Revisiting Noether's Theorem on constants of motion

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