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Looking at it I'm not sure why having $O(n)$ as isometry group implies local isometry to $\mathbb{R}^2\times S^{n-2}$, nor why this is equivalent to Birkhoff's theorem.
Indeed, I had looked into Birkhoff's theorem, but found no clear proof of it. Wikipedia, for example, only loosely states it and proceeds to discussion of its implications.
@Qmechanic this is not a duplicate, I'm not relating Hamiltonian and Lagrangian mechanics, I'm asking if their relation (the question you mentioned) is the same as that between conservation of energy and the equations of motion.
