An Action Principle for Action-dependent Lagrangians: toward an Action Principle to non-conservative systems

TL;DR

This paper introduces a generalized Action Principle for systems with Action-dependent Lagrangians, extending the Herglotz variational problem to multiple variables, enabling modeling of non-conservative classical and quantum systems.

Contribution

It generalizes the Action Principle to include Action-dependent Lagrangians with multiple variables, applicable to non-conservative systems, and recovers traditional principles for conservative cases.

Findings

Derived a necessary condition equivalent to Euler-Lagrange equations for the generalized principle.

Constructed physically meaningful Action-dependent Lagrangians for diverse non-conservative systems.

Showed the classical Action Principle is a special case when Action dependence is removed.

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.