Jorba A., Villanueva J.
Numerical Computation of Normal Forms around some Periodic
Orbits of the Restricted Three Body Problem
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ABSTRACT. In this paper we introduce a general methodology for computing
(numerically) the normal form around a periodic orbit of an autonomous
analytic Hamiltonian system. The process follows two steps. First,
we expand the Hamiltonian in suitable coordinates around the orbit and
second, we perform a standard normal form scheme, based on the Lie
series method. This scheme is carried out up to some finite order
and, neglecting the remainder, we obtain an accurate description of
the dynamics in a (small enough) neighbourhood of the orbit. In
particular, we obtain the invariant tori that generalize the elliptic
directions of the periodic orbit. On the other hand, bounding the
remainder one obtains lower estimates for the diffusion time around
the orbit.
This procedure is applied to an elliptic periodic orbit of the spatial
Restricted Three Body Problem. The selected orbit belongs to the
Lyapunov family associated to the vertical oscillation of the
equilibrium point $L_5$. The mass parameter $\mu$ has been chosen such
that $L_5$ is unstable but the periodic orbit is still
stable. This allows to show the existence of regions of effective
stability near $L_5$ for values of $\mu$ bigger that the Routh
critical value. The computations have been done using formal
expansions with numerical coefficients.