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$N$--body 
problem, nearly--integrable Hamiltonian systems, 
lower dimensional elliptic tori.
---------------0404081205617
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\title{ $N$-dimensional invariant tori for the planar \\
 $(N+1)$-body
problem
 \footnote{{\bf Acknowledgments.} Supported by MIUR {\sl Variational
Methods and Nonlinear Differential Equations}.
We thank Carlangelo Liverani for
fruitful suggestions.}
}



\author{
{Luca Biasco} \and {Luigi Chierchia} \and { Enrico
Valdinoci}}


\maketitle



\begin{abstract}{

\noindent For any $N\geq 2$ we prove the existence of quasi--periodic
orbits lying on $N$--dimensional invariant elliptic
tori for the planetary planar $(N+1)$--body problem. For concreteness, we
focus on a caricature of the outer solar system.}
\end{abstract}

\nl
{{\bf Keywords:}
$N$--body
problem, nearly--integrable Hamiltonian systems,
lower dimensional elliptic tori.
}

\noi
{{\bf MSC2000 numbers:}
70F10, 34C27, 37J40, 70K43.
}


{{\tableofcontents}}
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\pagina



\section {Introduction and results}
\setcounter{equation}{0}
\nopagebreak
\tool{intro}{Quasi--periodic motions in the many--body problem}
The existence of stable trajectories of the many body problem
viewed as a model for the solar system
has been the subject of researches of many distinguished
scientists both in the past
and in recent years. Despite  eminent contributions given, among others,  by
Poincar\'e [Poi1905], Arnold [A63] and Herman [H95] and some
numerical evidence of ``stability'' of the giant planets of the solar system
(compare [L96]), the literature is still lacking in rigorous
general proofs\footnote{See, however, [F03].}. The main difficulty consists
in the application of general tools (such as averaging and KAM theory) to particular
cases of interest in celestial mechanics, where strong degeneracies appear. The scope
of this paper is to make some progress in this direction, showing the existence of
quasi--periodic orbits lying on
$N$--dimensional invariant elliptic tori for the planar $(N+1)$--body problem.
Though the method exposed here is quite general, for concreteness
we will focus our attention on a caricature of the outer solar system.

More precisely,
our model will be given by a Sun  and $N$ planets
with relatively small masses (say, of order $\e$). All these $(N+1)$ bodies are
considered as point masses in mutual gravitational interaction. Two planets
(such as Jupiter and Saturn in the real world) will be assumed
to have mass considerably bigger than the other planets.
The bodies lie in a given plane and we assume that the initial
configuration is far from collisions. We also assume,
mimicking the case of the outer solar system, that the two big
planets have an orbit which is internal with respect to the orbits
of the small planets.
We will establish, for a large set of semi--axes,
the existence
of
quasi--periodic orbits with small eccentricities
filling up $N$--dimensional invariant elliptic
tori. Such orbits can be seen as continuations of ``limiting''
circular trajectories of the system obtained by
neglecting the mutual interactions
among the planets. A more precise statement is
given in Theorem \ref{THM} below.

The above ``outer model", which roughly mimics some traits of physically relevant
cases, has also the nice
feature of providing particulary simple expressions
in the related perturbing functions, as we will see in Section \ref{conto}
below. We stress, however, that many other situations (such as one
large planet plus  $N-1$ small planets; ``inner" or ``mixed" models, etc.) may be
easily dealt with using the techniques and results presented in this paper.

The proof of our result is based on techniques developed in [BCV03] and on
the explicit computation of the eigenvalues of the quadratic part of the so--called
principal part of the perturbation for the planar many--body problem.

\Giu

\noi
The first result on quasi--periodic orbits of interest in celestial mechanics goes
back to [A63], where quasi--periodic orbits lying on
$4$--dimensional tori are shown to exist for the planar three--body problem
(the general case
was discussed there, but no complete proof was given).
Related results were given in [JM66], which found
linearly unstable quasi--periodic orbits lying on
$2$--dimensional tori for the
non planar three--body problem. More recently,
[LR95]--[R95] and [BCV03] proved
the existence of
quasi--periodic orbits for the
non planar three--body problem,
lying on $4$--dimensional and linearly stable $2$--dimensional tori,
respectively.
Periodic orbits of the
non planar three--body problem
winding around invariant tori have been constructed in [BBV03].
The result presented below,
in the case $N=2$, has been obtained, with different methods,  in [F02].

The paper is organized as follows. In Section \ref{stat}, we give a
more precise statement of
our main result. In Section
\ref{app:nbody}
we write down the $(N+1)$--body problem Hamiltonian in
Delaunay--Poincar\'e variables. 
In Section~\ref{conto} (which, in a sense, is the crucial part of the paper) we
discuss degeneracies. In Section~\ref{proof} we give the proof of the
main result. The scheme of proof is similar to the one presented in [BCV03]
(see also [BBV03]) in the three--body case and it is based on a
``general''
averaging theorem and on KAM theory for lower--dimensional tori
([P96], [BCV03], [BBV03]).
For completeness, we
include a classical (but not--easy--to--find) description of analytical properties of
the Delaunay--Poincar\'e variables (see
\S~\ref{app:nbody} and Appendix~\ref{Poincare});  in   Appendix~\ref{linearalgebra}
we collect some simple linear algebra lemmata that are used in the arguments
given in Section~\ref{conto}.


\tool{stat}{Statement of results}
We denote the $N+1$ massive
points (``bodies") by
$P_0,\,\dots\,,P_N$ and let
$m_0,\,\dots\,,m_N$ be their masses interacting through  gravity
(with constant of gravitation 1). Fix $m_0>0$ and assume
that
\beq{ipot.m}
m_i =\e\mu_i\ ,\qquad i=1,\dots, N\ ,\qquad 0<\e<1\ .
\eeq
Here, $\e$ is regarded as a small parameter and $\mu_i$ is of order 1 in $\e$.
The point $P_0$ represents
the ``Sun'' and the points $P_i$, $i=1,\dots,N$, the ``planets''.
We assume that all the bodies lie on a  fixed plane, that will be
identified with $\real^2$. The phase space of this dynamical system (``the
planetary, planar
$(N+1)$--body system") has dimension $4N$.

We will state the result in terms of  orbital elements of the ``osculating
ellipses'' 
%(at time $t_0$)
of the two--body problems
associated to  $(P_0,P_j)$. 
Let $u^\ppz$ and $u^\ppj$ denote the coordinates of
$P_0$ and $P_j$ 
%at time $t_0$
(at a given time)
and let $\dot u^\ppz$ and $\dot u^\ppj$ denote the
corresponding velocities. By definition, the ``osculating ellipse"  is  the
ellipse described by the solution of the two--body problem
$(P_0,P_j)$
with initial data 
%at $t=t_0$
given by $(u^\ppz,u^\ppj,\dot u^\ppz, \dot u^\ppj)$.
Of course, such ellipses  describe the motions of the full $(N+1)$--body
problem only approximately;
%
%\footnote{For $\e=0$, the system decouples in $N$
%integrable two--body systems.}
%
nevertheless, it provides a nice set of
coordinates allowing, for example, to describe the true motions in terms of the
eccentricities $e_j$ and the major semi--axes $a_j$
of the osculating ellipses.
For further details and pictures of the orbital
elements, we refer to [Ch88] and [BCV03].

In this paper we consider a planetary (planar) model with planets evolving
from phase points corresponding to well separated nearly--circular ellipse
($e_i\ll 1$); here ``well separated" means that 
\beq{ipoa}
0<{a_i}<\th\  {a_{i+1}}\ ,\qquad\qquad1\le i\le N-1\ .
\eeq
for a suitable constant $0<\th<1$.
For concreteness, we shall focus on a caricature of the outer solar system,
i.e., we will assume that, for some
$m_0< \bar \m_i< 4 m_0$, 
\begin{eqnarray}\label{ipom}
\mu_i = \bar\mu_i 
\phantom{AAAA\,}
&\qquad{\rm for}&\quad i=1,2\,,\nonumber\\
\mu_i = \d\bar\mu_i 
\phantom{AAAa}
&\qquad{\rm for}&\quad i=3,\dots,N\,,\qquad 0<\d<1
\,.
\end{eqnarray}
In this setting, $P_1$ and $P_2$ imitate (in a very rough way, of course)
the physical features of the giant planets Jupiter and Saturn, while $P_3$ and
$P_4$  represent  Uranus and Neptune\footnote{The Jupiter/Saturn  mass ratio
is approximately 3.34, while the Neptun/Uranus mass ratio is
about 1.18.}.


\begin{theorem}\label{THM} Consider a planar, planetary $(N+1)$--body system
satisfying (if $N\ge 3$) \equ{ipom}. Let $A\subset \rN$ be a compact set of
semi--axes where \equ{ipoa}  hold. 
Then,
there exists $\d^\star>0$ and for any  $0<\d<\d^\star$ there exists
$\e^\star>0$  so that the following holds. For any
$0<\e<\e^\star$, the 
planetary, planar
$(N+1)$--body system possesses a family of
$N$--dimensional elliptic invariant tori, travelled with Diophantine
frequencies, provided that the osculating major semi--axes belong to a subset of
$A$ of density\footnote{Here and in the sequel, the ``density'' is intended with
respect to Lebesgue measure.}
$1-C_1\e^{c_1}$. These motions correspond to orbits with osculating
eccentricities bounded by $C_2\e^{c_2}$ and the variation in time of the
osculating major semi--axes of these orbits is bounded by $C_3 \e^{c_3}$.
\end{theorem}
A few comments:

\begin{itemize}

\item
The numbers  $\d^\star$ and $\th$  can be easily computed in the course
of the proof and are not ``very small"; in fact $\th$ is a
``universal" constant while $\d^\star$ depends only on $N$ and $A$.
On the other hand
$\e^\star$, which depend on $N$, $A$ and $\d$, is related to a KAM smallness
condition and rough estimates lead, as well known, to ridiculously small
quantities (for somewhat more serious KAM estimates, we refer to [CC03]). 
Finally, the positive constants $C_i$'s depend on $N$, $A$ and $\d$, while the
$c_i$'s depend ony on $N$ (and could, also, be easily calculated: see \equ{c1c2}).


\item
The  assumptions \equ{ipoa} and \equ{ipom} in the theorem
are used to check {\sl explicitely} suitable ``non degeneracy" conditions.
Howerver, giving up explicit constants and estimates, one can show that
the thesis of the theorem holds, essentially, with no hypotheses on the
semi--axes $a_j$ and the rescaled masses  $\m_j$ (provided
$a_i\neq a_j> 0$ and $\m_j>0$).

\item
The invariant tori found in Theorem~\ref{THM} are {\sl lower dimensional
elliptic tori} meaning that the dimension of the tori are strictly smaller
than (in fact, half of)  the dimension of the Lagrangian (maximal) tori, which
have dimension $2N$. ``Elliptic" means that the tori are linearly stable. It
is not difficult to show that such elliptic tori are surrounded by a set
of positive measure of maximal tori.


\item
The proof given below is based on a well known elliptic KAM theorem, which
works under ``non degeneracy" (or Melnikov) conditions. To check these conditions
one has to study the eigenvalues of the ``secular" (or averaged) quadratic part
of the Newtonian many--body interaction, which will be denoted $\bHd$;
``quadratic", here, referes to the symplectic Cartesian variables measuring the
eccentricity  and the orientation of the osculating ellipses. The diagonalization
of $\bHd$ is trivial (under the only assumption that $a_i\neq a_j$), while
conditions  
\equ{ipoa} and \equ{ipom} will be used to check that the associated eigenvalues
are non zero, simple and distinct so that Melnikov conditions are satisfied.
The proof is non inductive on $N$.



\item
A more detailed and quantitative version of Theorem~\ref{THM} is 
Theorem~\ref{thm:tori} below.
\end{itemize}

\section{Poincar\'e Hamiltonian setting
}\label{app:nbody}
\nopagebreak
The results described in this section are classical and go back to
Delaunay and Poincar\'e; however since they are not so easy to find, for
convenience of the reader,   we reproduce them  in
Appendix~\ref{Poincare}.

\nl
Consider $N+1$ bodies $P_0,\,\dots\,,P_N$, in a fixed (ecliptic) plane,
of masses $m_0,\,\dots\,,m_N$ interacting through gravity
(with constant of gravitation 1). We assume that the mass of $P_0$ (the ``star") is
much larger that the mass of the other bodies (the ``planets"), i.e., we assume
\equ{ipot.m}.
In heliocentric planar  (suitably rescaled) variables, the dynamics of the
planar $(N+1)$--body problem is governed (as explained in
Appendix~\ref{Poincare}) by the Hamiltonian

\beq{Kep2} \cH^{(N)}(X,x):= \cH_0^{(N)}(X,x) +\e \cH_1^{(N)}(X,x) \ ,\qquad
\eeq
where  $X:=(X^\ppu,...,X^\ppN)\in\real^{2N}$ and
$x:=(x^\ppu,...,x^\ppN)\in\real^{2N}$ are conjugated Cartesian symplectic variables
and
\beqa{defHi} &&
\cH_0^{(N)}:= \sum_{i=1}^N\Big( \frac{1}{2 {\rm m_i}}
|X^\ppi|^2-\frac{{\rm m}_i {\rm M}_i}{|x^\ppi|} \Big)\,,
\quad\nonumber\\
&& \cH_1^{(N)}:= \sum_{1\leq i<j\leq N}\,\Big(
X^\ppi\cdot  X^\ppj- \frac{\m_i
\m_j}{m_0^2}\,\frac{1}{|x^\ppi-x^\ppj|}\Big)\,;
\eeqa
here we have introduced the  dimensionless masses
\beq{masses2}
{\rm M}_i:= 1 + \e \frac{\m_i}{m_0}\ ,\qquad
{\rm m}_i:= \frac{\m_i}{m_0+\e\m_i}= \frac{\m_i}{m_0}\  \su{{\rm M}_i}\ .
\eeq
The Hamiltonian
$\cH_0^{(N)}$ is simply the sum of
$N$ un--coupled planar Kepler problems (formed by the star and the $i^{\rm
th}$ planet). Being interested in phase region where the un--coupled Kepler
problem describe nearly--circular orbits, we  introduce {\sl planar Poincar\'e
variables}, the construction of which are based on the following classical
$4$--dimensional symplectic map.

