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%%%%%%%%%%%%%%%%%%%%%%%%
%\baselineskip=10pt
%\centerline{\tenbf WORLD SCIENTIFIC PUBLISHING CO. PTE. LTD.}
%\vglue 12pt
\centerline{\tenbf ARNOLD INSTABILITY FOR NEARLY--INTEGRABLE}
\vglue 0.2cm
\centerline{\tenbf ANALYTIC HAMILTONIAN SYSTEMS}
\vglue 1.0cm
\centerline{\tenrm LUIGI CHIERCHIA}
\baselineskip=13pt
\centerline{\tenit Department of Mathematics, Terza Universit\`a di Roma}
\baselineskip=12pt
\centerline{\tenit via C. Segre 2, 00146 Roma, Italy}
\baselineskip=12pt
\centerline{\tenit Internet: luigi@matrm3.mat.uniroma3.it}
\vglue 0.8cm
\centerline{\tenrm ABSTRACT}
\vglue 0.3cm
{\rightskip=3pc
\leftskip=3pc
\tenrm\baselineskip=12pt%\parindent=1pc
\noindent
A notion of instability,
which is proposed as a definition for the so--called ``Arnold diffusion",
for one--parameter families of
nearly--integrable analytic Hamiltonian systems,
is introduced. An example of
unstable system is given.
\vglue 0.8cm }
\line{\elevenbf 1. Introduction \hfil}
\bigskip
\baselineskip=14pt
\elevenrm
One of the central questions
in the theory of nearly--integrable Hamiltonian systems
concerns the ``stability" of the ``action" variables, or, equivalently,
of the perturbed integrals of motions.
In the late fifties KAM
theory\nota*{
A.N. Kolmogorov, V.I. Arnold, J.K. Moser; see,
e.g., Ref.\ref{1} and references therein.}
gave a partial answer to that question,
ensuring, under general conditions, ``stability" for ``most" initial data.
In 1964, Arnold\ref{2} pointed out a mechanism for ``instability",
illustrating it on
an example containing two parameters.
The name ``Arnold diffusion" for related
phenomena was introduced in the physics literature by B.V. Chirikov.
Here, we introduce, in the context of one--parameter
families of real--analytic
Hamiltonian systems, the definition of
``Arnold instability",
which captures the main features of the mechanism discovered by Arnold.
After a brief review of some
known results, we discuss an explicit example
(containing only one parameter),
of an Arnold--unstable system.
The {\sl content} of the rest of the paper is:
\medskip
{\tenrm\baselineskip=12pt%\parindent=1pc
\item{\S 2} Nearly--Integrable Analytic Hamiltonian System
\item{\S 3} Definition of ``Arnold Instability"
\item{\S 4} A Brief Review of Some Known Result
\item{\S 5} An Arnold--Unstable System (Result and Scheme of Proof)
\itemitem{\S 5.1} Main Result
\itemitem{\S 5.2} Outline of Proof
\item{\S 6} On Two Technical Points
\itemitem{\S 6.1} Remarks on the KAM Construction
\itemitem{\S 6.2} Generalized Poincar\`e Integrals
\item{\S 7} References}
%
\vfill\eject
\vglue 0.6cm
\line{\elevenbf 2. Nearly--Integrable Analytic Hamiltonian Systems\hfil}
\vglue 0.4cm
%
Let $\cM$ denote the Cartesian product of some open connected set
$G\subset \rN$ times the standard $N$--torus $\tN\=\rN/(2\p\zN)$. On the
``phase space" $\cM$, endowed with the standard symplectic form
$\sum_{i=1}^N dI_i\wedge d\f_i$, consider one--parameter
families of Hamiltonian
functions $H(I,\f;\e)$. {\sl Throughout this paper we shall assume the
following
real--analyticity and near--integrability assumptions}: for some
$\bar\e>0$,
(i) $H$ is real-analytic on $\cM\times (-\bar\e,\bar\e)$;
(ii) $H(I,\f;0)\=H_0(I)$.
The study of the trajectories (or ``orbits") generated by such
$H(I,\f;\e)$, namely, the solutions of the Hamilton equations
$$
\dot I_i=-{\dpr H\over \dpr \f_i}\ ,\qquad\dot \f_i=
{\dpr H\over \dpr I_i}\ ,
\quad\big[i=1,...,N;\ \dot{()}\={d\over dt}()\big]\ ,
\eqno(1)
$$
was considered by H. Poincar\'e ``{\sl le probl\`eme g\'en\'eral de la
Dynamique}" (see Ref.\ref{3}, Tome I, chap. I, No. 13). For $\e=0$,
$I_1$,...,$I_N$ are
$N$ independent
integrals in
involution and all trajectories are
quasi--periodic\nota\dag{A function
on
$\cM$ is an integral for $H$ if it is constant
on the trajectories generated by $H$;
``independent" means ``functionally independent"; two (smooth) functions
$f$ and $g$ are in involutions if their Poisson bracket
$\{f,g\}=\sum_{i=1}^N
{ \dpr f\over \dpr I_i}
{ \dpr g \over \dpr \f_i}-
{ \dpr g\over \dpr I_i}
{ \dpr f\over \dpr \f_i}$ vanishes;
a trajectory is quasi--periodic if it is conjugated to a linear flow
on a torus.}:
this is equivalent to say that the system is {\sl completely integrable}.
The basic {\sl stability problem} is here whether, for $\e\neq 0$,
the evolution of the ``action" variables, $I_1(t)$,...,$I_N(t)$,
stays close, for all $t$, to their initial values $I_1(0)$,...,$I_N(0)$.
%
\vglue 0.6cm
\line{\elevenbf 3. Definition of ``Arnold Instability"\hfil}
\vglue 0.4cm
%
Let us denote phase points by $z=(I,\f)$ and by $\phi_H^t$
the Hamiltonian flow
generated by $H$ \ie $z(t)\=\phi_H^t(z)\=(I(t),\f(t))$ solves
Eq. (1) with initial data $z(0)\=z\=(I,\f)$.
A set $\cT\subset \cM$ will be called
{\sl invariant for $H$}
(or $H$--invariant)
if it is closed and $\phi_H^t(\cT)\subset \cT$ for
all $t\in\real$.
We now give the following definitions (compare with Ref.\ref{2}).
\giu
{\elevenbf Definition 1.}
{\sl An $n$--ple of trajectories $z^\ppu(t)$,...,$z^\ppn(t)$ is called a
``chain of heteroclinic trajectories" if there are $(n+1)$ invariant sets
$\cT^\ppu$,...,$\cT^{(n+1)}$ such that, for $1\le i \le n$,
$$
\lim_{t\to-\io} \dist\big(z^\ppi(t),\cT^\ppi\big)=0\ ,\qquad
\lim_{t\to+\io} \dist\big(z^\ppi(t),\cT^{(i+1)}\big)=0\ .
$$
The sets $\cT^\ppu$,...,$\cT^{(n+1)}$ are said to be
connected by the chain
$z^\ppu(t)$,...,$z^\ppn(t)$.
