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perturbation theory, averaging theorem, thermodynamic limit
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\title{ An averaging theorem for Hamiltonian dynamical systems in the
thermodynamic limit}
\author{ A. Carati\footnote{ Universit\`a di Milano, Dipartimento di Matematica
Via Saldini 50, 20133 Milano (Italy)
E-mail: {\tt carati@mat.unimi.it} }
}
\date{\today}
\begin{document}
\maketitle
\centerline {ABSTRACT}
It is shown how to perform some steps of perturbation
theory if one assumes a measure--theoretic point of view,
i.e. if one renounces to
control the evolution of the single trajectories, and the attention is
restricted to controlling the evolution of the measure of some
meaningful subsets of phase--space. For a system of coupled rotators,
estimates uniform in $N$ for finite specific energy can be obtained in
quite a direct way . This is achieved by making reference not to the
sup norm, but rather, following Koopman and von Neumann, to the much
weaker $L^2$ norm.
\noindent
\vskip 1truecm
\vfill
%%% DA METTERER IN FONDO
%\noindent PACS numbers: 03.50.D, 03.65.B, 02.30.M
%\par
%%% DA CANCELLARE
\noindent
Running title: An averaging theorem in the thermodynamic limit
\vskip 3.truecm
\noindent
\eject %%%% Da cancellare se non si vuole l'abstract in una pagina
\section{Introduction}
A very much discussed problem is the question
whether Nekhoroshev--type theorems can
have some relevance for the foundations of Statistical Mechanics. In
its heuristic formulation (''\textsl{actions remain frozen to their
initial values up to exponentially long times}'') this theorem seems to
grasp the essential feature of the Fermi--Pasta--Ulam phenomenon: the
energy remains confined to the low frequency modes, while the energies
(i.e., up to a factor, the actions) of the high frequency modes remain
frozen up to very large times. On the other hand, in Nekhoroshev
theorem the estimate for the
time of freezing is of the type $T\simeq T_0
\exp(\veps^*/\veps)^{1/N}$, where $\veps$ is the pertubative parameter
(for example the specific energy)
and $N$ is the number of degrees of freedom of the system. Thus, for
systems of interest to Statistical Mechanics, in which $N$ is very large,
the exponential estimate in the above formula disappears
(see \cite{cmp}, \cite{vanzini}).
These considerations seem to indicate that Classical Perturbation Theory
may be useless for the aims of statistical mechanics.
The following remark is however in order. The aim of Perturbation Theory, as
it was developed until now, is to give the most accurate
description of every trajectory of a dynamical system, and this enforces,
at a technical level, the use of the sup norm in the phase space for
the estimate of the relevant quantities.
On the other hand, for the aims of Statistical Mechanics a control
over any single trajectory is completely
irrelevant (the knowledge of the values of $10^{23}$ actions instant
by instant is enormously redundant for estimating, for example, the specific heat).
So one can limit oneself to control only the evolution of some significant
quantity, for example the energy of some subsystem of the complete
system. In this case the dependence on the total number $N$ of degrees of
freedom changes drastically, and in fact estimates
uniform in $N$ were obtained (see \cite{bgg}, \cite{dario});
however, such estimates turn out to be
valid only for finite total energy $E$, namely for vanishing
specific energy $E/N$ in the thermodynamic limit $N\to+\infty$.
Very recently, in the same spirit, estimates uniform in $N$ for
non--vanishing specific energy $E/N$ where given (see \cite{ponno})
in the Fermi--Pasta--Ulam problem, but only for a special class of initial data.
%The open problem is thus whether significant estimates uniform in $N$
%can be obtained which depend on the specific energy $E/N$ and not
%just on total energy $E$. This clearly requires some new idea because
%it appears that with
%the traditional approach not even a single step of perturbation theory
%can be performed in the thermodynamic limit $N\to\infty$, with finite
%specific energy $E/N$.
But, for the pourpose of the Statistical Mechanics, the author feels
that also this weakened approach is unnecessarily strong, because one
pretends to control some dynamical variable ``initial data by initial data'',
without taking into account any statistical aspect.
Instead, a measure--theoretic point of view ought to be taken, namely one
should renounce to control the evolution of the single trajectories, and
the attention should be restricted to controlling the evolution of the
measure of some meaningful subset of phase--space: actually, in this case,
estimates uniform in $N$ for finite specific energy can be obtained
in quite a direct way. This is achieved by making reference not to the sup
norm, but rather to some much weaker integral norm, tipically, following
Koopman and von Neumann (see \cite{koopman}, \cite{neumann}),
the $L^2$ norm, which will be the one used in the
present paper.
To this end, we first show how, for a generic
system, an estimate of the rate of mixing for any invariant
measure $\mu$ can be given. This is shown in Section~2.
Then, by considering a concrete example (a system of $N$ rotators with nearest
neighbour interaction), we show that at least three steps of the perturbative
construction can be performed. This will be obtained by making use of
the method of the direct construction of integrals
of motion. The corresponding estimates show that
the mixing rate is much smaller than the one estimated directly from
the equations of motion. To this second task Sections~3, 4 and 5
will be devoted. In particular, in Section~3 a normal form
theorem is given, from which the estimate of the mixing
rate follows as a Corollary.
Section~4 is devoted to an accurate discussion of the first step of
perturbation theory, while two further steps are performed in
Section~5, leading to the proof of the theorem.
A technical Lemma is proven in an appendix.
\section{The estimate of the mixing rate}
Consider a Hamitonian system with Hamiltonian function $H$ on a phase
space $\mathcal{M}$,
endowed with a finite invariant measure $\mu$
(so that we can suppose $\mu(\mathcal{M})=1$). It is well
known that the existence of a smooth integral of motion $f(\vett x)$
independent of the energy implies that the system is not ergodic
(on a single surface of constant energy). Indeed, obviously,
the two sets $A=\{x: |f(\vett x) -\bar f|2k\}$, where
$\bar f \defz \int f(\vett x) \diff \mu$ is the expectation of $f$ and
$k$ a positive constant,
are invariant disjoint nontrivial sets (considering for
example the Gibbs measure, and a not too large value of $k$ ).
Suppose now $f$ is only a quasi--constant of motion, in the sense that
(we denote by $[\cdot,\cdot]$ the Poisson bracket of two functions)
$[H,f]$ is small in $L^2$ norm; the problem of finding such a
function will be one of the main themes discussed later.
In such a case the sets $A$ and $B$ are no more invariant:
denoting by $A_t$ the set evolved from $A$ according to the dynamics,
one expects that $A_t\cap B \neq \emptyset$.
But if the evolution is slow (in the mean), one expects that it will
take some time in order that, at a given point of $A$, the value of
the function $f_t$, i.e. the evolution of $f$ (to be defined in a
moment), grows from the value (smaller than $k$) it has at time $t=0$
to the values (larger than $2k$) that $f$ has in $B$, i.e., in terms
of sets, the measure of $A_t\cap B$ is expected to remain
small up to a certain time.
In order to give a rigorous form to such rather vague reasoning,
we begin with introducing the notion of \emph{mixing time}.
We recall that one defines a system to be (strongly) mixing if $\mu(
A_t \cap B) \To \mu( A) \mu( B)$ as $t\To +\infty$. But,
as especially pointed out by von Neumann, it is also of interest to
have an estimate of the actual relaxation time, i.e. the time at which
the limit value is actually reached. This is particular relevant if
$\mu(A_t\cap B)$ grows slowly. This justifies the following definition
\begin{Definizione}[mixing time]
We define the mixing time $t_{\mathrm{mix}}$ for the two sets $A$ and $B$,
defined as above, by $t_{\mathrm{mix}}=\sup\,t^*$, where $t^*$ is
such that
\begin{equation}\label{def1}
\mu( A_t \cap B) < \frac 12 \mu( A) \mu( B)\qquad
\end{equation}
for all $0 2 k$ (since $\vett x\in
B$), and $|f_t(\vett x)- \bar f| < k$ (since
$f_t(x)=f(\Phi^{-t}x)$ and $ \Phi^{-t}x\in A$). So one has
\begin{equation*}
\begin{split}
k\mu( A_t \cap B) & \le \int_{A_t \cap B}
\Big |\, |f(\vett x) - \bar f| - |f_t(\vett x)- \bar f| \,
\Big | \diff \mu \le \\
& \le \int_{A_t \cap B} |f(\vett x) - f_t(\vett x) | \diff \mu
\le \Big (\int_{A_t \cap B} \diff \mu \Big )^{1/2}\,
\Big( \int_{A_t \cap B}
|f(\vett x) - f_t(\vett x) |^2\diff \mu \Big)^{1/2} \\
& \le \Big( \mu( A_t \cap B)\Big )^{1/2}\, \norm{f_t-f} \ .
