I've been reading a lot about path integrals lately, and I've found it fascinating to see at the quantum level how the extremal values of the lagrangian are basically the only ones that contribute when the action is large and therefore we get the classical path. Something that continues to puzzle me, though, is why the lagrangian function in particular is extremized. Why should T-V be the quantity that gets extremized instead of some other function? Is there any consensus view on this question? Thanks!