  We review some techniques from non-linear analysis in order to investigate
critical paths for the action functional in the calculus of variations applied
to physics. Previous attempts to analyse when these are minima ex- ist, but
mainly based on physical reasoning and only for a restricted class of models.
Our main intention in this regard is to develop precise mathematical conditions
for critical paths to be minimum solutions in a variety of situations. Our
claim is that, with a few techniques, a systematic analysis (including the
domain for which critical points are genuine minima) of non-trivial models is
possible. We present specific models arising in modern physical theories in
order to make clear the ideas here exposed.
