Bessel-Hagen on the extension of Noether's theorems and their application to classical electromagnetism

TL;DR

This paper analyzes Bessel-Hagen's 1921 extension of Noether's theorems, highlighting its application to symmetries in electromagnetism and its historical significance within the context of the Erlangen programme.

Contribution

It extends Noether's theorems to divergence symmetries and applies them to electromagnetism, clarifying historical and conceptual developments in symmetry and conservation laws.

Findings

Bessel-Hagen's work extends Noether's theorems to divergence symmetries.

Application to classical electromagnetism's conformal symmetries.

Historical analysis of Bessel-Hagen's contribution within the Erlangen programme.

Abstract

This work analyses the 1921 article by Erich Bessel-Hagen entitled \"Uber die Erhaltungss\"atze der Elektrodynamik ("On the conservation laws of electrodynamics"). The article is based on Noether's theorems, which were formulated by Emmy Noether in 1918 and concern consequences of symmetries of actions, including conservation laws. Bessel-Hagen firstly extends Noether's theorems to symmetries up to a divergence and next applies them to the n-body problem and to classical electromagnetism, especially to its conformal symmetries. The work explains the context in which Bessel-Hagen's article was written, the way his explanations proceeded and the significance of his results. It is argued in particular that Bessel-Hagen's article as much as Noether's are best considered as elements of Felix Klein's Erlangen programme. A close reading of Bessel-Hagen's text is also provided, as is its comparison with relevant works, including recent works by philosophers of physics Katherine Brading, Harvey Brown and Peter Holland. Such an approach helps to see where Bessel-Hagen's treatment is confused and to clarify it, as well as to estimate the advancement in our understanding of the relevant topics made within the last one hundred years.