\lemtwo{lem:P}{Planar Poincar\'e variables}
Let
\beq{def.Fi}
F_1(t):= \left(1-\frac{t}4\right)^{\su2}\ ,\qquad F_2(t):= \su2\ \left(1-
\frac{t}4\right)^{-1}\ ,\qquad \Big(|t|<1/4\Big)\ ;
\eeq
let $G_0(s,t)=t+st+\cdots$ be the function analytic in a neighborhood of
$(0,0)$ implicitely defined by
\beq{defG0}
G_0(0,0)=0\ ,\qquad G_0=s \sin G_0 + t \cos G_0\ ;
\eeq
define the following four functions of three variables $(\reta,\rxi,\l)$
real--analytic in a neighborhood of the set
$\{(\reta,\rxi)=(0,0)\}\times\torus$:
\beqa{defGE}
G(\reta,\rxi,\l)&:=& G_0\left( (\reta \cos \l - \rxi \sin \l) \ F_1(t)\ ,\
(\rxi \cos\l- \reta \sin \l)\  F_1(t) \right) \ ,\nonumber\\
{\cE}_{\rm s}(\reta,\rxi,\l)&:=&  \left( \rxi \cos(\l+G) + \reta \sin(\l+G)
\right)\   F_1(t)\ , \nonumber\\
\cC(\reta,\rxi,\l)&:=& \cos(\l+{\cE}_{\rm s}) - \reta\  F_1(t) - \rxi\,
{\cE}_{\rm s} \  F_1(t)\, F_2(t)\ ,
\nonumber\\
\cS(\reta,\rxi,\l)&:=& \sin(\l+{\cE}_{\rm s}) +\rxi\  F_1(t) - \reta\,
{\cE}_{\rm s} \  F_1(t)\, F_2(t)\ ,
\eeqa
where $t$ is short  for $t=\reta^2+\rxi^2$, $G$ is short for
$G(\reta,\rxi,\l)$ and ${\cE}_{\rm s}$ is short for ${\cE}_{\rm s}(\reta,\rxi,\l)$.
Fix $\e,\m,m_0>0$ and let
\beqa{def.varie}
&& {\rm M}:=1+\e\frac{\m}{m_0}
\ ,\qquad
{\rm m}:=  \frac{\m }{m_0}\  \su{{\rm M}}
\ ,\qquad
{\rm \overline m}:=\frac{\m}{m_0}\ \su{\sqrt{\rm M}}\ ,
\nonumber\\
&&
\s:=  \Big(\frac{\m}{m_0}\Big)^3
\su{{\rm M}}\ ,
\qquad
a=a(\L;\m,\e):=\frac{\L^2}{{\rm \overline m}^2}\ .
\eeqa
Then, for any $\L_+>\L_->0$, there exists a ball $B$ around the origin
in $\real^2$ such that the 4--dimensional map
$$\Psi_{\rm P}:(\L,\l,\eta,\xi)\in D:=(\L_-,\L_+)\times\torus\times B\to
(X,x)\in
\real^4$$
where
\beqa{defXx}
&& x_1=x_1(\L,\l,\h,\x;\m,\e):= a(\L;\m,\e)\
\cC\left( \frac{\h}{\sqrt{\L}},\frac{\x}{\sqrt{\L}},\l \right)\ ,\\
&&x_2=x_2(\L,\l,\h,\x;\m,\e):= a(\L;\m,\e)\
\cS\left( \frac{\h}{\sqrt{\L}},\frac{\x}{\sqrt{\L}},\l \right)\ ,\nonumber\\
&& X= X(\L,\l,\h,\x;\m,\e):=
\frac{{\rm \overline m}^4 }{\L^3}  \frac{\partial x}{\partial
\l}(\L,\l,\h,\x;\m,\e) =\frac{\rm \overline m}{a(\L;\m,\e)^{3/2}}\ 
\frac{\partial x}{\partial \l}\ ,\nonumber
\eeqa
is real-analytic in $D$ and symplectic:
$$
d\L\wedge d\l +d\h \wedge d\x = dX_1 \wedge dx_1+ dX_2 \wedge dx_2\ .
$$
Furthermore, if $\cH_0^{(1)}$ denotes the two-body Hamiltonian
$$\cH_0^{(1)}(X,x):=\frac{1}{2 {\rm m}}
|X|^2-\frac{{\rm m} {\rm M}}{|x|}\ ,
$$
then, on the phase region of negative energies
$\Big(\cH_0^{(1)}\Big)^{-1}\Big(-\frac{\s}{2\L_-^2},-\frac{\s}{2\L_+^2}\Big)$, one has
$$
\cH_0^{(1)}\circ \Psi_{\rm P}= - \frac
{\s }{2 \L ^2}\ ;
$$
in the planar coordinates $x\in\real^2$ the corresponding motion describes
an ellipse of major semiaxis $a=a(\L;\m,\e)$ and eccentricity
\beq{eccentricity}
e= \sqrt{\frac{\h^2+\x^2}{\L}}\ F_1\Big(\frac{\h^2+\x^2}{\L}\Big)=
\sqrt{\frac{\h^2+\x^2}\L\
\Big(1- \frac{\h^2+\x^2}{4\L}\Big)}\
\ .
\eeq
\elem
The proof of this lemma may be found in Appendix~\ref{Poincare}. Note
that
\beq{simple}
\cC(0,0,\l)=\cos \l\ ,\qquad \cS(0,0,\l)=\sin \l\ ,
\eeq
so that the $(a,\l)\to x$ transformation is, for $\h=\x=0$, just polar
coordinates. Let, now, $\Psi_{\rm P}^{(N)}$ be the $4N$--dimensional map,
parameterized by
$(\m_1,...,\m_N,\e)$, defined by
\beq{PSIN}
\Psi_{\rm P}^{(N)}:
\Big((\L_1,\l_1,\eta_1,\xi_1),...,(\L_N,\l_N,\eta_N,\xi_N)\Big)\in
\Big( (0,\io)\times\torus\times\real^2\Big)^N\to (X,x)
\eeq
with
\beqa{PSI}
&&
(X,x)=\Big((X^\ppu,...,X^\ppN),(x^\ppu,...,x^\ppN)\Big)
\ ,\nonumber\\
&&
(X^\ppi,x^\ppi)=\Psi_{\rm
P} (\L_i,\l_i,\eta_i,\xi_i;\m_i,\e)\ .
\eeqa
Then,  $\Psi_{\rm P}^{(N)}$ is symplectic and
\begin{equation}\label{xnalalall}
\cH_0^{(N)}\circ \Psi_{\rm P}^{(N)}=
-\su2\sum_ {i=1}^N\frac
{\s_i}{\L_i^2} =: \cH_0(\L)\ ,\qquad\qquad
\s_i       :=  \Big(\frac{\m_i}{m_0}\Big)^3
\su{{\rm M}_i}\ .
\end{equation}


\noi
In such Poincar\'e variables the full planar $(N+1)$--body
Hamiltonian $\cH^{(N)}$ becomes
\beq{Hamtb*} \cH(\L,\l,\eta,\xi)=\cH_0(\L)+ \e\cH_1(\L,\l,\eta,\xi)\ ,
\qquad
\cH_1:=\cH_1^{(N)}\circ \Psi^{(N)}_{\rm P}=:\cH^{{\rm compl}}_{1} +
\cH^{{\rm princ}}_{1}\ ,
\eeq
where
the so-called ``complementary part" $\cH^{{\rm compl}}_{1}$  and
the ``principal part" $\cH^{{\rm princ}}_{1}$ of the perturbation
are, respectively, the functions
\begin{equation}\label{ppart}
\sum_{1\leq i<j\leq N} X^\ppi\cdot\  X^\ppj\ \qquad{\rm and}\qquad
\sum_{1\leq i<j\leq N} \frac{\m_i
\m_j}{m_0^2}\,\frac{1}{|x^\ppi-x^\ppj|}\end{equation} expressed in Poincar\'e
variables\footnote{$X=(X_1,X_2)$  and $x=(x_1,x_2)$
denote here the functions definded in \equ{defXx}.}:
$$X^\ppi=X(\L_i,\l_i,\h_i,\x_i;\m_i,\e) \qquad {\rm   and}\qquad
x^\ppi=x(\L_i,\l_i,\h_i,\x_i;\m_i,\e)\ .$$

\nl
Notice that, since $X^{(i)}=({\rm \overline m}_i^4/\L_i^3) \partial_{\l_i}x^{(i)}$
the\label{vanishes}
$\l$--average of $\cH^{{\rm compl}}_{1}$ vanishes.
Moreover, as it is well known, the
$\l$--average of
$\cH_1$ is an even function of $(\eta,\xi)$;  see, also,
Appendix~\ref{Poincare}. Hence, we may split
the perturbation function
as


\beq{F***}
\cH_1={ \overline  \cH}_1+\tHp\ ,\qquad
\eeq
with
\beq{F****}
{ \overline  \cH}_1(\L,\h,\x):=\igl{\torus^N}{}\cH_1\frac{d\l}{(2\p)^N}\ ,\qquad
\qquad \int_{\torus^N} \tHp\,d\l=0\ .
\eeq
Furthermore, ${ \overline  \cH}_1$ may be written as
\beq{ordini}
{ \overline  \cH}_1(\L,\h,\x)=\bHo(\L)+\bHd(\L,\h,\x)+\bHs(\L,\h,\x)\ ,
\eeq
where $\bHo:={ \overline  \cH}_1(\L,0,0)$, $\bHd$ is the $(\h,\x)$--quadratic  part of
${\overline  \cH}_1$ while $\bHs$ is the ``remainder of order four":
$$ |\bHs(\L,\eta,\xi)|\leq {\rm
const}\,|(\eta,\xi)|^4\,.$$

\section{The averaged quadratic potential $\bHd$}
\label{conto}
\noi
In this section we analyze the function $\bHd$ (i.e., the $(\h,\x)$--quadratic
part of the
$\l$--average of the perturbation) defined  in \equ{ordini}, which may be
written as
\begin{equation}\label{quadraticformQ}
\bHd=\su2\ \sum_{1\leq i,j\leq N} Q_{ij}\left( {{\eta_j}\atop{\xi_j}}
\right)\cdot
\left( {{\eta_i}\atop{\xi_i}}
\right)
\eeq
where $Q_{ij}$ are $(2\times 2)$--matrices defined as

$$Q_{ij}:=\left.
\left({\matrix{
\dpr^2_{\h_i,\h_j} \bHd &\dpr^2_{\h_i,\x_j} \bHd\cr
\dpr^2_{\x_i,\h_j} \bHd& \dpr^2_{\x_i,\x_j} \bHd}}\right)
\right|_{(\L,0,0)}\ .$$
In view of the definition of the Poincar\'e variables, it is natural to look at
the rescaled variables $(\reta,\rxi)$ rather then $(\eta,\xi)$.
Therefore, we define
\beqa{defbarf}
&& \bar f_{ij}(\L,\reta,\rxi):=\\
&& \quad \su{(2\p)^N} \int_{\torus^N} \frac{d\l}
{|
x^\ppi (\L_i,\l_i,\sqrt{\L_i}\reta_i,\sqrt{\L_i}\rxi_i;\m_i,\e)
-
x^\ppj(\L_j,\l_j\sqrt{\L_j}\reta_j,\sqrt{\L_j}\rxi_j;\m_j,\e)
|}\,.\nonumber
\eeqa
Thus, letting\footnote{Recall \equ{def.varie} and \equ{masses2}.}
\beqa{cij}
&& a_i:=a(\L_i;\m_i,\e)\ ,\qquad \qquad \phantom{AAAAA\ \ }
c_{ij}:= \su{m_0}\left(\frac{{\rm
M}_i{\rm M}_j}{a_i a_j}\right)^{1/4}\ ,\nonumber\\
&&A_{ij} :=\left. \left(
{\matrix{ \partial^2_{\reta_i \reta_j}\bar f_{ij} &
\partial^2_{\reta_i \rxi_j}\bar f_{ij}\cr
\partial^2_{\rxi_i \reta_j}\bar f_{ij}&
\partial^2_{\rxi_i \rxi_j}\bar f_{ij}
}
 }\right)\right|_{\reta=\rxi=0}  ,\ \ 
B_{ij} :=\left. \left(
{\matrix{ \partial^2_{\reta_j \reta_j}\bar f_{ij} &
\partial^2_{\reta_j \rxi_j}\bar f_{ij}\cr
\partial^2_{\rxi_j \reta_j}\bar f_{ij}&
\partial^2_{\rxi_j \rxi_j}\bar f_{ij}
}
 }\right)\right|_{\reta=\rxi=0}
,
\eeqa
we find 
$$ Q_{ij} =\cases{\dst\sqrt{\mu_i\mu_j}\,c_{ij} \ A_{ij}\ ,
\phantom{\igl{A}{B}\frac21}} {i\neq j\ ,}
{\dst\sum_{k\neq j}{\sqrt{\mu_k\mu_j}\, c_{kj}\ B_{kj}\ ,}}
{i=j\ .}
$$
It is a remarkable fact that, {\sl for the planar planetary  $(N+1)$--body
problem the matrices
$A_{ij}$ and
$B_{ij}$ are proportional to the
$(2\times 2)$--identity matrix ${\bf 1}_2=\pmatrix{1&0\cr 0&1\cr}$} and have 
simple integral representation.
In fact, define, for $a\neq b$,
\begin{eqnarray*}
\cJ(a,b) &:=& \frac{1}{2\pi}\int_0^{2\pi} \frac{-17 ab\cos t
+8(a^2+b^2) \cos(2 t)+ab\cos(3t) }{(a^2+b^2-2ab\cos t)^{5/2}}\,dt\ ,
\\
\cI(a,b) &:=& \frac{1}{2\pi}\int_0^{2\pi} \frac{-7 ab+4(a^2+b^2)
\cos t-ab\cos(2t) }{(a^2+b^2-2ab\cos
t)^{5/2}}\,dt
\,,
\end{eqnarray*}
and denote, for $a_i\neq a_j$, 
\beq{defab}
\a_{ij}:=\frac{a_i a_j}{8} \cJ(a_i,a_j)\ ,\qquad
\beta_{ij}:=\frac{a_i a_j}{4} \cI(a_i,a_j)\ .
\eeq
Then, the following ``algebraic" result holds.


\begin{proposition}\label{proconto}
Assume $a_i\neq a_j$ for $i\neq j$. Then
$A_{ij}=\a_{ij} {\bf 1}_2$ and $B_{ij}=\b_{ij} {\bf 1}_2$.
\end{proposition}

\rem{rem:conto} (i)
An immediate corollary of this result is that, in the collisionless domain $\{a_i\neq
a_j\}$, $\bHd$  has the simple form
\beq{M}
\bHd=\su2\Big( M\eta\cdot\eta + M\xi\cdot\xi\Big)\,,
\eeq
$M$ being the real, symmetric $(N\times N)$--matrix with entries
\beq{M'}
M_{ij}=\cases{\sqrt{\m_i\m_j}\, c_{ij}\ \a_{ij}\ ,
}{i\neq j\ ,}
{\dst\sum_{k\neq j} \sqrt{\m_k\m_j}\, c_{kj}\ \b_{kj}\ ,
\phantom{\frac{\igl01A}{B}}
}{i=j\ .}
\eeq
The Hamiltonian \equ{M}
can be immediately put in symplectic normal form: if $U$ is the real
orthogonal matrix ($U^T=U^{-1}$) which diagonalize $M$ ($U^T M U
=\diag(\bar\O_1,...,\bar\O_N)$), then the map $p= U^T \h$, $q=U^T \x$ is symplectic
and, in such variables,  the new Hamiltonian takes the form
$$\su2 \sum_{i=1}^N \bar\O_i (p_i^2+q_i^2)\ .
$$
(ii)
The functions
$\cJ$ and  $\cI$ (which admit simple  representations in terms of Gauss
hypergeometric functions) are symmetric ($\cJ(a,b)=\cJ(b,a)$ and
$\cI(a,b)=\cI(b,a)$) and  satisfy
$$
\cJ(a,b)=b^{-3}\cJ(a/b,1)\ ,  \qquad \cI(a,b)=b^{-3}\cI(a/b,1)\ ,\qquad   a<b\ .
$$
The function of one real variables $s\in(-1,1)\to \cJ(s,1)$ and $s\in(-1,1)\to
\cI(s,1)$ are, respectively, even and odd in $s$, and satisfy, for small $s$,
the following asymptotics
\beq{asymp}
\cJ(s,1) =
-\frac{15}{8}s^2-\frac{105}{8}s^4+O(s^6)\ ,
\qquad \cI(s,1) =
3s+\frac{45}{8}s^3+O(s^5)\,.
\eeq
\erem
%%%%
\proof (of Proposition~\ref{proconto})
The computations we are going to perform are algebraic in character and it is
therefore enough to consider real variables.
Fix
$i\neq j$ and define
\beq{defcR}
\cR_{ij}(\L,\l,\hat \h,\hat \x)
:= \left|
x^\ppi (\L_i,\l_i,\sqrt{\L_i}\reta_i,\sqrt{\L_i}\rxi_i;\m_i,\e)
-
x^\ppj(\L_j,\l_j\sqrt{\L_j}\reta_j,\sqrt{\L_j}\rxi_j;\m_j,\e)
\right|^2\ ,
\eeq
so that (recall \equ{defbarf})
\beq{defbarf'}
\bar f_{ij}(\L,\reta,\rxi)=
\su{(2\p)^N} \int_{\torus^N} \frac{d\l}
{\sqrt{\cR_{ij}}}\ .
\eeq
By \equ{defXx} we find
\beq{cR'}
\cR_{ij}=a_i^2 \chi_i^2+ a_j^2 \chi_j^2 - 2 a_ia_j\Big(\cC_i \cC_j
-\cS_i\cS_j\Big)\ ,
\eeq
where $\cC_k$, $\cS_k$ and  $\chi_{{} _{k}}$ are short for, respectively,
$$
\cC_k=\cC(\reta_k,\rxi_k,\l_k) \ , \quad
\cS_k=\cS(\reta_k,\rxi_k,\l_k)\ \quad {\rm and}\quad \chi_{{} _{k}} =\sqrt{\cC_k^2+
\cS_k^2}\ .
$$
The proof will consist in
computing explicitely $\l$--averages of quantities of the form
\begin{equation}\label{deriva}
\r_{\!_{\zeta_i,\zeta_j}}(\l_i,\l_j):=
\left.\partial^2_{\zeta_i
\zeta_j}\frac{1}{\sqrt{\cR_{ij}}}\right|_{\reta=\rxi=0}= \left.\frac{3
(\partial_{\zeta_i}\cR_{ij}) (\partial_{\zeta_j}\cR_{ij})- 2 \cR_{ij}
(\partial_{\zeta_i \zeta_j}\cR_{ij})}{ 4\cR_{ij}^{5/2}} \right|_{\reta=\rxi=0}\,,
\end{equation}
where $\z_k$ denotes either of the variables $\reta_k$ or $\rxi_k$.
Thus, what we need to do is to  compute suitable orders in the variables
$(\reta_k,\rxi_k)$ of the function $\cR_{ij}$. For this purpose it will be
useful the following

