Inevitability of the Lagrangian and gauge potentials

TL;DR

This paper explores the fundamental role of Lagrangian and gauge potentials in Maxwell's equations, emphasizing their natural emergence in a minimal variational formulation and highlighting the significance of the minus sign in defining the Lagrangian.

Contribution

It provides a theoretical analysis showing the inevitability of Lagrangian and gauge potentials in the variational formulation of electromagnetism.

Findings

Scalar and vector potentials arise naturally as adjoint functions.

The minus sign in the Lagrangian is crucial and naturally emerges.

The formulation offers insights into the fundamental structure of Maxwell's equations.

Abstract

In seeking a minimal variational formulation of Maxwell's equations, one is led naturally to the scalar and vector potentials as "adjoint" functions in a well-defined sense and to the crucial minus sign that defines the Lagrangian.