Constants of motion for non-differentiable quantum variational problems

In the present work, we propose an action principle for action-dependent Lagrangians by generalizing the Herglotz variational problem for several independent variables. This action principle enables us to formulate Lagrangian densities for nonconservative fields. In particular, from a Lagrangian depending linearly on the action, we obtain generalized Einstein field equations for nonconservative gravity and analyze some consequences of their solutions for cosmology and gravitational waves. We show that the nonconservative part of the field equations depends on a constant cosmological four-vector. Depending on this four-vector, the theory displays damped/amplified gravitational waves and an accelerating Universe without dark energy.

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.