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\vspace*{45mm}
{\Large\bf
\noindent Effective Stability for Periodically
\bigskip
\noindent Perturbed Hamiltonian Systems
}
\vspace*{15mm}
\noindent\hspace*{28mm} \`Angel Jorba,$^{1}$ and Carles Sim\'o$^{2}$
\bigskip
\noindent\hspace*{26mm}$^{1}$Departament de Matem\`atica Aplicada I, ETSEIB
\noindent\hspace*{28mm}Universitat Polit\`ecnica de Catalunya
\noindent\hspace*{28mm}Diagonal 647, 08028 Barcelona (Spain)
\noindent\hspace*{26mm}$^{2}$Departament de Matem\`atica Aplicada i An\`alisi
\noindent\hspace*{28mm}Universitat de Barcelona
\noindent\hspace*{28mm}Gran Via 585, 08007 Barcelona (Spain)
\vspace*{15mm}
\begin{abstract}
In this work we present a method to bound the diffusion
near an elliptic equilibrium point of a periodically time-dependent
Hamiltonian system. The method is based on the computation of the normal
form (up to a certain degree) of that Hamiltonian, in order to obtain
an adequate number of (approximate) first integrals of the motion.
Then, bounding the variation of those integrals with respect to time
provides estimates of the diffusion of the motion.
The example used to illustrate the method is the Elliptic Spatial
Restricted Three Body Problem, in a neighbourhood of the points
$L_{4,5}$. The mass parameter and the eccentricity are the ones
corresponding to the Sun-Jupiter case.
\end{abstract}
\section{Introduction}
The study of the nonlinear stability of an elliptic equilibrium
point of a Hamiltonian system is a classical and difficult topic.
There are mainly two kind of results concerning this: results of KAM
type (perpetual stability on a Cantor set of initial conditions) and
results of Nekhoroshev type (stability for an exponentially long
time span, on an open set of initial conditions). A survey of both kind
of methods can be found in Arnol'd.$^1$
In this work we are going to focus on the results of Nekhoroshev type.
Our purpose will be to bound the diffusion of the motion near an elliptic
equilibrium point of a Hamiltonian system. The kind of methods we are
going to use is very similar to the ones used by Giorgilli et al.$^2$
and Sim\'o$^3$ for autonomous Hamiltonians.
Let us recall the definition of effective stability. Without loss
of generality, let us assume that the elliptic equilibrium point is
located at the origin. Let $B_{\rho}$ a ball of radius $\rho$ centered
at the origin. Let $\eta$ and $T$ be positive numbers, with $\eta<1$.
\begin{defi}
The origin is $(\eta,T)$-stable iff for all initial condition $z_0$
in $B_{\eta\rho}$, the corresponding solution is in the ball
$B_\rho$ during a time span of lenght $T$.
\end{defi}
The values of $\eta$ and $T$ depend on the application we are
interested in. For the Sun-Jupiter case, the usual values are
$\eta=0.9$ (we allow a diffusion of 10\%) and $T$ of the order of
the age of the Solar system.
In the next sections we are going to see an adapted version of the method
used by Sim\'o,$^3$ in order to bound the diffusion (near an elliptic
equilibrium point) of a time-periodic Hamiltonian system. For shortness,
the methodology will be directly presented on a concrete example. The
one used is the study of the nonlinear stability of
the $L_5$ point of the Elliptic Spatial RTBP (from now on, ESRTBP),
taking as eccentricity and mass parameter the ones
corresponding to the Sun-Jupiter case, that is, $e=0.048498458$ and
$\mu=0.95387536\times 10^{-3}$.
\section{Expansion of the Hamiltonian}
The first step of the process is to obtain a power expansion of the
Hamiltonian around the equilibrium point. The way to obtain that
expansion depends strongly on the form of the Hamiltonian we are dealing
with. In our example, we have used a recurrence based on the one of the
Legendre polynomials.
Now, let us start from the equations of motion of the ESRTBP.
