Action principle for action-dependent Lagrangians toward nonconservative gravity: Accelerating universe without dark energy

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.