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\title{On persistence of invariant tori \\ and a theorem by Nekhoroshev}
\author{Dario Bambusi\footnote{E-mail: bambusi@mat.unimi.it} \
and \ Giuseppe Gaeta\footnote{Supported by ``Fondazione CARIPLO per la
Ricerca Scientifica'' under project ``Teoria delle perturbazioni per
sistemi con simmetria''. E-mail: gaeta@berlioz.mat.unimi.it} \\ {\it
Dipartimento di Matematica, Universit\'a di Milano} \\ {\it via
Saldini 50, I--20133 Milano (Italy)} }
\date{\giorno}
\maketitle
\noindent
{\bf Summary.} {We give a proof of a theorem by N.N. Nekhoroshev
concerning Hamiltonian systems with $n$ degrees of freedom and $s$
integrals of motion in involution, where $1 \le s \le n$. Such a
theorem ensures persistence of $s$-dimensional invariant tori under
suitable nondegeneracy conditions generalizing Poincar\'e's condition
on the Floquet multipliers. We also deal in detail with perturbations
of systems having reducible tori: in this case persistence can be
ensured by a nonresonance condition expressed in terms of linear
combinations of determinants involving the frequencies of the motion
on the torus and the frequencies of small oscillations about the
torus.}
\bigskip\bigskip\bigskip\bigskip
\section*{Introduction.}
A Hamiltonian system in $n$ degrees of freedom having $n$ independent
integrals of motion in involution is integrable \cite{Arn1}. When the
system has a number $s$ of independent integrals of motion greater
than one, but smaller than the number of degrees of freedom, i.e. $1 <
s 0$, such that, for all $\eps\in
E_0$, $E_0:=(-\eps_*,\eps_*)$, the following holds true:
\noindent {\rm (1)} In a neighbourhood $U \sse W$ of $\La$ in $M$,
there is a family of symplectic submanifolds $N^\eps \ss U$, of
dimension $2s$, which are fibered over a domain $B \ss R^s$ having as
fibers $C^r$-differentiable tori $\La_\b^\eps = N^\eps \cap \Fb_\eps^{-1}
(\b) \simeq \toro^s$ (with $\b \in B$). For each $\eps \in E_0$, the
tori $\La_\b^\eps$ are $C^r$-diffeomorphic to $\La$ and invariant
under the $X^\eps_i$; the tori $\La_\b^\eps$ and the manifolds $N^\eps$
depend in a $C^r$ way jointly on
$(\eps,\b) \in E_0 \times B$.
\noindent {\rm (2)} There exist global symplectic action angle coordinates
$(I_1^\eps,...,I_s^\eps ; \phi_1^\eps , ... , \phi_s^\eps )$ in $N$,
such that $F_i^\eps \big\vert_{N^\eps} = F_i^\eps \big\vert_{N^\eps}
(I_1^\eps ,..., I_s^\eps )$, with $i=1,...,s$. The coordinates
$(I^\eps ; \phi^\eps )$ depend in a $C^r$ way on $\eps \in E$, and so
do the functions $F_i^\eps \big\vert_{N^\eps}$.
\end{theorem}
\sp {\bf Remark 5.} This result allows to continue invariant tori of a
given system ($\eps = 0$) to invariant tori of its perturbations
($\eps \in E_0$). \EOR
\sp {\bf Remark 6.} In the same situation one could try to apply KAM
theory for lower dimensional tori \cite{Eli,Kuk,Poe,Bou} (note that
this would require a stronger nondegeneracy condition), and this would
ensure persistence of a Cantor family of invariant tori; on the other
hand we ensure here existence of a {\it continuous} family of
invariant tori. In particular we ensure also persistence of the {\it
resonant} tori. Obviously this is due to the fact that the systems
admit some integrals of motions independent of the Hamiltonian, and
thus our situation is exceptional. \EOR
\sp {\bf Remark 7.} When studying infinite dimensional systems, one meets
cases where a continuous spectrum arises, and this makes KAM theory
non applicable at all. On the other hand theorem \ref{pert} extends
immediately to some infinite dimensional situations of this kind; in
particular it is used in \cite{BV} to construct quasiperiodic
breathers in infinite lattices. \EOR
In the perturbative case (i.e. in the case of theorem \ref{pert}) the
statement (2) is particularly useful in that it allows to characterize
the dynamics on the invariant tori of the perturbed system. In
particular one has the following
\begin{corollary}
\label{qp}
In the same hypotheses of theorem \ref{pert}, assume also that for some
$\k\in\left\{1,...,s\right\}$, and for some $m\in\Lambda$ one has
\begin{equation}
\label{nd1}
{\rm det}\left(\frac{\partial^2 F^0_{\k}}{\partial I_j\partial
I_k}\right)(m) \not=0\ ,
\end{equation}
and fix $\eps$ small enough; then there exists a neighbourhood $B_0$
of $\beta_0$ in $\Re^s$ such that for almost all $\beta\in B_0$ the
flow of $X_{\k}^{\eps}$ on $\Lambda_{\beta}^\eps$ is
quasiperiodic with $s$ frequencies independent over the rationals.
\end{corollary}
\sp {\bf Proof.} By smooth dependence of action angle variables and of
$F^\eps_{\k}$ on $\eps$, eq.\ref{nd1} holds also for $\eps$ small
but different from zero. It follows that the map from the actions to
the frequencies is a local isomorphism. \EOP
\section{Proofs}
In this section we will always {\it assume that all the hypotheses of
theorem \ref{main} hold}, without stating this explicitly in each
lemma.
