From the Feynman lectures:
>“In other words, the laws of Newton could be stated not in the form F=ma but in the form: the average kinetic energy less the average potential energy is as little as possible for the path of an object going from one point to another.
I wrote deliberately:
>The difference is extremized (on average).
and not "is minimized on average" (as Feynman says).
Why? - because it seems the action may not always be minimized.
Arnold has an Example, where Trajectories are Geodesics, and like the geodesics on the sphere those need not be shortest. Moreover I just checked an old physics undergraduate textbook and there it is even stated that if one has constraints then the action is always maximized. This "theorem" is there however stated without proof or reference so I dont know how trustworthy that is.