In the book of Classical Mechanics by Goldstein, at page 19, while deriving D'Alembert's principle, the author assumes that
$$\dot p = m \ddot r.$$
However, when the mass of the bodies also changes, is the principle still correct, or can be corrected? If so, how?
In mechanics theorems, the material particles considered are universally assumed to have constant mass. If you have a situation where a body or a particle changes its mass, you have to model it as a collection of smaller particles of constant mass which leave or become part of the control volume associated with the original body. A body has to have constant mass for the mainstream Newton or other mechanical laws to be directly applicable.
There is no general principle for systems where particles change their mass, because what such particles and systems of them will do depends on details of how the particles lose their mass. Does the particle eject a stream of smaller particles, all in a single one direction? If so, which direction then? Or does it eject them in all directions equally? Net reactive force on the original particle will depend on these details and so will the evolution of system of such particles.
Don't get me wrong, Newton's laws or D'Alembert principle can deal with variable mass systems. However, the way this is done is careful analysis of individual particle forming the system, each of which has to have constant mass.
The suggestion in the Wikipedia article https://en.wikipedia.org/wiki/D%27Alembert%27s_principle that one should differentiate mass of the particles is a very strange one. It is probably based on mistaken belief that $\mathbf F=\dot{\mathbf p}$ is universally valid expression of second law, even for variable mass systems. However, one does not differentiate mass with respect to time to apply Newton's second law, ever. See also my answer here:
