I remember asking my second year Mechanics teacher about why do the Lagrangians give the equations of motion. His answer was that there is no answer to that, it is an empirical fact, and that asking the why of the Lagrangians is the same as asking why Newton's laws hold.
I understood his point but I wonder if you have a better answer. Why do Hamiltonians and Lagrangians work?
Here's the thing, let's say I gave you an alleged answer to the question of "why". For concreteness, let's say my answer is: "Because of the Principle of Stationary Action".
Now, my question to you is this: would this answer satisfy your question of "why" or would it be the case that your curiosity would lead you ask me: "But why is nature such that the Principle of Stationary Action holds?".
It's possible that I might even give you an alleged answer to that. But, if I did, the "Principle" of Stationary Action would no longer be a principle.
The point is this: if something is fundamental or is a principle, asking "why" presumes that it isn't fundamental or a principle. For, if there were a genuine answer to "why", that answer would be the fundamental or principle instead.
At some point, the only "answer" is "It just is".
