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\title{\bf The periodic Melnikov vector, the continuation of
periodic orbits and a theorem on the non-integrability of perturbed
Hamiltonians of $n$ degrees of freedom}
\author{Simos Ichtiaroglou \& Efi Meletlidou\\
Department of Physics\\
University of Thessaloniki\\ 54006, Greece}
\begin{document}
\baselineskip=0.24in
\maketitle
%\titlepage
\begin{abstract}
The periodic Poincar\'e - Melnikov vector is defined for a perturbed
Hamiltonian system with $n$ degrees of freedom of the form
$H=H_0+ \varepsilon H_1$ and its relation to the bifurcation
of periodic orbits and the non-integrability of the perturbed
system is investigated.
\end{abstract}
\section{The Poincar\'e - Melnikov vector}
Consider the $n$-degrees-of-freedom nearly integrable Hamiltonian of the form
%eq1
\begin{equation}
H=H_0 + \varepsilon 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, such that
%eq 2
\begin{equation}
H_0 = H_0 ( J_i ).
\end{equation}
Let $\bf C$ be the symmetric matrix with elements
%eq 3
\begin{equation}
C_{ij} = \frac { \partial^{2} H_0 } { \partial J_{i} \partial
J_{j}}.
\end{equation}
We also suppose that $H_0$ is non-degenerate, i.e.
%eq 4
\begin{equation}
\det \left | \bf C \right | \neq 0.
\end{equation}
According to Liouville - Arnold theorem (e.g. [1], p.271), an open domain
of phase space of $H_0$ is foliated by $n$-dimensional tori, defined by
%eq 5
\begin{equation}
J_{i} = a_{i} = \mbox{const.} \: \: \: \: (i=1,...,n),
\end{equation}
which remain invariant under the motion. On any such torus,
the unperturbed solution is described by the angle
coordinates as
%eq 6
\begin{equation}
w_i = \omega_{i} t + \vartheta _{i} \: \: \: \: (i=1,...,n)
\end{equation}
where $w_i$ are mod$(2 \pi )$ and $ \vartheta_i $ are arbitrary initial
conditions on the torus. Consider a particular torus of $H_0$,
such that the resonant conditions hold
%eq 7
\begin{equation}
\frac{ \omega_1} {p _1} = \frac {\omega_2}{p_2} = ... = \frac{\omega_n}{p_n},
\end{equation}
where $ \omega_i =\frac{ \partial H_0} {\partial J_i}$ are the frequencies
of the integrable part and $p_i$ are non-zero integers with no common divisor.
This resonant torus is foliated by periodic orbits $\cal O$ with
period
%eq 8
\begin{equation}
T = \frac { 2\pi p_i } {\omega_i}.
\end{equation}
The quotient manifold $Q = T^{n} / \cal O$ of the resonant torus,
with respect to this foliation is a $(n-1)$-dimensional torus $T^{n-1}$
and every periodic orbit is mapped on this manifold to a point with coordinates
%eq 9
\begin{equation}
\varphi_i = p_{i+1} \vartheta_1 - p_1 \vartheta_{i+1} \:\:\:
\mbox{mod} (2\pi s_i) \: \: \: \: (i = 1,..., n-1),
\end{equation}
where $s_i$ are the common divisors of the pairs $p_1, p_{i+1}$.
Following the definition of the homoclinic Melnikov vector given
in [2-4],
a {\it periodic} Melnikov vector ${\bf M}$ on every periodic orbit of a
resonant torus of $H_0$ may be defined as
%eq 10
\begin{equation}
M_{i} = \int_{0}^{T} [H_1,\Phi_{i0}] \: dt \: \:\:\: (i=1,...,n),
\end{equation}
where $\Phi_{i0}$ is any set of independent involutive integrals of
$H_0$ and the integration is to be carried along the periodic orbit.
The above definition is not satisfactory for several
reasons:
(i) There is an ambiguity on the selection of the set of
integrals $\Phi_{i0}$.
(ii) By definition (10), every $M_i$ depends on the
initial conditions $\vartheta_i$ only through $\varphi_j$, i.e.
it is not a function of the torus $T^n$ but of the quotient
manifold $Q$.
