When deriving d'Alembert's Principle it must be assumed, that the total virtual work done by constraint forces vanishes. $$\sum_{j=1}^N\mathbf{C}_j\cdot\delta \mathbf{r}_j=0.$$ In the books I've read, the authors have made it a point to state, that the virtual work being done by individual constraints forces doesn't have to be zero. It isn't quite clear to me how that would be possible. I can't think of a single example where any constraint force does any virtual work.

So I'm searching for a system where: $\exists j\in\{1,\ldots,N\}:\quad\mathbf{C}_j\cdot\delta \mathbf{r}_j\neq0.$


1 Answer 1


Example. Consider two point-masses at positions ${\bf r}_1$ and ${\bf r}_2$ connected by a rigid massless rod, which is a constraint. If we (virtually) move the point-masses, the internal forces (=constraint forces) acting on the two point-masses will separately produce (virtual) work, but their sum will vanish due to Newton's 3rd law.

See also this related Phys.SE post.


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