Electromagnetic force and torque derived from a Lagrangian in conjunction with the Maxwell-Lorentz equations

TL;DR

This paper explores the derivation of electromagnetic force and torque from a Lagrangian framework, discussing its applicability and limitations in classical electrodynamics.

Contribution

It analyzes the conditions under which electromagnetic force and torque can be derived from a Lagrangian in conjunction with Maxwell's equations.

Findings

Lagrangian formulations can reproduce classical electromagnetic force and torque expressions

Existence of a Lagrangian depends on the specific formulation of the electromagnetic system

Some electromagnetic systems lack a Lagrangian description

Abstract

Electromagnetic force and torque are typically derived from a stress tensor in conjunction with Maxwell's equations of classical electrodynamics. In some instances, the Principle of Least Action (built around a Lagrangian) can be used to arrive at the same mathematical expressions of force and torque as those derived from a stress tensor. This paper describes some of the underlying arguments for the existence of a Lagrangian in the case of certain simple physical systems. While some formulations of electromagnetic force and torque admit a Lagrangian, there are other formulations for which a Lagrangian may not exist.