I've heard a few of my professors throw around the term "finding the Lagrangian of a theory". What exactly is this referring to. From what I understand it seems that you determine invariances (symmetries) and they give you a hint for what your Lagrangian is. Furthermore there is more to the story because I know:
$L=T-U$ is only one of the forms the Lagrangian can take in classical mechanics. So far I only learned about the Lagrangian in classical mechanics and might be building up to a limited knowledge of Feynman's path integral in my QM course.
What other theories have Lagrangians and how you can tell?
Are all Lagrangians of a given theory equivalent?