\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 108, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/108\hfil Integrability of Hamiltonian systems]
{Polynomial and rational integrability of polynomial Hamiltonian
systems}

\author[J. Llibre, C. Stoica, C. Valls \hfil EJDE-2012/108\hfilneg]
{Jaume Llibre, Cristina Stoica,  Cl\`audia Valls}  % in alphabetical order

\address{Jaume Llibre \newline
Departament de Matem\`{a}tiques,
Universitat Aut\`onoma de Barcelona, 08193 Bellaterra, Barcelona,
Catalonia, Spain}
\email{jllibre@mat.uab.cat}

\address{Cristina Stoica \newline
Department of Mathematics,
Wilfrid Laurier University, Waterloo, N2L 3C5, Ontario, Canada}
\email{cstoica@wlu.ca}

\address{Cl\`audia Valls \newline
Departamento de Matem\'atica, 
Instituto Superior T\'ecnico, 
Universidade T\'ecnica de Lisboa, Av. Rovisco Pais
1049--001, Lisboa, Portugal} 
\email{cvalls@math.ist.utl.pt}


\thanks{Submitted January 26, 2012. Published June 26, 2012.}
\subjclass[2000]{37J35, 37K10}
\keywords{Polynomial Hamiltonian systems; polynomial first
integrals; \hfill\break\indent rational first integrals; Darboux polynomial}

\begin{abstract}
 Within the class of canonical polynomial Hamiltonian systems anti-symmetric
 under  phase-space involutions, we  generalize some results on the existence of
 Darboux polynomial and rational first integrals for
  ``kinetic plus potential" systems to general  systems.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction and statement of the main results}

This note concerns the integrability of canonical polynomial Hamiltonian 
systems. Usually the integrability of these kind of Hamiltonian systems 
is considered using Ziglin's approach \cite{Z} or differential Galois 
theory \cite{Gt}, but here we  use the Darboux theory of integrability \cite{Da}.
Our findings are generalisations of some results presented by  
Maciejewski et al. in \cite{NMP, MP}, and
Garcia at el. in \cite{GGL}.

A natural class of canonical Hamiltonian systems  is given by systems expressed 
as  sum of the kinetic and potential terms
\begin{equation}\label{eq:1}
H(q,p)=\frac 1 2 \sum_{i=1}^m \mu_i p_i^2 +V(q),
\end{equation}
where $q,p \in \mathbb{C}^{m}$,  and $\mu_i \in \mathbb{C}$ for $i=1,\ldots,m$.
In what follows we observe that certain statements on polynomial Hamiltonians 
of the form \eqref{eq:1}   obtained in  \cite{GGL} generalize to time-reversible 
Hamiltonian systems  with an arbitrary polynomial Hamiltonian
$H(q,p)$. For such systems, under convenient assumptions,  we deduce
the existence of  a second polynomial first integral independent of
the Hamiltonian.

Further, we consider   polynomial Hamiltonian systems  together with 
anti-sym\-metric under involutions $(q,p) \to (-q,p)$. In this case we obtain
a second polynomial or rational first integral independent of the
Hamiltonian.

A canonical Hamiltonian system   with $m$ degrees
of freedom and Hamiltonian $H(q,p)$ is given by
\begin{equation}\label{eq:2}
\frac{d q_i}{dt} = \frac{\partial H(q,p)}{\partial p_i}, \quad
\frac{d p_i}{dt} = -\frac{\partial H(q,p)}{\partial q_i}, \quad
\text{for $i=1,\ldots,m$},
\end{equation}
where  $q=(q_1,\ldots,q_m) \in \mathbb{C}^m$ and $p=(p_1,\ldots,p_m) \in
\mathbb{C}^m$  are the generalized coordinates and momenta, respectively.

