  In this work, we propose an Action Principle for Action-dependent Lagrangian
functions by generalizing the Herglotz variational problem to the case with
several independent variables. We obtain a necessary condition for the extremum
equivalent to the Euler-Lagrange equation and, through some examples, we show
that this generalized Action Principle enables us to construct simple and
physically meaningful Action-dependent Lagrangian functions for a wide range of
non-conservative classical and quantum systems. Furthermore, when the
dependence on the Action is removed, the traditional Action Principle for
conservative systems is recovered.
