I'm currently reading "Classical Mechanics", by Herbert Goldstein (2nd
ed). In Section 2.4 (p. 45 ff.) he extends Hamilton's Principle to
nonholonomic constraints. The constraints he discusses give relations
between the differentials of the coordinates: sum_k a_lk d q_k + a_ll =
0 (Eq. 2-20)
(I assume that there is a typo; that the last term should be a_ll dt,
otherwise the dimensions wouldn't match.)

But then he says: "It would be expected that the varied paths, or
equivalently, the displacements constructing the varied paths, should
satisfy the constraints of Eq. (2-20)." And then he substitutes
"variation-deltas" (or "virtual deltas") for the d:s in Eq (2-20), and
solves this with Lagrange multipliers.

But this seems wrong to me. I see no reason to believe that these
displacements should satisfy Eq (2-20). On the contrary, it seems
wrong. If we apply this to a vertical wheel, that can turn, rolling on
a horisontal plane, an example Goldstein studies at page 14 ff (section
1.3, Eq (1-39), this is an example with the right kind of constraints)
all this would mean that the wheel could _roll_ from a position the
original path to the corresponding position on the varied path. And
this seems wrong to me. On the contrary, it seems to me that if the
wheel should be so displaced, it should be moved mostly laterally, thus
not satisfying the proposed equations.

So, am I wrong here, or is Goldstein wrong? All help will be
appreciated.


Regards,

Erland Gadde