We denote by $X_H$ the associated Hamiltonian vector field in
$\mathbb{C}^{2m}$ to the Hamiltonian system \eqref{eq:2}; i.e.,
\begin{equation}\label{eq:3}
X_H= \sum_{i=1}^m \frac{\partial H(q,p)}{\partial p_i}
\frac{\partial}{\partial q_i} - \sum_{i=1}^m \frac{\partial
H(q,p)}{\partial q_i} \frac{\partial}{\partial p_i}.
\end{equation}

Let $U$ be an open subset of $\mathbb{C}^{2m}$, such that its closure is
$\mathbb{C}^{2m}$. Then, a function $I:U \to \mathbb{C}^{2m}$ constant on the orbits
of the Hamiltonian vector field $X_H$ contained in $U$ is called a
{\it first integral} of $X_H$, i.e. $X_H I\equiv 0$ on $U$. 
It is immediate that $H$ is a first integral of the vector field $X_H$.


A non-constant polynomial $F \in \mathbb{C}[q,p]$ is a \emph{Darboux
polynomial} of the polynomial Hamiltonian vector field $X_H$ if there
exists a polynomial $K \in \mathbb{C}[q,p]$, called the \emph{cofactor} of
$F$, such that $X_H F =  K F$. We say that $F$ is a \emph{proper}
Darboux polynomial if its cofactor is not zero, i.e. if $F$ is not a
polynomial first integral of $X_H$.

One may check directly from the definition of a Darboux
polynomial $F$ that the hypersurface $F(q,p)=0$ defined by a Darboux
polynomial is invariant by the flow of $X_H$, i.e., if an
orbit of the vector field $X_H$ has a point on that
hypersurface, then the whole orbit is contained in it.


The Darboux polynomials where introduced by Darboux \cite{Da} in
1878 for studying the existence of first integrals in the polynomial
differential systems in $\mathbb{C}^m$. His original ideas have been
developed by many authors; see the survey \cite{Ll} and the paper
\cite{LZ} with the references therein  on the   recent result on
the Darboux theory of integrability.

We say that a function $G(q,p)$ is \emph{even} with respect to the
variable $q$ if $G(q,p)=G(-q,p)$, and we say that it is \emph{odd}
with respect to the variable $q$ if $G(q,p)=-G(-q,p)$. An analogous definition applies for $G$ being even or odd with respect to the variable $p$.


\section{Involutions with respect to momenta}

In general, a (smooth) involution is a (smooth) map $f$ such that 
$f \circ f=Id,$, where $Id$ is the identity. 
In our context, consider the involution  given by the diffeomorphism 
 $\tau : \mathbb{C}^{2m} \to \mathbb{C}^{2m}\,,$  $\tau(q,p):=(q,-p)$.
The vector
field $X_H$ on $\mathbb{C}^{2m}$ is said to be \emph{$\tau$--reversible} if
$\tau_{*}(X_H)=-X_H$,
where $\tau_{*}$ is the push--forward associated to the
diffeomorphism $\tau$.  This is the case when
\[
\frac{\partial H(q,-p)}{\partial p_i}= -\frac{\partial H(q,p)}{\partial p_i}\quad 
\text{and} \quad 
\frac{\partial H(q,-p)}{\partial q_i}= \frac{\partial H(q,p)}{\partial q_i}.
\]
For instance, systems of the form \eqref{eq:1} fulfill these conditions.

\begin{theorem}\label{t1}
Consider a polynomial Hamiltonian $H(q,p)$ such that its
corresponding Hamiltonian vector field \eqref{eq:3} is
$\tau$-reversible. Let $F(q,p)$ be a proper Darboux polynomial of
the Hamiltonian vector field $X_H$ with a cofactor $K(q,p)$ which
is an even function with respect to the variable $p$. Then
$F(q,p)F(q,-p)$ is a polynomial first integral of $X_H$.
\end{theorem}


To prove the above theorem, we need the following result.


