaje thanks Dr. Alessandro
Bravetti for his helpfull comments on some aspects of contact geometry.

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