Central Stable/Unstable Manifolds and the destruction of KAM tori in the
planar Hill problem

C. Sim\'o, Departament de Matem\`atica Aplicada i An\`alisi, Universitat
   de Barcelona, Barcelona, Spain

T. J. Stuchi, Instituto de F\'{\i}sica, Universidade Federal do Rio de 
   Janeiro, Rio de Janeiro, Brazil

Abstract

The classical Hill's problem is a simplified version of the restricted
three body problem (RTBP) where the distance of the two massive bodies
(say, primary for the largest one and secondary for the smallest)
is made infinity through the use of Hill¨s variables  and a limiting
procedure, so that a neighborhood of the secondary can be studied in detail.
In this way it is the zeroth--order approximation in powers of $\mu^{1/3}$.
The Levi--Civita regularization takes the Hamiltonian  into the form of two
uncoupled harmonic oscillators perturbed by the Coriolis force and the
Sun action, polynomials of degree four and six, respectively.

As it is well known the RTBP has five equilibrium points, two triangular, $L_4$
and $L_5,$ and three collinear, $L_1,\,L_2$ and $L_3.$ In the Hill's version
only two of the collinear, $L_1$ and $L_2,$ remain and they are symmetric with
respect to the secondary. The zero velocity curves ({\em zvc}) have a closed
component, implying that the motion is bounded, only up to the energy value of
these two libration points. For values of the energy larger than the one at
$L_1$ and $L_2$ the {\em zvc} are open, and persistence of confined motion in
that case is our main concern in this paper. We investigate the geometrical
behavior of the center--stable and center--unstable manifolds of the libration
points $L_1$ and $L_2$. Suitable Poincar\'e sections make apparent the relation
between these manifolds and the main periodic orbits and also the destruction of
the invariant KAM tori surrounding the secondary.