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Hamilton's Principle

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?


Before variations, T - V = L (Lagrangian) is a function of unknown independent variables. They are not the equation solutions yet, one cannot use them as solutions.

When the solutions are found, they can be used to calculate quantitatively some useful conserved quantities like the system energy, momentum, angular momentum expressed via L and its derivatives (see Noether theorems).


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