This question already has an answer here:
- Why the Principle of Least Action? 10 answers
- Motivation for the Euler-Lagrange equations for fields 1 answer
- Hamilton's Principle 6 answers
Whilst studying Field Theory and after checking numerous sources it appears that people always just state the action without providing some sort of motivation/intuition as to why we should/can use the Lagrangian formalism in field theory.
I understand the motivation for it in classical mechanics and I do see the analogy between the classic Lagrangian and the Lagrangian density by taking the limit where you want to see the dynamics of an infinite number of particles (ie. each one being a point of a field) however I still can't realize the full picture as to why the Principle of Stationary Action should be valid for a field.
Is the Lagrangian formalism used simply because the equations were already known and people just constructed a Lagrangian that worked and provided the already known equations? Or was it built from the ground up already assuming the Principle of Stationary Action is valid ? If so why?
Note: I known a very similar question was already asked before here however I felt that the question was not answered fully.
EDIT: As I said I known that the question has been asked before however in the links provided and the links therein (which I had read prior to asking this question) an explicit answer to this is not provided.