D'Alembert principle states that: $$ \sum_i \left (\mathbf F_i^{(a)}- \mathbf{\dot p}_i \right) \cdot \delta \mathbf r_i = 0 $$ but I'm not quite getting it. The derivation seems trivial for me, following from Newton's laws.
If: $$ \mathbf F_i^{(a)}- \mathbf{\dot p}_i = \mathbf 0 $$ Then taking the dot product would obviously be zero as well. I know that there it is an slight diference between $\mathbf F_i^{(a)}$ (the applied forces) and $\mathbf F_i$ (total forces, including constraints).
But my question is aren’t the constraints directly taken into account with $\delta \mathbf r_i$? The system would make no virtual work in the direction of the virtual displacement, right?
But here comes another question. The virtual displacement is kind of an imaginary displacement, done without violating the constraints. To my understanding (correct me if wrong) that’s basically the directon of velocity. The direction in which the system will evolve given the forces and the constraints.
If so, wouldn't the d'Alembert principle be equivalent to: $$ \sum_i \left (\mathbf F_i^{(a)}- \mathbf{\dot p}_i \right) \cdot \mathbf v_i = 0 $$
What are virtual displacements then? (I've searched quickly for this explanation but the definition via the work integral confused me more).
