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

Abstract

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.

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

“…In this section, we first present the Action Principle introduced by us in [16,17], and after we show that the Action is gauge invariant. This gauge invariance will play a fundamental role in the generalization of the Noether Theorem.…”

“…Although the Herglotz problem was introduced in 1930, a covariant generalization of (1) for several independent variables is not direct and was proposed only recently [16,17]. For a scalar field φ(x µ ) = φ(x 1 , x 2 , · · · , x d ) defined in a domain Ω ∈ R d (d = 1, 2, 3, · · · ), the classical problem of variational calculus deals with the problem to find φ that extremizes the functional…”

“…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.…”

“…In Section 2 we review the basic notions of Herglotz variational problem and our generalization for fields [16,17]. Furthermore, we discuss the gauge invariance in our Action Principle [16,17] and we introduce a Canonical Gauge. The Noether's theorem for Lagrangian density functions depending itself on an action-density field is obtained in Section 3.…”