\end{split}
\end{equation*}
Thus, by (\ref{eq1}) one gets
\begin{equation}\label{eq2}
\mu( A_t \cap B) \le \eta^2 t^2 \frac {\norm{f}^2}{k^2} \ .
\end{equation}
So, we have proved the following theorem (analogous to that of Chebyshev)
\begin{Teorema}
Let $\mu$ be an invariant finite measure, and $f\in L^2(\diff \mu)$
have the property $\norm{[H,f]}\le \eta \norm{f}$.
Define the sets $A$ and $B$ by
$$
A=\{\vett x: |f(\vett x) - \bar f|\le k\} \ , \quad
B=\{\vett x: |f(\vett x) - \bar f|\ge 2k\} \ ,
$$
with $\bar f =\int f\diff \mu$ and $k$ a positive constant.
Then the estimate (\ref{eq2}) holds.
\end{Teorema}
Relation (\ref{eq2}) allows one to give a lower bound to the mixing
time.
%We recall
%that one defines a system to be (strongly) mixing if $\mu( A_t \cap B) \To
%\mu( A) \mu( B)$ as $t\To +\infty$; now,
In fact
using (\ref{eq2}) one has
$\mu( A_t \cap B) < \frac 12 \mu( A) \mu( B)$
for all $t$ such that
\begin{equation}
t < \fraz {k\sqrt{2}}{\eta \norm{f}} \Big ( \,
\mu( A) \mu( B) \Big)^{1/2} \ ,
\end{equation}
so that one gets the estimate
\begin{equation} \label{eq4}
t_{\mathrm{mix}} \ge \fraz {k\sqrt{2}}{\eta \norm{f}} \Big ( \,
\mu( A) \mu( B) \Big)^{1/2} \ ,
\end{equation}
One sees that $t_{\mathrm{mix}}\To +\infty$ as $\eta\To 0$, so that a sort
of continuity is recovered. If $f$ is a constant of motion, the two sets $A$ and
$B$ remain separated for all times;
if the time derivative of $f$ is small, then $A$ and
$B$ remain ``quasi'' separated (at least in measure) for very long
times, which tend to infinity
with the vanishing of the derivative $\dot f$, namely of $\eta$.
A comment on relation (\ref{eq4}): up to now we have considered the
constant $k$ as a free parameter. But, as $k$ is a measure of the
deviation of $f$ from its expectation $\bar f$, it is meaningful to take it of the
same order of magnitude as the standard deviation of $f$,
$$
\delta_f \defz \left[\, \int (f(\vett x) -\bar f)^2 \diff \mu \,\right ]^{1/2}\ .
$$
Otherwise it could happen that the measure
of $A$ or that of $B$ be essentially zero and the estimate (\ref{eq4})
trivial. So, in the rest of the paper we fix
$k=\delta_f$, and our estimate (\ref{eq4}) becomes
\begin{equation} \label{eq4bis}
t_{\mathrm{mix}} \ge
\fraz {\sqrt{2}\,\delta_f}{\eta\norm{f}} \Big( \mu( A) \mu(
B) \Big)^{1/2} \ .
\end{equation}
%Another way to estimate the time needed to reach a complete mixing,
%is through the use of correlations,\footnote{This was
%suggested to my by prof. G.~Benettin.} which is, at the end,
%equivalent to the use of the sets.
%If a system is mixing it is known that correlations vanish as
%$t\to+\infty$, i.e., denoting with $< \cdot,\cdot>$ the scalar product
%in $L^{2}$, one has
%\begin{equation*}
% - {\bar f}^2 \to 0 \ ,
%\end{equation*}
%which, in terms of standard deviation, can also be written as
%\begin{equation*}
% + \delta_f^2 \to 0 \ .
%\end{equation*}
%Now, from inequality (\ref{eq1}), one finds
%$||\le \eta t \norm{f}^2$, so that for times
%\begin{equation*}
% t< \frac{\delta_f^2}{2\eta\norm{f}^2} \ ,
%\end{equation*}
%the correlation remains bigger than half of the standard deviation.
\section{The periodic chain of rotators}
In the rest of the paper we tackle the problem of constructing, for a
concrete system, a function $f$ which has a slow evolution, i.e.,
satisfies (\ref{eq1bis}) with a small $\eta$.
The system we consider
is a classical one, a chain of $2N$ rotators with nearest neighbour
trigonometric coupling and periodic boundary conditions,
i.e., the system with Hamiltonian
\begin{equation}\label{eq5}
H=\sum_{j=-N+1}^N \frac{p_j^2}2 - \sum_{j=-N}^N V_0 \cos (q_{j+1} - q_j)
\ , \qquad q_N=q_{-N} \ ,
\end{equation}
where $q_j\in\toro^1$, $p_j\in\reali$ and $V_0$ is a positive constant.
As an invariant measure we take the Gibbs one at inverse temperature $\beta$,
defined by
\begin{equation}\label{eq6bis}
\diff \mu = \fraz 1{Z} \exp(-\beta H) \diff\vett x \ , \quad Z=\int
\exp(-\beta H) \diff \vett x
\end{equation}
with $\vett x=(q_{-N+1},\ldots,p_{N})$, and $\diff \vett x
=\diff q_{-N+1} \ldots \diff p_{N}$.
For notational simplicity we will perform the (noncanonical) change of
coordinates $\tilde q_j=q_j$, $\tilde p_j=\beta^{1/2} p_j$, and a
change of time $\tau = \beta^{-1/2} t$. This being understood, we
drop tildes, and denote $\tilde q_j$ by $q_j$ and $\tilde p_j$ by $p_j$.
The resulting equations of
motion can be deduced from the Hamiltonian function
\begin{equation}\label{eq5bis}
H=\sum_{j=-N+1}^N \frac{ p_j^2}2 - \veps \sum_{j=-N}^N
\cos (q_{j+1} - q_j)
\ , \qquad q_N = q_{-N} \ ,
\end{equation}
where we have denoted $\veps = \beta V_0$.
Correspondingly, the Gibbs measure becomes
$$
\diff \mu = \fraz 1{Z} \exp(-H) \diff\vett x \ ,
\quad Z=\int \exp(-H) \diff \vett x \ .
$$
From the form of the Hamiltonian it is apparent that
$\veps$ is our small parameter, because for $\veps=0$ our
system is formally integrable, having, as constants of motions, all the functions
$p_j$. For small $\veps$, one has instead
$$
[H, p_j] = \veps \Big ( \sin(q_{j+1}-q_j) - \sin(q_j - q_{j-1}) \Big )
\ .
$$
From this it follows that the momenta $p_j$ themselves have a slow
evolution, or are quasi--integrals, because they satisfy the relation
(\ref{eq1bis}) with $\eta=2\veps$, i.e.,
$$
\norm{[H,p_j]}\le 2\veps = 2\veps \norm{p_j} \ .
$$
This follows making use of the facts that $|\sin x|\le 1 $
and that the $p_j$, being normally distributed with unit variance and
zero mean, have the property $\norm{p_j}=1$.
%% DIRE DEL TEOREMA DELLA MEDIA
So, applying the estimate (\ref{eq4bis}) one finds that the mixing
time is $\tau_{\mathrm{mix}}\sim \veps^{-1}$, which in terms of the
original, non--rescaled time, gives the estimate
\begin{equation}\label{eq6}
t_{\mathrm{mix}} \sim \veps^{-1/2} \ .