\bigskip
In order to prove the two theorems stated in the previous section, the
main point is to introduce suitable coordinates in a neighbourhood $U$
of an arbitrary point of $\Lambda$. In order to do that we introduce,
for any $m$ in $\Lambda$ a manifold $\Sigma_m$ of codimension $s$
passing through $m$. By the tubular neighbourhood theorem \cite{Lang},
the manifolds $\Sigma_m$ can be chosen in such a way to define a
foliation in $U$.
\begin{lemma}
\label{coo}
Let $m$ be an arbitrary point in $\La$. There exists a neighbourhood
$\V_m \subset W \subset M$ of $m$ and a coordinate map
$$
\V_m\ni
p\mapsto (\beta,\tau,y)\in\R^s\times\R^s\times\R^{2(n-s)}
$$
with the following properties (where we identify a point with its coordinates):
(i) $F_i (\beta,\tau,y)=\beta_i $,
(ii) $m=(\beta_0,0,0)$
(iii) $(\partial / \partial\tau_i ) \, = \, X_i$, $\forall i=1,...,s$
(iv) $p\in\Sigma_m\ \iff\ p=(\beta,0,y)$
(v) the coordinate map depends smoothly on $m$
\end{lemma}
\sp {\bf Proof.} First one has that $\Fb$ restricted to $\Sigma_m$ is
a submersion and therefore one can introduce coordinates $(\beta,y) $
in $\Sigma_m$ defined in an open set of $ \R^s\times\R^{2(n-s)}$ with
$\Fb(\beta,y)=\beta$ and such that the coordinates of $m\in\Sigma_m$
are $(\beta_0,0)$.
For $(\tau , z) $ in some subset of $\R^s\times \Sigma_m$, consider
the map $(\tau,z)\mapsto g^{\tau}(z)$. Using the coordinates
$(\beta,y)$ in $\Sigma_m$ this can be expressed as a map $(\tau,
\beta, y)\mapsto g^{\tau}(\beta,y)$. It is easily verified that its
differential at $m$ is an isomorphism and therefore, by the implicit
function theorem, the $(\beta,\tau,y)$ are a local coordinate system
at $m$, which by construction (and by commutativity of the flows
$g_i$) have the properties stated in the lemma. \EOP
The coordinates constructed in this lemma will be called
{\it adapted coordinates based at the point $m$}
Let us now fix a nontrivial homotopy class $\alpha\in\pi_1(\Lambda)$
and, as in the previous section, consider a vector field $X_\a =
\sum_i c_i X_i$ such that its trajectories are periodic of period 1 on
$\Lambda$ and belong to the homotopy class $\a$. For ease of writing,
we denote by $\Phi$ the time 1 flow of $X_\alpha$, i.e. $\Phi (p) =
g^{\cb}(p)$ where $\cb:=(c_1,...,c_s)$.
Consider now a point $m \in \La$ and introduce adapted coordinates
based at $m$; we write
$(\^\beta,\^\tau,\^y):=\Phi(\beta,\tau,y)$. Remark that since
$\Phi(\beta_0,0,0)=(\beta_0,0,0)\equiv m$, by smooth dependence of
solutions on initial data there exists a neighbourhood ${\widetilde
\V_m}$ of $m$ which is mapped under $\Phi$ in the domain $\V_m$ of
definition of the above coordinates. We restrict $\Phi$ to such a
neighbourhood.
\begin{lemma}
\label{str}
In the adapted coordinates, the map $\Phi: (\b , \tau , y) \to
(\^\beta,\^\tau,\^y) $ is described by
$$
\^\beta (\beta,\tau,y) \, = \, \beta \ \ ; \ \
\^\tau (\beta,\tau,y) \, = \, \tau \, + \, \tau_0(\beta,y) \ ,
$$
with $\tau_0$ a suitable function.
Moreover,
$$
\partial \^y / \partial \tau \ = \ 0 \ .
$$
\end{lemma}
\sp{\bf Proof.} The first equality is a trivial consequence of the way
the coordinates are defined. To prove the other two equalities fix
$\mu\in\Re^s$ small and
consider
$$
\^\beta (\beta,\tau+\mu,y)\ \ ; \ \
\^\tau (\beta,\tau+\mu,y) \, \ ; \ \ \^y(\beta,\tau+\mu,y)
$$
by definition these are the coordinates of the point
$$
\Phi(\beta,\tau+\mu,y)=\Phi\left(g^{\mu}(\beta,\tau,y)\right)=
g^{\mu}
\left(\Phi\left(\beta,\tau,y)\right)\right)
= (\beta,\hat\tau(\beta,\tau,y)+\mu,\^y(\beta,\tau,y))\ .
$$ From this
$$
\^\tau(\beta,\tau+\mu,y)=\^\tau(\beta,\tau,y)+\mu\ ,\quad \^
y(\beta,\tau+\mu, y)=\^ y(\beta,\tau,y)
$$
which shows that $\^ y$ is independent of $\tau$. To conclude the
proof just put $\tau_0(\beta,y):=\hat \tau(\beta,0,y)$. \EOP
It is useful to remark that the Floquet multipliers of the
periodic orbits of $X_{\alpha}$ do not depend
on the initial point of the orbit in $\Lambda$.
\begin{lemma}
\label{flo}
Let $m$ and $m_1$ be two points of $\Lambda$; then the Floquet
multipliers of $g^{t\cb}(m)$ and of $g^{t\cb}(m_1)$ coincide.
