Application of Herglotz's Variational Principle to Electromagnetic Systems with Dissipation

TL;DR

This paper extends Herglotz's variational principle using contact geometry to model dissipative electromagnetic systems, including electrons in conductors and damped electromagnetic fields, deriving new equations and the Poynting theorem.

Contribution

It introduces a generalized contact formalism for electromagnetic systems with dissipation, incorporating gauge modifications and field damping effects.

Findings

Modeling of dissipative electron dynamics in conductors.

Application of contact theory to damped electromagnetic fields.

Derivation of a generalized Poynting theorem for media with dissipation.

Abstract

This work applies the contact formalism of classical mechanics and classical field theory, introduced by Herglotz and later developed in the context of contact geometry, to describe electromagnetic systems with dissipation. In particular, we study an electron in a non-perfect conductor and a variation of the cyclotron radiation. In order to apply the contact formalism to a system governed by the Lorentz force, it is necessary to generalize the classical electromagnetic gauge and add a term in the Lagrangian. We also apply the k-contact theory for classical fields to model the behaviour of electromagnetic fields themselves under external damping. In particular, we show how the theory describes the evolution of electromagnetic fields in media under some circumstances. The corresponding Poynting theorem is derived. We discuss its applicability to the Lorentz dipole model and to a highly resistive dielectric.