Vertical extension of Noether Theorem for Scaling Symmetries

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 AdS3. 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.

A geometric approach to the generalized Noether theorem

“…From ( 20), ( 21) and ( 23) we get that Y = X F + Y t (X h + ∂ t ) = X F + Y t X t h , while equation ( 22) can be rewritten after some algebra as X t h (F ) = − ∂h ∂S F , that is, the dissipation equation (17). Theorems 3 and 4 are the main contributions of this work.…”

“…We hope that the analysis initiated in this work will be helpful to clarify the origin and structure of scaling symmetries and their related invariants, by putting them on the same footing as other standard Noether symmetries. As further developments, we will address the comparison of our results with the "Eisenhart-Duval lift" of mechanical systems employed in [36], with the "vertical extension" of the tangent space presented in [17], with the reduction in the pre-symplectic setting described in [27], and also with the "unit-free approach" to Hamiltonian mechanics introduced…”

“…An interesting approach to solve some of the above puzzles has been presented in [17], where the authors construct a Noether theorem for scaling symmetries from the variational principle in the vertical extension of the standard phase space. The advantages of such approach are its general scope and the fact that it effectively yields invariants associated with scaling symmetries which do not depend on computing the on-shell action.…”