They are (see Szebehely$^4$)
\begin{eqnarray*}
x^{\prime\prime}-2y^{\prime} & = & \frac{\partial\omega}{\partial x}, \\
y^{\prime\prime}+2x^{\prime} & = & \frac{\partial\omega}{\partial y}, \\
z^{\prime\prime}+z^{\prime} & = & \frac{\partial\omega}{\partial z},
\end{eqnarray*}
where
$$
\omega=\frac{\Omega}{1+e\cos f},\;\;\;\;
\Omega=\frac{1}{2}(x^2+y^2+z^2)+\frac{1-\mu}{r_1}+\frac{\mu}{r_2}+
\frac{\mu(1-\mu)}{2},
$$
and $r_1^2=(x-\mu)^2+y^2+z^2$, $r_2^2=(x-\mu+1)^2+y^2+z^2$. Moreover,
$e$ is the eccentricity of the two primaries and $^{\prime}$ stands for
the derivative with respect to the true anomaly $f$.
Defining the momenta $p_x=x^\prime-y$, $p_y=y^\prime+x$, $p_z=z^\prime$
and making the (canonic) change of variables $p_x\rightarrow p_x-\beta$,
$p_y\rightarrow p_y+\alpha$, $p_z\rightarrow p_z$,
$x\rightarrow x+\alpha$, $y\rightarrow y+\beta$ and $z\rightarrow z$,
where $(\alpha,\beta)$ are the $(x,y)$ coordinates of an equilibrium
point, the Hamiltonian of the system is
\begin{eqnarray}
H & = & p_f+\frac{1}{2}(p_x^2+p_y^2+p_z^2)+\frac{1}{2}(x^2+y^2+z^2)+
yp_x-xp_y- \nonumber \\
& & -\frac{1}{1+e\cos f}\left[\frac{1}{2}(x^2+y^2+z^2)+\alpha x+\beta y+
\frac{1-\mu}{r_{PS}}+\frac{\mu}{r_{PJ}}\right], \label{eq:he1}
\end{eqnarray}
where $r_{PS}^2=(x-x_S)^2+(y-y_S)^2+z^2$,
$r_{PJ}^2=(x-x_J)^2+(y-y_J)^2+z^2$, being $(x_S,y_S)=(\mu-\alpha,-\beta)$,
$(x_J,y_J)=(\mu-1-\alpha,-\beta)$. The term $p_f$ is the momentum
corresponding to $f$, and has been added to obtain an autonomous
Hamiltonian.
As we are interested in the points
$L_{4,5}$ we have $(\alpha,\beta)=(\mu-1/2,\pm\sqrt{3}/2)$, that is,
$x_S=1/2$, $y_S=\mp\sqrt{3}/2$, $x_J=-x_S$, $y_J=y_S$.
>From now on, we are going to focus on the $L_5$ case (the same results
will hold for $L_4$, due to the symmetry): $x_S=1/2$, $y_S=\sqrt{3}/2$.
Our purpose now is to expand this Hamiltonian in a power series of
$x$, $p_x$, $y$, $p_y$, $z$ and $p_z$:
$$
H(x,p_x,y,p_y,z,p_z,f,p_f)=p_f+\sum_{j\geq 2} H_j(x,p_x,y,p_y,z,p_z,f),
$$
where $H_j$ is an homogeneouos polynomial of degree $j$, whose
coefficients are Fourier series with respect to $f$.
To reduce the global computational effort, we will perform the
change of variables that puts $H_2$ in normal form, at the same time
that the Hamiltonian is expanded. For this reason, the first point
we are going to deal with is the normal form of $H_2$.
\subsection{Normal Form of $H_2$}
This corresponds to a linear differential equation with periodic
coefficients, that can be reduced to constant coefficients by means of
the classical Floquet Theorem. We have obtained the change of variables
numerically, and we have performed a Fourier analysis in order to
obtain an (approximate) analytical expression for the change. At
the same time, we want that $H_2$ be of the form:
$$
H_2=\frac{\omega_1}{2}(x^2+p_x^2)+\frac{\omega_2}{2}(y^2+p_y^2)+
\frac{\omega_3}{2}(z^2+p_z^2),
$$
in order to simplify the next steps. This is done by a linear
change of variables, that can be composed with the Floquet one
to obtain a single change making both things.
