What is the physically intuitive reason behind the fact that the action is stationary for the true path a particle takes? I understand that a path that satisfies the euler-lagrange equation minimizes (or maximizes or "saddle-points") the result of the action functional. I also understand that the euler-lagrange equation basically says F - F = 0 where F in one side is the force as it is defined (time change of momentum) and F on the other side is true simply by the mathematical definition of the potential energy (force integrated over distance). Neither of these things explain why the quantity of action has any significance whatsoever. Is there a significance or did someone out there just make a very astute observation that this particular mathematical quantity does something under "true conditions"? Explainations I have read of the lagrangian formulation of mechanics always simply state that the true path is that of least action and just show how the euler-lagrange equation arises from that. Sometimes they'll talk about the classic problem in calculus of variations that has to do with finding the shape of the ramp that will minimize the time taken for a marble to roll from starting point to ending point, but this appears to have no relation to the principle of least action, since most any ramp you can come up with will result in the marble going down a true, but different path. The marble could take any of those paths it has nothing to do with finding the marble's true path.