}
\giu
{\elevenbf Definition 2.} {\sl
A chain $z^\ppu(t)$,...,$z^\ppn(t)$ is called a ``transition chain" if,
for any $\r>0$, there exists a trajectory $z(t)$
and a time $T>0$ such that:
$z(0)$ is in a $\r$--neighbourhood of $\cT^\ppu$; $z(T)$ is in a
$\r$--neighbourhood of $\cT^{(n+1)}$; for all $0\le t\le T$,
$z(t)$ is in a $\r$--neighbourhood of
$Z\=\bigcup_{i=1}^n \{ z=z^\ppi(t):\ t\in\real\}$.
}
\giu
{\elevenbf Definition 3.}
{\sl A Hamiltonian system $(\cM,H)$ with $\cM$ and $H$ as in \S 2
is called ``Arnold unstable" if,
for some $0<\e_0\le \bar\e$,
the following two conditions are fulfilled:
\noindent
{\rm (i)} for all $\e\in (- \e_0,\e_0)$ there exist
invariant sets $\cT(\e)$
and $\cT'(\e)$, which are continuous in the Hausdorff
metric\nota{**}{
If $A$ and $B$ are closed subsets of $\cM$,
the Hausdorff metric $\r(\cdot,\cdot)$
is defined by
$$
\r(A,B)= \max \{ \sup_{x\in A} \dist(x,B)\ ,\
\sup_{x\in B} \dist(x,A)\}\ .
$$},
and
$p_1(\cT(0))\neq p_1(\cT'(0))$,
where $p_1(I,\f)=I$ denotes
the projection onto the ``action" variables;
\noindent
{\rm (ii)} for all $\e\neq 0$ in $(- \e_0,\e_0)$ there exists
a transition
chain connecting $\cT(\e)$ with $\cT'(\e)$.
}
\giu
In other words, an Arnold--unstable system posses,
for all $\e\neq 0$ small
enough, trajectories along which (some of )
the ``action" variables undergo a variation of order 1 and, furthermore,
such trajectories ``shadow" long chains of heteroclinic connections.
Notice that the above instability mechanism is
completely different from the
``linear drift" occurring along
so--called ``superconductivity channels". The classical example
(due to N. N. Nekhoroshev\nota*{See, e.g.,
\S 3.3, chapter 5 of Ref.\ref{1}.}) is the region $I_1=-I_2$ for
the Hamiltonian ${1\over 2} (I_1^2-I_2^2)+\e\sin(\f_1-\f_2)$,
which admits the
trajectory $I_2(t)=\e t=-I_1(t)$, $\f_1(t)=\f_2(t)=-{1\over 2}\e t^2$.
%
\vglue 0.6cm
\line{\elevenbf 4. A Brief Review of Some Known Results\hfil}
\vglue 0.4cm
%
Even though the above definition is strongly inspired by Arnold's
paper\ref{2},
it is not known whether
the Hamiltonian system considered there is
(in a suitable sense) Arnold unstable or not.
Let us be more
specific on this point. In Ref.\ref{2}, Arnold considered on
$\cM\=\real^3\times \torus^3$ the {\sl two--parameter}
family of Hamiltonians
given by
$$
H=I_1+ {1\over 2}\big( I_2^2+ I_3^2\big) + \e (\cos \f_3-1) +
\e\m (\cos \f_3-1) (\cos \f_1 + \sin \f_2)\ .
\eqno(2)
$$
Since $\dot\f_1=1$, it is
$\f_1=t_0+t$ so that the above Hamiltonian
describes, effectively,
a system with two ``degrees of freedom" depending explicitly
(and periodically)
on
time\nota\dag{
We prefer the ``autonomous" formulation which allows a unified
treatment; one should, however, bear in mind that
the ``action" $I_1$ is an auxiliary variable introduced
so as to make the system
autonomous and is not really relevant from a dynamical point of view.
In Ref.\ref{2} $\f_1$, $\f_2$, $t$ correspond here to
$\f_3$, $\f_2$, $\f_1$
and $I_1$, $I_2$ to $I_3$, $I_2$.}.
%
The ``resonant" 2--torus $\{I_1=$const, $I_2=$const, $I_3=0=\f_3$,
$(\f_1,\f_2)\in\torus^2\}$ is invariant for $H$ {\sl for
all $\e$ and $\m$}; such a torus is run by the linear flow
$(\f_1,\f_2)\to(\f_1+t,\f_2+ \o t)$ where $\o$
is the constant value of $I_2$: using
the frequency $(1,\o)$ to parametrize the above invariant torus we
shall denote it by $\cT(\o)$.
Of course the fact the $\cT(\o)$ is invariant for all values of the
parameters $\e$ and $\m$ is a very special feature of the particular
perturbation chosen by Arnold.
Using the language introduced in \S 3, the main result in Ref.\ref{2}
concerning the system with Hamiltonian as in Eq. (2), can
be formulated as follows
\thm{Theorem 1 (Arnold\ref{2})}{ For every $\e,r>0$ there
exists a $\m_0>0$
such that for all $0<\m<\m_0$ there are invariant tori $\cT(\o)$ and
$\cT(\o')$, with $|\o-\o'|>r$,
which are connected by a transition chain.}
%
Note that, for all $\e,\m,\o$, the projection onto the $I_2$--value of
$\cT(\o)$ is just $\o$.
The proof of this result is perturbative in $\m$ keeping
$\e>0$ fixed. For $\m=0$, although one can write down explicitly all
trajectories, the system is not ``completely integrable" in the sense
described in
\S 2. In fact (for $\m=0$) the trajectories with initial
values of $(I_3,\f_3)$
on the ``separatrix" ${1\over 2} I_3^2+\e
(\cos \f_3 -1)=0$ (and $I_3\neq 0$)
are not quasi--periodic.
The arguments used in Ref.\ref{2} indicate that $\m_0$ may be taken to be
of order
$$\exp\big(-{1\over \e^a}\big)\ ,
\eqno(3)$$
for some constant $a\ge {1\over 2}$, leaving, thus, open the following
\thm{Question}{ Does there exist a positive integer $n$ such that
the system with Hamiltonian as in Eq. (2) with $\m=\e^n$ is
Arnold unstable\nota{**}{
The choice $\m=\e^n$ in Arnold's model is somehow ``natural",
not only because it makes $H$ belong to the
``Poincar\`e category" described in \S 2, but also in view of the fact
that near a ``simple resonance" the dynamics
generated by a ``general" Hamiltonian $h(I)+\e f(I,\f)$
is equivalent to that of a Hamiltonian $\tilde h (I;\e)+\e g(I,\f_N;\e)+
\e^n\tilde f(I,\f;\e)$ (everything being real--analytic in a neighbourhood
of, say, $I_0\times \tN$ and of $\e=0$);
for more informations see Ref.\ref{1}.
Furthermore,
in case of models with several perturbation parameters,
it seems natural (at least from the applicative point of view)
to require that such parameters be related
by a power law.}?}
%
A result analogous to that of Arnold has been recently obtained by
Z. Xia\ref{4} in the elliptic restricted three--body problem of Celestial
Mechanics.
