  The goal of this paper is to present a methodology for the computation of
invariant tori in Hamiltonian systems combining flow map methods,
parameterization methods, and symplectic geometry. While flow map methods
reduce the dimension of the tori to be computed by one (avoiding Poincare
maps), parameterization methods reduce the cost of a single step of the derived
Newton-like method to be proportional to the cost of a FFT. Symplectic
properties lead to some magic cancellations that make the methods work. The
multiple shooting version of the methods are applied to the computation of
invariant tori and their invariant bundles around librational equilibrium
points of the Restricted Three Body Problem. The invariant bundles are the
first order approximations of the corresponding invariant manifolds, commonly
known as the whiskers, which are very important in the dynamical organization
and have important applications in space mission design.
