I've been learning about the Lagrangian formulation recently, and while I'm with the process, I am still struggling somewhat with the theory behind it. As I (rather poorly) understand it, the formulation is based on the principle of least action. The Euler-Lagrange equation is used to find the minimums of the integrals of the Lagrangian (the action), which is then used to solve for whatever value you are interested in.
What I DON'T understand is why the Lagrangian is equal to the difference between the total kinetic and potential energy of the system. The definition seems arbitrary. What is the significance of that particular difference? Every lecture I've found seems to brush past the point, and I've yet to find a satisfying explanation. Can anyone provide a better justification?