Hamilton's principle and virtual work by constraint forces

Question

I have a question about the following page 48 from the third edition of 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 $\delta q_k$?

Answer 1

Note that the use of Hamilton's principle (a.k.a. the principle of stationary action) for systems with semi-holonomic constraints in Ref. 1 is inconsistent with Newton's laws, and has been retracted on the errata homepage for Ref. 1. See Ref. 2 for details. See also this & this related Phys.SE posts.

For starters, Ref. 1 provides a wrong (or at best an incomplete) definition of semi-holonomic constraints, cf. eqs. (2.20) & (2.20'). However, the definition itself is the least of the problems with Ref. 1.

In conclusion, the arguments of Ref. 1 pertaining to OP's specific question are based on false assumptions, and therefore rendered moot.

References:

1. H. Goldstein, Classical Mechanics; 3rd ed; Section 2.4. Errata homepage. (Note that this criticism only concerns the treatment in the 3rd edition; the results in the 2nd edition are correct.)
2. M.R. Flannery, The enigma of nonholonomic constraints, Am. J. Phys. 73 (2005) 265.

Answer 2

First, assume that there are no constraints then with 2.32 derive 2.34. Now, add constraints then $Q_k$ becomes $Q_k+H_k$, where $H_k$ is force of constraint We can see that the additional force terms should do no work in order to preserve 2.34 for in unconstrained motion.

Answer 3

Insofar as learning Classical Mechanics is concerned, I found Taylor to be much better than Goldstein for initial forays.

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.