Such a system contains many parameters and
by ``similar" we mean the following. First,
transition chains are detected
in parameter regions where, to a positive value of
one of the perturbative parameter, say $\e$,
corresponds some ``reference" hyperbolic trajectory
(analogous to the separatrix
of the pendulum). Then, {\sl keeping fixed $\e$}, perturbation theory
in a second parameter (corresponding to $\m$ in Arnold's example)
is performed. Again, the argument indicates that the result holds
for $0<\m<\m_0(\e)$ with $\m_0(\e)$ as in Eq. (3) and the question
whether the elliptic restricted three--body problem is Arnold unstable
in parameter regions where the various
parameters are related by, say, power
laws is open.
As of now the only explicit system, which has been proven to be
Arnold unstable is the ``eccentric D'Alembert model" of Celestial Mechanics
considered in Ref.\ref{5}, which may be described as follows. Consider a
planet modelled by a ``homogeneous"
rotational ellipsoid whose center of mass revolves on
a Keplerian ellipse of eccentricity $\m$. The planet is subject to the
gravitational attraction of a fixed star occupying one of the foci of
the ellipse. If $(\th_1,\th_2,\th_3)$ denote
the Euler angle of the planet,
the Lagrangian\nota{*}{See, e.g., Ref.\ref{1} for
generalities.}
of the system is given by
$$
{J_3\over 2} (\dot \th_2\cos \th_1+ \dot \th_3)^2+{J_1\over 2}
(\dot \th_1^2
+\dot \th_2^2 \sen^2 \th_1)+ c \int_{E(t)}{ dx\over |x_0(t)+x|}
$$
where $J_3$ and $J_2\=J_1$ are the inertia moments of the planet, $c$ is
a physical constant\nota\dag{$c=\k m_0 m/V$ where $\k$ is Newton's
gravitational constant, $m_0,m$ are the masses of the star and
of the planet, and $V$ is the volume of the planet.},
$E(t)$ is the region occupied by the planet at time $t$ and $x_0(t)$
is the
position of the center of mass of the planet at time $t$. Using the
Andoyer variables (see Ref.\ref{1}) , one can find symplectic
coordinates on $\cM\=G\times \torus^3$
($G$ being some open set of $\real^3$)
so that the dynamics generated by the above Lagrangian is equivalent
to the dynamics generated by the Hamiltonian
$$
H=\o I_1+ \e {I_2^2\over 2} + {I_3^2\over 2} +\e \m f(I,\f;\m)
\eqno(4)$$
for a suitable function $f$
real--analytic (on $\cM\times (-\m_0,\m_0)$),
whose explicit expression can be found in Appendix 14 of Ref.\ref{5};
here $\e$ is essentially the ratio between the polar and equatorial
radii of the planet.
In Ref.\ref{5} it is proven the following
\thm{Theorem 2}{ There exists $n_0\ge 1$ such that for
all integers $n\ge n_0$ the ``eccentric D'Alembert problem"\nota{**}{
That is, the Hamiltonian system on $\cM$ with $H$ as in Eq. (4).}
with $\m=\e^n$ is Arnold unstable.}
%
We remark that in Ref.\ref{1,2,4} there is
no mathematical definition of ``Arnold diffusion" (to be precise
in Ref.\ref{2} such a terminology does not even appear while in
Ref.\ref{1} appears only in a note by the translator). In Ref.\ref{5},
a genearal and detailed terminology (including notions such
as ``diffusion paths") is introduced in the context of perturbations of
``a priori unstable systems", which, roughly speaking, are
``integrable systems possessing separatrices". Examples of
such systems include the above Arnold's model with $\e$ kept fixed and
positive and with $\m$ as perturbation parameter, and the three--body
problem in the parameter region considered by Xia.
Finally, we point out that Theorem~2 is
differently formulated in Ref.\ref{5}
(as the notions given in \S 3 are introduced in this paper).
\section{5}{An Arnold--Unstable System (Result and Scheme of Proof)}
%
\subsection{5.1}{Main Result}
Let $\cM\=\real^3\times \torus^3$ and consider
$$
H(I,\f;\e)\=\e\big({I_1^2\over 2}+ {I_2^2\over 2}\big) + {I_3^2\over 2} +
\e (\cos \f_3 -1) + \e^n \big(\cos (\f_1+\f_3)+\cos(\f_2+\f_3)\big)\ ,
\eqno(5)$$
where $n$ is an integer greater than 1. We shall prove
\thm{Theorem 3}{ There exists $n_0\ge 1$
such that for all integers $n\ge
n_0$ the above Hamiltonian system $(\cM,H)$ is Arnold unstable.}
\noindent
{\bf Remark} (i) In some
sense $(\cM,H)$ is the ``simplest" Arnold--unstable
system with a perturbation\nota*{
In Eq. (5) the perturbation is $H_1=\e\big\{{I_1^2\over 2}+ {I_2^2\over 2}
+(\cos \f_3 -1) + \e^{n-1}
\big(\cos (\f_1+\f_3)+\cos(\f_2+\f_3)\big)\big\}$,
the ``integrable part" is $H_0={I_3^2\over 2}$: $H=H_0+ H_1$.
As already mentioned, having a ``separation" (in terms of order in $\e$)
may be achieved near a ``simple resonance"
by means of symplectic transformations.
}
exhibiting the main feature of a ``general perturbation".
In particular the invariant sets $\cT(\e)$,
which will be shown to be connected, for $\e\neq 0$,
by heteroclinic trajectories, will
be non--trivial continuations of invariant sets
for $\e=0$
(i.e., $\cT(0)$ is {\rm not} invariant
for $\phi_H^t$ as it happens in Arnold's
example).
(ii) What is really ``not general"
in the Hamiltonian in Eq. (5) is the integrable part
$H_0\={I_3^2\over 2}$, which is highly degenerate (depending only on one of the
three action variables). Note that also the ``eccentric D'Alembert problem"
features a similar (although less dramatic) degeneracy, which is actually
quite common in problems arising in Celestial Mechanics.
(iii) {\sl No example is known of an Arnold--unstable
system on $G\times \real^N$}
($G$ open set in $\real^N$) {\sl
with ``non--degenerate" integrable
part} (of the form, say, $H_0\={1\over 2} \sum_{i=1}^N I_i^2$).
\medsk
\subsection{5.2}{Outline of Proof}
The system with Hamiltonian
$$
h(I,\f_3;\e)\= \e\big( {I_1^2\over 2}+
{I_2^2\over 2}\big) +{I_3^2\over 2} +\e (\cos
\f_3-1)
$$
may be explicitly solved as it decouples
into a bidimensional completely integrable
system times a pendulum.
The instability region will be sought in a neighbourhood
of the ($\e$--dependent) separatrix of the pendulum.
Since for $\e$ small and positive\nota\dag{During the proof
we shall always assume
$\e>0$. The case $\e<0$ can be easily recovered by a simple translation in
the $\f_3$ variable.},
$\e^n\ll \e$, it is natural to
treat the full Hamiltonian system as a perturbation of the system $(\cM,h)$.
To make this remark precise, we replace, in a first moment, $\e^n$ with an
{\sl auxiliary parameter} $\m$, and establish perturbative results in $\m$.
