|Abstract||This paper revisits Noether's theorem on the constants of motion for Lagrangian mechanical systems in the ODE case, attempting a clarification on both the theoretical and the applied side.
Noether's variational theorem requires some form of (infinitesimal) invariance of the Lagrangian with respect to some set of transformations, and provides conserved quantities as a result. First of all, we obtain both a simpler theory and new applications by allowing transforms that are not point functions. Then we compare the three known formulations of Noether's theorem, that involve respectively (1) invariance without gauge transform, under both dependent and independent variable transformation; (2) gauge-invariance under a transformation of dependent variable; (3) gauge-invariance under transformation of both dependent and independent variable. We show that, in the case of one independent variable, all three formulations are equivalent, in the sense that any conservation law, that can be deduced with one, can also be deduced with any other. In the application sections we work out several examples following a unified general scheme and using some newly devised transformations, most notably in the derivation of the Laplace-Runge-Lenz vector for Kepler's problem.|