\end{equation}
But actually the mixing time is much smaller, because perturbation
theory up to third order leads to the following
\begin{Teorema}[normal form construction]\label{teo3}
For any $j$, there exists a function $f_j$ of the form $f_j= p_j + \veps^{3/5}
X_j(\vett p,\vett q)$ having the properties
\begin{gather}
\label{eq7a} \norm{[H,f_j]}\le C_1 \veps^{1+\fraz 35} \\
\label{eq7b} \norm{X_j}\le C_2 \ ,
\end{gather}
with two positive constants $C_1$ and $C_2$ independent of $\veps$ and $N$.
\end{Teorema}
The construction of the function $f_j$ is performed in the next two Sections,
using the method of the direct construction of a first integral
(see for example \cite{giorgilli1}), and implementing three steps
of the perturbative construction.
It is clear that the estimate (\ref{eq7a}) leads to an estimate of the
mixing time of order
\begin{equation}\label{eq6ter}
t_{\mathrm{mix}}\simeq \veps^{-1/2-3/5} \ ,
\end{equation}
which is much larger than (\ref{eq6}). There remains open the question
of how many steps of the perturbative construction can be performed.
If one could prove that the construction can be performed to all orders,
one would obtain a mixing time exponentially large, thus recovering the
analog of Nekhoroshev theorem, with however a complete elimination of $N$.
For a proof of (\ref{eq6ter}) one has to estimate
the other quantities entering formula (\ref{eq4bis}). This is
provided by the following Lemma
\begin{Lemma}\label{lm1}
For any $j$,
consider the function $f_j= p_j + \veps^{3/5} X_j(\vett p,\vett q)$,
%% FARE CON POTENZA GENERICA
with $X_j\in L^2(\diff\mu)$. Then one has ($\delta$ denoting standard deviation)
\begin{description}
\item[i)] $\bar f_j = O(\veps^{3/5})$
\item[ii)] $\delta_{f_j}^2=\delta_{p_j}^2 + O(\veps^{3/5}) = 1+
O(\veps^{3/5})$
\item[iii)] for the sets $A_{\veps}=\{\vett x :| f_j-\bar f_j
|<\delta_{f_j}\}$ and $B_{\veps}=\{\vett x :| f_j-\bar f_j
|> 2 \delta_{f_j}\}$ one has
$$
\mu(A_{\veps})= \mu(A_0) + O(\veps^{2/5}) \ , \quad
\mu(B_{\veps})= \mu(B_0) + O(\veps^{2/5}) \ .
$$
\end{description}
\end{Lemma}
\vskip 3truemm
\textbf{Proof.} The proof goes as follows.
i) This is immediate. One has $\bar f_j= \bar p_j +\veps^{3/5} \bar
X_j$. On the other hand $\bar
p_j=0$, and $|\bar X_j | \le $ $ \int | X_j |\diff\mu $ $ \le
\norm{ X_j }$.
ii) This is obtained by a simple computation. Indeed one has
\begin{equation*}
\begin{split}
\delta_{f_j}^2 &=\int (f_j-\bar f_j)^2 \diff\mu \\
&= \int p_j^2 \diff\mu + 2\veps^{3/5}
\int p_j X_j\diff\mu + \veps^{6/5}
\int X_j^2 \diff\mu -(\bar f_j)^2 \ .
\end{split}
\end{equation*}
Now, $ \int p_j^2 \diff\mu =\delta_{p_j}^2$ since $\bar p_j=0$,
while, by the Schwarz inequality, $|\int p_j X_j \diff\mu | \le
\norm{p_j}\,\norm{X_j}$. The result is then obtained by estimating
$\bar f_j$ through i).
iii) We show only the first inequality, because the second one is proved in the
same way. We start noticing the trivial relation $\delta_{X_j}=\int
(X_j-\bar X_j)^2\diff\mu\le \norm{X_j}^2$, so that, introducing the set
$C=\{ \vett x: |X_j - \bar X_j| \ge \veps^{-1/5} \}$, by Chebyshev
theorem one gets
$$
\mu(C) \le \veps^{2/5} \norm{X_j}^2 = O(\veps^{2/5}) \ .
$$
Now, the complementary set $A_{\veps}/C$ is contained in the set
$A'= \{ \vett x : |p_j|\le \sigma_{f_j} +\veps^{2/5}\}$, because in
$A_{\veps}/C$ one has $|X_j - \bar X_j| \le \veps^{-1/5}$. The
measure of the set $A'$ can be readily evaluated, recalling that $p_j$ is
normally distributed, and that in addition, by ii),
one has $\delta_{f_j}=1+\veps^{3/5}$. One thus finds $\mu(A')=\mu(A_0) +
O(\veps^{2/5})$, and so one gets the thesis using
$\mu(A_{\veps}) = \mu(C) + \mu(A_{\veps}/C)\le \mu(C) + \mu(A')$.
\hfill \textbf{Q.E.D.}
\section{The first Perturbative Step}
We have now to show how the quasi--constants of motion $f_j$ entering
Theorem~\ref{teo3} are constructed. The first perturbative step is
performed in the present section.
From the Hamiltonian (\ref{eq5bis}) we obtain the following equations
of motion
\begin{equation}\label{eq9a}
\left \{
\begin{split}
\dot p_j &= \veps \Big(
\sin(q_{j+1} - q_j) - \sin(q_j - q_{j-1}) \Big) \\
\dot q_j &= p_j \ .
\end{split}
\right.
\end{equation}
It is well known (see \cite{giorgilli2}) how a normal form
(which is however formal for $N\gg1/\veps$) can be constructed for
this equation.
However, in this simple case
it is possible to find a first--order integral directly, avoiding
the use of the normal form techniques. This is obtained by recalling
that in virtue of the equation of motion one has the relation
\begin{equation*}
\begin{split}
\veps & \sin(q_{j+1} - q_j) = - \fraz {\diff~}{\diff t} \left(
\veps \fraz {\cos(q_{j+1} - q_j)}{p_{j+1}-p_j} \right) + \\
& + \veps^2\fraz {\cos(q_{j+1} -
q_j)}{(p_{j+1}-p_j)^2} \Big( \sin(q_{j+2} - q_{j+1})-2\sin(q_{j+1}
- q_j)+\sin(q_j - q_{j-1})
\Big) \ .
\end{split}
\end{equation*}
and the analogous one for $\veps \sin(q_{j} - q_{j-1})$.
So, in the region $p_j \ne p_{j\pm 1}$, if we define
$$
\tilde X_j^{(1)} \defz \fraz {\cos(q_{j+1} - q_j)}{p_{j+1}-p_j} -
\fraz {\cos(q_j - q_{j-1})}{p_j - p_{j-1}} \ ,
$$
we find that the function $\tilde f_j^{(1)}= p_j + \veps \tilde
X_j^{(1)}$ evolves with velocity of order $\veps^2$, i.e., is slower
than $p_j$. It is obvious that, due to the presence of the
denominators (the small divisors), the function $\tilde X_j^{(1)}$
is not in $L^2$, and so is useless for the estimates.
This example also shows in a very clear way the difficulty of applying
the standard perturbation techniques for large $N$.
Indeed, in order to have a slow evolution
(of order $\veps^2$), it is not enough to restrict the initial data to the
region $|p_j - p_{j\pm 1}|>\sigma$ (with $\sigma$ a positive parameter), but
one has to secure that such an inequality also holds for times or order
$\veps^{-2}$. On the other hand, this cannot be secured,
because $p_{j\pm 1}$ evolve in
general on a time scale of order $\veps^{-1}$. One way to secure that $p_{j\pm
1}$ do not evolve too much is to consider the functions $\tilde
X_{j\pm1}^{(1)}$, and choose initial data such that $|p_{j\pm 1} -
p_{j\pm 2}| > \sigma$. But then we have the problem of the evolution of
the variables $p_{j\pm 2}$. Thus, one is forced to iterate this
procedure, so that
our hypothesis can be secured only in a set of the form $C=\{ \vett x:
|p_j-p_{j+1}|>\sigma, \forall j\}$. On the other hand a
simple computation shows that
one has $\mu(C) \simeq (1-\sigma)^{N}$, i.e., that the set $C$ has
essentially a vanishing measure for $N$ large.
Instead, if we want
to control the evolution of the measure of the sets, and not the
single trajectories, this kind of problems is not met.
