I have a basic question. The classical Lagrangian is defined to be the difference between kinetic and potential energy. Then the actual paths of motion are found to be those that minimize the integral of the function over the over the path (the action).

So, this "works."

But is there an intuitive explanation for why it works? Does the difference between kinetic and potential energy -- at a point in time -- have a _physical_ meaning that is explainable?