\end{lemma}
\sp{\bf Proof.} Remark that there exists $\tau_1$ such that
$m_1=g^{\tau_1}(m)$. Then, by the commutation of flows we have
$g^{\cb}=g^{-\tau_1}\circ g^{\cb}\circ g^{\tau_1}$. Taking the
differential (with respect to the space variable) of this equality at
the point $m$ we get
$$
dg^{\cb}(m)=dg^{-\tau_1}(m_1)dg^{\cb}(
m_1)dg^{\tau_1}(m)\ ,
$$
which gives the relation between $dg^{\cb}(m)$ and $dg^{\cb}(m_1)$;
the eigenvalues of the former are the Floquet multipliers at $m$, and
the eigenvalues of the latter are the Floquet multipliers at
$m_1$. This shows that $dg^{\cb}(m)$ and $dg^{\cb}(m_1)$ are
conjugated and therefore have the same eigenvalues. \EOP
\begin{lemma}
\label{impl}
Fix $m\in\Lambda$ and introduce adapted coordinates based at $m$,
consider the map $\Phi: (\beta,\tau,y) \to (\^\b , \^\tau , \^y )$;
then there
exists $\beta_{*}$ independent of $m$ such that
$$
\^ y(\beta,\rho_m(\beta))=\rho_m(\beta)\ .
$$
defines a unique smooth map
$\rho_m(\beta)$ for all $\beta$ with $|\beta-\beta_0|<\beta_{*}$.
\end{lemma}
\sp {\bf Proof.} In order to be guaranteed that we can solve the
equation $\^y(\beta,y)=y$ by means of the implicit function theorem,
we show now that our assumption (iii) on the Floquet multipliers (see
theorems \ref{main}, \ref{pert}) implies that $1$ is not an eigenvalue
of the Jacobian of the map $y\mapsto \hat y$ at $(\beta_0,0,0)$.
To prove this fact remark that by lemma \ref{str} the Jacobian matrix
$J$ of $\Phi$ at $m$ takes the block form
\begin{equation}
\label{jac}
J= \pmatrix{\displaystyle{ \frac{\partial\hat \beta}{\partial\beta}}
&\displaystyle{ \frac{\partial\hat \tau}{\partial\beta}} &
\displaystyle{\frac{\partial\hat y}{\partial\beta}} \cr
\displaystyle{\frac{\partial\hat \beta}{\partial \tau} }
&\displaystyle{ \frac{\partial\hat \tau}{\partial \tau}}
&\displaystyle{ \frac{\partial\hat y}{\partial\tau}} \cr
\displaystyle{\frac{\partial\hat \beta}{\partial y}} &
\displaystyle{\frac{\partial\hat \tau}{\partial y}} &
\displaystyle{\frac{\partial\hat y}{\partial y}} \cr}= \pmatrix{ 1 &
\displaystyle{\frac{\partial \tau_0}{\partial\beta}} &
\displaystyle{\frac{\partial \hat y}{\partial\beta}} \cr \null &\null
&\null \cr 0 & 1 & 0 \cr \null&\null&\null \cr 0 &
\displaystyle{\frac{\partial \tau_0}{\partial y}} &
\displaystyle{\frac{\partial \hat y}{\partial y}} \cr} \ .
\end{equation}
It follows that the secular equation for $J$ takes the form (with $I$
the identity matrix)
$$
\det ( \lambda I -J ) \ = \ (\lambda-1)^{2s} \ \det \left(\lambda
I - \frac{\partial
\hat y}{\partial y}\right) \ = \ 0
$$
which gives the relation between the Floquet multipliers $\lambda$ and
the eigenvalues of the Jacobian matrix of the map $y\mapsto \^y$. As
$J$ has, by assumption, {\it exactly } $2s$ Floquet multipliers equal
to 1, it follows that 1 is not an eigenvalue of $(\partial \^y /
\partial y)$. Note that this also shows that we have always a
multiplicity at least $2s$ for the eigenvalue 1
We can thus apply the implicit function theorem, which ensures
existence uniqueness and smoothness of the map $\rho_m$. We can choose $\b_*$
independent of $m$: indeed by compactness of $\La$ the $C^r$ norm of $\^y$
can be bounded uniformly with respect to $m$ and the eigenvalues of
$(\pa \^y / \pa y)$
are uniformly bounded away from 1, see lemma 2.9. \EOP
Define now the map $\s_\b : \La \to U$ as $\sigma_\beta(m)\equiv
(\beta,0,\rho_m(\beta))$ where we used adapted coordinates based at
$m$. Remark that since $\sigma_\beta(m)\in\Sigma_m$, one has
$$
\sigma_\beta(m)\not=\sigma_\beta(m')\quad {\rm if }\quad m\not=m'.
$$
Define
$$
\Lambda_\beta:=\sigma_\beta(\Lambda)\ ,
$$
and remark that this set is in one to one correspondence with
$\Lambda$. We are going to prove that actually $\Lambda_\beta$ is a
smooth manifold, and that the above correspondence is a
diffeomorphism. To this end, having fixed $m$ and $\beta$ with
$|\beta-\beta_0|<\beta_{*}$, define
$$
\M_m^\beta:=\left\{p\in\widetilde\V_m\ :\ p=(\beta,\tau,y)\ ,\
y=\rho_m(\beta) \right\}
$$
where we use adapted coordinates based at $m$. Then
$\sigma_\beta(m)\in\M_m^\beta$, and moreover $\M_m^\beta$ is an
$s$--dimensional smooth submanifold of $M$.
\begin{lemma}
\label{minlambda}
One has $\M_m^\beta\subset\Lambda_\beta$.
