I've been trying to wrap my head around Lagrangian mechanics and Lagrangians in general, and I've found it difficult. After some thinking, I believe that the issue I have is with action principles. Why are they so consistently and widely useful and powerful?
Classical mechanics, classical field theories, quantum mechanics, quantum field theory, relativistic mechanics, and GR can all be formulated in terms of least action principles, as well as other emergent physical theories, and theories we could simply invent. This seems unreasonably effective, for a formulation that on the surface does not immediately strike a person as being extremely general.
So ultimately, I have two questions:
What would motivate you to even think of inventing the idea of a least action principle and using it to describe physics (I am not interested in the historical development of the idea, but rather, a sort of absurd hypothetical where we have all our physical theories but somehow without any least action formulations)?
Why would we then expect such a theory to be so wildly general as to cover so many different physical theories of such wildly different character (classical/quantum, particle/rigid-body/field, etc.)?
I also want to strike down immediately an answer I've seen before, which I think is a total non-answer, which is that "classical lagrangian mechanics is just the macroscopic limit of quantum lagrangian mechanics, which comes from a quantum least action principle, and that's why you can explain so much stuff in terms of a least action principle". My issue with that argument is that we can imagine living a world that was actually, legitimately classical, and we would not then be unable to use lagrangian formulations. We can also look at emergent physical theories which have no direct reason to inherit least-action descriptions from their constituents, and yet still have such descriptions, seemingly for their own reasons.