\begin{lemma}\label{L1}
Under the assumptions of Theorem \ref{t1}, we have that $F(q,-p)$ is
another proper Darboux polynomial of $X_H$ with cofactor
$-K(q,-p)$.
\end{lemma}

\begin{proof} Since
\[
X_H F(q,p)= K(q,p)F(q,p),\]
we have
\[
\tau_*(X_H F)(q,p)= \tau_*(K\cdot F)(q,p)\,.
\]
In the relation above, the left hand side is
\begin{equation}  \label{lhs}
\begin{aligned}
\tau_*(X_H F)(q,p)
&= \tau_*(X_H) \tau_*(F)(q,p)
= - X_H  F \left( \tau^{-1}(q,p)\right) \\
&=  - X_H  F \left( \tau(q,p) \right)= - X_H  F (q,-p)
\end{aligned}
\end{equation}
where we used  that $\tau^{-1}=\tau$. The right hand side is
\begin{equation}
\begin{aligned}
 \tau_*(K\cdot F)(q,p)
&=\left(  \left(K\cdot F\right) \circ \tau^{-1} \right)(q,p)
 =  \left(\left(K \cdot F\right) \circ \tau \right)(q,p) \\
 &= (K \cdot F)(q,-p) = K(q,-p) \cdot F(q,-p)  \label{rhs}
\end{aligned}
\end{equation}
Since \eqref{lhs} equals \eqref{rhs}  we obtain
\[
X_H F(q,-p)= -K(q,-p) F(q,-p).
\]
So $F(q,-p)$ is a proper Darboux polynomial of $X_H$ with cofactor
$-K(q,-p)\neq 0$, because $K(q,p)\neq 0$ due to the fact that
$F(q,p)$ is a proper Darboux polynomial.
\end{proof}



\begin{proof}[Proof of Theorem \ref{t1}]
Under the assumptions of Theorem \ref{t1} we have $X_H F(q,p)
= K(q,p) F(q,p)$ with $K(q,p)\neq 0$.
By Lemma \ref{L1} we have that $X_H F(q,-p)= -K(q,-p) F(q,-p)$.
Therefore,
\begin{align*}
X_H (F(q,p)F(q,-p))
&= X_H (F(q,p)) F(q,-p)+ F(q,p) X_H
(F(q,-p))\\
&= K(q,p) F(q,p)F(q,-p)+ F(q,p)( -K(q,-p) F(q,-p))\\
&= (K(q,p)-K(q,-p))F(q,p)F(q,-p).
\end{align*}
This last expression is zero due to the fact that the cofactor
$K(q,p)$ is an even function in the variable $p$. So $F(q,p)F(q,-p)$
is a polynomial first integral of Hamiltonian vector field $X_H$.
\end{proof}


\begin{corollary}\label{c0}
Consider a polynomial Hamiltonian $H(q,p)$ given by \eqref{eq:1}.
Let $F(q,p)$ be a proper Darboux polynomial of the Hamiltonian
vector field $X_H$. Then $F(q,p)F(q,-p)$ is a polynomial first
integral of $X_H$.
\end{corollary}

A proof of the above corollary can be found in \cite[Theorem 3]{GGL};
We omit it. 

A Hamiltonian system is called \textit{time-reversible} if for any 
integral curve $\left( q(t), p(t)\right)$ of $X_H$ we have
 $\left( q(-t), p(-t)\right)=\left( q(t), -p(t)\right)$. 
In the configurations space this means that whenever we have a trajectory 
$q(t)$ then $q(-t)$ is also a trajectory. Note that time-reversibility
 is equivalent to the invariance of the flow under  involutions 
acting on the independent variable (time) as well; i.e., $(q,p,t)  \to (q,-p,-t)$.
In this context, Theorem \ref{t1} may be extended  as follows:

\begin{theorem}\label{t2}
Let  $H(q,p)$ be a time-reversible polynomial Hamiltonian  system and 
assume that  $F(q,p)$ is a proper Darboux polynomial of the
Hamiltonian vector field $X_H$ with a cofactor $K(q,p)$ such that
$K\circ \tau= K$. Then $F\cdot(F\circ \tau)$ is a polynomial first integral
of $X_H$.
\end{theorem}

The proof of Theorem \ref{t2} is similar to the proof of Theorem \ref{t1}.
We omit it.

