The Lagrangian formulation of Classical Mechanics seem to suggest strongly that "action" is more than a mathematical trick. I suspect strongly that it is closely related to some kind of "laziness principle" in nature - Fermat's principle of "least time", for example, seems a dangerously close concept - but I cannot figure out these two principles are just analogous, or if there's something deeper going on. Am I missing something obvious? Why is action
$$A = \int (T-V) \, dt$$
and what interpretation does the $T-V$ term in it have?