\end{lemma}
\sp{\bf Proof.} Let $p\in\M_m^\beta$, $p\not=\sigma_\beta(m)$, then
there exists a unique $m_1$ such that $p\in\Sigma_{m_1}$. Consider
$\^p:=\Phi(p)$. Using adapted coordinates based at $m$, if $p = (\b ,
\tau , y)$ then, by definition of $\M_m^\beta$, the coordinates of
$\hat p$ are $(\beta,\^\tau,y)$. From this it follows $g^{-(\^\tau -
\tau)} (\^p)=p$. Introduce adapted coordinates $(\beta,\tau_1, y_1)$
based at the point $m_1$; one has $p=(\beta,0,y_1)$, and denote
$$
\widehat p=\left(\beta, \widehat \tau_1, \widehat y_1\right)\ .
$$
Applying $g^{-(\^\tau - \tau )}$ we obtain
$$
\left(\beta,\widehat\tau_1-(\widehat\tau - \tau), \widehat y_1\right)
= g^{-(\widehat\tau - \tau)} (\widehat p ) = p =(\beta,0,y_1)
$$
from which in particular one has $y_1=\widehat y_1(\beta, y_1)$. By
uniqueness of the solution to $y_1 = \widehat{y} (\b , y_1 )$, it
follows $y_1 = \rho_{m_1} (\b) $ and therefore
$p=\sigma_\beta(m_1)$. \EOP
\sp {\bf Remark 8.} The sets $\M_m^\beta$ are a covering of
$\Lambda_\beta$: indeed $\M_m^\beta$ contains at least
$\sigma_\beta(m)$.
\begin{lemma}
\label{smooth} $\Lambda_\beta$ is a smooth compact manifold,
$C^r$-diffeomorphic to $\Lambda$.
\end{lemma}
\sp{\bf Proof.} First remark that the map
$$ \psi^\beta_m: \U_m \to \M_m^\beta \ \ , \ \ \psi_m^\beta
(\beta_0,\tau,0) = (\beta,\tau, \rho_m(\beta)) $$ (where we used
adapted coordinates based at $m$) is smooth from a neighbourhood
$\U_m$ of $m$ in $\Lambda$ to a neighbourhood of $\sigma_\beta(m)$ in
$\Lambda_\beta$. Use this map to introduce the $\tau$ as local
coordinates in $\Lambda_\beta$. Then the maps $\psi^\beta_m$ yield a
$C^r$ atlas of $\Lambda_\beta$; the transition functions between the
coordinate system $\tau$ on $\M_{m}^\beta$ and the coordinate system
$\tau_1$ on $\M_{m_1}^\beta$ are given simply by the functions
$\tau_1(\beta,\tau,\rho_m(\beta))$, which are $C^r$-smooth. It follows
that $\Lambda_\beta$ is a $C^r$ manifold; it is also $C^r$
diffeomorphic (by the map $\sigma_\beta$) to $\Lambda$, and therefore
is compact. \EOP
\sp{\bf Remark 9.} By construction the fields $X_i$ are tangent to each
of the $\M_m^\beta$, and therefore the manifold $\Lambda_\beta$ is
invariant under the flows of all the $X_i$.
\begin{lemma}
\label{symp}
The $C^r$ manifold $\widetilde N$ obtained as the union of $\Lambda_\beta$ for
$\left|\beta-\beta_0\right|<\beta_*$ is symplectic and fibered in
isotropic tori $\toro^s$.
\end{lemma}
\sp {\bf Proof.} It is clear that $\widetilde N$ is $C^r$ and that it
is fibered in tori $\toro^s \equiv \Lambda_\beta$. The tori
$\Lambda_\beta$ are integral manifolds of the Hamiltonian vector
fields generated by the functions $F_i$, and lie in common level
manifolds of the $F_i$; thus we can choose a basis of variables
$\tau_i$ along $\Lambda_\beta$ such that $\Om (\pa / \pa F_i , \pa /
\pa \tau_j ) = \delta_{ij}$. This shows at once that the
$\Lambda_\beta$ are isotropic and that the restriction
$\Om_{\widetilde N}$ of the symplectic form $\Om$ defined on $M$ to
the submanifold $\widetilde N \ss M$ is non-degenerate. As $\d \Om =
0$ implies $\d \Om_{\widetilde N} = 0$, the proof is complete. \EOP
\begin{corollary}
\label{ciao}
There exist a $2s$ dimensional symplectic submanifold $N$,
$\Lambda\subset N\subset\widetilde N$, and global action angle
coordinates on $N$.
\end{corollary}
\sp{\bf Proof.} The restriction of the system to $\widetilde N$ is
integrable in the Arnold Liouville sense. Thus the standard
construction of action angle coordinates for integrable systems
applies \cite{Arn1} and allows to construct action angle coordinates
in a neighbourhood $N$ of $\Lambda$ in $\widetilde N$. \EOP
This concludes the proof of theorem \ref{main}. We leave to the reader
to check that the same proof, with minor modifications, also applies to
theorem \ref{pert}.