\section{Involutions with respect to coordinates}

Let    $\hat \tau \colon \mathbb{C}^{2m} \to \mathbb{C}^{2m}$ be the involution
$\hat \tau(q,p)=(-q,p)$. The vector
field $X_H$ or the Hamiltonian system \eqref{eq:2} on $\mathbb{C}^{2m}$ is
\emph{$\hat \tau$-equivariant} if the Hamiltonian system
\eqref{eq:2} is invariant under $\hat \sigma$, that is
$
 \hat \tau_*(X_H)=-X_H.
$ This is the case when
\[
\frac{\partial H(-q,p)}{\partial p_i}
= \frac{\partial H(q,p)}{\partial p_i}\quad \text{and} 
\quad \frac{\partial H(-q,p)}{\partial q_i}= -\frac{\partial H(q,p)}{\partial q_i}
\]

\begin{theorem}\label{main.1}
Consider a polynomial Hamiltonian $H(q,p)$ such that its
corresponding Hamiltonian vector field \eqref{eq:3} is
 $\hat \tau$-equivariant. Let $F(q,p)$ be a proper Darboux polynomial of
the Hamiltonian vector field $X_H$ with a cofactor $K(q,p)$. Then
the following statements hold.
\begin{itemize}
\item [(a)] If $K(q,p)$ is an even function with respect to 
$q$, then $F(-q,p)F(q,p)$ is a polynomial first integral of
$X_H$.
\item [(b)] If $K(q,p)$ is an odd function with respect to 
 $q$, then $F(-q,p)/F(q,p)$ is a rational first integral
of $X_H$.
\end{itemize}
\end{theorem}

To prove the above theorem we need  the following result:


\begin{lemma}\label{lem.5}
Under the assumptions of Theorem \ref{main.1} we have that $F(-q,p)$
is another proper Darboux polynomial of $X_H$ with cofactor
$-K(-q,p)$.
\end{lemma}

\begin{proof}
 From the definition of $\hat \tau_*$ it follows that $\hat \tau_*(X_H)=-X_H$. 
This implies that
\begin{equation}\label{eq:22}
\hat \tau_*(X_H F)= -X_H \hat \tau(F)=-X_H F(-q,p).
\end{equation}
Moreover, we have that $X_H F = KF$ and thus
\begin{equation}\label{eq:23}
\hat \tau_*(X_H F)= \hat \tau_*(K F)=\hat \tau_*(K) \hat
\tau_*(F)=K(-q,p) F(-q,p).
\end{equation}
Combining equations \eqref{eq:22} and \eqref{eq:23} we obtain
\[
X_H F(-q,p)=-K(-q,p) F(-q,p).
\]
Therefore, $F(-q,p)$ is a proper Darboux polynomial of $X_H$ with
cofactor $-K(-q,p) $. We note that $K(-q,p) \ne 0$ due to the fact
that $F(-q,p)$ is a proper Darboux polynomial and consequently
$K(q,p) \ne 0$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{main.1}]
Under the assumptions of Theorem \ref{main.1} we have 
 $X_H F(q,p)=K(q,p)F(q,p)$ with $K(q,p) \ne 0$.
By Lemma \ref{lem.5} we have that $X_H F(-q,p)=
-K(-q,p) F(-q,p)$.
Therefore,
\begin{align*}
X_H(F(-q,p) F(q,p)) 
&= X_H(F(-q,p)) F(q,p) + F(-q,p) X_H (F(q,p))\\
&= -K(-q,p) F(-q,p) F(q,p) +F(-q,p) K(q,p) F(q,p) \\
&= (-K(-q,p)+K(q,p)) F(q,-p)  F(q,p).
\end{align*}
If $K$ is an even function in the variable~$q$, the last expression
is zero. So, in this case,  $F(-q,p) F(q,p)$ is a polynomial first
integral of the Hamiltonian vector field $X_H$. This completes the
proof of statement (a).