In fact, one
can limit oneself to perform the normalization only in the non--resonant zone
$|p_j - p_{j\pm 1}|>\sigma$, and keep the action $p_j$ unaltered in the
resonant one. The idea is thus to define
\begin{equation*}
f_j^{(1)} \defz \left \{
\begin{split}
&~ p_j \quad\qquad\qquad \hbox{for} \quad |p_j - p_{j\pm 1}|<\sigma \\
&~ p_j + \veps \tilde X_j^{(1)} \quad
\hbox{for} \quad |p_j - p_{j\pm 1}|>\sigma \ ,
\end{split}
\right .
\end{equation*}
in such a way that the region where the derivative of $f_j^{(1)}$ is
large, has a small measure (of order $\sigma$). Now,
choosing in an appropriate way $\sigma$ as a function of $\veps$ (we
will take $\sigma=\veps^{2/5}$), one can obtain that the
$L^2$--norm of $\dot f_j^{(1)} $ becomes
less than the $L^2$--norm of $\dot p_j$, notwithstanding the fact that
these two functions have the same sup--norm.
In order to give this idea a clear mathematical content, we
need to introduce some objetcs. First of all we need a truncation function
$\zeta(x)$ of $\mathcal{C}^\infty$ class, i.e. a
function having the properties stated in
\begin{Lemma}\label{lm2}
For every sufficiently small (positive) constant $\sigma$,
there exists a $\mathcal{C}^\infty(\reali)$ function $\zeta(x)$ such that:
\begin{description}
\item[i)] one has $\zeta(x)=1$ for $|x|<\sigma$, and $\zeta(x)=0$
for $|x|>3\sigma$;
\item[ii)] for all $n\in\naturali$, one has
$|\partial_x^n \zeta(x)|3\sigma$;
\item[iii)] having defined $\Zeta(x) \defz \int \zeta(x)\diff x$ and
$\Zeta^{(2)}(x) \defz \int \Zeta(x)\diff x$, then for $|x|>3\sigma$
one has $\Zeta(x) = 0$ and $\Zeta^{(2)}(x) =0$. Moreover the estimates
$|\Zeta(x)|<4 |x| $ and $|\Zeta^{(2)}(x)|< |x|^2/2$ hold.
\end{description}
\end{Lemma}
\vskip 3truemm
These are standard properties of truncation functions, the only
unusual one being iii), the meaning of which will become clear in the
next Section, when we will go beyond the first order.
The proof of this Lemma is deferred to the Appendix.
Furthermore, we define the integer vectors $\vett e_j\in\interi^{2N}$ as the
standard basis vectors, i.e., those having all components vanishing
but the $j$--th one, which is equal to one.
Analogously we define the vectors
$\vett \delta_j \defz \vett e_{j+1}
-\vett e_j$. We finally define the set $M_j^1$ made up of the
four vectors $\pm \delta_j,\pm \delta_{j-1}$.
In order to have formulas
with the minimal number of indices, from now on we concentrate on the case
$j=0$, but it will be obvious that all formulas are valid for a generic
value of $j$. The equation of motion for $p_0$ can be rewitten as
\begin{equation*}
\dot p_0=\sum_{\vett k\in M_0^1} \fraz {\veps}{2i} c_{\vett k} \exp(i\vett
k\cdot\vett q) \ ,
\end{equation*}
where the constants $c_{\vett k}=\pm 1$ come from the expression of
the sine in terms of complex exponentials. Now, using the function
$\zeta(x)$ one can separate the resonant part from the non resonant one as
follows
\begin{equation*}
\dot p_0=\veps \sum_{\vett k\in M_0^1} c_{\vett k} \zeta(\vett k\cdot\vett p)
\fraz {\exp(i\vett k\cdot\vett q)}{2i} +
\veps \sum_{\vett k\in M_0^1} c_{\vett k} \big(1- \zeta( \vett k\cdot\vett p)
\big) \fraz {\exp(i\vett k\cdot\vett q)}{2i}
\ .
\end{equation*}
Thus, by integrating by parts the second term at the r.h.s., one gets
the identity
\begin{equation}\label{eq13}
\begin{split}
\frac {\diff~}{\diff t} & \left ( p_0 + \veps \sum_{\vett k\in M_0^1}
c_{\vett k} \frac {1- \zeta( \vett k\cdot\vett p)}{\vett k\cdot\vett p}
\; \fraz {\exp(i\vett k\cdot\vett q)}{2} \right ) =
\veps \sum_{\vett k\in M_0^1} c_{\vett k} \zeta(\vett k\cdot\vett p)
\fraz {\exp(i\vett k\cdot\vett q)}{2i} \\
& + \veps \sum_{\vett k\in M_0^1} \frac {c_{\vett k}}2 \partial_x
\left . \frac {1- \zeta(x)}x \right |_{\vett k\cdot\vett p}
\exp(i\vett k\cdot\vett q) \, \vett k\cdot \vett{\dot p} \ .
\end{split}
\end{equation}
Now the term $\vett k\cdot \vett{\dot p} $ is of order $\veps$, because
$$
\vett k\cdot\vett {\dot p} = \veps \sum_{j} \sum_{\vett k'\in M_j^1} \vett k'
\cdot \vett e_j \frac {c_{\vett k'}}{2i} \exp(i\vett k'\cdot\vett
q) \ ,
$$
and moreover, since $\vett k\in M_0^1$, only the terms with
$j=-2,\ldots,1$ do not vanish
(for the other values of $j$ one has $\vett k' \cdot \vett e_j=0$).
So, one gets
$$
\exp(i\vett k\cdot\vett q) \vett k\cdot\vett{ \dot p} =
\veps \sum_{\vett k'\in\{M_j^1\}} \vett k' \cdot \vett e_j \frac
{c_{\vett k'}}{2i} \exp\big(i(\vett k+\vett k')\cdot\vett q\big)
$$
where the summation is performed over all sets $M_j^1$ with
$j=-2,\ldots,1$. Now, introducing the function
\begin{equation}\label{eq10bis}
X_0^{(1)} \defz \sigma \sum_{\vett k\in M_0^1}
c_{\vett k} \frac {1- \zeta( \vett k\cdot\vett p)}{\vett k\cdot\vett p}
\fraz {\exp(i\vett k\cdot\vett q)}{2} \ ,
\end{equation}
we can rewrite (\ref{eq13}) in the form
\begin{equation}\label{eq10}
\frac {\diff~}{\diff t} \left( p_0 + \veps \sigma^{-1} X_0^{(1)} \right ) =
\veps \sum_{\vett k\in M_0^1} c_{\vett k} \zeta(\vett k\cdot\vett p)
\fraz {\exp(i\vett k\cdot\vett q)}{2i} + \mathcal{R}_1 \ ,
\end{equation}
where the remainder $\mathcal{R}_1$ has the form
\begin{equation}\label{eq10ter}
\mathcal{R}_1 \defz \veps^2 \sum_{\vett k_1\in M_0^1} \sum_{\vett
k_2\in\{M_j^1\}} \vett k_2 \cdot \vett e_j \fraz
{c_{k_1}c_{k_2}}{4i} \left . \partial_x \frac {1- \zeta(x)}x \right |_{\vett
k_1\cdot\vett p} \exp\big(i(\vett k+\vett k')\cdot\vett q\big)
\ .
\end{equation}
The estimate of the r.h.s of (\ref{eq10}) is then very
simple. Indeed one has
\begin{equation*}
\begin{split}
\int \Big| & \zeta(\vett k\cdot \vett p) \exp(i\vett k\cdot\vett q)
\Big|^2 \diff \mu
=\int \Big| \zeta(\vett k\cdot \vett p)\Big|^2 \diff \mu \\
&\le \int_{|p_0-p_{\pm 1}|<3\sigma} \exp\left(
-\fraz {p_0^2+p_{\pm 1}^2}{2}\right )\diff p_0\diff p_{\pm 1}
=6 \sigma \ ,
\end{split}
\end{equation*}
so that
\begin{equation}\label{eq11}
\norm{ \sum_{\vett k\in M_0^1} c_{\vett k} \zeta(\vett k\cdot\vett p)
\fraz {\exp(i\vett k\cdot\vett q)}{2i} } \le 4\sigma^{1/2} \ .
