A generalized Noether theorem for scaling symmetry

TL;DR

This work generalizes Noether's theorem to include scaling symmetries by introducing chrono-projective transformations and an action-term in the conserved charge, unifying nonrelativistic scaling with the classical action. The key result is a conserved quantity $Q = \frac{\partial L}{\partial \dot{q}_i}\delta q_i - H\delta t - \delta f - (\delta\Lambda)\int_0^t L\,d\tau$, which reproduces Schrödinger dilations for a free particle and, for homogeneous potentials with $V(\mu{\bm q})=\mu^k V({\bm q})$, yields a relation $c=a\bigl(1+\tfrac{k}{2}\bigr)$ and a virial-type theorem $\langle K\rangle = \tfrac{k}{2}\langle V\rangle$. The paper also connects this generalized symmetry to the Bargmann (Eisenhart–Duval) lift, showing how Kepler scaling and other cases arise as chrono-projective symmetries in a higher-dimensional spacetime, with concrete illustrations in exact plane gravitational waves and oscillator dynamics. These results provide a cohesive framework linking Noether symmetries, virial relations, and gravitational-wave geometry, with potential extensions to massive geodesics and field theories.

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.

Figures (3)

• Figure 1: The Eisenhart-Duval lifts to $3d$ Bargmann space of two motions of a 1d harmonic oscillator. Projected to the $(q,t)$ plane [which is here vertical] we get the familiar sinus curves related by position-alone scaling by $\lambda$. The Hamiltonian actions calculated along the trajectories, shown in the $(S \, , \, t)$ [here horizontal] plane oscillate with double frequency and are scaled by $\lambda^2$. The E-D lift of the scaling [in green], (\ref{['osciscalelift']}), the homothety carries the lifted curves into each other.
• Figure 2: The Brdička metric (\ref{['Brdmetric']}) provides us with the Bargmann description of a particle moving in a saddle potential. The trajectory combines oscillation in the attractive $q^1$ sector with exponentially escaping motion in the repulsive $q^2$ and $s$ sectors. The trajectories are carried into each other by the homothety (\ref{['homothety']}).
• Figure 3: Motion of a $2d$ oscillator unfolded to 4D Bargmann space [and dropping the non-relativistic time coordinate]. The homothety (\ref{['homothety']}) carries lifted oscillator-ellipses to lifted oscillator-ellipses.