# A generalized Noether theorem for scaling symmetry

## TL;DR

This paper extends Noether's theorem to include scaling symmetries, deriving conserved quantities for homogeneous potentials and connecting to the virial theorem and gravitational wave models.

## Contribution

It generalizes Noether's theorem to scaling symmetries by incorporating the classical action, providing new conserved quantities and insights into homogeneous potentials.

## Key findings

- Derived conserved quantities for scaling symmetries.
- Connected the generalized theorem to the virial theorem.
- Illustrated the framework with exact gravitational wave solutions.

## Abstract

The recently discovered conserved quantity associated with Kepler rescaling is generalised by an extension of Noether's theorem which involves the classical action integral as an additional term. For a free particle the familiar Schroedinger dilations are recovered. A general pattern arises for homogeneous potentials. The associated conserved quantity allows us to derive the virial theorem. The relation to the Bargmann framework is explained and illustrated by exact plane gravitational waves.