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

Abstract

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.

Braneworld gravity within non-conservative gravitational theory

“…According to what is shown in [3], s μ shall disappear during the extremisation of the action, so that it does not show up in the field equations. The coupling four-vector λ μ may be in general coordinate-dependent, although we shall concentrate in the simplest case, in which its components are constant.…”

“…In the covariant generalization of this formalism introduced in [3], the authors propose a gravitational theory given by the following extended Einstein-Hilbert Lagrangian…”

“…For instance, when considering a linear dependence on the action he has shown the appearance of a typical velocity-proportional frictional term in the corresponding equations of motion derived from such Lagrangian. Almost ninety years separated the pioneer Herglotz's contribution from a covariant extension of his formalism, which was just recently accomplished by Lazo et al [3]. From this covariant formulation the authors constructed a new theory of gravity consisting of a set of modified field equations along a e-mail: julio.fabris@cosmo-ufes.org b e-mail: trpcarames@gmail.com c e-mail: hoff@feg.unesp.br with a non-conservation for the energy-momentum tensor.…”

Noether theorem for action-dependent Lagrangian functions: conservation laws for non-conservative systems

“…Unfortunately, all these approaches either give us non-physically meaningful Lagrangian functions (in the sense that they provide non-physical relations for the momentum and Hamiltonian of the system) or make use of non-local differential operators with algebraic properties different from usual derivatives (see [14,15] for a detailed discussion). In recent works [16,17], in order to formulate an Action Principle for non-conservative systems, we take a different approach and we propose a physically meaningful Action Principle for dissipative systems by generalizing the variational problem [18,19,20] for Lagrangian density functions depending itself on an action-density field. In any physical theory, the Lagrangian function which defines the Action is constructed from the scalars of the theory, and from it, the corresponding dynamical equations can be obtained.…”

“…Only recently the Herglotz variational problem gained more interest in the literature [19,20,21,22,23,24,25,26,27] and, in particular, in a recent work [16] we formulated a covariant generalization for the Herglotz problem to construct a non-conservative gravitational theory from the Lagrangian formalism. Furthermore, by following the ideas we introduced in [16], in [17] we formulated a general Action Principle for non-conservative systems for Lagrangian density functions depending itself on an action-density field. We obtained a generalization of the Euler-Lagrange equation for this Action Principle and applied it in several classical and quantum systems.…”