\end{equation}
Instead, in order to estimate the term $\mathcal{R}_1$ given by
(\ref{eq10ter}),
one has to estimate a finite sum of integrals of the type
\begin{equation*}
\begin{split}
\int \bigg| & \partial_x \frac {1-\zeta(x)}{x} \Big
|_{\vett k\cdot \vett p}
\exp(i\vett k\cdot\vett q) \bigg|^2 \diff \mu =
\int \bigg | \partial_x
\frac {1-\zeta(x)}{x} \Big |_{k\cdot \vett p} \bigg |^2
\diff \mu \\
& \le \int_{|p_0-p_{\pm 1}|>3\sigma} \bigg | \partial_x \
\frac {1-\zeta(x)}{x} \Big |_{p_0-p_{\pm 1}} \bigg |^2
\exp\left ( -\fraz {p_0^2+p_{\pm 1}^2}{2}
\right )\diff p_0\diff p_{\pm 1} \ .
\end{split}
\end{equation*}
We can estimate the derivative appearing in the last term, using
ii) of Lemma~\ref{lm2} for the derivative of $\zeta(x)$, and the fact
that the denominator is bounded away from zero (since $|x|>\sigma$).
One has thus
\begin{equation*}
\left | \partial_x \left . \frac {1-\zeta(x)}{x} \right |_{p_0-p_{\pm 1}}
\right |^2 \le \mathsf{C} \sigma^{-4} \ ,
\end{equation*}
with a given constant $\mathsf{C}$, and using this bound in the above
formula one finds
\begin{equation*}
\int \left| \partial_x \left . \frac {1-\zeta(x)}{x} \right |_{k\cdot \vett p}
\exp(i\vett k\cdot\vett q) \right|^2 \diff \mu \le \mathsf{C} \sigma^{-4} \ .
\end{equation*}
Thus, there exists a numerical constant $\mathcal{C}_1$ such that
\begin{equation}\label{eq12}
\norm{ \mathcal{R}_1} \le \mathcal{C}_1 \sigma^{-2} \veps^2 \ .
\end{equation}
In exactly the same way one can show, from the expression
(\ref{eq10bis}) for the function $X_0^{(1)}$, that there exists a
numerical constant $\mathcal{C}_2$, such that
\begin{equation}\label{eq12bis}
\norm{ X_0^{(1)} } \le \mathcal{C}_2 \ .
\end{equation}
Now, taking $\sigma=\veps^{2/5}$, from estimates (\ref{eq12}) and (\ref{eq11})
one gets that the function
$f_0^{(1)} = p_0 + \veps^{3/5} X_0^{(1)}$ has a time derivative
of order $\veps^{1+1/5}$, i.e., it evolves more slowly than the action
$p_0$. To obtain a time--evolution as slow as the one implied by
(\ref{eq7a}) of Theorem~\ref{teo3}, one needs to perform two more
perturbative steps. This will accomplished in the next Section.
\section{ The next steps of Perturbative Theory and Proof of
Theorem~\ref{teo3}}
By symmetry, letting $(\vett k_1,\vett k_2) \To (-\vett k_1,-\vett
k_2)$, one can easily check that in the expression (\ref{eq10ter})
for $\mathcal{R}_1$ the terms with $\vett k_1+\vett k_2=0$ are
lacking. So, in complete analogy with what was done for the first step, we
can integrate by parts all the terms, obtaining, at least in the region
where $|(\vett k_1+\vett k_2)\cdot \vett p |>\sigma $, a remainder of
order $\veps^{3}/\sigma^4$, while the resonant zone
$|(\vett k_1+\vett k_2)\cdot \vett p |<\sigma $ will give a
contribution to the remainder of order $\veps^{2}/\sigma^{3/2}$. For
$\sigma=\veps^{2/5}$ the two contributions are of the same order
$\veps^{1+2/5}$. The question remains of understanding how to treat the first--order
resonant term of the remainder, because in this case the phase of the
terms $\exp(i\vett k\cdot\vett q)$ is slow, and so we cannot take the average.
But now one has to consider that slow angles appear only in
the time derivatives of slow actions (the quantities $\vett k\cdot\vett
p$). So one could argue that the terms with slow
angles may be replaced by the time derivatives of $\vett
k\cdot\vett p$ plus some terms containing fast angles.
In this case, the terms with the time derivative of the slow action would
give a total derivative, while the terms with fast angles could be
averaged away. Indeed, things go exactly in this way.
In fact, in our case in which $\vett k =\pm\vett \delta_j$, $j=0,-1$,
one has that $\vett k'=\vett e_{j+1} + \vett e_j$ is orthogonal
to $\vett k$, so that $\vett k'\cdot\vett p$ is ''fast''. In addition,
from the equations of motion one gets directly the relation
\begin{equation*}
\begin{split}
\veps \sin(\vett \delta_j\cdot\vett q) & = \frac 12 (\vett e_{j+1} +
\vett e_j)\cdot \vett {\dot p} -\frac 12 (\vett e_{j+1} -
\vett e_j)\cdot \vett {\dot p} +\veps \sin(\vett
\delta_{j-1}\cdot\vett q) \\
&= \vett k'\cdot\vett {\dot p} -\frac 12 \vett k \cdot \vett
{\dot p} + \veps \sin(\vett k''\cdot\vett q) \ .
\end{split}
\end{equation*}
where we have set $\vett k''=\delta_{j-1}$ (remember that $\vett k =\delta_{j}$).
From this it follows that the first term at the r.h.s of (\ref{eq10})
can be rewritten as
\begin{equation*}
\begin{split}
\veps \sum_{\vett k\in M_0^1} & \fraz {c_{\vett k} \zeta(\vett
k\cdot\vett p)}{2i}\exp(i\vett k\cdot\vett q) =
\frac 12 \sum_{\vett k\in M_0^1} - \zeta(\vett k\cdot\vett p)\vett k\cdot\vett
{\dot p} \\
& + \frac 12 \sum_{\vett k\in M_0^1} \zeta(\vett
k\cdot\vett p) \vett k' \cdot\vett {\dot p} +
\veps \sum_{\vett k\in M_0^1} \zeta(\vett k\cdot\vett p)
\sin(\vett k''\cdot \vett q)
\\
&= \fraz {\diff~}{\diff t} \left ( - \frac 12 \sum_{\vett k\in
M_0^1} \Zeta(\vett k\cdot\vett p) \right ) + \frac {\veps}{2}
\sum_{\vett k_1\in M_0^1}\sum_{\vett k_2\in M_{\vett k_1}^2}
c_{\vett k_2}^1 \zeta(\vett k_1\cdot\vett p)\exp(i\vett
k_2\cdot\vett q)\ ,
\end{split}
\end{equation*}
where $c_{\vett k_2}^1$ are numerical constants (less than 2 in
absolute value), and the sets $M_{\vett k_1}^2$
are made up of the vectors $\pm \vett \delta_{j-1}$, $\pm \vett \delta_{j-1}$
(remember that $\vett k_1= \pm\vett \delta_{j}$), as one can check
using the relation
$\vett k' \cdot\vett {\dot p}=\veps \Big(\sin(\delta_{j+1}\cdot\vett q)
- \sin(\delta_{j-1}\cdot\vett q) \Big)$.
From the expression of $M_{\vett k_1}^2$
one checks that $\vett k_2\ne\vett k_1$, so we have rewritten the
resonant term as a total derivative plus some non--resonant
terms. Now the non resonant terms can be integrated by parts to give
\begin{equation}\label{eq14}
\begin{split}
\veps \sum_{\vett k\in M_0^1} & \fraz {c_{\vett k} \zeta(\vett
k\cdot\vett p)}{2i}\exp(i\vett k\cdot\vett q) =
\fraz {\diff~}{\diff t} \left [ - \frac 12 \sum_{\vett k\in
M_0^1} \Zeta(\vett k\cdot\vett p) + \right. \\
-& \left .
