FWIW this did make Lagrangian mechanics feel more intuitive for me. I wish it showed the derivation of the classical version as a second order approximation though.
FWIW, if I were asked before reading this why you substract the potential energy from the kinetic energy, I wouldn’t have had a quick answer—I think “it’s minimizing the overall time dilation from gravity and speed” is a really neat way to think about it.
As to why time dilation would be relevant, if you’ve read QED by Richard Feynman he has a visualization where you think of each (version of a) particle of having a clock with a hand that goes around and around, and you add up the hands of all the clocks for all the particles that took all the different paths.
In the end only the (versions of the) particle that took the paths very close to where the time taken is extremized add up to the final result, everything else cancels because tiny differences in path lead to opposite directions of the clock hand.
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.
This article comes from trying to understand A Simple Introduction to Particle Physics, “notes…intended for a student who has completed the standard undergraduate physics and mathematics courses”.
Your definition of “intuitive” is unlikely to match that of most participants here.
FWIW this did make Lagrangian mechanics feel more intuitive for me. I wish it showed the derivation of the classical version as a second order approximation though.
FWIW, if I were asked before reading this why you substract the potential energy from the kinetic energy, I wouldn’t have had a quick answer—I think “it’s minimizing the overall time dilation from gravity and speed” is a really neat way to think about it.
As to why time dilation would be relevant, if you’ve read QED by Richard Feynman he has a visualization where you think of each (version of a) particle of having a clock with a hand that goes around and around, and you add up the hands of all the clocks for all the particles that took all the different paths.
In the end only the (versions of the) particle that took the paths very close to where the time taken is extremized add up to the final result, everything else cancels because tiny differences in path lead to opposite directions of the clock hand.
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.
Well, there’s always the classical derivation. :)
This article comes from trying to understand A Simple Introduction to Particle Physics, “notes…intended for a student who has completed the standard undergraduate physics and mathematics courses”.