  We set up a methodology for computer assisted proofs of the existence and the
KAM stability of an arbitrary periodic orbit for Hamiltonian systems. We give
two examples of application for systems with 2 and 3 degrees of freedom. The
first example verifies the existence of tiny elliptic islands inside large
chaotic domains for a quartic potential. In the 3-body problem we prove the KAM
stability of the well-known figure eight orbit and two selected orbits of the
so called family of rotating Eights. Some additional theoretical and numerical
information is also given for the dynamics of both examples.
