Why should physical theories always have a Lagrangian formalism?

Question

I've often heard that every physical theory has some kind of Lagrangian formalism, or a formalism in terms of a principle of stationary action. The Standard Model has one, General Relativity has one, Electromagnetism has one, and so on.

On the surface, these all seem like very different theories. What is it that unifies them so that they all end up having a Lagrangian (or other action-based) formalism? Why should physical laws be expressed in this way?

EDIT: My question has been identified as a possible duplicate of this question. That question specifically asks about quantizing Lagrangian, when is it possible, and how general it is. I'm asking what the salient properties of physical theories are that makes it possible to associate to them a Lagrangian formalism.

Closely related: Why the principle of least action? and QFTs that have no action and How general is the Lagrangian quantization approach to field theory? and How do I show that there exists variational/action principle for a given classical system?. — ACuriousMind, Aug 27 '15 at 23:41
Tip: Readers and answerers often don't read further than the title. If you want to claim this post is not a duplicate, make sure that the title question reflects this. — Qmechanic, Aug 30 '15 at 14:03
It doesn't let me remove the [duplicate] marker from the title. — FlagCapper, Aug 31 '15 at 01:33

Why should physical theories always have a Lagrangian formalism?

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