\lem{lem:ordini}  Define the following elementary functions
\begin{eqnarray*}
&&C_+ (\l) := 1+\cos^2\l \,=\,\frac{3+\cos(2\l)}{2}\,,\\
&&C_- (\l) :=1+\sin^2\l \,=\,\frac{3-\cos(2\l)}{2}\,,\\
&&S_0 (\l) :=\cos\l\sin\l\,=\,\frac{1}{2}\sin(2\l)\,,\\
&&\bar\chi(x,y,\l) := 1-2y\cos\l +2 x \sin\l\,\\
&&S(x,y,\l) := \sin\l + x C_+(\l)+y S_0(\l)\,,\\
&&C(x,y,\l) := \cos\l -y C_-(\l) -x S_0(\l)  \ ,
\end{eqnarray*}
and denote by $O_p(z_1,...,z_n)$ a  function of
the  variables $(z_1,...,z_n)$ (depending possibly on other variables) analytic in a
neighborhood of
$(0,...,0)$ and starting with a homogeneous polynomial of degree $p$
in $(z_1,...,z_n)$.
Then,
\begin{eqnarray}\label{3.44tris}
&&\chi_{{} _{k}}^2 =\cC(\reta_k,\rxi_k,\l_k)^2+\cS(\reta_k,\rxi_k,\l_k)^2=
\bar\chi(\reta_k,\rxi_k,\l_k) + O_2(\reta_k,\rxi_k)\,,\nonumber \\
\nonumber
&&\cC_k=
\cC(\reta_k,\rxi_k,\l_k) =  C(\rxi_k,\reta_k,\l_k)+O_2(\reta_k,\rxi_k)\
,\\
&&\cS_k=\cS(\reta_k,\rxi_k,\l_k)
= S(\rxi_k,\reta_k,\l_k)+O_2(\reta_k,\rxi_k)\ .
\end{eqnarray}
\elem
The proof of this lemma follows at once from the explicit expressions for $\cC$
and $\cS$ given in Lemma~\ref{lem:P} and is left to the reader.

\nl
We consider first the matrices $A_{ij}$ (which allow to compute
$Q_{ij}$ for $i\neq j$) and then we turn to the matrices $B_{ij}$
(which  allow to compute $Q_{jj}$).


\tool{part1}{Computation of the matrices $A_{ij}$}
\nopagebreak
First, observe that the two derivatives involved in the definition of
$A_{ij}$ are {\em always} mixed in the variables with indexes $i$ and $j$.
Thus, we can neglect the terms of third order in
$(\reta_i,\rxi_i,\reta_j,\rxi_j)$ and  the terms of {\em second order} of the
type $O_2(\reta_i,\rxi_i)$ and
$O_2(\reta_j,\rxi_j)$.

\nl
By Lemma ~\ref{lem:ordini}, the function $\cR_{ij}$ in \equ{cR'} has the form
\begin{eqnarray}\label{++R++}
\cR_{ij} &=& a_i^2 (1-2\reta_i\cos\l_i+2\rxi_i\sin\l_i)+a_j^2
(1-2\reta_j\cos\l_j+2\rxi_j\sin\l_j)\nonumber\\ &&\quad -2 a_i a_j
\Big[
(\cos\l_i-\rxi_iS_0(\l_i)-\reta_iC_-(\l_i))
(\cos\l_j-\rxi_jS_0(\l_j)-\reta_jC_-(\l_j))\nonumber\\&&\quad
+(\sin\l_i+\rxi_iC_+(\l_i)+\reta_iS_0(\l_i))
(\sin\l_j+\rxi_jC_+(\l_j)+\reta_jS_0(\l_j)) \Big]  \nonumber\\
&&\quad+O_2(\reta_i,\rxi_i)+O_2(\reta_j,\rxi_j)+
O_3(\reta_i,\rxi_i,\reta_j,\rxi_j)\,.
\end{eqnarray}
Therefore,  letting
$(\cdot)|_0$ be short for $(\cdot)|_{\reta_i=\rxi_i=\reta_j=\rxi_j=0}$, one finds
\beqa{list}
\cR_{ij} \big|_{0} &=& a_i^2+a_j^2-2a_i
a_j\cos(\l_i-\l_j)\,,\nonumber\\
\partial_{\reta_i}\cR_{ij} \big|_{0} &=& -2a_i^2\cos\l_i-2a_i
a_j[-C_-(\l_i)\cos\l_j+S_0(\l_i)\sin\l_j]\,,\nonumber\\
\partial_{\reta_j}\cR_{ij} \big|_{0} &=& -2a_j^2\cos\l_j-2a_i
a_j[-C_-(\l_j)\cos\l_i+S_0(\l_j)\sin\l_i]\,,\nonumber\\
\partial_{\rxi_i}\cR_{ij} \big|_{0} &=& 2a_i^2\sin\l_i-2a_i
a_j[C_+(\l_i)\sin\l_j-S_0(\l_i)\cos\l_j]\,,\nonumber\\
\partial_{\rxi_j}\cR_{ij} \big|_{0} &=& 2a_j^2\sin\l_j-2a_i
a_j[C_+(\l_j)\sin\l_i-S_0(\l_j)\cos\l_i]\,,\nonumber\\
\partial_{\reta_i\reta_j}^2\cR_{ij} \big|_{0} &=& -2a_ia_j
[C_-(\l_i)C_-(\l_j)+S_0(\l_i)S_0(\l_j)]\,,\nonumber\\
\partial_{\rxi_i\rxi_j}^2\cR_{ij} \big|_{0} &=& -2a_ia_j
[C_+(\l_i)C_+(\l_j)+S_0(\l_i)S_0(\l_j)]\,,\nonumber\\
\partial_{\reta_i\rxi_j}^2\cR_{ij} \big|_{0} &=& -2a_ia_j
[C_-(\l_i)S_0(\l_j)+S_0(\l_i)C_+(\l_j)]\,,\nonumber\\
\partial_{\rxi_i\reta_j}^2\cR_{ij} \big|_{0} &=& -2a_ia_j
[S_0(\l_i)C_-(\l_j)+C_+(\l_i)S_0(\l_j)]\,.
\eeqa
In particular, $\cR_{ij}$ and $\partial_{\reta_i}\cR_{ij}$ are even in
$(\l_i,\l_j)
\in\torus^2$, while $\partial_{\rxi_j}\cR_{ij}$ and $\partial_{\reta_i
\rxi_j}^2\cR_{ij}$ are odd. Thus, recalling the definition of
$\r_{\!_{\zeta_i,\zeta_j}}(\l_i,\l_j)$ in \equ{deriva}, we find that
$\r_{\!_{\reta_i,\rxi_j}}(\l_i,\l_j)$ is odd in $(\l_i,\l_j)$ and it has therefore
zero average. For the same reasons, also $\r_{\!_{\rxi_i,\reta_j}}(\l_i,\l_j)$ has
zero average. Hence, the off--diagonal terms of $A_{ij}$ are zero.
We now compute the diagonal terms of $A_{ij}$.
We begin with $\r_{\!_{\reta_i,\reta_j}}(\l_i,\l_j)$.
%%%%%%%%%%%%%
By
\equ{deriva} and the list in \equ{list}, we find
\beq{tmpreta}
\r_{\!_{\reta_i,\reta_j}}(\l_i,\l_j) :=\left.\partial^2_{\reta_i
\reta_j}\frac{1}{\sqrt{\cR_{ij}}}\right|_{0}=
\frac{\r_1(\l_i,\l_j)}{\r_2(\l_i,\l_j)}
\eeq
with

\beqa{defr1r2}
\r_1(\l_i,\l_j) &:=&
a_i a_j\cdot \Big[-24 a_i^2  \cos(2 \l_i)
- 24 a_j^2  \cos(2 \l_j)+8 (a_i^2+a_j^2)  \cos(2 (\l_i -
\l_j))
\nonumber\\
&& \qquad\quad- 3 a_i a_j \cos(\l_i - 3 \l_j) -
17 a_i a_j \cos(\l_i - \l_j)
\nonumber\\
&&\qquad\quad + a_i a_j \cos(3 (\l_i - \l_j)) -
3 a_i a_j \cos(3 \l_i - \l_j) \nonumber
\\
&&\qquad\quad + 54 a_i a_j \cos(\l_i + \l_j) \Big]\ ,\nonumber\\
\r_2(\l_i,\l_j) &:=&
8 \Big(a_i^2  + a_j^2  - 2 a_i a_j
\cos(\l_i - \l_j)\Big)^{5/2}\,.
\eeqa
Thus, changing variable of integration, one finds
\begin{eqnarray*}
&& \frac{1}{(2\pi)^2}\int_{\torus^2}
\left.\partial^2_{\reta_i
\reta_j}\frac{1}{\sqrt{\cR_{ij}}}\right|_{0}\,d\l_i\,d\l_j
\\ &&\\ &&
\qquad \qquad\quad= \frac{1}{2\pi}
\int_{\torus}
a_i a_i\cdot\frac{ -17 a_i a_j \cos t + 8( a_i^2 +  a_j^2 ) \cos(2 t) +
a_i a_j \cos(3 t)}{8  (a_i^2  + a_j^2  - 2 a_i a_j\cos t)^{5/2}}\,dt\,\\
&&\qquad \qquad\quad= \frac{a_i a_j}{8}\ \cJ(a_i,a_j)=:\a_{ij}\ .
\end{eqnarray*}







%%%%%%%%%%%%%

\nl
The case   $\r_{\!_{\rxi_i,\rxi_j}}(\l_i,\l_j)$ is very similar (and will yield the same
result). In place of \equ{tmpreta} one finds
\beq{tmprxi}
\r_{\!_{\rxi_i,\rxi_j}}(\l_i,\l_j) :=\left.\partial^2_{\rxi_i
\rxi_j}\frac{1}{\sqrt{\cR_{ij}}}\right|_{0}=
\frac{\r_3(\l_i,\l_j)}{\r_2(\l_i,\l_j)}
\eeq
with

\beqa{defr3}
\r_3(\l_i,\l_j) &:=&
a_i a_j\cdot \Big[24 a_i^2  \cos(2 \l_i)
+ 24 a_j^2  \cos(2 \l_j)+8 (a_i^2+a_j^2)  \cos(2 (\l_i -
\l_j))
\nonumber\\
&& \qquad\quad+ 3 a_i a_j \cos(\l_i - 3 \l_j) -
17 a_i a_j \cos(\l_i - \l_j)
\nonumber\\
&&\qquad\quad + a_i a_j \cos(3 (\l_i - \l_j)) +
3 a_i a_j \cos(3 \l_i - \l_j) \nonumber
\\
&&\qquad\quad - 54 a_i a_j \cos(\l_i + \l_j) \Big]\ .
\eeqa
Integrating one finds again
\begin{eqnarray*}
&& \frac{1}{(2\pi)^2}\int_{\torus^2}
\left.\partial^2_{\rxi_i
\rxi_j}\frac{1}{\sqrt{\cR_{ij}}}\right|_{0}\,d\l_i\,d\l_j
=\a_{ij}\ .\end{eqnarray*}
This proves
Proposition \ref{proconto} in the case of
$Q_{ij}$, with $i\neq j$.


\tool{part2}{Computation of the matrices $B_{ij}$}
Observe that the derivatives involved in the definition of $B_{ij}$  are
two derivatives with the same index
$j$. We can, therefore, neglect
the third order terms and set $\reta_i=\rxi_i=0$.

