\documentstyle{article}


\title{On the number of isolating integrals in perturbed Hamiltonian systems}
\author{Efi Meletlidou and Simos Ichtiaroglou\\
Department of Physics\\
University of Thessaloniki 54006 Greece}
\date{}


\begin{document}
\maketitle


Consider the n-degrees-of-freedom nearly integrable Hamiltonian of the form


\begin{equation}
H = H_0 + \epsilon H_1
\end{equation}

where $H_0$ is integrable, i.e, it possesses n-1, in addition to $H_0$,
single-valued, independent integrals in involution and suppose that we can define
action-angle variables $J_i$, $w_i$ (i=1,...,n), at least in an open domain of
phase space. We also suppose that $H_0$ is non-degenerate, i.e.

\begin{equation}
det\left [{\partial^2 H_0 \over \partial J_i \partial J_j}\right ]\neq 0
\end{equation}

On any torus of $H_0$, the unperturbed solution is described by the angle
coordinates as follows

\begin{equation}
w_i = \omega_i t + \vartheta _i , \hskip .5cm (i=1,...,n)
\end{equation}

where $w_i$ are $mod(2\pi)$ and $\vartheta _i$ arbitrary initial conditions on
the torus. We consider the periodic motions of a resonnant torus of $H_0$ with
frequency artio

\begin{equation}
{\omega _1 \over p_1} = {\omega _2 \over p_2} = ... = {\omega _n \over p_n}
\end{equation}
where $\omega_i = \partial H_0 /\partial J_i$ are the frequencies of the
integrable part and $p_i$ are non-zero integers with no common divisor. A
particular resonant torus is foliated by periodic orbits with period

\begin{equation}
T={2\pi p_i \over \omega _i}
\end{equation}

Every periodic orbit is mapped on the quotient manifold of the resonant torus
with respect to the foliation induced by the trivially integrable 1-forms
$p_{i+1}dw_i - p_1dw_{i+1} = 0 (i=1,n-1)$, to a point with coordinates

\begin{equation}
\varphi_i = p_{i+1}\vartheta _1 - p_1\vartheta _{i+1}  \hskip .3cm  mod(2\pi s_i), 
\hskip .3cm  (i=1, ..., n-1)
\end{equation}

here $s_i$ are the common divisors of the pairs $p_i, p_{i+1}$.
\newtheorem{guess}{theorem}
\begin{guess}
Consider the n-degrees-of-freedom Hamiltonian $H = H_0 + \epsilon H_1$ where $H_0$
is a non-degenerate integrable part for which action-angle variables can be
defined in an open domain of phase space. The quotient manifold $Q(J_1, ...,
J_n)$ of each resonant torus of $H_0$ with respect to the foliation induced by
the periodic orbits is the (n-1)-dimensional torus $T^{n-1}$. Let $<H_1>$ be the
average value of $H_1$ along these periodic orbits, which is a scalar function on
each $Q$. Then

a) If in this open domain of phase space, a dense set of resonant tori of $H_0$
is found, such that for at least one orbit on each one of them

\begin{equation}
{\partial\over \partial\varphi_i}<H_1>\neq 0
\end{equation}

for at least one $i$, $H$ does not possess a complete set of analytic integrals
for $\epsilon \in (-\epsilon _0, \epsilon _0)\ \{ O \} $

b) If, similarly,

\begin{equation}
rank\left | {\partial ^2 <H_1 > \over \partial\varphi _m \varphi _j}\right | \geq
r \hskip 1cm (m, j=1, ..., n-1)
\end{equation}

then $H$ cannot possess more than (n-r) independant integrals, including $H$,
which are analytic in the same interval of $\epsilon$.

We apply the previous results to the system of three quartic oscillators with a
weak coupling described by the Hamiltonian

\begin{equation}
H = H_0 + \epsilon H_1 = {1\over2} (P_1^2 + P_2^2 + P_3^2 ) + {1\over 4} (x_1^4
+x_2^4 + x_3^4 ) + \epsilon (x_1 x_2 + x_1 x_3)
\end{equation}

and prove that $H$ does not possess another integral of motion for $\epsilon \in
(-\epsilon _0, \epsilon _0)\ \{ O \} $. This result, however is valid for all
$\epsilon$, since equations of motion are invariant with respect to the
transformation

\begin{equation}
x_i \rightarrow cx_i, \hskip .3cm t \rightarrow .3cm c^{-1}t, \hskip .3cm \epsilon
 \hskip .3cm c^2\epsilon
\end{equation}

with arbitrary $c$.

If we select $H_1 = x_1 x_2$ we can still prove non-integrability, the Hessian,
however, is identically zero and this is due to the existence of a second
integral (i.e the energy along the $x_3$-axis).


\end{guess}





\end{document}
