Revisiting Noether's Theorem on constants of motion

TL;DR

This paper revisits Noether's theorem for Lagrangian systems, emphasizing BH-invariance and nonlocal constants, and provides new derivations of classical conserved quantities like the Laplace-Runge-Lenz vector.

Contribution

It introduces a new perspective on BH-invariance, showing it is not more general than original invariance, and offers simplified formulas and novel derivations of conserved quantities.

Findings

BH-invariance can be trivialized to recover Noether's original invariance

Nonlocal constants of motion can be deduced without BH-function as a total derivative

Two derivations of the Laplace-Runge-Lenz vector using space and time change

Abstract

In this paper we revisit Noether's theorem on the constants of motion for Lagrangian mechanical systems in the ODE case, with some new perspectives on both the theoretical and the applied side. We make full use of invariance up to a divergence, or, as we call it here, Bessel-Hagen (BH) invariance. By recognizing that the Bessel-Hagen (BH) function need not be a total time derivative, we can easily deduce nonlocal constants of motion. We prove that we can always trivialize either the time change or the BH-function, so that, in particular, BH-invariance turns out not to be more general than Noether's original invariance. We also propose a version of time change that simplifies some key formulas. Applications include Lane-Emden equation, dissipative systems, homogeneous potentials and superintegrable systems. Most notably, we give two derivations of the Laplace-Runge-Lenz vector for Kepler's problem that require space and time change only, without BH invariance, one with and one without use of the Lagrange equation.