Alternative expression for the electromagnetic Lagrangian

TL;DR

This paper revisits Livens' century-old alternative electromagnetic Lagrangian, demonstrating its equivalence to the traditional form and highlighting its simplicity and potential insights into quantum effects like the Aharonov-Bohm phenomenon.

Contribution

It introduces and validates an alternative Lagrangian formulation based on local superposition of fields, offering a more elegant and potentially insightful description of charged particle interactions.

Findings

Equivalent to traditional Lagrangian within its domain

Simpler and more elegant form for multiple particles

Provides a novel perspective on the Aharonov-Bohm effect

Abstract

We reintroduce an alternative expression for the Lagrangian density that governs the interaction of a charged particle with external electromagnetic fields, proposed by Livens about one century ago. This Lagrangian is written in terms of the local superposition of the particle fields with the applied electromagnetic fields, not in terms of the particle charge and of the electromagnetic potentials as is usual. Here we show that the total Lagrangian for a set of charged particles assumes a simple elegant form with the alternative formulation, giving an aesthetic support for it. We also show that the alternative Lagrangian is equivalent to the traditional one in their domain of validity and that it provides an interesting description of the Aharonov-Bohm effect.