P.S.

suppose you have a particle (in a gravitational field, for instance) which starts somewhere and moves to to some other point B by free motion. For example, let it be a particle that you throw upward, it would go up and down. It would go from point A to point B in a certain amount of time.

Now, suppose we have an imaginary path for the particle that looked as in fig 2. we change only the path, it is still happening in between t1 and t2.

If you calculate the kinetic energy at every moment on the path, take away the potential energy, and integrate it over the time during the whole path, you'll find that the number you'll get is bigger than that for the actual motion.

In other words, the laws of Newton could be stated not in the form of F=ma but in the form: the avergae kinetic energy less the average potential energy is as little as possible for the path of an object going from one point to another point.

we can calculate the kinetic energy minus the potential energy and integrate that value wrt time for such a path or for any other path we want. The miracle is that the true path is the one for which that integral is the least.

awesome.... but still don't get it... it is still a mere proven dogma for me

Feynman gave this example:
suppose we take the case of a free particle for which there is no potential energy at all. Then the rule says that in going from 1 point to another pint in a given amount of time the kinetic energy  integral is the least, so it must go at a uniform speed.

that's right. since there is no external force acting on the body, Newton's law would predict that the body move at uniform velocity.the path of particle is determined by the velocity. If it traverse curved line, then its velocity cannot stay uniform lo.. at some point it would have to turn over, and hence change the velocity vector.

But in that sense, what about uniform circular motion, the body could just go through circular motion with uniform speed. This motion needs external force though.

...
...
...

need to recall taylor's series
to be ctd