Verify Equation 5.20, [ m_{i} ddot{mathbf{r}}_{i} cdot frac{partial mathbf{r}_{i}}{partial q_{k}}=left[frac{d}{d t}left(frac{partial}{partial dot{q}_{k}} ight)-frac{partial}{partial q_{k}} ight]left(frac{1}{2} m_{i} dot{mathbf{r}}_{i}

Verify Equation 5.20,

\[ m_{i} \ddot{\mathbf{r}}_{i} \cdot \frac{\partial \mathbf{r}_{i}}{\partial q_{k}}=\left[\frac{d}{d t}\left(\frac{\partial}{\partial \dot{q}_{k}}\right)-\frac{\partial}{\partial q_{k}}\right]\left(\frac{1}{2} m_{i} \dot{\mathbf{r}}_{i} \cdot \dot{\mathbf{r}}_{i}\right) \]