Such result will be analytic in $\m$ and we shall prove that for a suitable $n$,
$\e^n$ is inside the $\m$--analyticity domain allowing to recover
the original system (by setting $\m=\e^n$).
Therefore, consider the {\sl modified system} on $\cM$ with Hamiltonian
$$
\widetilde H (I,\f;\e,\m)\= h+\m
f\ ,\qquad f\=\cos(\f_1+\f_3)+\cos(\f_2+\f_3)\ ,
$$
so that $\widetilde H(I,\f;\e,\e^n)=H(I,\f;\e)$.
For $\m=0$, $\e>0$ and $\o\in\real^2$, consider the $h$--invariant set
$$
\cT(\e,\o)\=\{ I_1=\o_1\ ,\quad I_2=\o_2\ ,\quad
I_3=\f_3=0\ ,\quad (\f_1,\f_2)\in\torus^2\}\ ,
$$
on which the flow is quasi--periodic with frequencies $\e\o$:
$$
\phi_h^t(I_1,I_2,0,\f_1,\f_2,0)=(I_1,I_2,0,\f_1+\e\o_1 t,
\f_2+\e\o_2 t,0)\ .$$
These tori admit ``asymptotic manifolds", called by Arnold ``whiskers",
given by
$$ W(\e,\o)\=\{ I_1=\o_1\ ,\quad I_2=\o_2\ ,\quad
{1\over 2} I_3^2 +\e(\cos\f_3-1)=0
\ ,\quad (\f_1,\f_2)\in\torus^2\}\ .
$$
Each $h$--trajectory $z(t)$ starting on $W(\e,\o)$
approaches, as $t\to \pm\io$, the torus $\cT(\e,\o)$; note that $W(\e,\o)
\supset \cT(\e,\o)$ and that if $z(0)\in W(\e,\o)\backslash \cT(\e,\o)$,
then $\dist\big(z(t),\cT(\e,\o)\big)=O\big(\exp(-|t|\sqrt{\e})\big)$.
As a set, $\cT(\e,\o)$ is {\sl independent of $\e$}
(hence, in particular, $\cT(\e,\o)$ is continuous in $\e\in\real$);
what changes with $\e$ are the frequencies $\e\o$. In the limit case
$\e=0$, $\cT(0,\o)$ is still an invariant 2--torus made up by fixed points.
On the other hand, in the limit case $\e=0$, the asymptotic manifolds
$W(0,\o)$ cease to exist in the sense that the set $W(0,\o)\=\{I_1=\o_1$,
$I_2=\o_2$, $I_3=0$, $\f\in\torus^3\}$ is, again, an invariant set
made up by fixed points hence a trajectory starting on $W(0,\o)\backslash
\cT(0,\o)$ {\sl does not} approach $\cT(0,\o)$.
The proof is based on three separate steps, which we proceed to describe.
{\bf (1)} Let, again, $\e>0$.
Roughly speaking,
the first step consists in constructing the analytic continuation
in $\m$ of the tori $\cT(\e,\o)$ together with their ``local" whiskers,
for ``many" values of the ``rescaled frequencies" $\o$; by ``local" we mean
``in a ($\e$--dependent) neighbourhood of $\cT(\e,\o)$". If $\e>0$
is small enough, the $\m$--analyticity domain contains a complex disk
of radius $\m_0\=\e^m$
(for $m$ suitable).
The set of $\o$'s for which the continuation holds (i.e. the set
of ``persistent tori") is a Cantor set of ``local arbitrarly high density"
and the whiskers can be smoothly interpolated.
A somewhat more precise formulation is given in the following:
\thm{Proposition}{
Given $R>0$ and an integer $s\ge 1$, there exist
$\e_0>0$, an integer $d\ge 2$, and an open set
$B\subset B_R^2$ (the bidimensional sphere of radius $R$) with
the following properties. For all $0<\e<\e_0$
and for all $m>\max\{2,d/s\}$
there exists
a Cantor set $\O_\e\subset B$ such that:
\newline
(i) $\O_\e\subset \O_{\e'}$ whenever $\e'\le\e$; if $\o\in\O_\e$ then
the ray $(1+a)\o\in \O_\e$ for all $0\le aFrom the above analysis it follows that $\d(\m)$ is analytic
for $|\m|\le \m_0\=\e^m$ and that $\d(0)=0$ i.e.
$\d(\m)=\d_1\m+\d_2\m^2 +\cdots$, the series being convergent
for $|\m|\le \m_0$. It is
clear that, in principle, one could compute $\d_k$ using the
strategy outlined above, which is based on KAM techniques.
In practice, however, it is much simpler to derive a
{\sl direct algorithm},
which we shall briefly describe in \S 6.2
below. {\sl The upshot is that $\d_1$ is not exponentially
small in $\e$}, contrary to what happens in Arnold's example
[where $\d_1\sim \exp(-\e^{-1/2})$],
but, as we shall see in \S 6.2, it is\nota{**}{Here $a\sim x$ means
that there exist two ($x$--independent) constants $c_1,c_2$
such that $c_1x\le a\le c_2 x$ for all $x$ small enough.}
$$\d_1\sim \e^{d_0}$$
%
for a suitable number $d_0$ {\sl depending only on $h$ and $f$}.
>From such relation and the analyticity properties in $\m$, it
follows easily that {\sl $\d$ can be exatly evaluated and is
of the size of $\e^{m_2}$ for a suitable $m_2$}, which we now compute.
In fact, since $\d(\m)$ is analytic in a disc of radius $\m_0=\e^m$
and can be bounded by an $\e$--independent constant,
one has that
$|\d_k|\le$ $\const\e^{-km}$. Thus, if one takes
$$
\m=\e^{m_1}\ ,\qquad {\rm with}\ m_1>
\max\{2m+d_0, m\}$$
one has, for all $k\ge 2$,
$$
|\d_k\m^k|\le \const \e^{k(m_1-m)}\le \const \e^{2(m_1-m)}<
\const \e^{d_0+m_1}\le |\d_1\m|$$
(as above $\const$'s denote positive constants, which depend only on $R$).
Thus
$$
\d(\e^{m_1},\e,\o)\sim \e^{m_2}\ ,\qquad
{\rm with}\quad
m_2\=d_0+m_1\ .$$
We finally point out that {\sl for the eccentric D'Alembert problem
$\d_1$ is indeed exponentially small in $\e$},
$\d_1\sim\exp(-\e^{-1/2})$, {\sl while it is $\d_2$ which
is a power of $\e$} (allowing to repeat the above argument).
{\bf (3)}
To construct transition chains we have to study {\sl heteroclinic
connections} i.e. orbits $z(t)$ approaching, as $t\to\pm\io$, {\sl
different} invariant tori. In other words, we have to study intersections
of $W^+_{\m,{\rm loc}}(\e,\o)$ with $W^-_{\m,{\rm loc}}(\e,\o')$
with $\o\neq \o'$. To do this, we fix $0<\e<\e_0$, $\m\=\e^{m_1}$,
$\bar\o\in\O_\e$, and let $\o$ and $\o+\a$ belong to $\O_\e(\bar\o,\m)$.
