Does anyone know a good heuristic motivation for the Lagrangian Formalism? I think most physicist just accept at one point that it works and thats that. I think I understand the historic origin. Lagrange was fascinated by the Fermat Principle and searched for a similar principle for massive objects. By guessing he ended up with the classical Lagrangian that reproduces the Newtons second law.
I would be interested in a modern viewpoint. Can in be maybe understood, in some way, from a quantum field theoretic or relativistic point of view? I stumbled upon the book chapter by stackexchange member Hans de Vries: http://physics-quest.org/Book_Chapter_Lagrangian.pdf , but he never elobarted on this. For example in What is the physical meaning of the action in Lagrangian mechanics? he just repeats his idea of "least action = least proper time", but does not give any further explanations for this claim.
Does anyone know some elaboration on this or some other illuminating idea for why the Lagrangian formalism works (not in classical mechanics, but the more fundamental case of Quantum Field Theory)?
PS: I dont think this is a dublicate of What is the physical meaning of the action in Lagrangian mechanics? because the question there is specifically about the physical meaning of the action