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It's my understanding that General Relativity abstracts away the concept of gravity as a force, and instead describes it as a feature of spacetime by which massive objects cause curvature. Then it follows that what we experience as a force is simply the difference between a geodesic on this curved surface and our perceived Euclidean space. What I am unsure of, exactly, is the implication of this.

$$S[q] \equiv \int L(q(t), \tfrac{ \delta q }{ \delta t }(t), t)dt$$ and Hamilton's Principle states that $$\tfrac{\delta S}{\delta q(t)} = 0.$$

If $$F(q(t)) = -\nabla U(q(t)) = \nabla (T(\tfrac{ \delta q }{ \delta t}(t)) - U(q(t))) = \nabla L,$$ which, as I understand is true for a conservative field like gravitation, then are these two statements not equivalent?

What additional insight do we gain by using principles of differential geometry versus classical potential theory?

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