\section{Reducible tori}
\def\hot{{\tt h.o.t.}}
\def\Z{{\bf Z}}
We want to apply the result proved above to the case of reducible
tori. It is clear that only the Floquet multipliers lying on the unit
circle in $\C$ can break the nondegeneracy condition required by
theorems \ref{main} and \ref{pert}. Thus the case of elliptic tori
contains all the difficulties met in the application of Nekhoroshev
theorem (and of its perturbative extension), so we will limit to
consider this. Actually, due to a result by Kuksin \cite{Kuk2},
elliptic tori are always reducible under the hypotheses assumed in
this paper.
\bigskip
Consider a Hamiltonian system with hamiltonian function $H$, and
assume that it has an invariant $s$-dimensional torus $\Lambda$. It is
known \cite{AKN} that if $s=1$ and the periodic orbit has Floquet
multipliers with modulus 1, then in a neighbourhood of the periodic
orbit there exists a system of canonical coordinates
$(I,\psi,p,q)\in\R\times T^1\times\R^{n-1}\times\R^{n-1}$, such that
the Hamiltonian takes the form
$$ \omega_1 I \ + \ \sum_{j=1}^{n-1} \
\nu_j \, \frac{p_j^2+q_j^2}2 + \hot \ .
$$
Here and below $\hot$
denotes higher order terms, i.e. terms which are at least quadratic in
$(I,p,q)$ if they depend on $I$, and terms which are independent of $I$
and at least cubic in $p,q$. Therefore the periodic orbit is just
$I=p=q=0$ and $\psi(t)=\omega_1 t+\psi_0$.
In the case of invariant tori of dimension $s \ge 2$, the existence of
such a system of coordinates is not guaranteed in generic systems (see
the discussion in \cite{Kuk}); if there exist variables in which the
Hamiltonian $H$ takes the form (here and below $r:= n-s$) $$ H \ = \
\sum_{j=1}^{s}\, \omega_j \, I_j \ + \ \sum_{j=1}^{r} \, \nu_{j} \,
\frac{p_{j}^2+q_{j}^2}2 \ + \ \hot $$ then the torus is said to be
{\it reducible} for $H$. We have denoted by $\om$ the frequencies
along the invariant torus, and by $\nu$ the transversal ones.
\bigskip
Here we want to apply Nekhoroshev's theorem to the case where the
invariant torus $\Lambda$ is reducible for each of the Hamiltonians
$F_1,...,F_s$ in the same system of coordinates. The main
point is that in this situation the assumption on the Floquet
multipliers takes a very simple and explicit form.
\sp {\bf Remark 10.} Kuksin proved that under the assumptions
considered in this paper, such coordinates exist \cite{Kuk2} \EOR
Since the discussion on the Floquet multipliers involves only the
linearized dynamics around the torus, and thus only the quadratic part
$\Phi_i$ of the expansion of $F_i$ around $\La$, and the linearization
$Y_i$ of the hamiltonian vector fields $X_i$ associated to the $F_i$ at
$\La$ (these also satisfy $Y_i \interno \Om = \d \Phi_i$), from now on
we neglect the higher order terms, and consider just $\Phi_i$ and
$Y_i$. The results we obtain hold, of course, for the complete
hamiltonian system and not just for the linearized one.
Thus our assumption of reducibility of $\La$ for all the $F_i$ means
that there exists a system of canonical coordinates $(I,\psi,p,q)$,
with $\psi\in\toro^s$ and $(I,p,q)$ defined in a subset of
$\Re^s\times\Re^{n-s}\times\Re^{n-s}$, in which the functions $\Phi_i$
take the form $$ \Phi_i \ = \ \sum_{j=1}^s \, \omega^{(i)}_j \, I_j \
+ \ \sum_{j=1}^r \, \nu^{(i)}_j \, \( {p_{j}^2+q_{j}^2 \over 2} \) \ .
$$
We write $\pa_j := (\pa / \pa \psi_j)$; we also introduce, for
ease of notation,
$$ \begin{array}{l}
J_j := \ (p_j^2+q_j^2) / 2 \\
\pa_j := \ p_j (\partial / \partial q_j) \, - \, q_j (\partial /
\partial p_j) \end{array} \quad \quad \quad j=s+1,...,n \ . $$
We also define, again for ease of notation, matrices $A$ and $B$ with
elements given by
$$ A_{ij} \ := \ \om^{(i)}_j \ \ ; \ \ B_{ij} = \nu^{(i)}_j \ . $$
Hence $A$ is a $(s \times s)$ matrix built with the frequencies of
motion in the invariant torus $\Lambda$, and $B$ is a $(s \times r)$
matrix built with the frequencies of small oscillations in the
transversal directions to the invariant torus.
