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 non-conservative fields. In special, from a Lagrangian depending linearly on the Action, we obtain a generalized Einstein's field equations for a non-conservative gravity and analyze some consequences of their solutions to cosmology and gravitational waves. We show that the non-conservative 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.The Action Principle was introduced in its mature formulation by Euler, Hamilton and Lagrange and, since then, it has become a fundamental principle for the construction of all physical theories. In order to obtain the dynamical equations of any theory, the Lagrangian defining the Action is constructed from the scalars of the theory. In this case, the action itself is a scalar. Consequently, we might ask: what would happen if the Lagrangian itself is a function of the Action? The answer to this question can be given by the Action Principle proposed by Herglotz [1][2][3]. The Herglotz variational calculus consists in the problem of determining the path x(t) that extremize (minimize or maximize) S(b), where S(t) is a solution oḟIt is easy to note that (1) represents a family of differential equations since for each function x(t) a different differential equation arises. Therefore, S(t) is a functional. The problem reduces to the classical fundamental problem of the calculus of variations if the Lagrangian function L does not depend on S(t). In this case we havė S(t) = L(t, x(t),ẋ(t)), and by integrating we obtain the classical variational problemwhereIt is important to notice from (2) that for a given fixed function x(t) the functional S reduces to a function of the domain boundary a, b. Herglotz proved [1, 2] that a necessary condition for a path x(t) to be an extremizer of the variational problem (1) is given by the generalized Euler-Lagrange equation:In the simplest case where the dependence of the Lagrangian function on the Action is linear, the Lagrangian describes a dissipative system and, from (4), the resulting equation of motion includes the well known dissipative term proportional toẋ. It should also be noticed that in the case of the classical problem of the calculus of variation (2) one has ∂L ∂S = 0, and the differential equation (4) reduces to the classical Euler-Lagrange equation.

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.

In the present work, we formulate a generalization of the Noether Theorem for action-dependent Lagrangian functions. The Noether's theorem is one of the most important theorems for physics. It is well known that all conservation laws, e.g., conservation of energy and momentum, are directly related to the invariance of the action under a family of transformations. However, the classical Noether theorem cannot be applied to study non-conservative systems because it is not possible to formulate physically meaningful Lagrangian functions for this kind of systems in the classical calculus of variation. On the other hand, recently it was shown that an Action Principle with action-dependent Lagrangian functions provides physically meaningful Lagrangian functions for a huge variety of non-conservative systems (classical and quantum). Consequently, the generalized Noether Theorem we present enable us to investigate conservation laws of nonconservative systems. In order to illustrate the potential of application, we consider three examples of dissipative systems and we analyze the conservation laws related to spacetime transformations and internal symmetries.

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