Now, define, for $i=1,2$,
$$
F_i(\f_1,\f_2,\a)\=I^-_i(\f_1,\f_2,\p,\e^{m_1},\e,\o+\a)-
I^+_i(\f_1,\f_2,\p,\e^{m_1},\e,\o)\ ,
$$
so that $F(\f_1,\f_2,0)=\tF(\f_1,\f_2;\e^{m_1})$.
Since, by our choice of $\o$ and $\a$,
the tori $\cT_\m(\e,\o)$ and $\cT_\m,(\e,\o+\a)$ lie in the same
energy manifold [which is that of $\cT_\m(\e,\bar\o)$],
heteroclinic connections correspond to solutions
$\f_1(\a)$, $\f_2(\a)$ satisfying
$F(\f_1(\a),\f_2(\a),\a)=0$.
By the Implicit Function
Theorem\nota{*}{We use a standard Implicit Function Theorem,
which guarantees continuous solutions $\a\in
B^2_a\to\f(\a)\in B^2_b$ of $F(\f,\a)=0$ [here $\f=(\f_1,\f_2)$]
provided $F(0,0)=0$, $F$ and ${\partial F\over \partial \f}$
are continuous in
$B^2_b\times B^2_a$,
the Jacobian ${\partial F\over \partial\f}(0,0)$ has an inverse $T$
and
$$
\sup_{|\a|\le a}|F(0,\a)|\le {1\over 2} \|T\|^{-1}\ b\ ,\qquad
\sup_{|\f|\le b,|\a|\le a} \|I- T{\partial F\over \partial \f}\|\le
{1\over 2}\ .
$$}
and the properties listed
in {\bf (1)} and {\bf (2)}, it follows that one has (transversal)
heteroclinic intersections whenever
$$
\o,\o+\a\in\O_\e(\bar\o,\e^{m_1})\ ,\qquad {\rm and}\quad
|\a|\le \const \e^{m_2}\ .
$$
To insure the existence of heteroclinic connections,
we have to guarantee that the gaps on
$\O_\e(\bar\o,\e^{m_1})$ are smaller that $\e^{m_2}$. In view of
point (v) of the Proposition in step {\bf (1)}, it is sufficient to require
that $(ms-d)/ 2>m_2$.
All conditions on $m$ and $s$ (which are still free)
may be collected in
$$
m>\max\{2,{d\over s}\}\ ,\qquad ms-d>2 \max\{m+d_0, 2(m+d_0)\}
$$
and it is clear that such conditions may always be met choosing
suitably $m$ and $s$ in terms of $d$ and $d_0$. The number $n_0$ in
Theorem 3, can be taken to be equal to $m_1$ which , in turn, may be fixed
so that
$$
\max\{m,2m+d_0\}a R\ ,\ \forall\ n\in \integer^2\ ,
\ 0<|n|\le s-1\}\cr
\O_\e &\=\{\o\in B:\ |\o\cdot n|\ge a\ {\e^{(ms-d)/2}\over |n|^2}\ ,\
\forall\ n\in \integer^2\ ,\ n\neq 0\}\ ,\cr}
$$
where $a>0$ is a suitable constant (depending on $s$) small enough and
`` $\cdot$ " detotes scalar product in $\real^2$.
The set $B$ is a disk of radius $R$ minus a finite number of strips
crossing the origin; the constant $a$ may be chosen
so that the measure of $B$ is close as we like to $\p R^2$. $\O_\e$ is a
Cantor set in $B$ (excluding, for all $\e$, the dense set of
``resonant" $\o$'s
for which $\o\cdot n=0$ for some $n\in\integer^2$)
that becomes of full measure as $\e$ approaches 0.
The definition of $B$, as we shall see below, allows to perform $(s-1)$
steps
of classical (Poincar\'e) perturbation theory, in a neighbourhood of
``resonant" tori $\cT(\e,\o)$ for which $\o\in B$. The definition of
$\O_\e$ is the classical (Kolmogorov) way of defining the frequencies
``simultaneously accessible" by KAM techniques (the $\e$--dependent
constant appearing in the definition of $\O_\e$ will be explained below).
Next, we introduce symplectic ``hyperbolic
coordinates" for the pendulum $P\= {1\over 2} I_3^2+\e(\cos \f_3 -1)$.
Such a pendulum has the origin $I_3=0=\f_3$ as unstable (hyperbolic) fixed
point. The quadratic part of $P$, given by ${1\over
2}\big(I_3^2-(\sqrt{\e}\f_3)^2\big)$, is transformed, via the symplectic
map
$$\pmatrix{I_3\cr \f_3\cr}={1\over \sqrt{2}} \pmatrix{ \e^{1/4} &
\e^{1/4}\cr -{1\over \e^{1/4}} & {1\over \e^{1/4}}\cr} \pmatrix{y\cr
x\cr}
\eqno(9)$$
into the ``normal hyperbolic" form $\sqrt{\e} yx$. It is therefore clear
that, near the origin, there exists a symplectic (real--analyitc)
transformation $S$, whose linear part is given by Eq. (9), transforming
the pendulum $P$ into the normal form
$$
P\circ S= g(yx, \sqrt{\e})=\sqrt{\e}(yx)+ O\big( (yx)^2\big)\ .
$$
The domain of definition of $S$ (and hence of $g$) can be taken to be
a 2--sphere centered at the origin of radius, say, $4b$ for a suitable
$b\sim\e^{1/4}$.
In the $(I_3,\f_3)$--coordinates, the sphere $B^2_{4b}$ correspond,
in a first approximation, to an ellipse with $I_3$--semiaxis of length
$\sim \sqrt{\e}$ and with $\f_3$--semiaxis of length $\sim 1$.
In the symplectic coordinates
$$
\bI\=(I_1,I_2)\ ,\quad \baf\=(\f_1,\f_2)\ ,\quad (y,x)\ ,
$$
the Hamiltonian $\wH$ takes the form
$$
H_1(\bI,\baf,y,x,\e,\m) \= h_1+\m f_1
\= {\e\over 2}\bI^2 + g(yx,\sqrt{\e})+ \m f_1(\baf, y,x, \sqrt{\e})\ ,
$$
where $f_1$ is a linear combination of $\cos \f_i$ and $\sin \f_i$,
with $i=1,2$, with coefficients which are real--analytic functions
of $\x=yx$. By classical perturbation theory, one can find a symplectic
transformation $\m$--close to the identity, conjugating $H_1$ to a
Hamiltonian
$$\eqalign{ H_2(\bI',\baf',y',x',\e,\m) &\= h_2+\m^s f_2\cr
&\= h_1(\bI', y'x', \sqrt{\e})+ \m g_1(y'x',\sqrt{\e},\m)+ \m^s
f_2(\bI', \baf', y',x',\sqrt{\e},\m)\ .
