I am reading Mechanics: From Newton's Laws to Deterministic Chaos by Florian Scheck Chap2 The Principles of Canonical Mechanics
Right now I have some doubts regarding the derivation process which makes me dizzy.
Firstly, since the Generalized Coordinates can uniquely determine the position of the entire particle system, why should $r_{i}$ not be $r_{i}(q_{1}(t),...,q_{s}(t))$ ?
Secondly,I don't understand how (2.10) is deduced, in which the author assumes that $\frac{\partial \frac{\partial r_{i}}{\partial q_{s}}}{\partial \dot{q_{k}}}=0$
I considered the simplest case,$r_{i}=q_{s}q_{k}e_{i}$,$\frac{\partial r_{i}}{\partial q_{s}}=\frac{\partial r_{i}}{\partial q_{s}}(q_{1},...,q_{s},t)$=$q_{k}e_{i}$ then $\frac{\partial \frac{\partial r_{i}}{\partial q_{s}}}{\partial \dot{q_{k}}}$can be non-zero if $q_{k}=t^2$