Revisiting Noether's Theorem on constants of motion

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 partic- ular, 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, dissi- pative 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

Lagrangian dynamics by nonlocal constants of motion

“…A previous work of ours [3] revisited Noether's Theorem from different points of view, including asynchronous perturbations (or "time change") and boundary terms, this last being a nomenclature recommended by Leach [8]. In the present 2752 GIANLUCA GORNI AND GAETANO ZAMPIERI paper we focus on the extension we obtained to constants of motion of the more general form N t, q(t),q(t) + t t0 M s, q(s),q(s) ds ,…”

“…• potentials with simple symmetries in Section 2 as basic motivation, • homogeneous potentials of degree −2 in Section 3, taken from [3], • viscous fluid resistance in Section 4, taken from [4], and two in the nonvariational case:…”

“…Homogeneous potentials of degree −2. In this section we are going to review a result in our previous work [3], Section 9. Consider the variational mechanical system of a point moving in a potential field:…”

Continuous and Symmetrical Transformations in the Framework of Classical Particle Systems

“…Nevertheless, if one considers a different and none-constant time transformation, say The main difference between a conservative system and a none-conservative system, from the point of view of the Noether's theorem, is that a conservative system is symmetrical under any continuous time transformation, and this transformation results in the conservation of total energy whereas a non-conservative system is symmetrical only under a selected group of continuous time transformation. In other words, Noether's theorem takes a strong and general form when it deals 13 The derivation of this conservation law is similar to the cases studied earlier: after a set of transportations is defined explicitly, one needs to find the explicit expression for K using the definition of invariance. Once K is determined, the Noether's conserve quantity can be easily found following equation IV.22.…”

“…xt is a solution of the Lane-Emden equation, then a rescaled version of () xt in the form of () e x e t  is also a solution [13]. Therefore, it is an educated guess to try a transformation in the form of () e x e t , then, up to the first order approximation,…”