Principle of least action states that the particle choose the path which optimizes its action.Where the action is defined by S=t1∫t2(L)dt and L is the Lagrangian of the system.This leads to δS=0 and it is a condition to optimize S. I will summarize what confuses me about this principle in a series of questions: 1-Is the principle of least action must be taken for granted and there is no proof for it?If there is a proof where I can find it and what are the mathematical prerequisites needed to grasp it? 2-Why the action is defined like this? I mean why the integrand is the Lagrangian? why it is not another function? 3-Finally why δS=0 is a condition to optimize S? and why we call it variational of S ?why we don't just write ds=0?