Nicely put, though I'd point out > It might break some usual physics intuition, It really shouldn't in light of Noether's theorem: The fact that time indep

Nicely put, though I'd point out > It might break some usual physics intuition, It really shouldn't in light of Noether's theorem: The fact that time independence of the action S gives rise to energy conservation in the first place tells us pretty clearly: If your action has time dependence, you don't have conservation of energy.