Finally, in order to obtain an adequate expression for the calculus
of the normal form it is convenient to complexify the
expansion. The reason is that, in suitable complex coordinates,
$H_2$ takes the form
$$
H_2=\sum_{j=1}^3 \sqrt{-1}\omega_j q_j p_j,
$$
that will be very useful (it will save time, memory and the process will
not be ill conditioned) during the computations.
For that purpose we perform the change of variables
\begin{eqnarray}
x=\frac{q_1+\sqrt{-1}p_1}{\sqrt{2}},\;\;\;\;\;\;\;
p_x=\frac{\sqrt{-1}q_1+p_1}{\sqrt{2}}, \label{eq:comp}
\end{eqnarray}
and similar expressions for $q_2$, $p_2$, $q_3$ and $p_3$.
The composition of this change with the previous ones provides
the final change we were looking for.
\subsection{Expansion}
In order to expand (\ref{eq:he1}), we are going to focus on the
expansion of $r_{PS}^{-1}$ ($r_{PJ}^{-1}$ is obtained with the same
procedure). It is known that
$$
\frac{1}{r_{PS}}=\frac{1}{\sqrt{1-2\rho\cos\psi+\rho^2}}=
\sum_{n=0}^{\infty}\rho^nP_n(\cos\psi),
$$
where $\psi$ is the angle between $(x_S,y_S,0)$ and $(x,y,z)$,
$\rho^2=x^2+y^2+z^2$ and $P_n$ is the Legendre polynomial of degree
$n$. Let us define $A_n$ as $\rho^nP_n(\cos\psi)$ (note that $A_n$ is
an homogeneous polynomial of degree $n$). Then, from the
recurrence of the Legendre polynomials is not difficult to obtain
a recurrence for the $A_n$:
\begin{equation}
A_{n+1}=\frac{2n+1}{n+1}(xx_S+yy_S)A_n-\frac{n}{n+1}(x^2+y^2+z^2)A_{n-1},
\label{eq:recu}
\end{equation}
being $A_0=1$ and $A_1=xx_S+yy_S$.
We have used this recurrence to obtain the expansion of the Hamiltonian,
using for $x$ and $y$ the expressions provided by the final
``Floquet" change.
Finally, the remaining terms of (\ref{eq:he1}) are computed directly.
\section{Normal form}\label{sec:nf}
In what follows, $k$ will be a multiindex, splitted
as $k=(k^1,k^2)$, where $k^1$ and $k^2$ correspond to positions and
momenta respectively.
The computation of the normal form is based on the following
proposition:
\begin{prop}
Let us consider the Hamiltonian
$$
H=p_f+H_2(q,p)+\sum_{i=3}^{r-1} H_i(q,p)+H_r(q,p,f)+
H_{r+1}(q,p,f)+\cdots,
$$
where $r>2$, $H_2(q,p)=\sum\omega_i q_ip_i$ ($\omega_i\in\mbox{\ccn C}$),
$H_i(q,p)=\sum_{|k|=i} h_i^k q^{k^1}p^{k^2}$,
$H_r(q,p,f)=\sum_{|k|=r} h_r^k(f) q^{k^1} p^{k^2}$ and $h_r^k(f)=\sum_j
h_{r,j}^k \exp{(jf\sqrt{-1})}$. Let us define $G_r=G_r(q,p,f)=
\sum_{|k|=r} g_r^k(f) q^{k^1} p^{k^2}$ as follows:
\begin{enumerate}
\item if $k^1\neq k^2$:
$$
g_r^k (f)=\frac{c_k-h_{r,0}^k}{<\omega,k^2-k^1>}+
\sum_{j\neq 0}\frac{h_{r,j}^k}{j\sqrt{-1}-<\omega,k^2-k^1>}
\exp{(jf\sqrt{-1})}.
