Can Lagrange's equation be used if the virtual work done by constraint forces is not zero?

Question

I'm learning analytical mechanics and was just introduced to d’Alembert’s principle, which I know is only valid when constraint forces' virtual work is zero. My question is, does this restriction also apply to Lagrange's equation?

Answer 1

It is possible to derive the following identity $$ \sum_{i=1}^N \left(\dot{\bf p}_i-{\bf F}^{(a)}_i\right)\cdot \delta {\bf r}_i ~=~ \sum_{j=1}^n \underbrace{\left(\frac{d}{dt} \frac{\partial T}{\partial \dot{q}^j} -\frac{\partial T}{\partial q^j}-Q^{(a)}_j\right)}_{\text{Lagrange equations}} \delta q^j,\tag{1}$$ cf. Ref. 1. Here the superscript $(a)$ stands for applied forces, cf. e.g. this Phys.SE post. Therefore d'Alembert's principle holds iff Lagrange equations hold. In other words, if a non-applied force produces virtual work, it violates d'Alembert's principle and Lagrange equations simultaneously.

References:

1. H. Goldstein, Classical Mechanics; Chapter 1, eqs. (1.45)+(1.52).