I can fairly understand some differential geometry concepts like curvature and covariant derivatives. I have been thinking about curves constrained by some distribution $B$, which leads to covariant derivatives spanned by this distribution annihilator $B^\perp$. My question is:
How do I interpret the influence of some potential map $\phi$ or even a vector $f$ like phisicist do in classical mechanics? The former result would lead to covariant derivative $\nabla_{\dot{\gamma}} \dot{\gamma}$ generated by opposite of map gradient vector $\mbox{grad}_{\mathcal{X}} \phi$ and the latter, to equality $\nabla_{\dot{\gamma}} \dot{\gamma} = f$.
I do not know the reason for them. Any help would help me understanding it. :-)