In this way we have
$$ \Phi_i \ = \ \sum_{k=1}^s \, A_{ik} \, I_k \ + \ \sum_{k=1}^r \,
B_{ik} \, J_k
$$
and the corresponding linear vector field $Y_i$ takes
the form
\begin{equation}
\label{decompo}
Y_i \ = \ \sum_{j=1}^s \ A_{ij} \, \pa_j \ + \ \sum_{j=1}^r \ B_{ij}
\, \pa_{j+s} \ \ \ \ (i=1,...,s)
\end{equation}
\def\wt#1{\widetilde{#1}}
We will denote by $\Om (k;j)$ the matrix obtained from $A$ by
substituting its $k$-th column with the $j$-th column of $B$. For ease
of notation, we also write $P := (A^T)^{-1}$. We will also denote by
$|M|$ the determinant of a matrix $M$.
\begin{theorem}
\label{redu}
Let $F_i,\Phi_i,Y_i$ and $\La$ be as above in this section.
Let $X_\a$ be the vector field $X_\a = \sum_i c_i Y_i$ having periodic
orbits $\gamma_\a$ on $\La$ with period one and in the homotopy class
$\a \in \Z^s$.
Then:
\noindent {\rm (1)} The periodic orbit $\gamma_\a$ has $2s$ Floquet
multipliers equal to 1,
and $2r$ ``transversal" Floquet multipliers given by $\exp [\pm 2 \pi i
Q_j (\a) ]$ ($j=1,...,r$) with
$$ Q_j (\a) \ = \ \sum_{i,k=1}^s \ B^T_{ji} P_{ik} \alpha_k \ . $$
\noindent {\rm (2)} The condition that the multiplicity of the Floquet
multiplier 1 does not exceed $2s$ is equivalent to $Q_j (\a) \not\in
{\bf Z}$.
\noindent {\rm (3)} The nondegeneracy assumption (iii) of theorem
\ref{main} holds if and only if there exists
$\alpha\equiv(\alpha_1,\alpha_2,...,\alpha_s)\in {\bf Z}^s$ such that
\begin{equation}
\label{complex}
Q_j (\a) \ \not= \ \ m \ \ \ \forall m \in \Z \ , \ \forall j=1,...,r
\ . \end{equation}
\noindent {\rm (4)} Condition \ref{complex} is equivalent to
\begin{equation}
\label{simple}
\sum_{k=1}^s \ \alpha_k \, | \Om (k;j) | \ \not= \ m \ |A| \ \ \ \
\forall m \in {\bf Z} \ ,\quad \forall j=1,...,r\ .
\end{equation}
\end{theorem}
\sp {\bf Corollary.} {\it Under the assumption \ref{complex}, or
equivalently \ref{simple}, the considered
invariant torus is part of an $s$ parameter family of invariant tori
and such a family persists under perturbation.} \sp
The corollary follows immediately from theorem \ref{main}.
\bigskip
We would like to stress that both the conditions given in the theorem
can be checked once we know the matrices $A$ and $B$; however
condition \ref{complex} requires to consider only one matrix $B^T
(A^T)^{-1}$ but also to perform the inversion of the $(s \times s)$
matrix $A$, while condition \ref{simple} requires to consider $s \cdot
(n-s)$ matrices $\Om (k,j)$, but does not require to consider any
inversion of matrices. Thus it can be more convenient to use one or
the other of them depending on the problem at hand; however condition
\ref{simple} can always be explicitly checked and requires only simple
linear algebra. As far as we know, condition \ref{simple} has not been
considered before.
\sp{\bf Remark 11.} In the case $s=1$ condition \ref{simple} reduces to
\begin{equation}
\label{lya}
\exists \alpha\in{\bf Z}\ :\ \alpha\omega_k\not=m\omega_1\ ,\
\forall m\in{\bf Z}\ ,\ \forall k=2,...,n \ ,
\end{equation}
which is easily seen to be equivalent to the standard condition under
which the Lyapunov center theorem holds, namely that
$\omega_k/\omega_1\not\in\ra $, $\forall k=2,...,n$. Indeed, if
$\omega_k/\omega_1\in\ra$ then, for any choice of $\alpha\in\ra$ one
has that $\alpha\omega_k/\omega_1\in\ra$, and therefore \ref{lya} is
also violated. Conversely, if $\omega_k/\omega_1\not\in\ra$ then
simply choose $\alpha=1$ and \ref{lya} also holds. \EOR
\sp{\bf Proof of theorem \ref{redu}.} It is obvious that $(1)
\Rightarrow (2) \Rightarrow (3)$; so we have only to prove (1) and (4).
Fix $\alpha\in\pi_1(\Lambda)$;
first of all we want to determine the vector field $X_\alpha$, or,
more precisely, its linearization $Y_\alpha$ about the torus. To this
end we remark that the projection $Z_c$ of a general vector field $Y_c
= \sum_{i=1}^s c_i Y_i$ to invariant tori is just the projection onto
$\{ \pa_1 , ... , \pa_s \}$. Hence, see \ref{decompo},
$$ Z_c \ = \ \sum_{i=1}^s \ \sum_{k=1}^s \ c_i \, A_{ik} \, \pa_k \ . $$
When we require that this has closed orbits with period 1 and winding
number $\alpha_i$ around the cycles of $\La$, we are requiring
$$ Z_c \ = \ \sum_{k=1}^s \ (2 \pi)\, \alpha_k \, \pa_k \ . $$
Thus we are asking $\sum_i c_i A_{ik} =2\pi \alpha_k$, i.e. $A^T \cb =
2\pi\alpha$, and we get (recall $P := (A^T)^{-1}$)
$$ c_i \ = \ \sum_{k=1}^s \ (2\pi) \, P_{ij} \, \alpha_j , $$
which explicitly defines $Y_{\alpha}:=\sum_{j=1}^s c_i Y_i$.
