[lucozade]

> why we define the Action as this object and why we should expect it to be minimised for the physical trajectory in the first place.

The most coherent explanation I've heard was from Feynnman [0]. As far as I understand it (and I may well not have understood it at all well), at the quantum level, all paths are taken by a particle but the contributions of the paths away from the stationary point tend to cancel each other. So, at a macroscopic level, the net effect appears to be be that the particle is following the path of least action.

> a proof of the equivalence to Newtonian mechanics

The Lagrangian method isn't really equivalent to Newton's method. Again, Feynman talks about this in [0]. It's that for a certain class of action, the Euler-Lagrange equations are equivalent to Newton's laws.

It's perfectly plausible to come up with actions that recover systems that represent Einsteinian relativity or quantum mechanics. This is the main reason (as I understand it) why it's considered a more powerful formalism.

[0] https://www.feynmanlectures.caltech.edu/II_19.html