Vertical extension of Noether Theorem for Scaling Symmetries

TL;DR

This paper introduces a novel vertical extension of Noether's theorem to derive constants of motion for scaling symmetries that are not symmetries of the action, with applications to Schwarzian mechanics and KdV equations.

Contribution

It develops a new approach extending Noether's theorem to scaling symmetries not preserving the action, including inverse theorem analysis and practical examples.

Findings

Constructed constants of motion for non-Lagrangian scaling symmetries.

Extended Noether's theorem via a vertical space approach.

Applied method to Schwarzian mechanics and KdV equations.

Abstract

The aim of this paper is to present a new approach to construct constants of motion associated with scaling symmetries of dynamical systems. Scaling maps could be symmetries of the equations of motion but not of its associated Lagrangian action. We have constructed a Noether inspired theorem in a vertical extended space that can be used to obtain constants of motion for these symmetries. Noether theorem can be obtained as a particular case of our construction. To illustrate how the procedure works, we present two interesting examples, a) the Schwarzian Mechanics based on Schwarzian derivative operator and b) the Korteweg-de Vries (KdV) non linear partial differential equation in the context of the asymptotic dynamics of General Relativity on AdS$_3$. We also study the inverse of Noether theorem for scaling symmetries and show how we can construct and identify the generator of the scaling transformation, and how it works for the vertical extended constant of motion that we are able to construct. We find an interesting contribution to the symmetry associated with the fact that the scaling symmetry is not a Noether symmetry of the action. Finally, we have contrasted our results with recent analysis and previous attempts to find constants of motion associated with these beautiful scaling laws.