Revisiting Noether's Theorem on constants of motion

DOI

10.1080/14029251.2014.894720How to use a DOI?

Keywords

Noether; invariance; constants of motion; nonlocal constants of motion; Bessel-Hagen

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.

Copyright

© 2014 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Gianluca Gorni
AU  - Gaetano Zampieri
PY  - 2021
DA  - 2021/01/06
TI  - Revisiting Noether's Theorem on constants of motion
JO  - Journal of Nonlinear Mathematical Physics
SP  - 43
EP  - 73
VL  - 21
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2014.894720
DO  - 10.1080/14029251.2014.894720
ID  - Gorni2021
ER  -