$$
\item if $k^1=k^2$:
$$
g_r^k (f)=\sum_{j\neq 0}\frac{h_{r,j}^k}{j\sqrt{-1}}
\exp{(jf\sqrt{-1})}.
$$
\end{enumerate}
Then, the new Hamiltonian $H^{\prime}$ obtained from $H$ by means of the
change of variables given by the generating function $G_r$,
$$
H^{\prime} = H+\{H,G_r\}+\frac{1}{2!}\{\{H,G_r\},G_r\}+\cdots,
$$
satisfies that
$$
H^{\prime} = p_f+H_2(q,p)+\sum_{i=3}^{r-1} H_i(q,p)+
H_r^{\prime}(q,p)+H_{r+1}^{\prime}(q,p,f)+\cdots,
$$
where $H_r^{\prime}(q,p)=\sum (h^{\prime})_r^k q^{k^1}p^{k^2}$ and
$$
(h^{\prime})_r^k=\left\{
\begin{array}{ll}
c_k & \mbox{if $k^1\neq k^2$} \\
h_{r,0}^k & \mbox{if $k^1=k^2$}
\end{array}\right.
$$
\end{prop}
\noindent {\bf Remark:} The value $c_k$ that appears in the function
$G$ can be selected according to several criteria. In our case, as we
want to obtain an integrable normal form, we have chosen that value
equal to 0. It is also possible to chose $c_k=h_{r,0}^k$ for the
``quasiresonant" values of $k$, to
alleviate the effect of the small divisors, but what we obtain in
this case is a (in general non-integrable) seminormal form.
\subsection{The resonant case}
In the elliptical problem, we have a (exact) resonance between
the frequency of the true anomaly and the frequency of the vertical
mode of the RTBP. Due to that resonance, it is not possible to
obtain an autonomous normal form, because some of the divisors
$j\sqrt{-1}-<\omega,k^2-k^1>$ appearing in the generating function
are zero. In that case, we select the corresponding $g_{r,j}^k$
equal to zero.
\subsection{Realification}
The last step of the procedure is to return to real coordinates
and to introduce action variables,
in order to study the dynamics of the final Hamiltonian.
The first part is done by using the inverse of the map given in
(\ref{eq:comp}). We denote again by $x$, $p_x$, etc. the real
coordinates. The second one by using the Poincar\'e change
$x=\sqrt{2I_1}\cos\varphi_1$, $p_x=-\sqrt{2I_1}\sin\varphi_1$, and
similar for the other variables. In this way, the global transformation
preserves the real character of variables and Hamiltonian.
With these changes, we obtain a final Hamiltonian of the form:
$$
H=N+R^{(m+1)}+I_f,
$$
where $m$ is the degree of the normal form and
$$
N=N(I_1,I_2,I_3,\varphi_3-f),
$$
that depends on $\varphi_3-f$ because of the $1:1$ resonance between
the true anomaly $f$ and the angle $\varphi_3$ (corresponding to the
action $I_3$) of the vertical mode of the RTBP. Here $I_f$ is the
actual momentum conjugated to $f$.
All the previous computations have been effectively carried out up to
order 20. The Fourier coefficients which appear in all the process
with amplitude less than $10^{-15}$ have been dropped.
\subsection{An approximation to the dynamics}\label{sec:dynam}
To study the flow of this Hamiltonian, let us perform the following
change of variables
$$
\begin{array}{cccc}
I_f = J_0-J_3, & I_1 = J_1, & I_2 = J_2, & I_3 = J_3, \\
f = \psi_0, & \varphi_1 = \psi_1, & \varphi_2 = \psi_2, &
\varphi_3 = \psi_3+\psi_0,
\end{array}
$$
to obtain
$$
H=N+R^{(m+1)}+J_0,
$$
where
$$
N=N(J_1,J_2,J_3,\psi_3)=\omega_1 J_1+\omega_2 J_2+O_2(J_1,J_2,J_3),
$$
is an integrable Hamiltonian and $R^{(m+1)}=\sum_{j>m} R_j$.
