Noether's theorem for finite-dimensional Lagrangian systems can be pushed past point-function first integrals: if the Bessel-Hagen (gauge) function is allowed not to be a total time derivative, every one-parameter family of trajectories yields a conserved quantity that mixes the canonical momentum term with an integral of the Lagrangian's variation over the elapsed motion. Students learn to derive these nonlocal constants using only the Hamiltonian action and the Lagrange equation, without vector fields or manifold language, and to recognize when the total derivative condition collapses them back into ordinary first integrals. The same elementary toolbox shows that space change, time change and gauge function are interchangeable, so invariance up to a divergence is no more general than Noether's original invariance.
This arxiv paper by Gianluca Gorni and Gaetano Zampieri comprehensively revisits Noether's theorem in classical mechanics, focusing on the relationship between continuous symmetries and constants of …