In order to compute the transversal Floquet multipliers we have to
study the dynamics of $Y_\alpha$ in the transversal direction. In
particular, recalling again \ref{decompo}, the flow generated by
$Y_{\alpha}$ in the planes $p_j,q_j$ ($j=1,...,r)$ take the form
$$
\pmatrix{p_j\cr q_j} \mapsto \pmatrix{p_j\cos(2 \pi \, Q_{j}(\a) \, t)
-q_j \sin(2 \pi \, Q_{j} (\a) \, t)\cr q_j \cos(2 \pi \, Q_{j} (\a) \,
t) + q_j\sin(2 \pi \, Q_{j} (\a) \, t)}\ ,$$ with
$$
Q_j (\a) := \ \sum_i c_i \nu_{ij} \ := \ \sum_i (B^T)_{ji}
c_i \ = \ \sum_{i,k=1}^s \ B^T_{ji} \, P_{ik} \, \alpha_k .
$$
This shows that the transversal Floquet exponents are indeed the $Q_j
(\a)$, as claimed in the statement, and proves (1). As remarked above,
(2) and (3) are then obvious.
It remains to prove (4), i.e. to reformulate the condition $Q_j (\a)
\not\in \Z$ in a convenient way. Let us consider the matrix
$$ M \ := \ B^T \ P \ \equiv \ B^T (A^T)^{-1} $$ and the matrices $\Om
(k,j)$ introduced above. We have to show that the elements $m_{kj}$ of
$M$ can be written as a ratio of determinants, $m_{kj} = |\Om
(k,j)|/|A|$.
Let us introduce a useful notation: given a matrix $M$ of
elements $m_{ij}$, we denote by $\wt{M}$ the matrix of its algebraic
complements; this means that the element $(\wt{m})_{ij}$ is the
algebraic complement of $m_{ij}$ in $M$.
We recall that $\sum_j m_{ij} (\wt{m})_{kj} = \delta_{ik} |M|$, that
$\wt{(M^T)} = \(\wt{M}\)^T$, and that by Cramer's theorem $M^{-1}$ is
obtained as $(M^{-1})_{ij} = | M |^{-1} \(\wt{M}\)^T$.
For ease of writing, we will think of $k$ and $j$ as fixed and write
simply $R$ for $[\Om (k,j)]^T$. By elementary linear algebra, as
recalled above, $ |R| = \sum_\ell R_{i\ell} (\wt{R})_{i\ell}$, for any
$i=1,...,s$; we consider $i=k$ and compute the determinant in this
way. By definition, however, $R_{k\ell} = (B^T)_{j\ell}$. As for
$(\wt{R})_{k\ell}$, we note that $R$ differs from $A^T$ only on the
row $k$; thus the $k$-th rows of $\wt{R}$ and of $\wt{A^T}$ are
identical, i.e. $\wt{R}_{k\ell} = \wt{A^T}_{k\ell} = \wt{A}_{\ell k}$.
In this way we obtain that
$$ |R| \ = \ \sum_{\ell} \, R_{k \ell} \, (\wt{R})_{k \ell} \ = \
\sum_\ell \, (B^T)_{j \ell} \, \wt{A}_{\ell k} \ = \ \sum_\ell \,
(B^T)_{j \ell} \, [(A^T)^{-1}]_{\ell k} \, (|A|) \ . $$ Using $|R^T| =
|R|$ and writing again in full $R^T \equiv \Om (k,j)$, we have shown
that
$$ M_{jk} \ := \ (B^T P)_{jk} \ \equiv \ (B^T (A^T)^{-1})_{jk} \ = \
{|\Om (k,j) | \over |A|} \ , $$ hence that $ Q_j (\a) = \sum_k \a_k
(|\Om (k,j) | / |A|)$; this completes the proof of (4) and thus of the
theorem. \EOP
\def\bx{{\bf x}}
\section{Applications}
\subsection{A three body problem}
Consider the system composed by a heavy star and two light planets
lying in a fixed plane. For simplicity we consider the case where the
star is fixed at some point and the two planets move in the plane
under the action of the gravitational force due to the star and of the
gravitational force due to the other planet. If the interaction
between the two planet is neglected then it is clear that there exist
solutions in which both planet move on circular orbits around the star
at distances given respectively by $R_1$ and $R_2$; we will show that,
provided the masses of the two planets are small enough and a suitable
nonresonance condition is fulfilled, such solutions persist in the
coupled case (where the interaction is also considered).
Denote by $M$ the mass of the star, by $\eps m_1$ and $\eps
m_2$ the masses of the two planets. Take the position of the star as
the origin of the coordinates and introduce polar coordinates
$r_1,\theta_1$, $r_2,\theta_2$ for the position of the two
planets. Then the dynamics of the system is described by the
Hamiltonian
\begin{equation}
\label{h.3}
H^\eps=\frac{P_1^2}{2m_1}+\frac{\pi_1^2}{2m_1r_1^2}-\frac{GMm_1}{r_1}
+ \frac{P_2^2}{2m_2}+\frac{\pi_2^2}{2m_2r_2^2}-\frac{GMm_2}{r_2}-\eps
K_1
\end{equation}
where $P_i$ is conjugated to $r_i$, $\pi_i$ is conjugated
to $\theta_i$, and the function $G_1$ is given by
$$
K_1:=\frac{Gm_1m_2 }{\sqrt{r_1^2+r_2^2-2r_1r_2\cos(\theta_1-\theta_2)
}} \ ;
$$
$G$ is the gravitational constant.
