Why can't d'Alembert's Principle be derived from Newton's laws alone?

Question

The wiki article states that D'Alembert's Principle cannot derived from Newton's Laws alone and must stated as a postulate. Can someone explain why this is? It seems to me a rather obvious principle.

Answer 1

Superficially, d'Alembert's principle

$$\tag{1} \sum_{i=1}^N ( {\bf F}_i^{(a)} - \dot{\bf p}_i ) \cdot \delta {\bf r}_i~=~0 $$

may look like a trivial consequence of Newton's 2nd law, but the devil is in the detail. Here the detail is the superscript $(a)$ on the force, which stands for applied forces. The term applied forces refers to that we have divided all forces (such as, e.g., gravity force, constraint force, etc.) into two bins:

1. The applied forces, and

2. the rest.

It is important to realize that d'Alembert's principle may be true or false, depending on how the above division is made. See also this Phys.SE post. The standard example of a force that one cannot put into the second bin is a sliding friction force.

In particular, if one puts all forces into the first bin, then indeed, d'Alembert's principle would be a trivial consequence of Newton's 2nd law. But in practice one would not like to do that. One would instead like to minimize the type of forces that one puts in the first bin in order to simplify the equations as much as possible.