More concretely, up to second order in the
$J_j$ variables, $N$ is given by
\begin{eqnarray*}
N_{(2)} & = & \omega_1 J_1+\omega_2 J_2+\alpha_{1,1} J_1^2+\alpha_{1,2}
J_1 J_2+\alpha_{2,2} J_2^2+ \\
& & +J_3[aJ_1+dJ_2+(bJ_1+eJ_2)\cos 2\psi_3+(cJ_1+fJ_2)\sin 2\psi_3]+
g J_3^2,
\end{eqnarray*}
where the numerical values obtained for the constants in the current
example are given in Table 1.
\begin{table}
\begin{center} Table 1 \end{center}
\begin{center}
\begin{tabular}{|rcrrcr|} \hline
$\omega_1$ & = & $-$.8080513430831042E$-$01 &
$\omega_2$ & = & $ $.9967588604945699E$+$00 \\
$\alpha_{1,1}$ & = & $ $.5842667892951852E$+$00 &
$\alpha_{1,2}$ & = & $-$.1551131477053751E$+$00 \\
$\alpha_{2,2}$ & = & $ $.5641749211325000E$-$02 &
$a$ & = & $ $.5439031817085827E$-$01 \\
$b$ & = & $ $.8735695861611382E$-$04 &
$c$ & = & $ $.2162237916664492E$-$03 \\
$d$ & = & $ $.5793623535537272E$-$02 &
$e$ & = & $-$.5265749230961082E$-$05 \\
$f$ & = & $-$.7753646646134309E$-$05 &
$g$ & = & $-$.1787604308798145E$-$03 \\ \hline
\end{tabular}
\end{center}
\end{table}
The dynamics described by this
Hamiltonian is very simple: as $J_1$ and $J_2$ are first integrals,
we can take $J_1=C_1$, $J_2=C_2$ ($C_{1,2}$ are constants) to obtain
an one-degree of freedom Hamiltonian. In our case this is, essentially,
a pendulum depending on the parameters $J_1$ and $J_2$. Using $N_{(2)}$
as an approximation, the fixed points are located at
$$
\cos 2\psi_3=\pm n/\sqrt{n^2+p^2},\;\;\;
\sin 2\psi_3=\pm p/\sqrt{n^2+p^2},\;\;\;
J_3=(m \pm \sqrt{n^2+p^2})/(-2g),
$$
where $m=aJ_1+dJ_2$, $n=bJ_1+eJ_2$ and $p=cJ_1+fJ_2$. The points with
the $+$ sign are hyperbolic and the ones with the $-$ sign are elliptic.
If $J_1=J_2=0$ the angle $\psi_3$ is moving quite slowly, with limit
frequency equal to zero when $J_3$ goes to zero. When $J_1$ and/or $J_2$
move away from zero a bifurcation occurs. This is related to the 1 to 1
resonance between the vertical mode and the frequency of the elliptic
motion. In particular the existence of normally hyperbolic tori of
dimensions 2 and 3 follows. However, this will not affect the existence
of effective stability, as we shall show.
\section{Bounding the diffusion}
To bound the diffusion speed it is enough to bound $J_1^{\prime}$,
$J_2^{\prime}$ and $N^{\prime}$, because $N$ is also an approximate
first integral of the motion. To do it, we recall that
\begin{equation}
J_j^{\prime}=\{J_j,H\}=\{J_j,R^{(m+1)}\},\;\; j=1,2, \;\;\;\;
N^{\prime}=\{N,R^{(m+1)}\}, \label{eq:dif1}
\end{equation}
and this can be bounded easily if we have bounds for $\|R_{m+1}\|$
and for the Poisson bracket.