\newcommand\average[3]{\left\langle \, #3\, \right\rangle_{#1,#2}}

\nl
Recalling \equ{simple} we see that $\chi_i|_{\hat \h_i=\hat \x_i=0}=1$ and
\beq{tmp1}
\cR_{ij}\Big|_{\reta_i=\rxi_i=0}=
a_i^2 + a_j^2 \chi_j^2 - 2 a_ia_j\chi_j\left((\cos \l_i)\
\frac{\cC_j}{\chi_j} -(\sin \l_i)\ \frac{\cS_j}{\chi_j}\right)\ .
\eeq
Defining $\f_j=\f_j(\L_j,\l_j,\reta_j,\rxi_j)$ through the
relation\footnote{Physically, $\f_j$ coincide with $v_j+g_j$ where $v_j$ and
$g_j$ are, respectively, the  true anomaly and the argument of the
perihelion of the osculating ellipse associated to the star and the $j^{\rm th}$
planet; compare Appendix~\ref{Poincare}.}
\beq{tmp2}
\cos \f_j =\frac{\cC_j}{\chi_j}\ ,\qquad\qquad
\sin \f_j =\frac{\cS_j}{\chi_j}\ ,
\eeq
we find
\beq{tmp3}
\cR_{ij}\Big|_{\reta_i=\rxi_i=0}=
a_i^2 + a_j^2 \chi_j^2 - 2 a_ia_j\chi_j \cos(\f_j-\l_i)\ .
\eeq
Denote by $\average{\th}{\t}{f}$ the average of a function $f$ over the angles
$\th$ and $\t$. Integrating {\sl first} with respect to $\l_i$ and changing
variable of integration ($t=\l_i-\f_j$) one gets
\beq{alk} \left.\average{\l_i}{\l_j}{
\frac{1}{\sqrt{\cR_{ij}}}\right|_{\reta_i=\rxi_i=0}}= \average{t}{ \l_j}{
\frac{1}{\sqrt{{\tilde\cR_{ij}}}}  }\eeq with
\beq{tildr}
{\tilde\cR_{ij}}:= a_i^2+a_j^2
\chi_j^2-2a_i a_j \chi_j \cos
t\,.\eeq
At this point, the argument is completely analogous to that used above.
First, we observe that
\begin{equation}\label{deriva2}
\average{\l_i}{\l_j}{
\left.\partial^2_{\zeta_h
\zeta_k}\frac{1}{\sqrt{\cR_{ij}}}\right|_{0}}
=
\average{t}{ \l_j}{
\left.\frac{3 (\partial_{\zeta_h}{\tilde\cR_{ij}})
(\partial_{\zeta_k}{\tilde\cR_{ij}})- 2 {\tilde\cR_{ij}}
(\partial_{\zeta_h\zeta_k}^2{\tilde\cR_{ij}})} {4{\tilde\cR_{ij}}^{5/2}}
\right|_{0}}\,,
\end{equation}
where $\z_\ell$ denotes here any of the variables $\reta_j,\rxi_j$.
From Lemma~\ref{lem:ordini} it follows that $\tilde \cR_{ij}$ can be written as
$$ {\tilde\cR_{ij}} = f(t) +g(t)\Big( h_1-h_2\Big)
+a_j^2 h_1^2
+O_3 (\reta_j,\rxi_j)\,,$$
with
\beqano
&& f(t):= a_i^2+a_j^2-2a_i a_j \cos t\ ,\qquad
g(t):= -2a_j^2+2a_i a_j \cos t\ ,\\
&& h_1:= \reta_j\cos\l_j-\rxi_j\sin\l_j\,\qquad h_2:= \rxi_j^2\cos^2
\l_j+\reta_j^2\sin^2\l_j +
\reta_j\rxi_j\sin
(2\l_j)\ .
\eeqano
Thus, since $h_1$ is of order one in $(\reta_j,\rxi_j)$
and $h_2$ is of order two in $(\reta_j,\rxi_j)$,
\beqano
{\tilde\cR_{ij}} \Big|_{0} &=& f(t)\,\\
\partial_{\eta_j}{\tilde\cR_{ij}} \Big|_{0}  &=&
g(t)\,\cos\l_j\,,\\
\partial_{\rxi_j}{\tilde\cR_{ij}} \Big|_{0}  &=&
-g(t)\,\sin\l_j\,,\\
\partial_{\reta_j\reta_j}^2{\tilde\cR_{ij}} \Big|_{0} &=&
-2g(t)\,\sin^2\l_j+2a_j^2\cos^2\l_j\,,\\
\partial_{\reta_j \rxi_j}^2{\tilde\cR_{ij}} \Big|_{0} &=&
-(g(t)+a_j)^2\,\sin(2\l_j)\,,\\
\partial_{\rxi_j\rxi_j}^2{\tilde\cR_{ij}} \Big|_{0} &=&
-2g(t)\,\cos^2\l_j+2a_j^2\sin^2\l_j\,.
\eeqano
Therefore, using \equ{deriva2}, one finds
\beqano
\average{\l_i}{\l_j}{
\left.\partial^2_{\reta_j\rxi_j}\frac{1}{\sqrt{\tilde\cR_{ij}}}
\right|_0}
=\ \ \
\left.\average{t}{\l_j}{
\frac{-\left(\frac{3}{2}g^2+2(a_j^2-g)f \right)\sin(2\l_j) }{4 f^{5/2}}
\right|_0}
\,=\,0
\eeqano
(since the integrand is odd in $\l_j$), showing that also $B_{ij}$ is a
diagonal matrix.
To compute the diagonal elements we calculate
\beq{trhh}
\left.\partial^2_{\reta_j\reta_j}\frac{1}{\sqrt{\tilde\cR_{ij}}}
\right|_0 = \frac{\tilde \r_1(\l_j,t)}{\tilde \r_2(\l_j,t)}\qquad
{\rm and}\qquad
\left.\partial^2_{\rxi_j\rxi_j}\frac{1}{\sqrt{\tilde\cR_{ij}}}
\right|_0 = \frac{\tilde \r_3(\l_j,t)}{\tilde \r_2(\l_j,t)}
\eeq
with
\beqano
\tilde\r_1 &=&
- 7\,{a_i^2}\,{a_j^2}
+ (9\,{a_i^2}\,{a_j^2}
+ 8\,{{a_j}^4})\,\cos (2\,\l_j)
+ \frac72 \,{a_i^2}\,{a_j^2}\,\cos (2\,\l_j - 2\,t)
\nonumber\\&&
- (2\,{{a_i}^3}\,a_j+ 10\,a_i\,{{a_j}^3})\,\cos (2\,\l_j - t)
+ 4 ({{a_i}^3}\,a_j
+ a_i\,{{a_j}^3})\,\cos (t)
\nonumber\\&&
- {a_i^2}\,{a_j^2}\,\cos (2\,t)
- (2\,{{a_i}^3}\,a_j
+ 10\,a_i\,{{a_j}^3})\,\cos (2\,\l_j + t)
+ \frac72 \,{a_i^2}\,{a_j^2}\,\cos (2\,\l_j + 2\,t)\ ;
\nonumber\\
\tilde\r_2&=& 4 (a_i^2+a_j^2-2a_ia_j \cos
t)^{5/2}\ ;\\
\tilde\r_3 &=&
- 7\,{a_i^2}\,{a_j^2}
- (9\,{a_i^2}\,{a_j^2}
+ 8\,{{a_j}^4})\,\cos (2\,\l_j)
- \frac72 \,{a_i^2}\,{a_j^2}\,\cos (2\,\l_j - 2\,t)
\nonumber\\&&
+ (2\,{{a_i}^3}\,a_j+ 10\,a_i\,{{a_j}^3})\,\cos (2\,\l_j - t)
+ 4 ({{a_i}^3}\,a_j
+ a_i\,{{a_j}^3})\,\cos (t)
\nonumber\\&&
- {a_i^2}\,{a_j^2}\,\cos (2\,t)
+ (2\,{{a_i}^3}\,a_j
+ 10\,a_i\,{{a_j}^3})\,\cos (2\,\l_j + t)
- \frac72 \,{a_i^2}\,{a_j^2}\,\cos (2\,\l_j + 2\,t)\ ;
\eeqano
%%%%%%%
taking the $\l_j$--average, one finds immediately
\beqano
&&
\average{\l_i}{\l_j}{ \left.\partial^2_{\reta_j\reta_j}
\frac{1}{\sqrt{\tilde\cR_{ij}}}
\right|_0}\,=\,
\left.\average{\l_i}{\l_j}{\partial^2_{\rxi_j\rxi_j}\frac{1}{\sqrt{\tilde\cR_{ij}}}
\right|_0}=\frac{a_i a_j}4 \cI(a_i,a_j)=:\b_{ij}\ .\qquad\qedeq
\eeqano
%%%%%%%%%%%%%%%%%%%%%
Next result shows that, for $\d$ and $\e$ small, generically  the eigenvalues
of $M$ in \equ{M}--\equ{M'} are non vanishing, simple and distinct. 
We formulate the result regarding the semi--axis $a_j$ as {\sl independent
variables}. Recall the definitions of 
$\a_{ij}$ and $\b_{ij}$   in \equ{defab} and let (if $N\ge 3$)
\beq{defdj}
\b_j:= \sum_{k=1,2} \frac{\sqrt{\bar\mu_k\bar\mu_j}}{m_0} \,\su{\sqrt[4]{a_k a_j}}
\ \b_{kj}\
,\qquad j\ge 3\ .
\eeq

\begin{proposition}\label{proeige}
Assume that $a_j$ and $\bar \m_j$ verify\footnote{Clearly, if $N=2$, the
statements regarding the $\b_j$ and the eigenvalues $\bar \O_j$ for $j\ge 3$
have to be omitted.}
\beq{ND}
\a_{12}\neq 0\quad {\rm and}\quad \b_{12}\neq \pm \a_{12}\ ,\qquad 
\b_i\neq 0\quad {\rm and}\quad 
\b_i\neq
\b_j
\
\  {\rm for}\ \ i\neq j\ .
\eeq 
Then, there exist $0<\d^\star<1$ and $0<\e_0<1$ such that for all 
$0<\d<\d^\star$ and $0\le \e<\e_0$  the eigenvalues
$\{\bar
\O_1,...,\bar
\O_N\}$ of the matrix
$M$ are non vanishing, simple and distinct. Furthermore the following
asymptotics  holds\footnote{We use the standard notation: $a=O(\e)$ $\iff$
$\exists$ a constant $c>0$ (independent of $\e$) and $0<\e_0<1$  s.t.
$|a|\le c |\e|$ for all $|\e|\le \e_0$; $O(\s,\e)=O(\s)+ O(\e)$.}:
\begin{eqnarray}\label{eigM}
&& \bar \O_1=\frac{\sqrt{\bar\mu_1\bar\mu_2}}{m_0}\ \
\ \frac{\b_{12}+\a_{12}}{\sqrt[4]{a_1 a_2}}\,+\,O(\sqrt{\d},\e)\ ,
\phantom{\frac{\igl10 A}{B}}
\nonumber\\
&&\bar \O_2=\frac{\sqrt{\bar\mu_1\bar\mu_2}}{m_0}\ \
\ \frac{\b_{12}-\a_{12}}{\sqrt[4]{a_1 a_2} }\,+\,O(\sqrt{\d},\e)\ ,
\phantom{\frac{\igl10 A}{B}}
\nonumber\\
&& \bar \O_j=\sqrt{\d}
\sum_{k=1,2}\frac{\sqrt{\bar\mu_k\bar\mu_j}}{m_0}\, 
\,\,\,\frac{\b_{ij}}{\sqrt[4]{a_k a_j}}\,+\,\sqrt\d\ O(\sqrt\d, \e)\ ,
\quad \quad3\le j\le N\,.
\phantom{\frac{\igl10 A}{B}}
\end{eqnarray}
\end{proposition}

\proof 
First of all, from the  definition of $c_{ij}$ (see \equ{cij} and \equ{masses2})
it follows that
\beq{cijasy}
c_{ij}= \su{m_0} \ \frac{1}{\sqrt[4]{a_i a_j}} + O(\e)\ .
\eeq
Thus, 
by definition of $M$, by  definition of $\b_j$ and $\a_{ij}$  and by the
hypothesis on the masses
$\m_i$ (see \equ{ipom}) we find the following asymptotics:
\beqano
&\dst\qquad\qquad
\left({\matrix{
M_{11} & M_{12}\cr
M_{21} & M_{22}
}}
\right) = M^\star + O(\sqrt\d,\e)\ ,\ 
&\quad {\rm where}\quad M^\star:=
\frac{\sqrt{\bar \mu_1 \bar
\mu_2}}{m_0\sqrt[4]{a_1 a_2}}\ 
\left({\matrix{
\b_{12} 
&  \a_{12} \cr \a_{12} & \b_{12} }}
\right) 
\ ,\\
&\dst M_{jj} = \sqrt{\d} \b_j 
+O(\d,\e)\ ,\,
&\quad{\mbox{for $j\geq 3$, }}
\\
& \dst M_{ij} =
O(\sqrt\d)\,,
\phantom{AAAAa\ }
&\quad{\mbox{for $i=1,2$ and $j\geq 3$, or $j=1,2$ and $i\geq 3$,}}
\\
& M_{ij} =O(\d)\,,
\phantom{AAAAAa\ \,}
&\quad{\mbox{for $i,j\geq 3$, with $i\neq j$}}
\,.
\eeqano
Therefore\footnote{The $O(\sqrt\d)$ in the upper--right part of $M$ is
a $(2\times (N-2))$ matrix while the $O(\sqrt\d)$ in the lower--left  part of
$M$ is a
$((N-2)\times 2)$ matrix.}
$$ M= \left({\matrix{
M^\star + O(\sqrt\d,\e)& O(\sqrt{\d})\cr &&\cr O(\sqrt{\d})  & \sqrt{\d}
M_\star + O(\d,\e)}}\right) \,,$$
where
\beqano 
&&
M_\star:=\diag( \b_3,...,\b_N)\in
\Mat((N-2)\times(N-2))\ .
\eeqano
The eigenvalues of  $M^\star$
are
$$ \frac{\sqrt{\bar \mu_1 \bar
\mu_2}}{m_0\sqrt[4]{a_1 a_2}}\ 
(\b_{12}+\a_{12})
\qquad{\rm and}\qquad
\frac{\sqrt{\bar \mu_1 \bar
\mu_2}}{m_0\sqrt[4]{a_1 a_2}}\ 
(\b_{12}-\a_{12})\,,$$
which by the first two requirements in \equ{ND}, are
non zero, simple and distinct. The matrix $M_\star$ is diagonal and its
eigenvalues $\b_j$ are, also, non zero, simple and distinct by \equ{ND}.
The claim, now, follows by elementary linear algebra
(compare,e.g.,  Lemma~\ref{linalglem} in Appendix~\ref{linearalgebra}).
\qed

\rem{rem:asympt} (i)
The hypotheses \equ{ND} of Proposition~\ref{proeige} are easily
checked, for example, if
$a_j$  verifies
\equ{ipoa} for a suitable $\th>0$. In fact
the asymptotics for 
$\cJ(s,1)$ and $\cI(s,1)$ (see \equ{asymp}) yields immediately
\beqano
&&\a_{12} = -\frac{15}{64}\  \su{a_2}\ \Big(\frac{a_1}{ a_2}\Big)^{\! 3}\ \
\Big[ 1 +O\Big(\Big(\frac{a_1}{ a_2}\Big)^{\! 2}\Big)\Big]\,,\\ 
&&
\b_{12} =\frac{3}{4}\ 
\su{a_2}\ \Big(\frac{a_1}{ a_2}\Big)^{\! 2}\ \ 
\Big[
1 +O\Big(\Big(\frac{a_1}{ a_2}\Big)^{\! 2}\Big)\Big]\,,\\ 
&&
\b_j =
\frac{3}{4}\ \frac{\sqrt{\bar\mu_j\bar\m_2}}{m_0} \ \ 
\su{a_j^{3/2}}\ \ 
\Big(\frac{a_2}{ a_j}\Big)^{\! 7/4}\ \ 
\Big[1+
O\Big(\Big(\frac{a_1}{ a_2}\Big)^{\! 7/4}\Big)+
O\Big(\Big(\frac{a_2}{ a_j}\Big)^{\! 2}\Big)
\Big]
\ ,\qquad j\ge 3
\ ,\\
&& \b_{12}\mp \a_{12}= \frac{3}{4}\ 
\su{a_2}\ \Big(\frac{a_1}{ a_2}\Big)^{\! 2}\ \ 
\Big[
1 +O\Big( \frac{a_1}{ a_2}\Big)\Big]\,,\\ 
&& \b_j-\b_i=
\frac{3}{4}\ \frac{\sqrt{\bar\mu_j\bar\m_2}}{m_0} \ \ 
\su{a_j^{3/2}}\ \ 
\Big(\frac{a_2}{ a_j}\Big)^{\! 7/4}\ \ \times\\
&&\phantom{\b_j-\b_i=}
\times
\Big[1+
O\Big(\Big(\frac{a_1}{ a_2}\Big)^{\! 7/4}\Big)+
O\Big(\Big(\frac{a_2}{ a_j}\Big)^{\! 2}\Big)
+
O\Big(\Big(\frac{a_i}{ a_j}\Big)^{\! 13/4}\Big)
\Big]\ ,\qquad i>j\ge 3\ .
\eeqano
Thus, if $\th$ is small enough and if
\equ{ipoa} holds, one sees that 
\beqano
&& \a_{12}<0\ ,\\
&&\b_{12}\mp \a_{12}>0\ ,\\
&&\b_j>0\ ,\qquad \forall\ j\ge 3\ ,\\
&&\b_j-\b_i>0\ ,\qquad \forall\ i>j\ge 3\ ,
\eeqano
and the hypotheses \equ{ND} are verified as claimed. 

\nl
(ii) The $O(\cdot)$'s appearing in \equ{eigM} (and in the proof of
Proposition~\ref{proeige}) depend on the $a_j$'s (and on\footnote{Recall that
$m_0<\bar \m_j<4 m_0$; compare line before \equ{ipom}.}
$m_0$). Thus, it is important the order in fixing the various parameters. One
way of proceeding is  as follows. First determine $\th$ as explained in the
previous point (i). Then, let
$\bar a_i$, $1\le i\le N$, be positive numbers such that \equ{ipoa} holds, i.e.,
$\bar a_i/\bar a_{i+1}< \th$ for any $1\le i\le N-1$; (the $\bar a_i$ may
be physically interpreted as observed mean major semi--axis). Now, consider a compact
order--one neighborhood $A\subset \{0<a_1<\cdots<a_N\}$  of $(\bar a_1,...,\bar
a_N)$ for which 
\equ{ipoa} continues to be valid (such neighborhood exists simply by
continuity). Finally, fix
$\d^\star$ and
$\e_0$ so that Proposition~\ref{proeige} holds: such numbers will depend only
on
$\bar a_j$'s and the (order--one) size of the choosen neighborhood $A$.
\erem



\section{Existence of $N$--dimensional elliptic invariant tori}
\label{proof}
In this Section we prove the existence of $N$--dimensional elliptic
invariant tori for the
$(N+1)$-body problem Hamiltonian $\cH$
in \equ{Hamtb*} for any  $N\ge 2$.

\nl
Let $m_0<\bar \m_j<4 m_0$, let $\th$, $A$,  $\d^\star$ and $\e_0$ be as in
point (ii) of Remark~\ref{rem:asympt} and fix $0<\d<\d^\star$, which henceforth
will   be kept fixed. In the rest of the paper only $\e$ is regarded a free
parameter: at the moment, $\e$ is   assumed not to exceed $\e_0$ but later
will be required to satisfy  stronger smallness conditions. The  semi--major
axis map
\beq{amap}
\vec a:\L=(\L_1,...,\L_N)\mapsto
\big(a(\L_1;\m_1,\e),...,a(\L_N;\m_N,\e)\big)
\eeq
is a real--analytic diffeomorphysm and we define 
$$\Igot={\vec a}^{\,-1}(A)\ ,$$
then the Hamiltonian $\cH$ is real--analytic (and bounded) on the domain
$\Igot\times\torus^N\times B^{2N}_R$ for a suitable $R>0$ (here $B^n_r$ denotes
the $n$--ball of radius $r$ and center $0\in\real^n$).

