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I'm simulating the principle of least action for simple object motion and reading from Feynman Vol. 2, Chpt 19 -- The Principle of Least Action. He states (with my paraphrasing) that the true path of a trajectory is the one for which the integral over all points of kinetic energy minus potential energy is least.

I created a curve of an object shot straight up at velocity=500m/s from x0=0, g=9.8 and t=80s. It goes up and comes partway down and ends up 8919m above x0 (again, after 80s). I then calculated the integral of KE - PE from 0 to 80 and got a value of -4969173.33. I did the same numerically and came up with a value very close to that. My understanding is that parabolic curve should have the lowest value for the action from (x0,t0) to (x1,t1).

Then I generated a bunch of other curves and sure enough, they all had larger values. Then I drew a straight line from (x0,t0) to (x1,t1) and calculated the integral and came up with a smaller value! Namely, -6490206.667. Hey! The principle of least action didn't hold; a smaller value represents a non-true path. A little more work led me to other curves that "violated" the principle of least action.

The straight line is completely non-physical, of course. It doesn't rise as high a real object, it's not parabolic, etc. However, I think this should be allowable because the principle of least action is used to

*the trajectory motion. It should have given a larger value and I could have kept plugging away at curves infinitely and found that the parabola was the smallest.*

**define**My guess is that there is more to a valid path that just the starting and ending points being the same as the true path. What are those?

Thanks in advance.