\subsection{Norms}
The norm used to produce the bounds mentioned above is
$$
\|U_r\|=\sum_{|k|=r} \|u_r^k(f)\|=
\sum_{|k|=r}\sum_{j} |u_{r,j}^k|,
$$
where $U$ is an homogeneous polynomial of degree $r$, $u_r^k$ are its
coefficients, and $u_{r,j}^k$ are the corresponding Fourier coefficients.
This norm has two interesting properties:
\begin{enumerate}
\item Let us denote by $\|U_r\|_{B_\rho}$ the sup norm of the homogeneous
polynomial $U_r$ of degree $r$ over the ball
centered in the origin and with radius $\rho$, and for all (real)
time $f$. Then
$$
\|U_r\|_{B_\rho}\le\|U_r\|\rho^r.
$$
\item Let $U_r$ and $V_s$ be homogeneous polynomials of degree $r$ and
$s$. Then
$$
\|\{U_r,V_s\}\|\le rs\|U_r\|\|V_s\|.
$$
\end{enumerate}
Using the norms defined above it is possible to estimate the remainder
$R^{(m+1)}$ and also to bound $\|J_j^{\prime}\|_{B_{\rho}}$ and
$\|N^{\prime}\|_{B_{\rho}}$ over a ball of radius $\rho$. More
accurate results can be obtained using different norms (see Sim\'o$^3$).
\subsection{Final results}
>From (\ref{eq:dif1}), and using the norms mentioned above, it is
not difficult to obtain
$$
|J_j^{\prime}|\le\sum_{l>m}l\|R_l\|\rho^l,\;\;\;\; j=1,2,
$$
where the values $\|R_l\|$ can be estimated of the following form:
>From the recurrence (\ref{eq:recu}), we obtain
bounds for the norms of the homogeneous polynomials $H_r$ that
appear in the Hamiltonian expansion. Then, we bound
the norms of the homogeneous components of the successive Hamiltonians
obtained (using $G_3$ to $G_{20}$). In particular we get $\|R_l\|$.
We have used the recurrent relations given by the Lie transforms
to obtain the bounds up to order 800. Note that, with
this, we have bounded the diffusion with respect to the planar
variables.
Now we would like to bound the diffusion in the vertical direction.
To do that, instead of bounding $|N^{\prime}|$, let us define first
$\widetilde{N}$ as $N$ minus the part of $N$ depending only on $J_{1}$,
$J_2$ (obviously, $\widetilde{N}$ is also an approximate first integral).
Now, we bound $\widetilde{N}^{\prime}$ (as it was done with
$J_{1,2}^{\prime}$ by means of (\ref{eq:dif1})). From this bound
we need to derive
a bound for the diffusion in the vertical direction. At this point we
want to remark a difficulty in this approach: $J_3$ is not
an approximate first integral but a fast variable of the system.
The $J_3$ variable can oscillate around the elliptic equilibrium
point mentioned in Section~\ref{sec:dynam}.
We have used the following approach: to ensure that if we start at a
ball of radius $\eta\rho$ then we end, at most, at the boundary of
a ball of radius $\rho$ we should have
$$
\min_{(z_i,z_f)\in\partial B_{\eta\rho}\times\partial B_{\rho}}
|\widetilde{N}(z_f)-\widetilde{N}(z_i)|
\geq T|\widetilde{N}^{\prime}|_{B_{\rho}}.
$$
We have used $\eta=0.5$ (see later) and $T$ the approximate age
of the Solar system ($T=4.5\times 10^{9}$ years $\approx$
$2.4\times 10^{9}$ adimensional units). The maximum value of
$\rho$ satisfying the previous condition is then obtained.
Notice the following fact about this method:
even if the diffusion for $\widetilde{N}$ were exactly 0, we would have
``diffusion" in the vertical direction, due to the (possible) libration
of $J_3$. Obviously, this diffusion is not a real one. Otherwise,
this ``false" diffusion is a fast phenomenon that can be observed,
and it has a physical meaning.