\nl
By Proposition~\ref{proconto}, the quadratic part $\bHd$ of the averaged
Newtonian interaction $\cH_1$ has the simple form \equ{M}, $M$ being the
symmetric matrix defined in \equ{M'}. As already pointed out in 
Remark~\ref{rem:conto}, the matrix $M$ can be diagonalized with eigenvalues,
which, thanks to our assumptions and to Proposition~\ref{proeige}, have the
form in 
\equ{eigM} and, therefore, satisfy
\beq{hold} \inf_\Igot|\bar{\O}_j|>\bar{c}\ ,\qquad
\inf_\Igot \Big|\bar{\O}_i-\bar{\O}_j\Big| >\bar{c}
\,, \eeq for any $i\neq j=1,\dots,N$
and for a suitable positive constant $\bar c$ independent of $\e$.
If
$U:=U(\L)$ is the symmetric matrix which diagonalize $M$, $U^T MU=\diag (\bar
\O_1,...,\bar \O_N)$, then the map
\beq{Xi} 
\Xi:(I,\f, p, q)\mapsto (\L,\l,\eta,\xi)  \ ,\qquad {\rm where}\qquad
\left\{
\begin{array}{l}
p=U^T\h\,,\qquad 
q=U^T \x\,,\cr
I=\L\,,\cr
\dst\f =\l +\sum_{h,k,\ell} (\partial_{\L} U_{k\ell}) U_{h\ell}\, \h_k\,
\x_\ell\,,\cr
\end{array}
\right.
\eeq
is symplectic (and real--analytic) and 
\begin{equation}
\bHd\circ\Xi=\su2 \sum_{i=1}^N \bar{\O}_i(I)(p_i^2 +q_i^2)\,.
\end{equation}
Thus, the $(N+1)$-body problem Hamiltonian $\cH$
in \equ{Hamtb*}, in the case we are
considering, can be written as
\beq{113}
\cH\circ\Xi(I,\f,p,q;\e)= h(I)+f(I,\f,p,q;\e)
\eeq
with
$$
h:=\cH_0\ ,\qquad f:=\e f_1(I,p,q;\e)+\e f_2(I,\f,p,q;\e)\ ,
$$
$$
f_1:=f_{1,0}(I)+\su2 \sum_{i=1}^N\bar{\O}_i(I)(p_i^2 +q_i^2)+\tilde
f_1(I,p,q;\e)\ ,
$$
$$
f_{1,0}:=\bHo\ ,\qquad
\tilde f_1:=\bHs\circ\Xi\ ,\qquad
f_2:=\tHp\circ\Xi\ .
$$
Here $h$ is uniformly strictly concave,
$$
|\tilde f_1|\leq {\rm const}\,|(p,q)|^4\ ,\qquad
{\rm and}\quad  \int_{\torus^N} f_2 \,d\f=0\,.
$$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55
\Giu
\noindent
The construction of elliptic invariant tori for the Hamiltonian \equ{113} is
based on four steps, which we proceed to describe.


\tool{pf1}{Averaging}
Fix  $\t>N- 1\ge 1$ and pick two numbers $b_1$, $b_2$ such that
\beq{b1b2}
0<b_1<\su2\ ,\qquad\qquad 0<b_2<\Big(\su2 -
b_1\Big)\su{\t+1}\ .
\eeq
Since the integrable Hamitlonian $h$
depends only on the action $I$, the conjugated variable $\f$ is a ``fast" angle
and, in ``first approximation", the $(h+f)$-motions are governed
by the averaged Hamiltonian $h+\e f_1$, which posses an elliptic  equilibrium
at $p=q=0$. As we, now, proceed to describe, one may remove  the
$\f$-dependence of the perturbation function $f$ up to
high   order in
$\e$ by using averaging theory; for detailed information on averaging theory
in similar situations, see Proposition A.1 of [BCV03] or Proposition 7.1 of
[BBV03]. 

\nl
Denote by $D^n_R$   the complex $n$--ball of center zero and  radius $R>0$
and, for any $V\subset \rN$, denote  by  $V_R$   the complex neighborhood of
radius $R>0$ of the set $V$ given by $V_R:=\cup_{x\in V} D_R(x)$. 
Next, define  the set $\hat\Igot$ as the following
``Diophantine subset" of 
${\Igot}$:
\beq{defIg}
\hat\Igot:= 
\left\{ I\in  {\Igot}\,:\,
| \dpr_I h(I)\cdot k|\geq\frac{\bar\gamma}{|k|^\tau}\,,\;
\forall k\in\integer^N\setminus
\{0\}\right\}
\, \quad {\rm with}\ \ \  \bar\gamma:={\rm const\,}\, \e^{b_1}\ .\eeq
%with $c_*>0$  to be chosen (large enough) later.
Notice that (as it is standard to prove)
\begin{equation}\label{diofmes}
\meas\Big(
 {\Igot}\setminus
\hat\Igot
\Big)\leq {\rm const\,} \bar\gamma=
{\rm const\,}
 \e^{b_1}
\ .
\end{equation}
The Hamiltonian $h+f$  in \equ{113} is real--analytic on the  complex
domain 
\beq{domainU}
\Dgot_{{\rm r},s,\r} := \hat\Igot_{\rm r}\times \torus^N_s\times
D^{2N}_{\rho}\subset
\complex^{4N}\ ,
\eeq
with
\beq{rsrho}
{\rm r}:= {\rm const}\, \sqrt\e\ ,\qquad
s:= {\rm const}\ ,\qquad
\rho:= {\rm const}\, \e^{b_2}\ .
\eeq
The definition of $\hat\Igot$ is motivated by the necessity to
have an estimate on small divisors.
In fact, let $I\in \hat\Igot_{\rm r}$ (and $\e$ small
enough) and let  $I_0\in
\hat\Igot$ be a point at distance less than $r$ from $I$. Then,  for
any
$k\in\integer^N\backslash\{0\}$ such that $|k|\le K:=\const \e^{-b_2}$, by
the second relation in \equ{b1b2}, by \equ{defIg}  and by Cauchy estimates, one
finds
\beqa{stipiccden}
|\dpr_I h(I)\cdot k|&\ge& |\dpr_I h(I_0)\cdot k|- |\dpr_I h(I_0)-\dpr_I h(I)|\
|k|\nonumber\\ &\ge & \frac{\bar \g}{K^\t} - \max |\dpr_{I}^2 h| \ r\
K\nonumber\\ &\ge & \frac{\bar \g}{2 K^\t}=:\a=\const \e^{b_1+\tau b_2}
\ ,\qquad \Big(\, 0<|k|\leq K:= {\rm const\,}\e^{-b_2}\Big)\ .
\eeqa
In order to apply averaging theory so as to
remove the $\e$--dependence up to order $\exp(-\const K)$, one has to verify
the following ``smallness condition" (compare  condition (A.2), p. 110 in
[BCV03])
\beqno
\| f\|_{{\rm r},s,\rho}
\leq {\rm const\,}\frac{\a \min\{{\rm r}
s,\rho^2\}}{K}\ ,
\eeqno
where the norm $\|\cdot\|_{{\rm r},s,\rho}$ is defined  as the
standard ``sup--Fourier norm" 
\beq{normf}
\| f\|_{{\rm r},s,\rho}:=\sum_{k\in\zN}\Big(\sup_{(I,p,q)\in  \hat\Igot_{\rm
r}\times D^{2N}_\rho} |f_k(I,p,q)| \Big)e^{|k|s}\ ,
\eeq
($f_k(I,p,q)$ denotig Fourier coefficients of the multi--periodic, real--analytic
function $\f\mapsto f(I,\f,p,q)$).
Such condition, in view of \equ{b1b2}, can be achieved by taking $\e$ small
enough since, by \equ{stipiccden} and \equ{rsrho}, one has that
\beqno
\| f\|_{{\rm r},s,\rho}=O(\e)\ \qquad{\rm and}\qquad\ 
\frac{\a \min\{{\rm r}
s,\rho^2\}}{K}=O(\e^{b_1+(\tau+1)b_2+1/2})\ .
\eeqno
Hence, there exists a close--to--identity (real--analytic) symplectic change of
variables
$
(I',\f',p',q')$ $\mapsto$ $ (I,\f,p,q)
$
verifying (compare formulae (2.16) and (A.7) of [BCV03])
\beq{small1}
|I'-I|\leq \const \e^{\frac{1}{2}+b_2}\ ,\qquad {\rm and }\qquad
|p'-p|\,,|q'-q|\leq {\rm const\,}\sqrt\e \ ,
\eeq
and such that the Hamiltonian  expressed in the new
symplectic variables becomes
\beqa{44bis}
h(I') + \hat  g(I',p',q')+  \hat f
(I',\f',p',q')\,,\qquad \hat g := \e f_1(I',p',q')+{\e}\hat
f_1(I',p',q')
\eeqa 
with $\hat f_1$ and $\hat f$ real--analytic on the complex
domain 
$\Dgot_{ {\rm r}/2,{s}/6,{\r}/{2} }$ and satisfying
\beqa{spec}
&&\|\hat f_1\|_{ {\rm r}/2,{s}/6,{\r}/{2} } \le \frac{\e}{\a {\rm r}}=
\const \e^{b_2+b_3}\ ,\quad{\rm with}\quad b_3:=\su2-b_1-(\t+1)b_2>0\ ,
\nonumber\\
&& \|\hat f\|_{ {\rm r}/2,{s}/6,{\r}/{2} }\le \const e^{-\const
K}\ll \const \e^3\ .
\eeqa

\tool{pf2}{New elliptic equilibrium}
Due to the (small) term $ \hat f_1$ in \equ{44bis},  zero is no longer
an elliptic equilibrium for the ``averaged"  (i.e. $\f$-independent) Hamiltonian
$h + \hat g$.
Using the Implicit Function Theorem,
we can find a  new  elliptic equilibrium for
$h + \hat g$, which is $\e^{b_2+b_3}$ close to zero.
Hence we construct a real--analytic symplectic  transformation
\beq{small2}
(J',\psi',v',u')\mapsto
(I',\f',p',q')\ ,\quad {\rm with}\ \ \  I'=J'\quad
\mbox{and $\e^{b_2+b_3}$-close-to-the-identity, }
\eeq
such that in the  new symplectic variables $(J',\psi',v',u')$ the Hamiltonian
takes the form
$$h(J')+ \tilde{g}(J',v',u')+\tilde{f}(J',\psi',v',u')\,,$$
with $\tilde g$ having  $v'=u'=0$ as elliptic equilibrium; 
the functions $\tilde g$ and $\tilde f$ are real--analytic on a slighty smaller
complex domain, say $\Dgot_{ {\rm r}/7,{s}/7,{\r}/{7} }$, where they satisfy
bounds similar to those in \equ{spec}.
Furthermore, for $j=1,\dots,N$, 
the eigenvalues $\tilde{\O}_j (J')$  of the symplectic quadratic part of
$\tilde{g}$ are purely imaginary and
$\e^{1+b_2+b_3}$--close to
$\e\bar{\O}_j (J')$.

\tool{pf3}{Symplectic diagonalization of the qua\-dra\-tic term}
Using a well known result on
the symplectic diagonalization of qua\-dra\-tic
Hamiltonians, we can find a  real--analytic, symplectic transformation
\beq{small3}
(\tJ,\tpsi,\tv,\tu)\mapsto
(J',\psi',v',u')\ ,\quad {\rm with}\ \ \  J'=\tJ\quad
\mbox{and $\e^{b_2+b_3}$-close-to-the-identity, }
\eeq
such that
the quadratic part of $\tilde g$ becomes, simply,
$\sum_{i=1}^N \tilde{\O}_i (\tJ)\,(\tu_j^2+\tv_j^2)$.
Whence, the new Hamiltonian becomes 
(compare formula (2.22) of [BCV03])
\beq{hamKAM}
\widetilde\cH:=h_0(\tJ)+ \sum_{i=1}^N\tilde{\O}_i
(\tJ)\,(\tu_i^2+\tv_i^2)+
\tilde{g}_0 (\tJ,\tv,\tu)+ \tilde f_0(\tJ,\tpsi,\tv,\tu)\,,
\eeq
where
\beq{h0***}
h_0(\tJ):= h(\tJ)+ \e\tilde g(\tJ,0,0)\ ,
\eeq
$\tilde g_0$, $\tilde f_0$, $\tilde\O_j$ are real-analytic
and
\begin{equation}\label{ordine3}
|\tilde g_0
(\tJ,\tv,\tu)|\leq \const \,\e\,|(\tv,\tu)|^3\ ,\qquad |\tilde \O|\le \const \e\
,\qquad \|\tilde f\|_{{\rm r}/8,{s}/8,{\r}/{8}}\le \const \e^3
\,.\end{equation}
Finally, because of \equ{hold},
\beqa{eigenst**}
\inf |\tO_i|\geq\const\, \e>0\ ,\qquad \inf
|\tO_2-\tO_1|\geq\const\,\e>0
\,.
\eeqa

\Giu\giu\giu\giu

\tool{pf4}{Applying KAM Theory}We rewrite, now, the Hamiltonian
$\widetilde\cH$ in
\equ{hamKAM} in a  form suitable for  applying
(elliptic) KAM theory.
%%%%%
% (compare, e.g., Appendix~\ref{kam:sec}).
%%%%%
Introducing translated variables $y:=\tilde J-\pgot$ and complex
variables $z,\bar z$, we define
\beq{Hdritta0}
H=\widetilde\cH \Big(
\pgot+y,\psi,\frac{z+\bar z}{\sqrt 2},\frac{z-\bar z}{ {\rm i}\sqrt 2}
\Big)\ ,
\eeq
here $\pgot$ is regarded as a parameter and 
the symplectic form is $\sum_{j=1}^N
dy_j\wedge d\psi_j+{\rm i}
\sum_{j=1}^N dz_j\wedge d\bar{z}_j
$ with ${\rm i}:=\sqrt{- 1}$. The Hamiltonian $H$ is then seen to have the form
$$H={\cal N}+P$$
%(see\equ{HN***})
with
%\footnote{See formulae (2.27) and (2.28) of [BCV03].}
\beq{normalf}
{\cal N}= e+
\o\cdot y +
\sum_{j=1}^N \O_j  z_j\bar z_j\ ,
\qquad e:=h_0(\pgot)\ ,\quad
\o:=\dpr_{\tilde J}h_0(\pgot)\ ,
\quad \O:= \tilde\O(\pgot)
\eeq
and
$P$  a perturbation, which can be naturally be splitted in four terms:
$$P=\sum_{1\leq k\leq 4}P_k\ ,$$
with
\begin{eqnarray}\label{P1P2P3P4}
P_1 &=& h_0 (\pgot+y)-h_0(\pgot)-\dpr_{\tilde J}h_0(\pgot)\cdot y\sim
y^2\,,\nonumber\\ P_2 &=& \sum_{j=1}^n
\Big( \tilde\O_j(\pgot+y) -\tilde\O_j (\pgot) \Big)\,z_j\bar z_j
\sim y|z||\bar z|\,,\nonumber\\
P_3&=& \tilde{g}_0
\Big(
\pgot+y,\frac{z+\bar z}{\sqrt 2},\frac{z-\bar z}{ {\rm i}\sqrt 2}
\Big)
\sim \e (|z|+|\bar z|)^3
\,, \qquad \Big({\rm by}\ \ \equ{ordine3}\Big)\ , \nonumber\\
P_4 &=& \tilde{f}_0\Big(
\pgot+y,\psi,\frac{z+\bar z}{\sqrt 2},\frac{z-\bar z}{ {\rm i}\sqrt 2}
\Big)=O(\e^3)\,.
\end{eqnarray}
The parameter
 $\pgot$ 
runs over the
Diophantine set $\hat\Igot$ defined in
\equ{defIg}.
Notice that the integrable Hamiltonian ${\cal N}$
affords, for any given value of the parameter $\pgot$, the {\sl $N-$dimensional
elliptic torus}

\beq{elltor}
\{y=0\}\times  \torus^N \times\{z=\bar z=0\}\,,
\eeq
which is invariant for the Hamiltonian flow generated by 
${\cal N}$, the flow being, simply, the Diophantine translation $x\mapsto
x+\o t$, with $\o$ as in \equ{normalf}.

