An action principle for action-dependent Lagrangians: Toward an action principle to non-conservative systems

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.