To have an idea of the amount of this diffusion, we can compute from
the normal form a bound $A$ of the maximum size of the above-mentioned
libration motion. This size depends on the value $J_{3}^{*}$
for which we have the elliptic equilibrium point (see Section
\ref{sec:dynam}). It is not difficult to obtain that
$A/J_{3}^{*} < 0.27$ (this follows, essentially, from a careful
analysis of $N_{(2)}$). This is the reason why we have taken a small
value of $\eta$ (0.5 instead of something close to 1).
Finally, using $\eta=0.5$ and $T=2.4\times 10^{9}$, as stated before,
the radius of the ball of $(\eta,T)$-stability is found to be
$\rho=0.571\times 10^{-3}$.
Now we need to send that ball back to the original coordinates.
We are going to split this process in two steps: the normal form
change (the nonlinear part of the change obtained in
Section~\ref{sec:nf}) and the Floquet change.
The deformation produced by the normal form change can be bounded
from the norm of the coefficients of this change of variables.
This shows that this ball could be reduced in a factor of $5/6$,
that is, to the value $\widehat{\rho}=0.476\times 10^{-3}$.
As the Floquet change does not modify the vertical coordinates, we
are going to focus only on the planar ones. Note that the final region
produced by this change depends on the true anomaly $f$ in a periodic
way. To obtain a region of effective stability independent of $f$
we have intersected (for all $f$) all these regions. This produces
a domain with a ``banana" shape, having a minimum diameter of
$0.11\widehat{\rho}$ and a maximum one of $9.8\widehat{\rho}$. This
implies that the largest ball contained in that domain and centered
at the origin ($L_{4,5}$) has a radius of $0.52\times 10^{-4}$
adimensional units. On the other hand, in some ``good" direction,
the distance of stability is multiplied by a factor of
$9.8/0.11\approx 89$.
\subsection{Remarks}
Numerical simulations of the motion close to $L_{4,5}$ in the
same problem show that there exist points close to $L_{4,5}$ (but
outside the region mentioned before) that go away in a very short
time (see G\'omez et al.$^5$; Sim\'o$^6$). This is due to the effect of
the 1:1 resonance between the true anomaly and the vertical mode of the
RTBP (this phenomenon is now being studied (see Jorba and Sim\'o$^7$)).
This effect is more
evident when the value of $\mu$ is bigger (i.e. Earth-Moon
system), because the (angle of) splitting between the stable and
unstable manifolds of the hyperbolic tori mentioned above grows
very fast when $\mu$ increases.
There are also other regions of stability that are even larger, allowing
for large values for the inclinations (see G\'omez et al.$^5$). These
regions are the ones containing the Trojan asteroids.
\bigskip
\noindent {\bf Acknowledgements.} The authors have been partially
supported by the CICYT grant ESP91-0403.
\bigskip
{\Large\bf\noindent References}
\newcounter{bbb}
\begin{list}{\arabic{bbb}.}{\usecounter{bbb}
\setlength{\itemindent}{-3mm}
\setlength{\leftmargin}{\labelwidth}}
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\item A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and
C. Sim\'o: Effective stability for a Hamiltonian system near an
elliptic equilibrium point, with an application to the Restricted
Three Body Problem, {\em Journal of Differential Equations} 77:1
(1989).
\item C. Sim\'o: Estabilitat de sistemes Hamiltonians, Mem.
de la Real Acad. de Cienc. i Art. de Barcelona, Vol. XLVIII, no. 7
(1989).
\item V. Szebehely: ``Theory of Orbits", Academic Press (1967).
\item G. G\'omez, A. Jorba, J. Masdemont and C. Sim\'o:
{\sl Study of Poincar\'e maps for Orbits near Lagrangian Points},
ESOC contract 9711/91/D/IM(SC), Second Progress Report (1992).
\item C. Sim\'o: Stability regions for the elliptic RTBP near
the triangular points (in progress).
\item A. Jorba and C. Sim\'o: Hyperbolic tori close to the
equilateral points of the elliptic RTBP (in progress).
\end{list}
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