Ambiguity in d'Alembert's principle

Question

It seems to me that many different momenta $\dot{\bf p}_j $ can satisfy d'Alembert's principle:

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

in a constrained system.

For example, take two particles connected by a rod, with an applied non-zero force of the same size on each particle, along the line of the rod and in opposite directions (i.e. forces trying to pull the particles apart). In this case, any pair of $\dot{\bf p}_1, \dot{\bf p}_2$ of the same size and in opposite directions along the line of the rod seems to satisfy the d'Alembert's principle (taking into account the dependencies between $\delta {\bf r}_1$ and $\delta {\bf r}_2$ imposed by the rod), even though some of these do not satisfy the constraint.

Am I misunderstanding something, or does d'Alembert's principle not provide any meaningful information in this example?

Answer 1

1. There is no ambiguity in the definition of the momentum ${\bf p}_j$ of the $j$'th point particle.

2. Rather the ambiguity is in the definition of the $j$'th applied force ${\bf F}_j^{(a)}$ among all forces (such as, e.g., gravity force, spring force, constraint force, etc.) that act on the $j$'th point particle, cf. e.g. this Phys.SE post.

Answer 2

d'Alembert's principle is mostly used with independent generalised coordinates( with the dependent coordinates eliminated using the constraint equations). We can also use dependent coordinates but then we have to also take into account the constraint equations and incorporate them using lagrange multipliers in the zero virtual work principle to get the constraint forces. As for the rod the only possible momentum is p1=p2=0. As nothing else satisfies the constraint(else the rod cannot be rigid and would get deformed). this paper on virtual displacement and Lagrangian dynamics has a very good description of this

Also in this case the only allowed virtual displacement seems to be $\delta r1=\delta r2$. So only choice of values for the $\dot{\bf p}_j $'s satisfying the d'Alembert principle is $\dot{\bf p}_j $=0