\frac {\veps}{2} \sum_{\vett k_1\in M_0^1}\sum_{\vett k_2\in M_{\vett k_1}^2}
c_{\vett k_2}^1 \fraz{ \zeta(\vett k_1\cdot\vett p)\Big( 1 -
\zeta(\vett k_2 \cdot\vett p) \Big) }{\vett k_2\cdot\vett p}
\exp(i\vett k_2\cdot\vett q) \right ] + \\
& +\frac {\veps}{2} \sum_{\vett k_1\in M_0^1}\sum_{\vett k_2\in M_{\vett k_1}^2}
c_{\vett k_2}^1 \zeta(\vett k_1\cdot\vett p)\zeta(\vett k_2 \cdot\vett p)
\exp(i\vett k_2\cdot\vett q) + \mathcal{R}_2 \ ,
\end{split}
\end{equation}
with
\begin{equation}\label{eq14bis}
\begin{split}
\mathcal{R}_2 & = \frac {\veps}{2} \sum_{\vett k_1\in M_0^1}\sum_{\vett
k_2\in M_{\vett k_1}^2} \frac {c_{\vett k_2}^1}4
\left [ \fraz{\partial~}{\partial x} \zeta(x)|_{\vett
k_1\cdot\vett p} \fraz{ \Big( 1 - \zeta(\vett k_2 \cdot\vett p)
\Big) } {\vett k_2\cdot\vett p} \vett k_1 \cdot \vett {\dot p} \right . \\
& + \left .
\zeta(\vett k_1\cdot \vett p)
\fraz{\partial~}{\partial x}\fraz{ \Big( 1 - \zeta(x)\Big)
}{x}\Big |_{\vett k_2\cdot\vett p} \vett k_2 \cdot \vett {\dot p}
\right ] \exp(i\vett k_2\cdot\vett
q) \ .
\end{split}
\end{equation}
The remainder $\mathcal{R}_2$ can be simply estimated using the
estimate for the derivatives of $\zeta$ (the estimate ii) of
Lemma~\ref{lm2}) in the same way in which $\mathcal{R}_1$ was
estimated in Section~4. The only difference is the presence of the terms
$\zeta(\vett k_1\cdot \vett p)$ which restrict the computation of the
integrals to a region of measure $\sigma$, giving a smaller value. One
has indeed
\begin{equation}\label{stima1}
\norm{\mathcal{R}_2}\le C_3 \veps^2\sigma^{-3/2} \ ,
\end{equation}
where $C_3$ a positive numerical constant.
Now, we turn to the remainder $\mathcal{R}_1$ at the r.h.s of expression
(\ref{eq10}). As already said, every term can be integrated by
parts (outside of the resonant region), and so, using the function
$\zeta\big( (\vett k_1 + \vett k_2) \cdot\vett p \big)$ to split the
phase space in the resonant region and the non resonant one, we get
\begin{equation*}
\begin{split}
\mathcal{R}_1 & = \veps^2 \sum_{\vett k_1 \in M_0^1}
\sum_{\vett k_2 \in \{ M_j^1\} } \vett k_1\cdot \vett e_j \fraz
{c_{k_1}^1}{4i} \left . \fraz {\partial~}{\partial x} \fraz
{1-\zeta(x)}{x} \right |_{\vett k_1\cdot\vett p} \zeta\big( (\vett k_1
+\vett k_2) \cdot\vett p \big) \\
& \exp\big( (\vett k_1 +\vett k_2) \cdot\vett q \big)
+ \fraz {\diff ~}{\diff t} \left [ - \veps^2 \sum_{\vett k_1 \in M_0^1}
\sum_{\vett k_2 \in \{ M_j^1\} } \vett k_1\cdot \vett e_j \fraz
{c_{k_1}^1}4 \left . \fraz {\partial~}{\partial x} \fraz
{1-\zeta(x)}{x} \right |_{\vett k_1\cdot\vett p} \right . \\
& \left.\fraz{1 - \zeta\big( (\vett k_1 +\vett k_2) \cdot\vett p
\big)}{ (\vett k_1 +\vett k_2) \cdot\vett p } \exp\big( (\vett k_1
+\vett k_2) \cdot\vett q \big )
\right ] + \mathcal{R}_3 \ ,
\end{split}
\end{equation*}
with
\begin{equation*}
\begin{split}
\mathcal{R}_3 & \defz \veps^2 \sum_{\vett k_1 \in M_0^1}
\sum_{\vett k_2 \in \{ M_j^1\} } \vett k_1\cdot \vett e_j \fraz
{c_{k_1}^1}4 \left . \fraz {\partial~}{\partial x} \fraz
{1-\zeta(x)}{x} \right |_{\vett k_1\cdot\vett p}
\left . \fraz {\partial~}{\partial x} \fraz{1 -
\zeta(x)}{x}\right|_{(\vett
k_1 +\vett k_2) \cdot\vett p } \\
& (\vett k_1 +\vett k_2) \cdot\vett {\dot p}\, \exp\big( (\vett k_1
+\vett k_2) \cdot\vett q \big ) \ .
\end{split}
\end{equation*}
Now, defining
\begin{equation*}
\begin{split}
X_0^{(2)} & = X_0^{(1)} + \fraz {\sigma}{2\veps} \sum_{\vett k\in
M_0^1} \Zeta(\vett k\cdot\vett p) + \\
& + \sigma \sum_{\vett k_1\in
M_0^1} \sum_{\vett k_2\in M_{\vett k_1}^2}
c_{\vett k_2}^1 \fraz{ \zeta(\vett k_1\cdot\vett p)\Big( 1 -
\zeta(\vett k_2 \cdot\vett p) \Big) }{\vett k_2\cdot\vett p}
\exp(i\vett k_2\cdot\vett q) + \\
& + \veps \sigma \sum_{\vett k_1 \in M_0^1}
\sum_{\vett k_2 \in \{ M_j^1\} } \vett k_1\cdot \vett e_j \fraz
{c_{k_1}^1}4 \left . \fraz {\partial~}{\partial x} \fraz
{1-\zeta(x)}{x} \right |_{\vett k_1\cdot\vett p} \\
& \fraz{1-\zeta\big( (\vett k_1 +\vett k_2) \cdot\vett p \big)}
{(\vett k_1 +\vett k_2) \cdot\vett p } \exp\big( (\vett k_1
+\vett k_2) \cdot\vett q \big ) \ ,
\end{split}
\end{equation*}
we find
\begin{equation}\label{eq15}
\begin{split}
\fraz{\diff~}{\diff t} & \Big( p_0 +X_0^{(2)} \Big) = \fraz
{\veps}{2} \sum_{\vett k_1\in M_0^1}\sum_{\vett k_2\in M_{\vett k_1}^2}
c_{\vett k_2}^1 \zeta(\vett k_1\cdot\vett p)\zeta(\vett k_2 \cdot\vett p)
\exp(i\vett k_2\cdot\vett q) + \\
& + \veps^2 \sum_{\vett k_1 \in M_0^1}
\sum_{\vett k_2 \in \{ M_j^1\} } \vett k_1\cdot \vett e_j \fraz
{c_{k_1}^1}{4i} \left . \fraz {\partial~}{\partial x} \fraz
{1-\zeta(x)}{x} \right |_{\vett k_1\cdot\vett p} \zeta\big( (\vett k_1
+\vett k_2) \cdot\vett p \big) \\
& \exp\big( (\vett k_1 +\vett k_2) \cdot\vett q \big) +
\mathcal{R}_2 + \mathcal{R}_3
\ .
\end{split}
\end{equation}
The second step is then accomplished.
The estimate can be performed in a very
simple way, by estimating the $L^2$--norm as the sup of the function
times the measure (to the power $1/2$) of its support (i.e. of the region
in which the function does not vanish). One finds in this way
\begin{equation*}
\begin{split}
& \norm{\Zeta(\vett k\cdot\vett p)} \le const \, \sigma^{3/2} \\
& \norm{\veps \zeta(\vett k_1\cdot\vett p) \zeta(\vett k_2\cdot\vett p)
\exp(i\vett k_1\cdot\vett p)} \le const\, \sigma\veps \\
& \norm{\mathcal{R}_2} \le const\, \sigma^{-3/2}\veps^2 \\
& \norm{\mathcal{R}_3} \le const\, \sigma^{-4}\veps^3 \ ,
\end{split}
\end{equation*}
i.e. that, for $\sigma=\veps^{2/5}$, all terms are of order
$\veps^{1+2/5}$. One has then
\begin{equation*}
\begin{split}
\norm{ p_0 +\fraz {\veps}{\sigma} X_0^{(2)}} &\le C_3
\veps^{1+ \fraz 25} \\
\norm{X_0^{(2)}} &\le C_4 \ ,
\end{split}
\end{equation*}
with certain numerical constants $C_3$ and $C_4$.