(iii) If we choose in particular $\Phi_{n0}=H_0$, then it can be easily
shown that $M_n=0$. If, on the other hand, we select the action variables
$J_i$ as $n$ intependent integrals in (10), the projection of
${\bf M}$ along the
orbit is zero and ${\bf M}$ belongs to the tangent space of $Q$.
Indeed, in this case
\[ M_{i} = T \: \frac{\partial } {\partial \vartheta_i} \]
where
%eq 11
\begin{equation}
= \frac{1} {T} \int_{0}^{T} H_1 dt
\end{equation}
is the average value of $H_1$ along the periodic orbit, which
depends on $\vartheta_i$ through $\varphi_j$, so that
\[ M_1 = \sum_{j=1}^{n-1} p_{j+1} \frac{\partial }
{\partial \varphi_{j}} \]
\[ M_{j+1} = -p_1 \: \frac{\partial } {\partial
\varphi_{j}} \: \: \: \:
(j=1,...,n-1). \]
Let $\omega$ be the frequency vector, tangent to the orbit.
Then, by taking into account (7), it is easy to show that
%eq 12
\begin{equation}
{\bf M}^T {\omega} = 0.
\end{equation}
We define the {\it Poincar\'e - Melnikov} vector ${\bf P}$ on
$T_{\varphi}Q$ as
%eq 13
\begin{equation}
P_{i} = \frac{\partial } {\partial \varphi_i} \: \:\:\:(i=1,...,n-1).
\end{equation}
For the above definition, any regular linear combination of
$\varphi_i$ can be used. Poincar\'e [5,6] actually used as
coordinates in $Q$ the $\vartheta_2$,...,$\vartheta_n$ when
$\vartheta_1 = 0$. Let $\Phi_{10},\: \Phi_{20},...,\: \Phi_{(n-1)0}, \:H_0$
be a complete set of integrals for $H_0$, independent on the particular resonant
torus. Then the corresponding Melnikov vector $\bf M$ is related to
$\bf P$ by
%eq 14
\begin{equation}
M_i = T \sum_{j=1}^{n-1} D_j^{(i)} P_{j} \: \:\:\: (i=1,...,n-1),
\end{equation}
where
%eq 15
\begin{equation}
D_j ^{(i)}(J_1,...,J_n) = p_{j+1} \frac {\partial
\Phi_{i0}}{\partial J_1} - p_1 \frac{\partial
\Phi_{i0}}{\partial J_{j+1}}
\end{equation}
while $M_n=0$. Note that for every integral $\Phi_{i0}$ of $H_0$ there
exists a corresponding vector ${\bf D}^{(i)}$, defined by equation (15).
Suppose now that $s$ integrals $\Phi_{k0} \: (k=1,..,s)$ produce linearly
dependent vectors ${\bf D}^{(k)}$, i.e.
%eq 16
\begin{equation}
\sum _{k=1}^s \lambda _k D_j ^{(k)}=0 \: \:\:\: (j=1,..., n-1).
\end{equation}
Then the gradients of the integrals $\Phi_{10},.., \Phi_{s0}, H_0$
are also linearly dependent, i.e.
%eq 17
\begin{equation}
\sum_{k=1}^{s} \mu _k \frac{ \partial \Phi_{k0}} { \partial J_r} +
\mu_{0} \frac {\partial H_0}{\partial J_r} = 0 \:\:\:\: (r=1,...,n),
\end{equation}
with
\[ \mu_k = - \lambda_k \frac { \partial H_0 } {\partial J_1},
\:\:\: \mu_0 = \sum _{k=1} ^s \lambda_k \frac {\partial \Phi
_{k0}} {\partial J_1}. \]
If, on the contrary, the selected integrals are linearly
independent, the transformation (14) is regular. The
Poincar\'e - Melnikov vector, as defined in (13), is strongly
related to the non-integrability of the perturbed system and to
the continuation of the non-isolated periodic orbits of $H_0$
with respect to $\varepsilon$.