On the other hand,
\begin{align*}
X_H(F(-q,p)/F(q,p)) 
&= \frac{ X_H(F(-q,p)) F(q,p) - F(-q,p) X_H (F(q,p))}{F(q,p)^2} \\
&= \frac{-K(-q,p) F(-q,p) F(q,p) -F(-q,p) K(q,p) F(q,p)}{F(q,p)^2} \\
&= -(K(-q,p)+K(q,p)) \frac{F(-q,p)}{F(q,p)}.
\end{align*}
If $K$ is an odd function in the variable~$q$, the last expression
is zero. So, in this case, $F(-q,p)/F(q,p)$ is a rational first
integral of the Hamiltonian vector field $X_H$. This completes the
proof of the theorem.
\end{proof}

It is natural  to extend Theorem \ref{main.1} to involutions acting 
on the independent variable (time) of the form $(q,p,t)  \to (-q,p,-t)$, 
under which the flow is invariant.  In this case, whenever
 $\left( q(t), p(t)  \right)$ is an integral curve, so is 
$\left(  -q(-t),p(-t) \right)$.

\begin{theorem}\label{main.5}
Consider a polynomial Hamiltonian $H(q,p)$ such that its
flow is invariant under $(q,p, t) \to (-q,p, -t)$. Let $F(q,p)$ be a 
proper Darboux polynomial of
the Hamiltonian vector field $X_H$ with a cofactor $K$. Then the
following statements hold.
\begin{itemize}
\item [(a)] If $K$ is such that $K \circ {\hat \tau} =K$, then
$F\cdot (F \circ {\hat \tau})$ is a polynomial first integral of $X_H$.
\item [(b)] If $K$ is such that $K\circ {\hat \tau} =-K$,
then $(F\circ {\hat \tau})/ F$ is a rational first integral of $X_H$.
\end{itemize}
\end{theorem}

The proof of Theorem \ref{main.5} is the same as the proof of
Theorem  \ref{main.1}. we omit it.

\begin{proposition}\label{main.3}
Consider a polynomial Hamiltonian $H(q,p)$ given by \eqref{eq:1},\\
where   $V(q)$ is even. Let $F(q,p)$ be a proper Darboux polynomial of the
Hamiltonian vector field $X_H$ with cofactor $K$. Then the following
statements hold.
\begin{itemize}
\item [(a)] If $K$ is an even function in the variable $q$, then
$F(-q,p)F(q,p)$ is a polynomial first integral of $X_H$.
\item [(b)] If $K$ is an odd function in the variable $q$, then
$F(-q,p)/ F(q,p)$ is a rational first integral of $X_H$.
\end{itemize}
\end{proposition}


To prove Proposition \ref{main.3} we recall the following result whose
proof can be found in \cite{GGL}.

\begin{lemma}\label{lem.6}
Let $F(q,p)$ be a proper Darboux polynomial of the Hamiltonian
vector field $X_H$ associated to the Hamiltonian $H$ given by
\eqref{eq:1}. Then its cofactor is a polynomial of the form $K(q)$.
\end{lemma}

\begin{proof}[Proof of Proposition \ref{main.3}]
If $F(q,p)$ is a proper Darboux polynomial of the Hamiltonian vector
field $X_H$, by Lemma \ref{lem.6} we have that its cofactor is of
the form $K(q)$. Then, if $K$ is an even function in the variable
$q$ then the Hamiltonian vector field $X_H$ satisfies all the
assumptions of Theorem \ref{main.1}(a), and consequently
$F(-q,p)F(q,p)$ is a polynomial first integral of $X_H$. On the
other hand, if $K$ is an odd function in the variable $q$ then the
Hamiltonian vector field $X_H$ satisfies all the assumptions of
Theorem \ref{main.1}(b), and consequently $F(-q,p)/ F(q,p)$ is a
rational first integral of $X_H$. This completes the proof.
\end{proof}


\subsection*{Acknowledgements}
The first author was partially supported by  grants
MTM 2008--03437 from the MICINN/ FEDER,  2009SGR-410 from AGAUR,
and from ICREA Academia. The
second author was partially supported by 
a NSERC Discovery Grant and by
the grant MTM2008--03437 during her visit to Universitat Aut\`{o}noma de
Barcelona. 
The third author is supported by grant PIV-DGR-2010 from AGAUR,
and by FCT through CAMGDS, Lisbon.

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\end{document}