[prof-dr-ir]

Meh :) in classical mechanics one cares little about maximization versus minimization; it's really just (stationarity of L) <---> (equations of motion) that matters. Practically speaking it's just a cute gimmick to quickly find the equations of motion, especially for constrained systems.

In quantum mechanics everything is different, but for that you should study path integrals. To re-establish the connection with classical mechanics you have to learn about the saddle point approximation; it might also help to read Feynman's book called QED.