\section{The continuation of periodic orbits}
In this section we will present a theorem by Poincar\'e, relating the
Poincar\'e - Melnikov vector to the continuation of periodic
orbits to the perturbed system.
Consider a Hamiltonian system possessing a $T$-periodic solution.
Let ${\bf G}(t)$ be a fundamental matrix of solutions of the
system of first order variational equations around this periodic
solution, such that ${\bf G}(0) = {\bf I}$, where ${\bf
I}$ is the identity matrix. The eigenvalues $\lambda_j$ of the {\it monodromy
matrix} ${\bf G}(T)$ are the {\it characteristic multipliers}
and the quantities
%eq 18
\begin{equation}
\alpha_j = \frac {1} {T} \: \mbox {ln} \lambda_j \:\:\:\: \mbox
{mod} \: \frac {i2 \pi} {T} \:\:\:\:i= \sqrt{-1},\:\: (j=1,...,2n)
\end{equation}
are the {\it characteristic exponents} of the aforesaid periodic
solution. Because of the symplecticity of ${\bf G}(T)$, the
characteristic exponents in a Hamiltonian system appear in
opposite pairs. It is known ([6], \S64) that, in an autonomous
system, a pair of zero exponents always exists. The periodic
orbit is linearly stable, if all other pairs are purely
imaginary. If a periodic orbit possesses at least one pair of
non-zero $\alpha_j$, it is called {\it isolated}. If the system
possesses $p$ integrals in involution whose gradients are
linearly independent on the periodic solution, there exist $2p$ zero
characteristic exponents ([6], \S71).
Consider the integrable Hamiltonian $H_0$. If the non-degeneracy
condition (4) holds, the frequencies $\omega_i$ vary
continuously with the actions $J_i$. In this case, the set of resonant
tori is dense in the phase space of $H_0$. The
periodic orbits on a resonant torus where eqs. (7) hold, are
non-isolated, i.e. they possess $2n$ zero characteristic exponents.
Let $\bf A$ be the $(n-1) \times (n-1)$ matrix
%eq 19
\begin{equation}
A_{ij} = \frac { \partial^2 } {\partial \varphi _i
\partial \varphi_j }.
\end{equation}
Concerning the continuation of the non-isolated periodic orbits
of $H_0$ to the perturbed system, the following Theorem is known
\vskip12pt
{\bf Theorem 1} (Poincar\'e, [5], [6] \S42,79): If on a resonant
torus of $H_0$, for a particular periodic orbit defined on $Q$ by the point
$\varphi_i^{\ast}$, the Poincar\'e - Melnikov vector has a
simple zero in the vector sense, i.e.
%eq 20
\begin{equation}
{\bf P}(\varphi_i^{\ast}) = 0, \:\:\:\: \mbox {det} \left | {\bf A}(\varphi_i^
{\ast}) \right | \neq 0,
\end{equation}
then, for $\varepsilon$ in an open interval around zero,
a) the perturbed Hamiltonian (1) has a $T$-periodic solution,
which depends analytically on $\varepsilon$ and, for $\varepsilon= 0$,
coincides with the unperturbed solution $\varphi_i^{\ast}$.
b) The characteristic exponents of this solution can be
expanded in powers of $\sqrt {\varepsilon}$,
%eq 21
\begin{equation}
\alpha_i = \sqrt {\varepsilon} \: \alpha_{i1} + O(\varepsilon).
\end{equation}
The $\alpha_{i1}^2$ are eigenvalues of the $n \times n$ matrix
%eq 22
\begin{equation}
{\bf E} = {\bf BC},
\end{equation}
where ${\bf C}$ is defined in (3) and ${\bf B}$ is the $n \times
n$ matrix
%eq 23
\begin{equation}
B_{ij}= \frac {\partial^2 } {\partial \vartheta_i \partial \vartheta_j},
\end{equation}
evaluated at the periodic solution $\varphi_i^{\ast}$ of $H_0$.
\vskip12pt
Differentiating (12) with respect to $\vartheta_j$, we obtain
%eq 24
\begin{equation}
{\bf B} \: \omega = 0
\end{equation}
and since $\omega$ is non-zero, $\mbox {det} \left | {\bf B}
\right | =0$, so that a pair of zero $\alpha_{i1}$ always exists.
