  We show that a foliation of equilibria (a continuous family of equilibria
whose graph covers all the configuration space) in ferromagnetic models are
ground states.
  The result we prove is very general, and it applies to models with long range
interactions and many body. As an application, we consider several models of
networks of interacting particles including models of Frenkel-Kontorova type on
$\mathbb{Z}^d$ and one-dimensional quasi-periodic media.
  The result above is an analogue of several results in the calculus variations
(fields of extremals) and in PDE's. Since the models we consider are discrete
and long range, new proofs need to be given. We also note that the main
hypothesis of our result (the existence of foliations of equilibria) is the
conclusion (using KAM theory) of several recent papers. Hence, we obtain that
the KAM solutions recently established are minimizers when the interaction is
ferromagnetic (and transitive).
