A geometric derivation of Noether's theorem

TL;DR

This paper presents a straightforward geometric derivation of Noether's theorem using differential forms, simplifying the understanding of the connection between symmetries and conserved quantities in mechanics.

Contribution

It introduces a novel, geometric approach to derive Noether's theorem that avoids common complexities faced by students.

Findings

Provides a clear geometric derivation of Noether's theorem

Simplifies the understanding of symmetries and conservation laws

Uses differential forms to connect symmetries with conserved quantities

Abstract

Noether's theorem is a cornerstone of analytical mechanics, making the link between symmetries and conserved quantities. In this article, I propose a simple, geometric derivation of this theorem that circumvents the usual difficulties that a student of this field usually encounters. The derivation is based on the direct use of the differential form $pdq -Hdt$, where $p$ is the momentum and $H$ the Hamiltonian, integrated over a simple curve.