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.

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.