I have been wondering about coordinate vector field: It is somewhat common sense in differential geometry, that we have one or more charts $(\varphi_i, U_i)$, and a manifold $\mathcal{M}$, such that we cover each point with some parametrization $p = \varphi_i(u_i)$. There may be intersections between charts such that there are $u_i \in U_i$ and $u_j \in U_j$ such that $p = \varphi_i(u_i) = \varphi_j(u_j)$, but we are not handling such in this post.
I want to ask about the normalization of coordinate vector fields: We know, we can build a coordinate vector field basis $X_k$ from transformation map $\varphi_i$, given by partial derivative $(\varphi_i)_{;k}$.
In case I am interested in using this basis for real-world physical velocity $\dot{\gamma}$, some unit inconsistency arises: we know a body kinetic velocity arises from formula $\langle \dot{\gamma}^k X_k, \dot{\gamma}^l X_l \rangle = g_{kl} \dot{\gamma}^k \dot{\gamma}^l $: for the spherical coordinate vector map $\varphi$, derivative $\varphi_{;k}$ does not lead to a unit vector, only $\frac{\varphi_{;k}}{\sqrt{\varphi_{;k}^\intercal \varphi_{;k}}}$ does. We have an issue here: $\dot{\gamma}^k = \lvert\lvert\varphi_{;k}\rvert\rvert \, \dot{u}^k$, kinetic energy given by $g_{kl} \dot{\gamma}^k \dot{\gamma}^l$, but I wish to obtain $\dot{u}$ instead. I can concatenate these two systems, but it is to hacky for me to handle. Do you think, I am overthinking?
$$ \begin{cases} \dot{u}^k = \frac{1}{\lvert\lvert\varphi_{;k}\rvert\rvert} \dot{\gamma}^k \\ \ddot{\gamma}^k = - \Gamma_{ij}^k \dot{\gamma}^i \dot{\gamma}^j \end{cases} $$