I’ve read the post a couple of times over, and I still don’t have an intuitive understanding of why one would subtract potential energy from kinetic, despite having done graduate work in general relativity. Yes, extremizing action comes from the stationary phase approximation of the path integral, and yes, following a path with “low potential energy” makes you arrive to the destination younger, just like moving faster does (yet arriving at the same instant as those moving slower), but first, it’s not obvious why the former is so, and second, why it would matter in non-gravitational physics, especially in classical mechanics. I would like to see an intuitive argument where the difference between kinetic and potential energies makes sense.
I’ve read the post a couple of times over, and I still don’t have an intuitive understanding of why one would subtract potential energy from kinetic, despite having done graduate work in general relativity. Yes, extremizing action comes from the stationary phase approximation of the path integral, and yes, following a path with “low potential energy” makes you arrive to the destination younger, just like moving faster does (yet arriving at the same instant as those moving slower), but first, it’s not obvious why the former is so, and second, why it would matter in non-gravitational physics, especially in classical mechanics. I would like to see an intuitive argument where the difference between kinetic and potential energies makes sense.