I know basic differential geometry for general Relativity and classical mechanics. But an interesting fact was revealed in my calculations, namely, that I discovered that I didn't realize the difference between the spherical coordinate system and a rotational system.
The problem arose when I tried to calculate the "generalized" force as $$m\frac{d^{2}x^{i}}{dt^{2}} = - m\Gamma^{i}_{ij}\frac{dx^{i}}{dt}\frac{dx^{j}}{dt} \tag{1}$$ in spherical coordinates $(r,\theta,\phi)$, to calculate the fictitious forces, namely, the centrifugal, Coriolis and (I think) the Euler forces. But in fact you reach those fictitious forces just in rotational coordinates like:
$$\begin{cases} x' = x cos\theta - y sin\theta \\ y' = x sin\theta + y cos \theta \end{cases} \tag{2}$$
I am now confused because, when you are spinning a ball in a circular motion, you use polar coordinates to describe the physical fact that something is under rotation where the polar coordinates are associated with a non-inertial frame.
My doubt is why, using spherical coordinates metric tensor, I didn't get fictitious forces but in the rotational coordinates I did?