We note that, from the explicit form (\ref{eqa3}) of $\Zeta(x)$ given in
appendix, for $|\vett k\cdot\vett p| < \sigma $ one has $\Zeta(x)=x$, so
that in the resonant region one has
\begin{equation*}
p_0 + \fraz 12 \Zeta(\vett k\cdot\vett p) = \fraz 12 \vett k'\cdot\vett
p \ ,
\end{equation*}
i.e. in the resonant region our function coincides with the fast action.
At this point one can ask whether it is possible to perform more steps of the
perturbative construction, or even an infinite number of them. It is well
known that, in the process of the direct construction
of an integral of motion,
insurmountable difficulties are found in the resonant case. An example
of these difficulties was met at the second step, when we had
to deal with terms of the type $\zeta(\vett k\cdot\vett p)\sin(\vett
k\cdot\vett q)$. To perform the third step one analogously has to deal
with terms of the type
\begin{equation*}
\mathcal{N}_1 \defz \zeta(\vett k_1\cdot\vett p)\zeta(\vett k_2\cdot\vett p)\sin (\vett
k_2\cdot\vett q) \ ,
\end{equation*}
and
\begin{equation*}
\mathcal{N}_2 \defz \zeta\big((\vett k_1+\vett k_2)\cdot\vett p\big)
\frac{\partial~~}{\partial \vett k_1 \cdot\vett p} \frac{1 - \zeta(\vett k_1
\cdot\vett p)} {\vett k_1 \cdot\vett p}
\sin \big( (\vett k_1 + \vett k_2)\cdot\vett q\big) \ ,
\end{equation*}
in the remainder at the r.h.s of relation (\ref{eq15}). At the succesive steps
we will find other resonant terms having a form always different from
those of the previous steps, and at present we were
unable to find a recurrent scheme to perfom an arbitrary number of steps. We
limit ourselvws to show briefly how the resonant term at the r.h.s of (\ref{eq15})
can be dealt with, and so how the third step of the construction can be
performed. In fact the terms $\mathcal{R}_2$ and $\mathcal{R}_3$ are non
resonant and thus can be integrated by parts (giving rise, at fourth
order, to other resonant terms).
We begin considering the terms of the type $\mathcal{N}_1$. Using the
explicit form of the vectors $\vett k_1$ and $\vett
k_2$, one can check that
\begin{equation*}
\sin (\vett k_2\cdot\vett q) = \alpha_1 \vett k_1\cdot\vett {\dot p} +
\alpha_2 \vett k_2\cdot\vett {\dot p} +\sum_{\vett k_3\in M_{k_1,K_2}^2}
\beta_{k_3} \sin(\vett k_3\cdot \vett q) \ ,
\quad \vett k_3 \ne \vett k_1,\vett k_2 \ ,
\end{equation*}
$\alpha_i$ and $\beta_{k_3}$ being numerical constants, and
$M_{k_1,K_2}^2$ a given (finite) set of integer vectors. One thus gets
\begin{equation*}
\begin{split}
\zeta(\vett k_1\cdot\vett p)\zeta(\vett k_2\cdot\vett p) & \sin
(\vett k_2\cdot\vett q) = \alpha_1 \zeta(\vett k_2\cdot\vett p)
\fraz{\diff~}{\diff t}\Zeta(\vett k_1\cdot\vett p) + \\
& + \alpha_2 \zeta(\vett k_1\cdot\vett p)
\fraz{\diff~}{\diff t} \Zeta(\vett k_2\cdot\vett p) +
\mathcal{R}_4 \ ,
\end{split}
\end{equation*}
where $\mathcal{R}_4$ is non--resonant. Finally we have the
relation
\begin{equation*}
\begin{split}
\zeta(\vett k_1\cdot\vett p) & \zeta(\vett k_2\cdot\vett p)\sin (\vett
k_2\cdot\vett q) = \fraz{\diff~}{\diff t} \bigg ( \alpha_1
\zeta(\vett k_2\cdot\vett p) \Zeta(\vett k_1\cdot\vett p)
+ \\
& + \alpha_2 \zeta(\vett k_1\cdot\vett p) \Zeta(\vett
k_2\cdot\vett p) \bigg )
- \alpha_1 \zeta'(\vett k_2\cdot\vett p) \Zeta(\vett k_1\cdot\vett p)
\vett k_2\cdot\vett {\dot p} + \\
& - \alpha_2 \zeta'(\vett k_1\cdot\vett p) \Zeta(\vett k_2\cdot\vett p)
\vett k_1\cdot\vett {\dot p} + \mathcal{R}_4 \ .
\end{split}
\end{equation*}
At this point one can check that the terms at the r.h.s. outside the
time--derivative are non--resonant, and thus can be integrated by
parts. The resulting terms are of order $\veps\sigma^{3/2}$ (the terms
which are triply resonating) and of order $\veps^2\sigma^{-1}$
(the ones which are integrated by parts in a doubly resonating region of
measure $\sigma^2$).
The other resonant term $\mathcal{N}_2$ can be treated in a similar
way. One can check (using the explicit expressions of $\vett k_1$ and
$\vett k_2$) that
\begin{equation*}
\veps \exp \big( i(\vett k_1 + \vett k_2)\cdot\vett q\big) =
i(\vett k_1 + \vett k_2)\cdot\vett {\dot p} \exp ( i\vett
k_1\cdot\vett q ) + \veps\mathcal{P}_1 \ ,
\end{equation*}
where $\veps\mathcal{P}_1$ is a non--resonant trigonometric
polinomial. One has then
\begin{equation*}
\begin{split}
\zeta\big( &(\vett k_1+\vett k_2)\cdot\vett p\big)
\frac{\partial~~}{\partial \vett k_1 \cdot\vett p} \frac{1 - \zeta(\vett k_1
\cdot\vett p)} {\vett k_1 \cdot\vett p}
\exp \big( i(\vett k_1 + \vett k_2)\cdot\vett q\big) = \\
&~ \frac{\partial~~}{\partial \vett k_1 \cdot\vett p} \frac{1 - \zeta(\vett k_1
\cdot\vett p)} {\vett k_1 \cdot\vett p}
\exp(i\vett k_1\cdot\vett q)
\fraz{\diff~}{\diff t} \Zeta\big((\vett k_1+\vett k_2)\cdot\vett
p\big) + \mathcal{R}_5 \ ,
\end{split}
\end{equation*}
where $\mathcal{R}_5$ is non--resonant and can be integrated by parts.
