ask ChatGPT “from least action of Lagrangian we get equations of motion, from equation of motion’s space translation symmetry, we can arrive conservation of momentum, how to infer?”

ChatGPT’s reasoning is so beautiful and concise.

in the same vein, can you infer how symmetry of time leads to energy conservation?

in summary, stationary/least action of path (e.o.m, integral of Lagrangian) principle is an axiom observed and believed, like speed of light is constant is an axiom and believed by all physicists, this leads to the famous Euler-Lagrangian Equation, then if there are symmetry properties, be it be symmetry of space translation, rotation, time and others, the conservation of certain quantity can be inferred mathematically.