\cr}
$$
The ``small divisors" appearing in such a symplectic
transformation\nota{\dag}{The transformation can be found, e.g.,
by solving up to order $\m^{s-1}$ the Hamilton--Jacobi equation
for a generating function of the form $\bI'\cdot \baf+y'x+$ polynomial in
$\m$ of order $(s-1)$ vanishing at $\m=0$.},
have the form $i\e \bI'\cdot k+ j \sqrt{\e} y'x'$ with
$k\in\integer^2$, $j\in\natural$, $|k|+j>0$ and $|k|\le s-1$. Thus the
domain of definition (and analyticity) of $H_2$ is given, for $0<\e<\e_0$,
by $B\times\torus^2\times B^2_{2b}\times B^1_{\bar\m}$ and $\bar \m$ may
be choosen proportional to $\e^2$.
In these coordinates, the tori
$\cT(\e,\o)$ are given by $\bI'=\o$, $\baf'\in\torus^2$, $y'=x'=0$ and the
stable/unstable local whiskers are given by, respectively,
$\bI'=\o$, $\baf'\in\torus^2$, $|y'|\le 2b$, $x'=0$,
and
$\bI'=\o$, $\baf'\in\torus^2$, $|x'|\le 2b$, $y'=0$.
At this point one can use a KAM ``fast--converging" scheme
to construct, simultaneously for $\o\in\O_\e$,
invariant tori with flow conjugated to $t\to\e\o t$,
and their
local whiskers, which will be $\m^s$ close to the
local whiskers for $h_2$.
As it is well known, the smallness requirement
(or ``KAM condition") necessary
in order to carry out the scheme, is of the form
$$
\const\ {\m_0^s \over \e^d \g^2}<1\ ,
\eqno(10)$$
where the analyticity domain in $\m$ will contain the disk
$|\m|\le \m_0$ and $\g$ is the ``rescaled Diophantine constant"
associated to the persistent tori $\cT(\e,\m)$, i.e., $\o$ is such that
$$
|\o\cdot n|\ge {\g\over |n|^2}\ ,\qquad
\forall\ n\neq 0\ ,
\eqno(11)
$$
[hence the small divisors appearing in the KAM scheme satisfy
$|\e\o\cdot n|\ge (\e\g)/|n|^2$];
$d$ is a constant (depending only on the
structure of $h$) which is greater or equal than two since, as already
mentioned, the actual (``non--rescaled") Diophantine constant is $\e\g$
and such a constant appears quadratically in KAM conditions.
All the frequencies $\o$ satisfying the classical condition Eq. (11),
may be taken care of simultaneously and the complementary set
$\{\o\in B:$ Eq. (11) does not hold $\}$
has a measure of the order of $\g$.
Thus, choosing $\m_0=\e^m$, with $m>\max\{2,d/s\}$, we see that $\g$
may be taken to be [in order to meet the KAM condition Eq. (10)]
$\g=\const \e^{(ms-d)/2}$.
The conclusion is that, with the above choices, one can construct $C^3$
functions $\z$ and $\l$,
$$
\eqalign{
& \z: (\th,(y,x),\o,\m)\in\torus^2\times B^2_b\times B^2_R \times
B^1_{\m_0} \to \z(\th,y,x,\o,\m)\in\cM\ ,\cr
& \l:(\x,\o,\m)\in B^1_b\times B^2_R\times B^1_{\m_0}\to
\l(\x,\o,\m)\in\real\ ,
\cr}$$
such that the following holds.
For $\o\in \O_\e$, $\z$ and $\l$ are real--analytic in
all their remaining variables;
$\l(0,\o,0)=\sqrt{\e}$; $\z(\th,0,0,\o,0)=(\o,0,\th,0)$;
$$
\sup_{\torus^2\times B^2_b\times B^2_R
\times B^1_{\m_0}}\ |\partial^j_\th \partial^k_\o
\z|\le
\const\ ,\qquad\ \forall\ |j|\le 2\ ,\ |k|\le 1\ ;
$$
for $\o\in \O_\e$ and $|\m|\le \m_0$ one has
$$\eqalign{
&\cT_\m(\e,\m)= \{z=\z(\th,0,0,\o,\m): \ \th\in \torus^2\}\ ,\cr
&W^+_{\m, {\rm loc}}(\e,\o)= \{z=\z(\th,y,0,\o,\m);\ \th\in\torus^2\ ,
|y|**0\}$ crosses any
manifold which intersects transversally $W^-_{\m, {\rm loc}}$ in $z_-$].
As for the final point (v) of the KAM Proposition,
observe that from the definition of the set $\O_\e$, it follows
that the ray
$(1+a)\o$, with $0\le a< R-|\o|$, belongs to $\O_\e$, for any $\o\in \O_\e$.
Also, the set $\O_\e(\bar\o,\m)$ coincides with
$$
\O_\e(\bar\o,\m)\=\big\{\o\in\O_\e:\ \wH\big(\z(0,0,0,\o,\m)\big) =
\wH\big(\z(0,0,0,\bar\o,\m)\big)\big\}
$$
and the set
$$\G_\m=
\big\{\o\in B:\ \wH\big(\z(0,0,0,\o,\m)\big) =
\wH\big(\z(0,0,0,\bar\o,\m)\big)\big\}
$$
is a finite union of $C^1$ curves, which are
$|\m|$--close [by the smoothness of $\z$ in $\o\in B^2_R$ and the fact
that $p_1\big(\z(0,0,0,\bar\o,0)\big)=\bar\o$] to
$\{|\o|=|\bar\o|\}$ intersected with $B$. Thus, since $\O_\e$ has a measure
not smaller than, approximately,
$R^2 \big(1-\e^{(ms-d)/2}\big)$,
it is clear that, in general, $\O_\e\cap \G_\e(\bar\o,\e)$
cuts on $\G_\e(\bar\o,\e)$
a region of length not smaller than
$|\bar\o| \big(1-\e^{(ms-d)/2}\big)$.
\vglue0.4cm
\subsection{6.2}{Generalized Poincar\'e Integrals}
\medskip
Here we discuss with some detail {\sl
a recursive algorithm that allows to
compute all the $\m$--derivatives of the function $\d(\m)$} introduced in
Eq. (8). In particular, we shall prove that, for the Hamiltonian in
Eq. (5), it is
$$\d_1= {4\over \e}\big(1+O(\e)\big)\ ,\qquad {\rm i.e.}\quad d_0=-1\ .
\eqno(13)$$
Actually, to prove Eq. (13) is rather easy, but we shall spend some
time in order to discuss,
in some generality, the above mentioned
algorithm, which, we believe, is quite interesting in itself.
As already mentioned in the previous section, the derivatives of $\d$
could be computed by using the KAM results, however we prefer, following
seminal ideas of Poincar\'e\nota{**}{See also Ref.\ref{2} and, for
generalities and further references, Ref.\ref{1}.},
a more direct approach.
Keeping fixed $0<\e<\e_0$ and $\o\in\O_\e$ [and using the above
conventions: $\bI\=(I_1,I_2)$, $\baf\=(\f_1,\f_2)$], we denote
$\wH$--trajectories starting at $W^\pm_\m\cap\{\f_3=\p\}$ by
$$\eqalign{
z^\pm(t;\baf,\m)&\= \big( I^\pm(t;\baf,\m),\f^\pm(t;\baf,\m)\big)
\=\big( \bI^\pm(t;\baf,\m),p^\pm(t;\baf,\m),\baf^\pm(t;\baf,\m),
q^\pm(t;\baf,\m)\big)\cr
&\=\phi_\wH^t\big(I^\pm(\baf,\p,\m),\baf,\m\big)
\cr}
$$
where $I^\pm(\f,\m)$ are the functions introduced in {\bf (2)} of
\S 5.2 (in the first line of this definition, we have introduced different
symbols for the same object, which is then defined in the second line). In
particular,
$I^\pm(0;\baf,\m)=I^\pm(\baf,\p,\m)$,
$\f^\pm(0;\baf, \m)=(\baf, \p)$.
