Invariance of the Noether charge

TL;DR

This paper highlights the often-overlooked invariance property of the Noether charge under its symmetry transformation, providing a proof that serves as a valuable educational exercise in Lagrangian formalism.

Contribution

It presents a detailed proof of the invariance of the Noether charge, a property rarely discussed in standard textbooks and articles.

Findings

Noether charge is invariant under its symmetry transformation

The proof enhances understanding of symmetry properties in classical mechanics and field theory

Educational value for students learning Lagrangian formalism

Abstract

Surprisingly, an interesting property of the Noether charge that it is by itself invariant under the corresponding symmetry transformation is never discussed in quantum field theory or classical mechanics textbooks we have checked. This property is also almost never mentioned in articles devoted to Noether's theorem. Nevertheless, to prove this property in the context of Lagrangian formalism is not quite trivial and the proof, outlined in this article, can constitute an useful and interesting exercise for students.