\nl
Since det $ \dpr_{\tilde J}^2 h_0  \neq 0$, we can use the frequencies $\o$ as
parameters rather than the actions $\pgot$. We, therefore, set
\beq{wO}
{\cal O} :=\dpr_{\tilde J}h_0\big(\hat\Igot\big)=
\left\{\,
\o=\dpr_{\tilde J}h_0 (\pgot)\ : \  \pgot\in \hat\Igot\,
\right\}.
\eeq
Notice that, by\equ{defIg}, we have
\beq{measIwO}
{\rm meas}\Big(\dpr_{\tilde J}h_0(\Igot)\setminus{\cal O}
\Big)\leq {\rm const}\ \e^{b_1} \ .
\eeq
Now, if we put 
$ \pgot = \pgot ( \o ) := (\dpr_{\tilde J}h_0)^{-1} ( \o ) $ in
\equ{Hdritta0}, we can rewrite the  $(N+1)$--body Hamiltonian  in the form
\begin{equation}\label{Hdritta}
H( y, \psi ,z, \overline z; \o ) :=
{\cal N} (y, z, \bz; \o ) +  P (y,\psi , z, \bz; \o)
\end{equation}
where
\beq{Hdritta'}
 {\cal N}(y,z,\overline z;\o) :=
e(\o ) + \o \cdot y + \sum_{j=1}^N \O_j(\o)  z_j\bar z_j\
,
\qquad e ( \o ) :=  h_0 ( \pgot (\o ) )  \ ,
\qquad \O ( \o ) :=  {\widetilde{\O}} ( \pgot ( \o ) ) \
\eeq
and the perturbation
$ P (y,\psi , z, \bz; \o) $ is obtained by replacing $ \pgot $ with
$ \pgot ( \o )$ in \equ{P1P2P3P4}.
Recalling \equ{rsrho}
the Hamiltonian $ H $ in \equ{Hdritta} is real--analytic in
\beq{domD}
(y,\psi ,z,\bz; \o ) \in
\Dgot_{\rgot^2,\sgot,\rgot,\dgot}:= D^N_{\rgot^2} \times \torus^N_{\sgot}
\times D^{2N}_{\rgot} \times  {\cal O}^N_{\dgot }
\eeq
with
\begin{equation}\label{dominio1}
\rgot:={\rm const\,} \e^{3/4}\ ,\qquad
\sgot:={\rm const}\   ,
\qquad  \dgot:= {\rm const}\, \sqrt{\e}\ .
\end{equation}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\noi
We recall, now, 
a well known  KAM result concerning the persistence of lower dimensional
elliptic tori for nearly--integrable Hamiltonian systems ([M65], [E88], [K88]).
The version we present here is, essentially, a reformulation of 
P\"oschel's Theorem in [P89] (compare, also, with Theorem 5.1 of [BBV03]).


\begin{theorem}\label{pro:KAM} %{\rm {\bf (P\"oschel [P\"o1])}}
%Fix $ \tau > n - 1 $.
Let $H$ have the form  in \equ{Hdritta}, \equ{Hdritta'} and let it be real--analytic on
a domain
$\Dgot_{\rgot^2,\sgot,\rgot,\dgot}$ of the form  \equ{domD} for some 
$\rgot$, $\sgot$ and $\dgot$ positive.
Assume that
\beq{eMMe}
\sup_{\o \in {\cal O}_{\dgot} } |
\partial_\o \O ( \o ) | \leq \frac{1}{4}
\eeq
and that the non-resonance (or Melnikov) condition
\beq{nonresKAM}
| \O(\o)
\cdot  k | \geq {{\gamma_0}} \ ,\qquad \forall\ 1\leq |k|\leq 2, \ k \in
\integer^N, \qquad \forall\ \o \in {\cal O},
\eeq
is satisfied for some
$ {\gamma_0} > 0 $. Then, if
$\dgot \geq \gamma_0$ and
 $ P $ is sufficiently
small, i.e., 
\beq{KAMcond} {\tnor} P {\tnor}_{\rgot,{\sgot},{\dgot}}
:=
\sup_{\o\in {\cal O}_{\dgot}} \| P(\, \cdot\, ;\o)\|_{{\rgot}^2,\sgot,\rgot}
\leq {\rm const\,} {\gamma_0\,} \rgot^2\ ,
\eeq
then
 there exist
a normal form ${\cal N}_*:=e_*(\o) +\o\cdot y_*+\O_*(\o) z_*\bz_*$,
a Cantor set ${\cal O}({\gamma_0})\subset{\cal O}$
with
\beq{misuraomegagamma0}
{\rm meas} \Big( {\cal O}\setminus {\cal O}({\gamma_0}) \Big)\leq
{\rm const} \, \gamma_0\ ,
\eeq
%on which
%\beq{nonresKAM2}
%| \o \cdot \ell + \O_*(\o) \cdot h | \geq
%\frac{{\gamma_0}}{2(1+|\ell|^\t)}\ ,\quad \forall\ell\in\integer^n\ ,\
%|h|\leq 2\ ,\   |\ell|+|h|\neq 0\ ,\  \o\in{\cal O}({\gamma_0})\ ,
%\eeq
and a transformation
\beqano
\cF:\,
D^N_{\rgot^2/4} \times \torus^N_{\sgot/2} \times D^{2N}_{\rgot/2}
 \times {\cal O}(\gamma_0) &\longrightarrow&
D^N_{\rgot^2} \times \torus^N_{\sgot} \times D^{2N}_{\rgot}
 \times {\cal O}_{\dgot}\\
(y_*,\psi_*, z_* ,\bar z_*;\o)
&\longmapsto&
(y,\psi, z, \bar z;\o)
\eeqano
real--analytic and symplectic
for each $ \o $ and Whitney smooth in $ \o $, such that
\beq{eqKAM}
H\circ\cF={\cal N}_*+R_* \qquad with \quad
\partial_{y_*}^j
\partial_{z_*}^h
\partial_{\bar{z}_*}^k
R_*=0\ ,\quad  if\ \ \   2|j|+|h+k|\leq 2\ .
\eeq
In particular, for each $\o \in {\cal O}(\g_0)$, the torus $\{y_*=0\}\times
\torus^N\times \{z_*=\bar z_*=0\}$ is an $N$--dimensional, linearly elliptic, 
invariant torus run by the flow $\psi_*\to\psi_*+\o t$.
Finally
\beq{perideKAM}
| y_*-y |\ ,\  \ \rgot | z_*-z |\ , \ \ \rgot | \bar{z}_*-\bar{z} |
\leq \const \frac{{\tnor} P {\tnor}_{\rgot,{\sgot},{\dgot}}}{{\gamma_0}}\
.
\eeq
\end{theorem}

%By Theorem \ref{KAMPOESCHL} the Hamiltonian system generated
%by $ H $ possesses a family of $n$-dimensional elliptic invariant tori
%$ {\cal T} := $ $ \{ (y, \psi, z) =$ $ ( Y (0,\psi_*, 0,0), \  \psi_*, \
%z ( \psi_*, 0,0 ))\} $
%traveled with frequencies $ \o$,
%for each frequency vector in the Cantor set $ {\cal O}_{\gamma_0} $.


\noi
In this section we have shown that the many--body Hamiltonian
\equ{Hamtb*} (under the hypotheses spelled out at the beginning the section) has
indeed the form assumed in the KAM Theorem~\ref{pro:KAM}. Furthermore,
by \equ{eigenst**}, the elliptic frequencies $\O_i$ verify the
Melnikov conditions  \equ{nonresKAM} with
%\footnote{See formula (2.30) of [BCV03].}
\beq{somc1}
\gamma_0 = {\rm const }\  \e\ ,
\eeq
and, by \equ{P1P2P3P4}
and \equ{dominio1},
the   perturbation $P$
verifies, for small $\e$, 
the   KAM condition \equ{KAMcond}, since
\beq{KAM}
{\tnor} P {\tnor}_{{\rgot},\sgot,\dgot}=
O(\rgot^4+\e \rgot^3 +\e^3)=O(\e^3)
\leq {\rm const\,}\gamma_0 \,{\rgot}^2=O(\e^{5/2})\ .
\eeq
Thus, the
existence of
the desired quasi-periodic orbits follows at once from Theorem~\ref{pro:KAM}. We may
summarize the final result as follows.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\thm{thm:tori}
Let $N\ge 2$ and let $\cH$ be the $(N+1)$--body problem Hamiltonian in Poincar\'e
variables defined in
 \equ{Hamtb*}. Let $m_0<\bar \m_j<4 m_0$, let $\th$, $A$,   $\d^\star$ and
$\e_0$ be as in point (ii) of Remark~\ref{rem:asympt}. Fix $0<\d<\d^\star$ and
let
$\Igot=\vec a^{-1}(A)$ where $\vec a$ is the semi--major axis map defined in
\equ{amap}. Let
$\tau>N-1$ and pick
$b_1,$
$b_2$ as in
\equ{b1b2}. Finally, let
$0<\e^\star<\e_0$ be such that \equ{KAM} holds for any $\e\le
\e^\star$ and such that all conditions on
$\e$ required for constructing   the  symplectic transformations introduced in
\S~\ref{pf1}$\div$\S~\ref{pf3} are satisfied for $\e<\e^\star$. Then, for any
$\e<\e^\star$, there exist a Cantor set
$\Igot_*\subset  {\Igot}$, with
\beq{measI*}
\meas( {\Igot}\setminus \Igot_*)\leq
{\rm const\,}\e^{b_1}\ ,
\eeq
and a
Lipschitz continuous family of tori embedding
$$
\phi\, :\,
(\vartheta,\pgot)\in \torus^N\times\Igot_*
\mapsto \Big( \,
\L(\vartheta; \pgot),\,
\l(\vartheta; \pgot),\,
\eta(\vartheta; \pgot), \,
\xi(\vartheta; \pgot)\,
\Big)\in
{\Igot}
\times
\torus^N
\times B^{2N}_{\r_*}\,,
$$
with $\r_*:={\rm const\,}{\e^{b_2}}$ such that,
for any $\pgot\in \Igot_*$, $\phi(\torus^N;\pgot)$
is a real-analytic elliptic $\cH$-in\-va\-riant torus,
on which the $\cH-$flow is analytically conjugated
to the linear flow $\vartheta\to \vartheta+ \o_* t$, $\o_*$ being
$(\g,\t)$--Diophantine with
$\g=O(\e^{b_1})$. Furthermore,
the following bounds hold  uniformly on $\torus^N
\times\Igot_*$:
\beqa{thesis1}
|\L (\vartheta;\pgot) -\pgot| &\leq &
{\rm const\,} \e^{\frac{1}{2}+b_2}\ ,  \\
|\eta (\vartheta;\pgot)| + |\xi (\vartheta;\pgot)| &\leq &
{\rm const\,}{\e^{b_2}}\,.\label{ecc}
\eeqa
\ethm

\giu
Theorem \ref{THM}
follows, now,  by taking (recall the definitions of $b_k$ in \equ{b1b2})
\beq{c1c2}
c_1:=b_1 \ , \qquad c_2:=b_2\ ,\qquad  c_3:=b_2+\frac{1}{2}\ .
\eeq
In particular
the statements on the
density of the set of
the osculating major semi--axes, on the bound on the
osculating
eccentricities and on
the variation of the
osculating major semi--axes,
follows  from 
\equ{measI*},
\equ{eccentricity}, \equ{ecc},
\equ{def.varie} and  \equ{thesis1}.




\Giu
\Giu
\appendix

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\renewcommand{\theequation}{A.\arabic{equation}}

\section{Poincar\'e variables for the planar $(N+1)$--body 
problem}
\label{Poincare}

We briefly recall in this appendix the classical derivation
of the
Poincar\'e variables for the planar $N$--body problem,
showing also the validity of Lemma \ref{lem:P}.
Though the methods exposed here
essentially go back to [Poi1905], we believe that this sketchy
presentation can be of use for pointing out some less known
features of this set of variables that have been
employed in this paper\footnote{For a review of the
Poincar\'e variables in the non-planar case, see, for instance, [Ch88]
and [BCV03].}.
The question of the Poincar\'e variables for the $N$--body problem
boils down to
the two--body problem case. The set of coordinates thus obtained
will then
be extended to the one claimed in Lemma \ref{lem:P},
just by taking the Poincar\'e variables given by the interaction
between the sun and any of the planets.

\tool{A.1}{Canonical variables for the two--body problem}
Consider two bodies ${\rm P_0},\,{\rm P_1}$ of masses ${\mgot_0
},\,{\mgot_1}$ and position $u^\ppz,\,u^\ppu\in\real^2$
(with respect to an inertial frame). We assume that
${\rm P_0}$ and ${\rm P_1}$
interact through gravity, with gravitational constant 1.
By Newton's laws, the equations of motion for
such two-body problem are
\beqano
\ddot{u}^{(0)} &=& {\mgot_1}\, \frac{(u^\ppu-u^\ppz) }{
|u^\ppu-u^\ppz|^3}\,,\\
\ddot{u}^{(1)} &=& {\mgot_0}\,\frac{ (u^\ppz-u^\ppu) }{
|u^\ppz-u^\ppu|^3}\,.
\eeqano
Let
\beq{masses} {\Mgot}:={\mgot_0}+{\mgot_1} \ , \quad {\mgot}:=\frac{{\mgot_0} {\mgot_1}}{{
\Mgot
}}\ , \quad x:=u^\ppu-u^\ppz\ , \quad X:={\mgot} \dot x\ . \eeq
Then, the above equations of motion become
$$
\ddot{x} =\frac{\Mgot x}{|x|^3}\,,
$$
and
the motion of the two bodies is governed by the Hamiltonian
\begin{equation}
\label{kep} \cK (X, x)=
\frac{1}{2 \mgot}|X|^2 -\frac{\mgot \Mgot}{|x|}\,,
\end{equation}
with $(X,x)\in\real^2\times\real^2$ conjugate
variables, i.e., the equations of motion are $\dot{x} = \partial_{X} \cK$,
$\dot{X}=-\partial_x \cK$.\Giu

\nl
As well known, such system is integrable and  for $\cK<0$
the orbits are ellipses.
More precisely, one has

\pro{keplero}
Fix $\L_->0>\cK_0$ and let $\L_+:= \Big( \frac{{\mgot}^3
{\Mgot}^2}{-2\cK_0}\Big)^\su2 >\L_-$. Then, there exist
$\hat\r>0$ and a real--analytic symplectic trans\-for\-ma\-tion
$$\Psi_{\rm DP}: \Big(
(\L,\eta),(\l,\xi)
\Big)\in  \Big([\L_-,\L_+]\times
[-{\hat\r},\hat\r] \Big) \times\Big(\torus\times
[-{\hat\r},\hat\r]\Big)
\mapsto (X, x) \in \Big\{|x|\ge \frac{\hat\r^2}{{ \mgot}^2 {\Mgot} }\Big\}\ ,$$
casting \equ{kep} into the integrable
Hamiltonian
$(-{\mgot}^3 {\Mgot}^2)/(2 \L^2)$.
\epro
This classical proposition is a planar version of the classical one,
due to Poincar\'e (see [Poi1905], chapter III) and
the variables $(\L,\eta,\l, \xi)$
are, usually,  called\label{pqattent}
(planar)   Poincar\'e variables.
The proof of Proposition \ref{keplero}
is particularly interesting from the physical point
of view and rests
upon the introduction of three different (famous) change of variables,
which we, now, proceed to describe briefly.

\nl
Let  $\ell$ and $g$ denote, respectively,
the mean anomaly and the argument of the perihelion.

\giu{\em Step 1.} The system is
set in {\em ``symplectic" polar variables}: namely, we consider
the symplectic map $\Psi_{\rm spc}: \Big((R,\Phi),(r,\f)\Big)\mapsto
(X,x)$  (where $r>0$ and $\f\in\torus$)
given by
\beq{spc}
\Psi_{\rm spc}:\left\{ \begin{array}{l}
x_1=r\ \cos \f\\
\\
x_2=r\ \sin \f
\end{array}
\right.\ ,\qquad
\quad
X= \pmatrix{
{ \cos\f}  %11
&
-\displaystyle\frac{\sin \f }{r }\cr       %13
&&\cr
{ \sin\f}    %21
&
\displaystyle\frac{\cos \f }{r }}                                         %33
\pmatrix{ R\cr  \cr \Phi}
\eeq
and consider the new Hamiltonian $\cK_{\rm spc}:=\cK\circ\Psi_{\rm spc}$.

