The Herglotz variational principle for dissipative field theories

TL;DR

This paper extends the Herglotz variational principle to dissipative field theories, enabling new analysis of non-conservative systems in physics with applications to examples like the damped string and Korteweg-De Vries equation.

Contribution

It introduces a generalized Herglotz variational principle for first-order and higher-order field theories, broadening the framework for dissipative systems.

Findings

Developed a variational principle applicable to dissipative field theories.

Applied the formalism to the damped vibrating string and Korteweg-De Vries equation.

Highlighted differences between non-holonomic and vakonomic variational principles.

Abstract

In the recent years, with the incorporation of contact geometry, there has been a renewed interest in the study of dissipative or non-conservative systems in physics and other areas of applied mathematics. The equations arising when studying contact Hamiltonian systems can also be obtained via the Herglotz variational principle. The contact Lagrangian and Hamiltonian formalisms for mechanical systems has also been generalized to field theories. The main goal of this paper is to develop a generalization of the Herglotz variational principle for first-order and higher-order field theories. In order to illustrate this, we study three examples: the damped vibrating string, the Korteweg-De Vries equation, and an academic example showing that the non-holonomic and the vakonomic variational principles are not fully equivalent.