Note that by (9), $B_{ij}$ and thus $\alpha_{i1}$ can be
expressed as functions of $A_{ij}$.
An attempt to prove non-integrability of $H$ by using this
theorem is made in [5], but is demolished by the fact that one
cannot prove the existence of an infinity of isolated periodic
orbits, for small, but fixed $\varepsilon$ [7,8].
\section{A theorem on the non-integrability of the perturbed system}
Assume that the perturbed Hamiltonian $H$ possesses
$n$ independent integrals $\Phi_1,..,$ $\Phi_{(n-1)},\: H$,
analytic in an open interval around $\varepsilon=0$, i.e. $\Phi_i$ are
expandable as
%eq 25
\begin{equation}
\Phi_{i} = \Phi _{i0} + \varepsilon \Phi _{i1} + O
(\varepsilon ^{2}) \: \: \: \: (i=1,...,n-1).
\end{equation}
Since $\Phi_i$ are integrals of $H$, it must hold
$[H,\Phi_i]=0 \:\: (i=1,..,n-1)$ and
equating terms of the same order in $\varepsilon$, we obtain
%eq 26
\begin{equation}
[H_0, \Phi_{i0} ] = 0,
\end{equation}
%eq 27
\begin{equation}
[H_{0}, \Phi_{i1}] + [H_{1}, \Phi_{i0}] = 0.
\end{equation}
Equations (26), (27), which have been derived by Poincar\'e, hold
identically in phase space. Equations (26) indicate that $\Phi_{i0}$ are
integrals of $H_0$. It is known ([6], \S81,82) that
(i) if $H$ possesses $k \leq n-1$ independent integrals
$\Phi_i$, they can always be selected such that $ \Phi_{10},..., \:
\Phi_{k0}, \: H_0$ are independent and also,
(ii) if the non-degeneracy
condition (4) holds, then $\Phi_{i0}$ do not depend on
the angles.
We are interested in disproving the existence of any possible
integrals of $H$, so we will only use the above two properties and
the identity (27), which is a necessary condition for all
possible independent integrals $\Phi_i$.
Parametrizing equations (27) along the orbits of the unperturbed motion,
we derive that
%eq 28
\begin{equation}
\frac {d \Phi _{i1}} {dt} = [ H_{1}, \Phi _{i0} ],
\end{equation}
where both sides are evaluated along a particular solution of $H_0$.
By integrating equations (28) along any periodic orbit on the
resonant torus obeying the conditions (7) for one period $T$
and demanding $\Phi_{i1}$ to be single-valued functions, we deduce
the equations
%eq 29
\begin{equation}
M_i = 0 \: \:\:\: (i=1,...,n-1),
\end{equation}
where $M_i$ is the component of $\bf M$ corresponding to $\Phi_{i0}$,
which must hold on every solution on this resonant torus. Since
$\Phi_{i0}$ are unknown, we need further investigate the above
equations, which, by (14), may be written in the form
%eq 30
\begin{equation}
\sum_{j=1} ^{n-1} D_{j}^{(i)} P_j = 0
\:\:\:\: (i=1,..., n-1).
\end{equation}
If, on at least one orbit, the Poincar\'e - Melnikov vector $\bf
P$ is non-zero, then from equation (30) we get
%eq 31
\begin{equation}
\det \mid D_j ^{(i)} \mid = 0
\end{equation}
and, as shown in section 1, this leads to
%eq 32
\begin{equation}
\frac { D ( H_0, \Phi_{10}, ..., \Phi_{(n-1)0}) } {D (J_1, J_2,..., J_n)} = 0.
\end{equation}
If the vector $P$ is different from zero on a dense set of resonant
tori of $H_0$, on at least one orbit on each torus, equation
(31), according to (i), contradicts the fact
that $\Phi_i$ are independent integrals of $H$, i.e. $H$ does not
possess a complete set of integrals, analytic in an open interval around zero.
In what follows, we will obtain conditions for the non-existence of a certain
number $s