I accept the definition of a derivative as properly motivated because it helps me make physical predictions in classical mechanics. I'm looking for something similar with manifolds.
From what little symplectic geometry I have studied, it mostly dealt with generalizing Hamilton's equations to symplectic manifolds. However, I never saw it being used to make physical predictions. Is there a nice example here?
Similarly, I have heard that pseudo-Riemannian geometry allows one to make predictions in general relativity. Can someone give a quick example for someone that knows next to nothing about relativity?