An important remark, based on the KAM analysis, is that, since trajectories on
$W^\pm_\m$ approach (exponentially fast) the invariant tori $\cT_\m(\e,\o)$
on which the flow is conjugated to quasi--periodic,
{\sl the functions}\nota{*}{If not needed explicitly, we shall drop the
dependence on $\baf$ and $\m$.}
$I^\pm(t)$, $\f^\pm(t)$
{\sl are ``asymptotically (as
$t\to\pm\io$) quasi--periodic functions" with frequencies $\e\o$}.
Here by ``$f^\s$ asymptotically quasi--periodic", we mean that $f^\s=\psi+g$
with $\psi$ quasi--periodic with frequencies $\e\o$ and $g$ converging
to 0 {\sl exponentially fast}, as $t\to\s\io$.
The function $z^\pm(t)\=\big(I^\pm(t), \f^\pm(t)\big)$ is analytic in
$|\m|\le \m_0$ and has a convergent power--series representation
$$
I^\pm(t;\baf,\m)=\sum_{k=0}^\io I^{k\pm}(t,\baf)\m^k\ ,\qquad
\f^\pm(t;\baf,\m)=\sum_{k=0}^\io \f^{k\pm}(t,\baf)\m^k\ ,
$$
where $\Big(I^{0\pm}(t),\f^{0\pm}(t)\Big)\=z^0(t)$ is the unperturbed
$h$--trajectory given by
$$ z^0(t)\=\big(\o,p^0(t),\baf+\e\o t, q^0(t)\big)\ ,
$$
$(p^0,q^0)$ being the separatrix motion of the pendulum ${1\over
2}I_3^2+\e(\cos\f_3-1)$ starting at $\f_3=\p$:
$$q^0(t)=4\arctan e^{-\sqrt{\e}t}\ ,\quad p^0(t)={2\sqrt{\e}\over \cosh
\sqrt{\e} t}\ ,\quad
\cos q^0=1-{2\over (\cos \sqe t)^2}\ , \quad
\sin q^0=2{\sinh \sqe t\over (\cos \sqe t)^2}\ .
$$
(and, for definitess, we have choosen the upper branch, $I_3>0$).
Letting $(\bI^{k\pm},p^{k\pm})\=I^{k\pm}$, $(\baf^{k\pm},q^{k\pm})\=
\f^{k\pm}$ and
inserting the power series into the $\wH$--Hamilton
equations and equating the coefficients of $\m^k$ we get, for $k\ge 1$,
the hierarchy of equations
$$
\eqalign{ &\dot \bI^{k\pm}=\bar\F^{k\pm}\ ,\qquad\qquad\qquad\
\dot\baf^{k\pm}=\e\bI^{k\pm}\ ,\cr
& \dot p^{k\pm}=(\e\cos q^0) q^{k\pm}+\Phi_3^{k\pm}\ ,\quad \dot
q^{k\pm}=p^{k\pm}\cr}
\eqno(14)$$
where the functions $\Phi^{k\pm}(t;\baf)
\=(\Phi_1^{k\pm},\Phi_2^{k\pm},\Phi_3^{k\pm})
\=(\bar\Phi^{k\pm},\Phi_3^{k\pm})$
can be easy computed by Taylor
expansion and depend only on $z^{j\pm}$ with $0\le j\le k-1$.
For example, $\bar\Phi^{1\pm}$, $\Phi_3^{1\pm}$, which
are actually independent of the $+$ or $-$ sign, are given by
$$
\eqalign{
\bar\Phi^1(t,\baf)&\=-{\partial f\over \partial \baf}\big(\baf+\e\o
t,q^0(t)\big)\=
\Big(\sin\big(\f_1+\e\o_1 t+q^0(t)\big),
\sin\big(\f_2+\e\o_2 t+q^0(t)\big)\Big)\
,\cr
\Phi^1_3(t,\baf)&\=-{\partial f\over \partial \f_3}\big(\baf+\e\o
t,q^0(t)\big)\=
\sin\big(\f_1+\e\o_1 t+q^0(t)\big)+ \sin\big(\f_2+\e\o_2 t+q^0(t)\big)
\ .\cr}
$$
Observing that the fundamental solution (or ``Wronskian")
of the linearized pendulum equation $\dot p=(\e\cos q^0)q$,
$\dot q=p$, is given by
$$ W(t)=\pmatrix{ (1-{w(t)\over4}{\sinh \sqrt{\e}t\over\cosh^2\sqe t})
\cosh \sqe t&-\sqe{\sinh \sqe t\over\cosh^2 \sqe t}\cr
{w(t)\over4\sqe}&{1\over\cosh \sqe t}\cr},\qquad{w(t)}\={2\sqe t+\sinh
2\sqe t\over\cosh \sqe t}\ ,$$
integrating Eq. (14), one can express $z^{k\pm}(t)$
in terms of $z^0(t)$,...,$z^{(k-1)\pm}(t)$:
$$
\eqalign{
&\bI^\kp(t)=\bI^\kp(0)+\int_0^t\bar\F^\kp(\t)d\t\ ,\quad
\baf^\kp(t)=\e\bI^\kp(0)t + \e\int_0^t(t-\t) \bar\F^\kp(\t)d\t\ ,\cr
& p^{k\pm}(t)= w_{11}(t)q^{k\pm}(0)+ w_{11}(t)
\int^t_0 w_{22}(\t) \Phi^{k\pm}_3(\t) d\t- w_{12}(t)\int^t_0
w_{21}(\t) \Phi^{k\pm}_3(\t)\,d\t\ ,\cr
& q^{k\pm}(t)=
w_{21}(t)q^{k\pm}(0)+ w_{21}(t)
\int^t_0 w_{22}(\t) \Phi^{k\pm}_3(\t) d\t- w_{22}(t)\int^t_0
w_{21}(\t) \F^{k\pm}_3(\t)\,d\t\ ,\cr}
\eqno(15)$$
where $w_{ij}$ are the entries of $W(t)$ and we have used the fact that,
for $k\ge 1$, $\baf^\kp(0)=0=q^\kp(0)$.
Now, {\sl the key idea is to integrate from $0$ to $\pm\io$
and, using the fact that $z^\kp(t)$ is aymptotically (as $t\to\io$)
a quasi--periodic function, to try to identify the ``initial value"}
$I^\kp(0)\={1\over k!} \left.{d\over
d\m}\right|_{\m=0}I^\pm(\baf,\p,\m)$
{\sl in terms of integrals of the $\Phi$'s functions}.
In order to perform with ease improper integrations, we introduce the
following {\sl generalization of standard improper integration}\nota{\dag}{
The operations discussed here are introduced in Ref.\ref{5}.}.