This Hamiltonian system has an additional integral of motion given by
the total angular momentum
$$
F_2:=\pi_1+\pi_2\ .
$$
We will apply theorem \ref{redu} with $F_1^\eps:=H^\eps$ and
$F_2$ given by the total angular momentum.
In order to identify
$\Lambda$, fix two
positive numbers $R_1$ and $R_2$ and define
$$
L_1:=m_1\sqrt{GMR_1}\ ,\quad L_2:=m_2\sqrt{GMR_2}\ ;
$$
then we have
$$
\Lambda \ := \ \left\{ r_i=R_i , \, P_i =0 ,\,
\pi_i =L_i ,\, \theta_i \in S^1 \right\}\ .
$$
Denote by
$$
\omega_i:=\frac{L_i}{m_iR_i^2}\equiv \sqrt{\frac{GM}{R_i^3}}\ ,\quad i=1,2
$$
the frequencies of revolution of the two planets, and introduce
new coordinates $I_1,I_2$ (conjugated to the angles $\theta_1,\theta_2$)
and $(p_1,q_1)$, $(p_2,q_2)$ by
$$
I_i:=\pi_i-L_i\ ,\quad p_i:=\frac{P_i}{\sqrt{m_i \omega_i}}\
,\quad q_i:=(r_i-R_i)\sqrt{m_i\omega_i}\ ,\quad i=1,2
$$
then a straightforward computation shows that the two function $F^0_1$
and $F_2$ take the form
$$
F^0_1=\omega_1 I_1+\omega_2
I_2+\omega_1\frac{p_1^2+q_1^2}{2}+\omega_2\frac{p_2^2+q_2^2}{2}
+\hot\equiv\Phi_1+\hot
$$
$$
F_2=I_1+I_2\equiv\Phi_2\ ,
$$
which have a form suitable for the application of theorem \ref{redu}.
\begin{theorem}
\label{3body}
Assume that the frequencies fulfill the nonresonance condition
\begin{equation}
\label{nr3}
\frac{\omega_1}{\omega_1-\omega_2}\not\in\ra
\end{equation}
then, provided $\eps$ is small enough the conclusions of theorem
\ref{pert} hold.
\end{theorem}
In particular there exists a family of two dimensional invariant tori
which can be parametrized by the radii $R_1,R_2$ which continue in the
coupled case the above solutions. Remark that since in the unperturbed
case the frequencies of revolution are in one to one correspondence
with the parameters $R_1$, $R_2$, the same holds for the coupled case,
and in particular on most of the tori the motion will be quasiperiodic
with two nonresonant frequencies, while on the remaining tori the
motion will be periodic.
\sp{\bf Proof.} We have just to check that \ref{nr3} is equivalent to
the nonresonance condition \ref{simple}. We have
$$
A=\pmatrix{\omega_1 &\omega_2\cr 1&1}\ ,\quad B=\pmatrix{\omega_1
&\omega_2 \cr 0 &0 }
$$
from which
$$ \begin{array}{ll}
\Omega{(1;1)}=\pmatrix{\omega_1 & \omega_2\cr 0&1}\ ,& \
\Omega{(1;2)}=\pmatrix{\omega_2 & \omega_2\cr 0&1} \\
\Omega{(2;1)}=\pmatrix{\omega_1 & \omega_1\cr 1&0}\ ,& \
\Omega{(2;2)}=\pmatrix{\omega_1 & \omega_2\cr 1&0}\ . \end{array}
$$
Computing the determinants and inserting these in \ref{simple},
the nonresonance condition takes the form
$$ \exists (\alpha_1,\alpha_2)\in\ra^2 \ : \ \forall \, (m_1,m_2) \in
\ra^2 \ , \ \cases{ \omega_1(\alpha_1-\alpha_2)\not =
m(\omega_1-\omega_2) \cr \omega_2(\alpha_1-\alpha_2)\not =
m(\omega_1-\omega_2) \cr} $$ and this is easily seen to be equivalent
to \ref{nr3}. \EOP
\sp{\bf Remark 12.} It is possible to generalize theorem \ref{3body}
to the case of unperturbed elliptic (rather than circular) orbits, but
the computations would be more involved. \EOR
The result of theorem \ref{3body} can be obtained also by first
passing to the reduced system obtained by reducing the angular
momentum and then applying standard Poincar\'e theory. In particular
the reduction of the angular momentum is elementary in the case of the
system \ref{h.3}: just perform the change of variable $J_1:=I_1+I_2$,
$J_2=I_2$. However there are cases in which the reduction of the
angular momentum is not elementary and however Nekhoroshev's theory
works very well. In the next subsection we present an example of this
kind.
\subsection{A four body problem}
Consider three particles lying in a plane attracted by a fourth
particle lying at a fixed point of the plane (that we choose as the
origin of the coordinates) through a spherically symmetric force. The
particles also interact by a small spherically symmetric
potential. The system is described by
\begin{equation}
\label{lat}
F^\eps_1\equiv
H:=\sum_{k=1,2,3}\left(\frac{\norma{\bp_{k}}^2}2+V(\norma{\bq_k})
\right)+\eps \sum_{1\leq i