A geometric approach to the generalized Noether theorem

TL;DR

This paper extends the Noether theorem geometrically to include scaling symmetries, providing a natural, comprehensive framework applicable to dissipative systems and a broader class of symmetries.

Contribution

It introduces a geometric extension of the generalized Noether theorem that encompasses a wider class of symmetries and applies to dissipative systems, with a direct proof and inverse theorem.

Findings

Provides a natural geometric framework for generalized Noether theorem

Automatically yields an inverse Noether theorem

Applicable to dissipative systems and broader symmetries

Abstract

We provide a geometric extension of the generalized Noether theorem for scaling symmetries recently presented in \cite{zhang2020generalized}. Our version of the generalized Noether theorem has several positive features: it is constructed in the most natural extension of the phase space, allowing for the symmetries to be vector fields on such manifold and for the associated invariants to be first integrals of motion; it has a direct geometrical proof, paralleling the proof of the standard phase space version of Noether's theorem; it automatically yields an inverse Noether theorem; it applies also to a large class of dissipative systems; and finally, it allows for a much larger class of symmetries than just scaling transformations which form a Lie algebra, and are thus amenable to algebraic treatments.