Hamilton's principle and virtual work by constraint forces

I have a question about the following pages(pg 47 and 48) from Goldstein's "Classical Mechanics"

I do not understand how (2.34) shows that the virtual work done by forces of constraint is zero. How does the fact that "the same Hamilton's principle holds for both holonomic and semiholonomic systems" show that the additional forces of semiholonomic constraint do no work in the $\delta q_k$?

• Goldstein hasn't even defined what is a virtual displacement in the book. It is quite murky in describing D'Alembert's principle. I recommend reading Lectures in Analytical mechanics by F Gantmacher. It is quite mathematical but is very precise in dealing with virtual displacements. The book begins with this. – BoundaryGraviton Jul 28 '16 at 16:34
• @BoundaryGraviton: Would please provide the links? :) – H. R. Sep 22 '16 at 13:41

A force of a holonomic or semi-holonomic constraint is one that does work only in the direction of a conserved coordinate. One way to find such coordinates is to test the derivative of the "momentum" term in the Lagrangian $\frac{dL}{dq'_k}$ (where $q'_k$ is the time derivative of $q_k$).
If this quantity is a constant, then $q_k$ is a conserved (or ignorable) coordinate, and the force in that coordinate direction is $0$ (as force is defined in Newton's Second Law as the time derivative of momentum). This coordinate is therefore eliminated from your equations of motion. The remaining coordinates $q_k$ stay in your equations, but your forces of constraint $f_c$ have no effect on their motions.