# The Noether charges of all analytic Lagrangians associated with a scale invariant action

## TL;DR

This paper characterizes all analytic Lagrangians in classical particle mechanics that exhibit scale invariance, deriving the associated Noether charges and clarifying the conditions for scale symmetry in such systems.

## Contribution

It provides a complete classification of analytic Lagrangians with scale invariance and explicitly derives their Noether charges, expanding understanding of symmetries in classical mechanics.

## Key findings

- Identified conditions for scale invariance in Lagrangian systems.
- Derived the explicit form of all scale-invariant analytic Lagrangians.
- Computed the corresponding Noether charges for these systems.

## Abstract

Here we consider scale invariant dynamical systems within a classical particle description of Lagrangian mechanics. We begin by showing the condition under which a spatial and temporal scale transformation of such a system can lead to a symmetry by leaving the action unchanged. We then derive the form of all analytic Lagrangians that possess such a symmetry under change of scale. Finally, we write down the Noether charges of all analytic Lagrangians that are associated with a scale invariant action.

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Source: https://tomesphere.com/paper/1905.00714