\giu{\em Step 2.}
There is a symplectic map
$ \Psi_{\rm D}: \Big( (L,G),
(\ell,g)\Big)\mapsto
\Big((R,\Phi),(r,\f)\Big)$
that integrates the system: $\Psi_{\rm D}$ is obtained via the generating
function
\beq{Dvariables}
S(L,G,r,\f)= \int \sqrt{-\frac{{\mgot}^4 {\Mgot}^2}{L^2} + \frac{2 {\mgot}^2
{\Mgot}}{r} -
\frac{G^2}{r^2}  }\  dr\ + \ G\, \f \,.
\eeq
The variables $\Big( (L,G),
(\ell,g)\Big)$ are known as (planar) {\sl Delaunay variables}.
In such variables, the new Hamiltonian becomes $$\cK_{\rm D}:=
\cK_{\rm spc}\circ
\Psi_{\rm D} =-\frac{{
\mgot
}^3 {
\Mgot
}^2}{2 L^2}\,.$$
Also, if $C$ is the angular momentum
of the planet and $a$ is the major semi--axis, by
construction, one has that
$$ G=|C|\,,\quad\qquad
{\rm and}\quad\qquad
L= {
\mgot
}\sqrt{{
\Mgot
} a}\,.
$$
{\em Step 3.}
We need now to remove singularities, which appear fro samll eccentricity. To this
aim, we first introduce (planar) {\em Poincar\'e  action--angle
variables} by means of the linear symplectic transformation
$$\Psi_{\rm P_{aa}}:\Big((\L,H) ,(\l,h)\Big)\mapsto
\Big( (L,G),
(\ell,g)\Big)
$$
given by
\beq{Paa}
\Psi_{\rm P_{aa}}:\quad\left\{
\begin{array}{l}
\L = L\,,\quad H=L-G\,,\\
\l= \ell +g\,,\quad h=-g\,.
\end{array}\right.
\eeq
Then, we let
$\Psi_{\rm
P}:\Big((\L,\eta) ,(\l,\xi)
\Big)\mapsto
\Big((\L,H) ,(\l,h)\Big)$
be the symplectic map defined by
\begin{equation}\label{equa2}
\sqrt{2 H}\,\cos h = \eta\,,
\qquad \sqrt{2H}\,\sin h= \xi
\  .
\end{equation}
As Poincar\'e showed (see [Poi1905], [Ch88],
[BCV03]), the symplectic map $$\Psi_{\rm DP}: \Big(
(\L,\eta),(\l,\xi) \Big)\mapsto (X, x)$$ with
\begin{equation}\label{PHI}
\Psi_{\rm DP}:=\Psi_{\rm spc}\circ \Psi_{\rm D}\circ \Psi_{\rm
P_{aa}}\circ\Psi_{\rm P}\end{equation} is real--analytic in a
(complex) neighborhood of $$
\L\in\Big[\L_-,\L_+]\,,\qquad|\eta|,|\xi|\leq {\rm
const}\,\sqrt{\L_-},\,,\qquad\l\in\torus\,.$$ Also, the two--body
Hamiltonian, in Poincar\'e variables,  is $\cK\circ
\Psi=-\frac{{
\mgot
}^3 {
\Mgot}^2}{2 \L^2 }$.

\rem{rem:C.DP} (i) If we denote $(X,x)=\Phi_{\rm DP }\Big(
(\L,\eta,{\rm p}),(\l,\xi,{\rm q}) \Big)$ then,
\begin{equation}\label{C7bis}
X= \frac{{\mgot
}^4 {\Mgot}^2}{{\L}^3}\frac{\partial
x}{\partial\l}\, .
\end{equation}
Indeed, from the Hamilton equations
one sees that : $ \dot \l=\partial_{\L} \Big(
-\frac{{\mgot
}^3 {
\Mgot
}^2}{2 {\L}^2} \Big)=\frac{{
\mgot}^3 {\Mgot}^2}{{\L}^3}$, and $ \dot \L= \dot \xi =\dot \eta =\dot {\rm
p}=\dot {\rm q} =0$. Thus, by the chain rule, $X=\mgot\dot
x=\mgot (\partial_{\l} x)\,\dot \l= \frac{{\mgot}^4 {
\Mgot
}^2}{{\L}^3}\frac{\partial x}{\partial\l}$, proving
\equ{C7bis}.

\nl
(ii) We  collect
some useful relations among the above introduced quantities.
Let, as usual, $e$ denote the eccentricity of the Keplerian ellipse and let
$a$ denote
the major semi--axis.
Then, by construction, one sees that
\begin{eqnarray}\label{incecc}
&& \L={
\mgot
}\sqrt{{
\Mgot
}a}\,,\quad \qquad\sqrt{\xi^2 +\eta^2}=\sqrt{\L} \,e\,(1+O(e^2))\,.
\end{eqnarray}
Also, if $C$ is the angular momentum of
the system, one infers that \beqa{defh2} && |C| =
\L\,\sqrt{1-e^2}= \L (1+ O(e^2))\ . \eeqa The angular variables
$\ell$ and $g$ have the astronomical meaning of the mean anomaly
and the longitude of the perihelion, respectively (see, for
instance, [Ch88] and [BCV03] for pictures of the orbital elements
and for some geometric insight).

\nl
(iii)
A proof of the
analyticity of Poincar\'e variables will also follow
by directly inspecting the formulae given in
Lemma~\ref{lem:P}, which is proved in the coming section.
%can be also obtained by
%looking at the explicit computations in subsection \ref{41}.
%In fact
%we deduce from \equ{incecc} and
%\equ{ecosu,esinu} that $r$ is analytic in terms of
%$\L,\l,\reta,\rxi$. Since the position $x$ of the planet is given
%in polar coordinates by $(r\cos\f,r\sin\f)$ (recall \equ{spc}), we
%deduce from this, \equ{senfi} and \equ{cosfi} that the position of
%the planet can be analytically represented in terms of
%$(\L,\l,\reta,\rxi)$ for $\L$ bounded and bounded from zero,
%$\l\in\torus$ and $|\reta|,\,|\rxi|\leq{\rm const}$. From
%\equ{C7bis} it also follows that the momentum $X$ of the planet is
%an analytic function of $(\L,\l,\reta,\rxi)$, hence the above
%manipulations also give a proof of the analyticity of the map
%$\Psi_{\rm DP}$ introduced in \equ{PSI} (and defined  in \equ{PHI}). \label{pagi2}
\end{remark}

\tool{41}{Orbital elements}
We now sketch a way to explicitly represent some
quantities in terms of Poincar\'e variables.\label{anal}
This will also lead to the proof of Lemma \ref{lem:P}.

\nl
Let $u$ and $v$ denote the eccentric anomaly and the true anomaly,
respectively. By geometric considerations,
\begin{equation}\label{defu}
u=\ell+ e\sin u
\end{equation}
and\footnote{Such relations are classical and we refer to [Ch88] and [BCV03]
for a geometric interpretation of these anomalies.}
\begin{equation}\label{defv}
\cos v =\frac{\cos u-e}{1-e\cos u}\,.
\end{equation}
where
\begin{equation}\label{elle}
\ell = \l + h\,.
\eeq
Also, by \equ{equa2},
\begin{equation}\label{ccc}
H=\frac{\h^2+\x^2}{2}
\  .\end{equation}
An explicit expression taking into account
$H$, the eccentricity and
the major semi--axis is given by
\beqa{defh}
&& H=\L\,(1-\sqrt{1-e^2})= \L\ \frac{e^2}{2} (1+ O(e^2))\ ,\\
&& e(H,\L)=\sqrt{\frac{H}{\L} \,\Big(2- \frac{H}{\L}
\Big)}\,.\label{c1} \eeqa
In light of \equ{defv},
\begin{equation}\label{v}
\sin v=\frac{\sqrt{1-e^2}\sin u}{1-e\cos u}\,.
\end{equation}
By means of \equ{defu}, we have
\begin{eqnarray*}
u-\ell
=e\,\sin (u-\ell+\ell)= e\,\cos\ell\,\sin (u-\ell) +e\,\sin\ell\,\cos (u-\ell)\,.
\end{eqnarray*}
Thus, in the notation of Lemma \ref{lem:P},
if $G_0$ is implicitely defined by
$$ G_0(x,y)= x\sin G_0(x,y) +y\cos G_0(x,y)\,,$$
with $G_0(0,0)=0$,
we have that $G_0$ is real--analytic,
$G_0(x,y)=y+xy+O_3(x,y)$ and
\begin{equation}\label{u-ell} u-\ell = G_0 (e\cos\ell,
e\sin\ell)\,.\end{equation}
Therefore, we deduce from \equ{u-ell} and (\ref{elle}) that
\begin{equation}\label{u}
u = \l+h+G_0( e\cos h\cos \l-e\sin h\sin\l,\, e\sin h\cos \l+e\cos
h\sin\l)\,.
\end{equation}
Moreover, denoting
\beq{retarxi}
\reta= \eta/\sqrt{\L}\ ,\quad\qquad\rxi=
\xi/\sqrt{\L}\ ,
\eeq
 we deduce from \equ{equa2} and \equ{c1} that
\begin{eqnarray}\label{esinh} e \sin h
&=&\sqrt{\frac{2H}{\L}}\cdot\sqrt{1-\frac{H}{2\L}}\sin h= \rxi\,F_1
(\reta^2+\rxi^2)\,,\end{eqnarray} where
$F_1(t)=\sqrt{1-(t/4)}$ is real--analytic for $|t|<4$
(and agrees with the one introduced in Lemma \ref{lem:P}).
Analogously,
\begin{equation}\label{ecosh} e\cos h =
\reta\,F_1 (\reta^2+\rxi^2)\,.\end{equation} Therefore,
substituting \equ{esinh} and \equ{ecosh} in \equ{u}, we can write
$G_0$ as an analytic expression of $(\reta,\rxi, \l)$: more
formally, there exists a real--analytic $(\reta,\rxi, \l)\mapsto G(\reta,\rxi, \l)$
(which  agrees with the one introduced in Lemma by \equ{esinh} and
\equ{ecosh}), so that
$$
G_0( e\cos h\cos \l-e\sin h\sin\l,\, e\sin h\cos \l-e\cos h\sin\l)=
G(\reta,\rxi, \l)\,.$$ Hence, from of \equ{u},
\begin{eqnarray}\label{ecosu,esinu}
e\cos u &=& e\cos h\cos(\l+G)-e\sin h\sin(\l+G)\,,\nonumber\\
e\sin u &=& e\sin h\cos(\l+G)+e\cos h\sin(\l+G)\,,
\end{eqnarray}
with $G=G(\reta,\rxi, \l)$.
Notice also that, from the formulae in
\equ{c1} and \equ{ccc},
$$ \frac{1-\sqrt{1-e^2}}{e^2}=F_2 (\reta^2+\rxi^2)\,,$$
for a suitable real--analytic function $F_2$ (actually, $F_2(t)=
\frac{1}{2} \Big( 1-\frac{t}{4}\Big)^{-1}$, which agrees with
the notation in Lemma \ref{lem:P}). Thus, if we set
$\f=\l+v-\ell=v-h$, recalling also \equ{defv} and \equ{v}, we
have
\begin{eqnarray}\label{senfi}
\sin \f &=& \sin v\cos h-\cos v\sin h\nonumber \\
&=& \frac{1}{1-e\cos u}[\sqrt{1-e^2}\sin u\cos h -\cos u\sin
h+e\sin
h]\nonumber\\
&=& \frac{1}{1-e\cos u}[\sin (u- h) +e\sin h-F_2 \cdot(e\sin
u)\cdot(e\cos h) ]\nonumber\\&=& \frac{1}{1-e\cos u}[\sin
(\l+e\sin u) +e\sin h-F_2 \cdot(e\sin u)\cdot(e\cos h)] \,,
\end{eqnarray}
for $F_2=F_2(\reta^2+\rxi^2)$ and analogously
\begin{equation}\label{cosfi}
\cos\f = \frac{1}{1-e\cos u}[\cos (\l+e\sin u) -e\cos
h-F_2\cdot(e\sin u)\cdot(e\sin h)]\,.
\end{equation}
Hence, from \equ{esinh}, \equ{ecosh}, \equ{ecosu,esinu},
\equ{senfi} and \equ{cosfi}, it follows that $\sin\f$ and $\cos\f$
are real--analytic functions in $\l,\reta,\rxi$, for $\l\in\torus$ and
small $\rxi,\reta$.
In particular, if $\cC$, $\cS$ and ${\cE}_{\rm s}$ are as defined in
Lemma \ref{lem:P}, we deduce from \equ{ecosu,esinu}, \equ{esinh}
and \equ{ecosh} that
\begin{equation}\label{aux1} e\sin u ={\cE}_{\rm s}\,,\end{equation}
and then from \equ{cosfi} and \equ{senfi} that
\begin{equation}\label{aux2} (1-e\cos u)\cos\f =\cC
\qquad{\rm and}\qquad (1-e\cos u)\sin\f =\cS\,.
\end{equation}
Finally,
by geometric considerations, we have
\begin{equation}\label{r}
r=a(1-e\cos
u) \,
\end{equation}
where $r$ is the distance between the planet and the sun.
Thus, the formulae in Lemma \ref{lem:P} follow at once
by \equ{aux1}, \equ{aux2}, \equ{spc} and \equ{C7bis}.

\tool{60}{Hamiltonian setting for the planar
many body problem}
Consider $(N+1)$ bodies $P_0,\,\dots\,,P_N$
of mass $m_0,\, \dots,\,m_N$, all lying in the same plane,
interacting through gravity
(with constant of gravitation 1). 
Denote by
$u^\ppi$ the position of
$P_i$ in a given inertial frame of $\real^2$, with origin in the center
of mass of the system.
By Newton's laws, we have that
\begin{equation}\label{neN}
\ddot{u}^{(i)} =\sum_{0\leq j\neq i\leq N}\frac{m_j (u^{(j)}-
u{(i)})}{|u^{(j)}-u^{(i)}
|^3}\,.
\end{equation}
Thus, if $U^{(i)}:=m_i u^{(i)}$ denotes the momentum
of $P_i$, we see that the equations of motion \equ{neN}
come from the
Hamiltonian
$$ 
\sum_{i=0}^N \frac{1}{2
m_i}|U^\ppi|^2-\sum_{0\leq i<j\leq N}
\frac{m_i m_j}{|u^\ppi-u^\ppj|},$$
where
$U=(U^{(0)},\dots, U^{(N)}
)\in \real^{2(N+1)}$
and
$u=(u^{(0)},\dots, u^{(N)})\in \real^{2(N+1)}$
are conjugate symplectic variables.

We now consider $P_0$ as the ``sun'' and
introduce canonical heliocentric variables
via the linear symplectic
transformation 
\beqa{RUU}
&&
u^\ppz=r^\ppz\ ,\quad
u^\ppi=r^\ppz+r^\ppi\ ,\nonumber\\
&&
U^\ppz=R^\ppz-R^\ppu-\dots-R^{(N)}\ ,\quad
U^\ppi=R^\ppi\ ,\quad \quad{\rm for}\quad i=1,\dots,N\ .
\eeqa
Notice that, by the conservation of the center of mass
and our choice of coordinates, $R^{(0)}=0$. Thus, the planar
many body problem is governed by the $(2N)$-degree-of-freedom
Hamiltonian
$$
\sum_{i=1}^N\Big(
\frac{m_0+m_i}{2 m_0 m_i} |R^\ppi|^2-\frac{m_0 m_i}{|r^\ppi|}
\Big)+
\sum_{1\leq i< j\leq N}^N
\Big(\frac{R^{(i)}\cdot  R^{(j)}}{m_0}-
\frac{m_i m_j}{|r^{(i)}-r^{(j)}|}\Big)\ .
$$
If $m_i=\e \mu_i$ for $i=1,\dots, N$, i.e., if the ``planets'' are
way smaller than the ``sun'',
the momenta $R^{(i)}$ 
are of order $\e$. Therefore, it is convenient to
introduce the following rescaled
symplectic variables:
\beq{Sssy}
X^\ppi= \frac{R^\ppi}{\e m_0^{5/3}}\ , \qquad
x^\ppi=\frac{r^\ppi}{ m_0^{2/3}}\ ,\qquad i=1,\dots,N.
\eeq
In such variables, after a time scale of factor $\e m_0^{7/3}$
we obtain the Hamiltonian in \equ{Kep2}.
Notice that, in that setting, the Hamiltonian $\cH^{(N)}_0$
corresponds to the sum of $N$ integrable Hamiltonians of the form 
\equ{kep},
with $\mgot$ and $\Mgot$ replaced by
${\rm m_i}$ and ${\rm M_i}$, respectively.

