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Haven't seriously tackled this problem myself because it's been awhile since I've done any hard mathematics and I'm a bit rusty. However, you needn't spare the math in your answers.
I've been thinking of how many features of physics are general, in the sense of applying to some large class of (non-physical) dynamical systems. Notably, Noether's theorem tells us some quantity we can call "energy" will be conserved in any dynamical system or field equation which (a) arises from a local action, i.e. one that comes from integrating an action density, and (b) is time-symmetric. I am aware from the Wikipedia article that there is a generalization to non-local actions, but it wasn't clear to me whether it would still give a sensible notion of "energy" (again: a bit rusty).
The natural next question, which I pose to you, is: which dynamical systems and field equations arise from a (local) action? Other than the tautological answer, is there some way to characterize them in terms of properties of the corresponding dynamics?