Let $\s$ denote the $+$ or $-$ sign. Let $\cH_\s$ be the class of
measurable functions $f:\real\to\complex$ such that there exist $a>0>b$
for which, given $t\in \real$, the function
$$u\to F(u,t)\=\int_{\s\io}^t e^{-u\s \t} f(\t)d\t
$$
is analytic on the complex domain $\{u\in\complex:\Re\
u>a\}$ and admits an analytic continuation to
$\{u\in\complex:\Re\ u****0$,
then $f\in\cH_\s$ and $\cJ^t_\s f=\int_{\s\io}^t f$.
It is also easy to check that $f\=t^j e^{c t}\in\cH_\s$
for any $j,\s$ and any {\sl non vanishing} complex
number $c$. Instead, the polynomials are not contained in $\cH_\s$.
Nevertheless, we extend, by linearity,
the operator $\cJ$ on $\wcF_\s\=\cH_\s\oplus$ ring of
polynomials in $t$, by defining
$$
\cJ_\s^t\t^j= {t^{j+1}\over (j+1)}\ .
$$
Such an extension implies the following useful property of $\cJ$:
$$
\cJ_\s^tf-\cJ_\s^sf=\int_s^t f(\t)d\t\ ,
\eqno(16)$$
for any $f\in\wcF_\s$ and any $s,t$.
An easy recursive argument (either based on the KAM analysis of \S 6.1
or direct) shows that {\sl $z^\kp$, and hence $\F^\kp$,
belong to $\cH_\pm$ for all $k\ge 1$}.
>From Eq. (15) and Eq. (16) it then follows
$$
\bI^\kp(t)-\cJ_\pm^t\bar\F^\kp=
\bI^\kp(0)-\cJ_\pm^0\bar\F^\kp\ .
\eqno(17)$$
We, now, note the following two facts (which are easily checked): (i)
The quasi--periodic average, $\lim_{T\to\io}(\s T)^{-1}\int_0^T f$,
of a function $f\=\psi+g$,
which is asymptotically (as $t\to\s \io$) quasi--periodic,
coincides with the quasi--periodic average of the limiting
function $\psi$, and we shall denote such a number by $\langle f\rangle$;
(ii) if $f$ is asymptotically quasi--periodic with $\langle f\rangle=0$, then
($f\in\cH_\s$ and) so is $\cJ_\s^t f$. Thus,
taking the quasi--periodic average in the first line of Eq. (14),
one sees that both $\bar\F^\kp$ and $\bI^\kp$, which are asymptotically
quasi--periodic, have vanishing quasi--periodic average. Therefore,
taking the quasi--periodic average in Eq. (17), we see that
$$
\bI^\pm(0)=\cJ_\pm^0\bar\F^\kp\ ,\qquad{hence}\quad
\bI^\pm(t)=\cJ_\pm^t\bar\F^\kp\ ,\qquad \forall\ t\ .
$$
With similar arguments one finds also that
$$\eqalign{ & p^{k\s}(t)= w_{11}(t)\int_{\s\io}^t
w_{22}(\t)\F^{k\s}_3(\t)d\t- w_{12}(t)\int^t_0 w_{21}(\t)
\F^{k\s}_3(\t)d\t\ ,\cr
& q^{k\s}(t)=
w_{21}(t)\int_{\s\io}^t w_{22}(\t) \F^{k\s}_3(\t)d\t-
w_{22}(t)\int^t_0 w_{21}(\t) \F^{k\s}_3(\t)d\t\ .\cr}
$$
Let us now go back to the computation of $\d_1$ and let us introduce the
following notation: if $f^\pm$ denotes {\sl two} functions $f^+\in\cH_+$ and
$f^-\in\cH_-$, we let
$$
\cJ f^\pm\=\cJ_-^0f^- - \cJ_+^0 f^+\ .$$
Thus, recalling the definition of $\bar\F^1$, we have
$$\eqalign{
\bI^{1-}(\baf)-\bI^{1+}(\baf)&\=\left.{d\over d\m}\right|_{\m=0}
\big(\bI^{1-}(\baf,\p,\m) -\bI^{1+}(\baf,\p,\m)\big)\cr
&=
\Big(\cJ
\sin\big(\f_1+\e\o_1 t+q^0(t)\big), \cJ \sin\big(\f_2+\e\o_2
t+q^0(t)\big)\Big)\ .\cr}
$$
A straightforward computation, based upon the definition of $\cJ$,
symmetries\nota{**}{If $f\in\cH_\s$ is an even/odd function of $t$, then
$\cJ^t_\s f$ is odd/even. }
and residue theory, yields
$$\eqalign{
\cJ \sin(\f_i+\e\o_i t+q^0)&=
\sin\f_i\cJ\Big(\cos\e\o_i t\cos q^0-\sin \e\o_i t\sin q^0\Big)\cr
&=\sin \f_i \int_{-\io}^\io\{\cos \e\o_i t (\cos q^0 -1)\}- \sin\f_i
\int_{-\io}^\io \sin \e\o_i t\sin q^0\cr
&= -2\sin \f_i \int_{-\io}^\io
\Big\{
{\cos\e\o_i t\over (\cosh \sqe t)^2} +
{\sin\e\o_i t \senh \sqe t\over (\cosh\sqe t)^2}\Big\}\cr
&=-\p\o_i\sin\f_i \Big({1\over \sinh{\p\sqe \o_i\over 2} }+
{1\over \cosh{\p\sqe \o_i\over 2}}\Big)\cr}
$$
from which Eq. (13) follows at once.
\vglue 0.6cm
\line{\elevenbf 7. References \hfil}
\vglue 0.4cm
\medskip
\item{1.} V.I. Arnold (Ed.) , {\elevenit Encyclopaedia
of Mathematical Sciences} {\elevenbf III} {\elevenit ``Dynamical Systems"}
(Springer--Verlag, 1988) .
\item{2.} V. I. Arnold, {\elevenit Instability of Dynamical Systems with
Several Degrees of Freedom}, in {\elevenit Sov. Math. Dokl.}
{\elevenbf 6} (1964) p. 581--585.
%\item{3.} B. V. Chirikov,
%{\elevenit A Universal Instability of Many--Dimensional
%Oscillator Systems}, in {\elevenit Phys. Rep.} {\elevenbf 52} (1979),
%p. 263--379.
\item{3.} H. Poincar\'e, {\elevenit Les M\'ethodes Nouvelles de la M\'ecanique
C\'eleste} Vol. 1--3 (Paris, Gauthier--Villars, 1892, 1893, 1899).
\item{4.} Z. Xia, {\elevenit Arnold Diffusion in the Elliptic Restricted
Three--Body Problem}, in {\elevenit J. Dyn. Differ. Equations} {\elevenbf 5},
No. 2, (1993), p. 219--240.
\item{5.} L. Chierchia and G. Gallavotti, {\elevenit Drift and Diffusion
in Phase Space}, in {\elevenit Ann. H. Poincar\`e} (Physique Theorique),
{\elevenbf 60}, No. 1 (1994), p. 1--144.
\bye
**