This question already has an answer here:
Give some Lagrangian we use the principle of stationary action to find the desired euqations of motion for something (e.g. a field).
A lot of modern physics seems to be based on the principle of stationary action. I read it works for classical mechanics, general relativity, Quantum chromodynamics, quantum electrodynamics, weak interactions and more stuff.
However, is there a field where the principle of stationary action does not yield the desired result or where it fails (contradicts some experimental result)?