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But now that I've started to study a little QFT I see that there the Lagrangian seems to be simply constructed as "the function that after applying Hamiltons principle gives the correct equation of motion". In other words, I don't see any obvious physical interpretation outside of being a lucky guess giving the correct result.

Have I missed something important here or is the Lagrangian just a mathematical tool and not something with a distinct physical interpretation like the Hamiltonian?