The first term at the r.h.s gives instead, as usual,
\begin{equation*}
\begin{split}
\frac{\partial~~}{\partial\vett k_1 \cdot\vett p} & \frac{1 - \zeta(\vett k_1
\cdot\vett p)} {\vett k_1 \cdot\vett p}
\exp( i\vett k_1\cdot\vett q)
\fraz{\diff~}{\diff t} \Zeta\big((\vett k_1+\vett k_2)\cdot\vett
p\big) = \\
&~ \fraz{\diff~}{\diff t} \left (
\frac{\partial~~}{\partial\vett k_1 \cdot\vett p} \frac{1 - \zeta(\vett k_1
\cdot\vett p)} {\vett k_1 \cdot\vett p}
\exp ( i\vett k_1\cdot\vett q)
\Zeta\big((\vett k_1+\vett k_2)\cdot\vett p\big) \right ) + \\
& + \veps \left . \left (x\fraz{\partial~}{\partial x}
\fraz {1-\zeta(x)}{x}
\right )\right |_{\vett k_1 \cdot\vett p} \big((\vett k_1+\vett
k_2)\cdot\vett p\big) \exp(i\vett k_1
\cdot\vett p) + \mathcal{R}_6
\ ,
\end{split}
\end{equation*}
where $\mathcal{R}_6$ are non resonant terms. Instead, the second term
at the r.h.s is again a resonant one, but it can be transformed into a
total time--derivative (plus some non--resonant terms) using again a
relation of the kind
\begin{equation*}
\veps \exp(i\vett k_1 \cdot\vett p) = (\vett k_1 + \vett
k_2)\cdot\vett{\dot p} +\veps\mathcal{P}_2 \ ;
\end{equation*}
where again $\mathcal{P}_2$ is a non-resonant trigonometrical
polinomial. With some simple algebra one finally gets
\begin{equation*}
\begin{split}
\zeta \big( & (\vett k_1+\vett k_2)\cdot\vett p\big)
\frac{\partial~~}{\partial \vett k_1 \cdot\vett p} \frac{1 - \zeta(\vett k_1
\cdot\vett p)} {\vett k_1 \cdot\vett p}
\sin \big( (\vett k_1 + \vett k_2)\cdot\vett q\big) = \\
& ~
\frac{\diff~}{\diff t} \left [ \Zeta\big((\vett k_1+\vett
k_2)\cdot\vett p\big)
\frac{\partial~~}{\partial\vett k_1 \cdot\vett p} \frac{1 - \zeta(\vett k_1
\cdot\vett p)} {\vett k_1 \cdot\vett p}
\exp \big( i(\vett k_1 + \vett k_2)\cdot\vett q\big)
\right . \\
& + \left . \vett k_1 \cdot\vett p \frac{\partial~~}{\partial\vett k_1
\cdot\vett p} \frac{1 - \zeta(\vett k_1\cdot\vett p)} {\vett k_1
\cdot\vett p} \Zeta^{(2)}\big((\vett k_1+\vett
k_2)\cdot\vett p\big)
\right ] + \mathcal{R}_8 \ ,
\end{split}
\end{equation*}
where $\mathcal{R}_8$ is a non--resonant term. In this way it is clear
how is it possible to perform three steps of the construction, and at the
same time how complicated becomes the procedure of performing further
steps.
In any case, performing the estimate and putting $\sigma=\veps^{2/5}$, the
estimate of Theorem~\ref{teo3} is obtained
\section{Conclusions}
In this paper a perturbative scheme is introduced which may be called
``\emph{uniform}'', in the sense that it can be applied at the same
time both in the nonresonant region and in the resonant ones.
While this approach does not allow to control individual
trajectories, it is well suited to study the ergodic properties of the
dynamics (at least for the case of rotators), through the estimate of
an integral norm of suitable funtions. A distintive feature of
this approach is the fact that it can be applied to systems with an arbitrarily
large number of degrees of freedom (at least in the case of rotators),
so that it can be of use for systems of interest in statistical
mechanics.
The idea of using in perturbation theory an $L^p$ integral norm
instead of the familiar sup norm, which is a fundamental ingredient
of the present paper, was apparently introduced for the first time by
A.~Neishtadt in the paper \cite{neis}, where a non--Hamiltonian system
with a small number of degrees of freedom was considered.
I thank A.~Neishtadt for kindly pointing this out to me,
after the reading of a preliminary version of the present paper.
\appendix
\section*{Appendix: proof of Lemma~2}
The proof of Lemma~2 is quite standard (apart from property iii))
and can be found in any text--book in partial differential equations.
Consider a $C^{\infty}([-1,1])$ function $y(x)$ having
all derivatives vanishing for $x\to\pm 1$ (take for example
$y(x)=\exp(1/(x^2-1))$). Such a function can obviously be extended
smoothly to the whole real line by setting it equal to zero outside that
interval. Introduce now the auxiliary function
\begin{equation*}
\tilde \zeta(x)=\frac 1C \int_{-2}^{2} y(t-x)\diff t \ , \qquad
C \defz \int_{-1}^{1} y(t)\diff t \ .
\end{equation*}
In terms of $z=t-x$ one equivalently can write
\begin{equation*}
\tilde \zeta(x)=\frac 1C \int_{[-2-x,\,2-x]\bigcap[-1,1]} y(z)\diff
z \ ,
\end{equation*}
from which it is apparent that for $|x|<1$ one has $\tilde \zeta(x)=1$
(because in such a case one has $[-2-x,2-x]\bigcap[-1,1] = [-1,1]$\,), while
for $|x|>3$ one has $\tilde \zeta(x)=0$ (in such a case one has instead
$[-2-x,2-x]\bigcap[-1,1] = \emptyset$\,). If in addition one has
$y(x)\ge0$ one also gets $0\le \tilde \zeta(x) \le 1$. It is obvious that
$\tilde \zeta(x)$ is a $C^{\infty}$ function, and one can define the
constants
\begin{equation}\label{eqa1}
C^{\prime}_n \defz \sup_{|x| \le 3} \left| \fraz { \textrm{d}^n\ }
{\textrm{d}x^n} \tilde \zeta(x) \right | \ .
\end{equation}
In the case of the exponential function $y(x)=\exp(1/(x^2-1))$,
simple (numerical) estimates for the first three constants are
\begin{equation}\label{eqa2}
C^{\prime}_0=1\ , \quad C^{\prime}_1 < 2 \quad \textrm{and} \quad
C^{\prime}_2 < 21\ ;
\end{equation}
the other ones growing quite rapidly. It also obvious that all the
derivatives vanish for $|x|<1$ and $|x|>3$.
We take now
\begin{equation*}
\Zeta^{(2)}(x)\defz \frac {x^2}2 \tilde \zeta\left(\frac
{x}{\sigma}\right) \ ,
\end{equation*}
and consequently, $\Zeta(x)$ being the derivative of $\Zeta^{(2)}(x)$
and $\zeta(x)$ the derivative of $\Zeta(x)$, one gets
\begin{equation}\label{eqa3}
\begin{split}
\Zeta(x) & = x\, \tilde \zeta\left(\frac
{x}{\sigma}\right) +\frac {x^2}{2\sigma} \tilde \zeta^{\prime}\left(\frac
{x}{\sigma}\right) \\
\zeta(x) & = \tilde \zeta\left(\frac
{x}{\sigma}\right)+ \frac {2x}{\sigma} \tilde \zeta^{\prime}\left(\frac
{x}{\sigma}\right) +\frac {x^2}{2\sigma^2} \tilde
\zeta^{\prime\prime} \left(\frac
{x}{\sigma}\right) \ .
\end{split}
\end{equation}
The functions $\zeta(x)$, $\Zeta(x)$ and
$\Zeta^{(2)}(x)$ vanish for $|x|>3\sigma$, while for $|x|<\sigma$
they reduce to $\zeta(x)=1$, $\Zeta(x)=x$ and
$\Zeta^{(2)}(x)= x^2/2$ (recall that the derivatives of $\tilde \zeta$
vanish for $|x|<1$). So i) of Lemma~2 is proved. To prove ii), one remarks
that
\begin{equation*}
\begin{split}
\frac {\textrm{d}^n \zeta(x)}{\textrm{d}x^n\ } & =
\frac {\textrm{d}^{n+2}\ }{\textrm{d}x^{n+2}}\left( \frac {x^2}2 \tilde
\zeta \left(\frac{x}{\sigma}\right) \right) \\
& = \fraz {1}{\sigma^n} \left( (n^2+3n+2)
\frac {\textrm{d}^n \tilde\zeta }{\textrm{d}x^n\ }+ \frac {(n+2)x}{\sigma}
\frac {\textrm{d}^{n+1} \tilde\zeta}{\textrm{d}x^{n+1}\ } +
\frac {x^2}{2\sigma^2}\frac {\textrm{d}^{n+2} \tilde\zeta}
{\textrm{d}x^{n+2}\ } \right) \ ,
\end{split}
\end{equation*}
so that, recalling the bound (\ref{eqa1}),
the constant $c_n$ can be taken equal to
\begin{equation*}
c_n= 5 C^{\prime}_{n+2} + 6(n+2)C^{\prime}_{n+1} + (n+3n+2)
C^{\prime}_n \ .
\end{equation*}
Part iii) of Lemma~2 follows directly from the definition and from the
explict bound (\ref{eqa2}) for the constants $C^{\prime}_0$,
$C^{\prime}_1$ and $C^{\prime}_2$.
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