D'Alembert's Principle

Using Newton’s II Law and the principle of virtual work $$ \begin{align} F_i-p_i’=0\\ \sum_{i=1}^N (\vec{F_i}-\vec{p_i’})\cdot \delta \vec{r_i}=0\\ \Rightarrow \sum_{i=1}^N (\vec{F_i^a}-\vec{p_i’})\cdot \delta \vec{r_i}+\sum_{i=1}^N \vec{f_i}\cdot \delta \vec{r_i}=0\\ \\ \sum_{i=1}^N (\vec{F_i^a}-\vec{p_i’})\cdot \delta \vec{r_i}=0 \end{align} $$ (∵ Virtual work of constraints = 0)