\tool{42}{A parity property}
We recall here the well known fact that the $\l$--average of
$\cH_1$ (as in \equ{Hamtb*})
is even in $(\eta,\xi)$. The proof of this
will be accomplished by a $180$ degree rotation of the perihelia:

\pro{dise}
Let
$$
f_1(\L,\eta,\xi)
:=\su\e
\int_{\torus^2} \cH_{1}(\L,\eta,
\l,\xi)\
d\l\,
.$$
Then, $f_1(\L,-\eta,-\xi)=f_1(\L,\eta,\xi)$.
\epro
The rescaling by $\su\e$ is made so that   $f_1$ is a (real--analytic) uniformly
bounded (by an order--one constant) function.


\proof
The eccentricity $e_i$, the semiaxis $a_i$
and the mean anomaly $\ell_i$
of the osculating ellipse of $P_i$
are
invariant under the map $(\L_i, \l_i, H_i, h_i)\mapsto
(\L_i, \l_i-\pi, H_i, h_i+\pi)$, while
the argument of the perihelion $g_i$
changes of $\pi$. Let us denote
$\vec\pi:=(\pi,\dots,\pi)\in\real^N$.
In light of the consideration above we have that the map
\beq{transformation}
(\L, \l, H, h)\mapsto
(\L, \l-\vec\pi, H, h+\vec\pi)
\eeq
leaves $|r^\ppi-r^\ppj|$ invariant, for any $i, j=1,\dots, N$.
The map in \equ{transformation} corresponds to
$$ (\L,\l,\xi,\eta)\mapsto (\L,\l-\pi,-\xi,-\eta)\,,$$
usually referred to as ``space inversion''.\qed


%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\renewcommand{\theequation}{B.\arabic{equation}}


\section{Simple eigenvalue perturbations}
\label{linearalgebra}

\begin{lemma}\label{dettr}
Let $M^\star\in\Mat(m\times m)$, $M_\star\in \Mat(k\times k)$,
$M^\sharp\in\Mat(m\times k)$. Then,
$$ \det \left({\matrix{
M^\star & M^\sharp\cr{\bf 0}_{k\times m} & M_\star }}\right)\,=\,
\det(M^\star)\det(M_\star) \,.$$
\end{lemma}

\proof Induction over $m$.\qed

\begin{lemma}\label{linalglem}
Let $M^\star\in\Mat(m\times m)$, $M_\star\in \Mat(k\times k)$,
$M^\sharp\in\Mat(m\times k)$, $M_\sharp\in\Mat(k\times m)$. Let
$$ M_\epsilon := \left({\matrix{
M^\star+O(\epsilon) & \epsilon M^\sharp+O(\epsilon^2)\cr & \cr \epsilon
M_\sharp + O(\epsilon^2) &
\epsilon M_\star + O(\epsilon^2)}}\right)\,.$$ Then:
\begin{itemize}
\item if $\bar\l\neq 0$ is a simple eigenvalue of $M^\star$,
then there exists $\bar\l_\epsilon=\bar\l+O(\epsilon)$ which is an
eigenvalue of $M_\epsilon$, provided $|\epsilon|$ is suitably
small;
\item if $\tilde\l$ is a simple eigenvalue of
$M_\star$ and $\det(M^\star)\neq 0$, then there exists
$\tilde\l_\epsilon=\tilde\l+O(\epsilon)$ so that
$\epsilon\tilde\l_\epsilon$ is an eigenvalue of $M_\epsilon$,
provided $|\epsilon|$ is suitably small.
\end{itemize}
\end{lemma}

\proof For the first claim, apply the Implicit Function Theorem to
$$ \cF_1(t,\epsilon) :=\det (M_\epsilon-t{\bf 1}_{(m+k)})\,,$$
noticing that $\cF_1(t,0)=(-1)^k t^k\det(M^\star-t{\bf 1}_m)$. For
the second claim, apply the Implicit Function Theorem to
$$ \cF_2(t,\epsilon) :=\det \left( {\matrix{ M^\star -\epsilon
t{\bf 1}_m& M^\sharp\cr\epsilon M_\sharp &  M_\star-t{\bf 1}_k }}
\,. \right)\,,$$ noticing that
$\epsilon^{k} \cF_2(t,\epsilon)= \det (M_\epsilon-\epsilon
t{\bf 1}_{(m+k)})$ and that, by Lemma \ref{dettr},
$\cF_2(t,0)=\det(M^\star)\det(M_\star-t{\bf 1}_k)$. \qed




\bigskip
\renewcommand\thesection{}
\section*{References}

\footnotesize

\noindent[A63] Arnold, V. I.: {\sl Small denominators and problems
of stability of motion in classical and celestial mechanics}, Uspehi
Mat. Nauk {\bf 18} (1963), no. 6 (114), 91--192.
\smallskip

\noindent [BBV03] Berti, M.; Biasco, L.; Valdinoci, E.: {\sl
Periodic orbits in Hamiltonian systems with elliptic invariant
tori and applications to the planetary three--body problem},
to appear in Ann. Scuola Norm. Sup. Pisa,
Cl. Sci. 2004
\footnote{A preliminary version is available on--line at {{\tt
http://www.math.utexas.edu/mp$\!\!\!\_$arc}}}.
\smallskip

\noindent [BCV03] Biasco, L.; Chierchia, L.; Valdinoci, E.: {\sl
Elliptic two--dimensional invariant tori for the planetary
three--body problem}, Arch. Rational Mech. Anal.  {\bf 170}
(2003), no. 2, 91--135.
\smallskip

\noindent[CC03]  Celletti, A.;  Chierchia, L.: {\sl KAM Stability and Celestial
Mechanics\/},  Preprint  (December 2003);

\noi
http://www.mat.uniroma3.it/users/chierchia/PREPRINTS/SJV$\!$\_03.pdf


\smallskip

\noindent[Ch88] Chenciner, A.: {\sl Int\'egration du probl\`eme de
Kepler par la m\'ethode de Hamilton--Jacobi: coordonn\'ees
``action-angles'' de Delaunay and coordonn\'ees  de Poincar\'e},
Notes scientifiques et techniques du B.D.L., {\bf S026} (1988).
\smallskip

\noi[E88] Eliasson, L.: {\sl Perturbations of stable invariant tori
for Hamiltonian systems}, Ann. Scuola Norm. Sup. Pisa, Cl. Sci.,
{\bf 15} (1988), 115--147.
\smallskip

\noindent[F02] F{\'e}joz, J.: {\sl Quasiperiodic motions in the
planar three--body problem}, J. Differential Equations {\bf 183}
(2002), no. 2, 303--341.
\smallskip

\noindent[F03] F\'ejoz, J.: {\sl D\'emonstration du ``th\'eor\`eme d'Arnold"
sur la stabilit\'e du syst\`eme plan\'etaire (d'apr\`es Michael Herman)}; preprint
available on http://www.math.jussieu.fr/$\sim$fejoz/
\smallskip

\noi [H95] Herman M.: {\sl Proof of Arnold's theorem on the existence of
maximal tori for planetary systems}, Private communication (1995).
\smallskip

\noindent[JM66] Jefferys, W. H.; Moser, J.: {\sl Quasi--periodic
solutions for the three--body problem}, Astronom. J. {\bf 71} (1966)
568--578.\smallskip

\noindent [K88] Kuksin, S. B.: {\sl Perturbation theory of conditionally
periodic solutions of infinite--dimensional Hamiltonian systems and
its applications to the Korteweg--de Vries equation}. (Russian) Mat.
Sb. (N.S.) {\bf 136 (178)} (1988), no. 3, 396--412, 431; translation
in Math. USSR--Sb. {\bf 64} (1989), no. 2, 397--413.
\smallskip

%
%\noindent[L88] Laskar, J.: {\sl Les variables de Poincar\'e et le
%developpement de la fonction perturbatrice}, Notes scientifiques et
%techniques du B.D.L., {\bf S028} (1988).
%\smallskip

\noindent[L96]
Laskar, J.: {\sl Large scale chaos and marginal stability
in the solar system}, Celestial Mechanics and Dynamical Astronomy
{\bf 64} (1996), 115--162.

\noindent[LR95] Laskar, J.; Robutel, P.: {\sl Stability of the
planetary three--body problem. I. Expansion of the planetary
Hamiltonian}, Celestial Mech. Dynam. Astronom. {\bf 62} (1995),  no.
3, 193--217.
\smallskip

\noi [M65] Melnikov, V. K.: {\sl On certain cases of conservation of
almost periodic motions with a small change of the Hamiltonian
function}. (Russian) Dokl. Akad. Nauk SSSR {\bf 165} (1965),
1245--1248.
\smallskip

\noi[N77] Nekhoroshev N. N.: {\sl An exponential estimate of the
time of stability of nearly--integrable Hamiltonian systems}, I,
Usp. Mat. Nauk. {\bf 32}  (1977), 5--66; Russ. Math. Surv. {\bf 32}
(1977), 1--65.
\smallskip

\noi[Poi1905] Poincar\'e, H.: {\sl Le{\c{c}}ons de M\'ecanique
C\'eleste}, Tome 1, Gauthier--Villars, Paris
(1905)\footnote{Available on--line at

{{\tt
http://gallica.bnf.fr/scripts/ConsultationTout.exe?E=0\&O=N095010}}}.
\smallskip
%
%\noi [P93] P\"oschel J.: {\sl Nekhoroshev estimates for quasi--convex
%Hamiltonian Systems} Math. Zeitschrift {\bf 213},    (1993), 187--216.
%\smallskip

\noindent[P89] P\"oschel, J.:
{\sl
On elliptic lower dimensional tori in Hamiltonian system},
Math. Z. {\bf 202} (1989),
559--608.
\smallskip

\noindent[P96] P\"oschel, J.: {\sl A KAM--theorem for some nonlinear
PDEs},  Ann. Scuola Norm. Sup. Pisa, Cl. Sci., {\bf 23} (1996),
119--148.
\smallskip

\noindent[R95] Robutel, P.: {\sl Stability of the planetary
three--body problem. II. KAM theory and existence of quasi--periodic
motions}, Celestial Mech. Dynam. Astronom. {\bf 62} (1995),  no. 3,
219--261.
\smallskip

%\bigskip\bigskip
%centerline{\rule{30mm}{.5pt}}



\begin{center}
Luca Biasco \& Luigi Chierchia\\
  Dipartimento di Matematica, Universit\`a  ``Roma Tre", \\
  Largo S. L. Murialdo 1,\\
  I-00146 Roma (Italy)\\
  {e-mail:} {biasco, luigi@mat.uniroma3.it}
\end{center}
\smallskip

\begin{center}
Enrico Valdinoci\\
  Dipartimento di Matematica, Universit\`a di Roma Tor Vergata,\\
  Via della Ricerca Scientifica, 1\\
  I-00133 Roma (Italy)\\
  {e-mail:} {valdinoci@mat.uniroma2.it}
\end{center}



\end{document}


============teoria della misura

We now address the non-degenracy on the principal part of the
perturbation, which will be needed towards the proof of our
results. We define $$ \cD_N (\L;m_0,\dots,m_N):= \det
\partial^2_{\eta,\xi} f_{1}\Big|_{\eta=\xi=0 }\,.$$
We have:

\pro{det non zero} $\cD_N$ is not identically zero, i.e., there
exists $(\L^\star;m_0^\star,\dots,m_N^\star)\in
\real^N\times\real^{N+1}$ for which $\cD_N
(\L^\star;m_0^\star,\dots,m_N^\star)\neq 0$.\epro

\rem{rem:measure of zeros} Since $\cD_N$ is analytic in any of its
variables, by Fubini's Theorem it follows that the
$(2N+1)$-dimensional Lebesgue measure of the set on which $\cD_N$
vanishes is zero.\end{remark}

\proof The proof is by induction on the number of the planets. The
base of the induction goes as follows. ........

We now make the inductive step. Assume, by contradiction, that
$\cD$ vanishes identically. Then, for any
$(\L;m_0,\dots,m_N)\in\real^{2N+1}$, using the form of the
principal part of the perturbation and noticing that the position
of the $i$th planet does not depend on the Poincar\'e variables of
the $j$th planet, if $i\neq j$, we have that

\begin{eqnarray*} 0 & =&
\lim_{m_N\longrightarrow 0}\frac{|\cD_N (\L;m_0,\dots,m_N)
|}{m_N^{2}} \,=\\ & =& \Big|\det\left(
\int_{\torus^N}\sum_{i=1}^{n-1}\frac{m_i}{\e^2
m_0^2}\partial^2_{\eta_N,\xi_N} (|x^\ppi-x^\ppn|^{-1})
 \,d\l\,\right)\,\Big|_{\eta=0=\xi}\Big|
\,\cdot\,\\ &
&\cdot\,|\cD_{n-1}(\L_1,\dots,\L_{n-1};m_0,\dots,m_{n-1})|\,.
\end{eqnarray*}

Here above the position $x^\ppn$ has to be consider as a function
of the Poincar\'e variables, as discussed in Section I.3. Since
$\cD_{n-1}$ does not vanish identically for inductive hypothesis,
using also Remark \ref{rem:measure of zeros}, we have that $$
\det\left(\int_{\torus^N} \sum_{i=1}^{n-1}\frac{m_i}{\e^2
m_0^2}\partial^2_{\eta_N,\xi_N} (|x^\ppi-x^\ppn|^{-1})\,d\l\,
 \right)\,\Big|_{\eta=0=\xi}=0\,,
$$ for any $(\L;m_0,\dots,m_{n-1})\in\real^{2N}$. In particular, for
$m_2=\dots=m_{n-2}=0$, we have $$\det\left(\int_{\torus^N}
\partial^2_{\eta_N,\xi_N}
(|x^\ppu-x^\ppn|^{-1})\,d\l\,
 \right)\,\Big|_{\eta=0=\xi}=0\,,
$$ for any $(\L_1,\L_N) \in\real^{2}$. Thus, .......
\qed

==================================== generalizzazione (Enrico)

Cari ragazzi,
mi pare che il caso del sistema solare completo
(con Giove e Saturno non necessariamente
dentro o fuori) venga quasi gratis da (3.29),
con una piccola ipotesi
ulteriore:
basterebbe definire, che so,

$$
\bar\rho_{i,j} := \min \{a_i/a_j, a_j/a_i\}
$$
e

$$
\bar a_{i,j} := \max \{a_i,a_j\}
$$
e notare che comunque vadano le cose

$$a_i a_j = \bar\rho_{i,j} a_{i,j}^2$$

per cui, ci sono due casi:

nel primo, se il pianetone piu' vicino a i e j e'
lo stesso k

$$\b_j - \b_i =
3\frac{\sqrt{\bar\mu_k}{4 m_0}} (
\sqrt{\bar\mu_j}
\bar\rho_{k,j}^{7/4}\bar a_{k,j}^{-3/2}-
\sqrt{\bar\mu_i}
\bar\rho_{k,i}^{7/4}\bar a_{k,i}^{-3/2}+....
)$$

che e' molto positiva o negativa, a seconda che i o j sia
piu' vicino a k.

Secondo caso: il pianetone piu' vicino a i e' j e'
diverso, allora ci puo' essere un po' di competizione
tra due termini, tipo:

$$\b_j - \b_i =
3\frac{1}{4 m_0} (
\sqrt{\bar\mu_1\bar\mu_j}
\bar\rho_{1,j}^{7/4}\bar a_{1,j}^{-3/2}-
\sqrt{\bar\mu_2\bar\mu_i}
\bar\rho_{2,i}^{7/4}\bar a_{2,i}^{-3/2}+....
)$$

e qui bisognerebbe chiedere che questo termine
dominante sia nonzero.
Non so se ve ne importa molto o se vale la pena
farlo in dettaglio, magari potremmo dire qualcosa,
decidete voi.
Ciao,
E.

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