Hamilton's principle with nonholonomic constraints in Goldstein I am studying from Goldstein's Classical Mechanics, 3rd intl' edition, 2013. In section 2.4, he discussed Hamilton's principle with nonholonomic constraints. The constraints can be written in the form $$f_\alpha(q_1,...,q_n;\dot{q_1},...,\dot{q_n};t)~=~0\tag{2.24}$$ where $\alpha=1,...,m$. Using variational priciple, we get
$$\delta\int_{t_1}^{t_2}\left(L+\sum_{\alpha=1}^m \mu_\alpha f_\alpha\right)\mathrm dt=0 \tag{2.26}$$
where $\mu_\alpha=\mu_\alpha(t)$.
But how can he get the formula
$$\frac{d}{dt}\frac{\partial L}{\partial\dot{q_k}}-\frac{\partial L}{\partial q_k}=-\sum_{\alpha=1}^m \mu_\alpha \frac{\partial f_\alpha}{\partial\dot{q_k}} \tag{2.27}$$
for $k=1,...,n$ from the previous formula?
When I go through the steps as in section 2.3, I get
$$\frac{dI}{d\beta}=\int_{t_1}^{t_2}\sum_{k=1}^n\left(\frac{\partial L}{\partial q_k}-\frac{d}{dt}\frac{\partial L}{\partial\dot{q_k}}+\sum_{\alpha=1}^m \mu_\alpha\left(\frac{\partial f_\alpha}{\partial q_k}-\frac{d}{dt}\frac{\partial f_\alpha}{\partial\dot{q_k}}\right)\right)\frac{\partial q_k}{\partial\beta}\mathrm dt$$
where $\beta$ denotes the parameter of small change of path:
\begin{align}
q_1(t,\beta)&=q_1(t,0)+\beta\eta_1(t)\\
q_2(t,\beta)&=q_2(t,0)+\beta\eta_2(t)\\
&\ \,\,\vdots
\end{align}
Using the same argument as in the part of holonomic constraint in section 2.4, I get
$$\frac{\partial L}{\partial q_k}-\frac{d}{dt}\frac{\partial L}{\partial\dot{q_k}}+\sum_{\alpha=1}^m \mu_\alpha \left(\frac{\partial f_\alpha}{\partial q_k}-\frac{d}{dt}\frac{\partial f_\alpha}{\partial\dot{q_k}}\right)=0$$
for $k=1,...,n$.
What am I missing?
 A: TL;DR: Note that the treatment of Lagrange equations for non-holonomic constraints in Refs. 1 & 2 is inconsistent with Newton's laws, and has been retracted on the errata homepage for Ref. 2. See Ref. 3 for details.
Longer explanation:

*

*The main point of Goldstein's section 1.4 was to start from d'Alembert's principle (DAP) and derive Lagrange equations (LE) for holonomic constraints$^1$.


*Therefore (although Goldstein does admittedly not state this clearly$^2$), the main point of section 2.4 should be to start from DAP and derive LE for affine non-holonomic constraints (=semi-holonomic constraints).


*In fact, more generally, for independent
non-holonomic one-form constraints $$\begin{align}\omega_{\ell}~\equiv~& \sum_{j=1}^na_{\ell j}(q,\dot{q},t)\mathrm{d}q^j+ a_{\ell 0}(q,\dot{q},t)\mathrm{d}t~=~0, \cr \ell~\in~&\{1,\ldots, m\}, \end{align} \tag{NH1C}$$
one may show that DAP leads to LE
$$\begin{align} \frac{d}{dt}\frac{\partial T}{\partial \dot{q}^j}-\frac{\partial T}{\partial q^j}~=~&Q_j+\sum_{\ell=1}^m\lambda^{\ell} a_{\ell j}, \cr 
j~\in ~&\{1,\ldots, n\}.\end{align} \tag{LE}$$


*Now Refs. 1 & 2 use instead independent
non-holonomic constraints $$ f_{\ell}(q,\dot{q},t)~=~0, \qquad \ell~\in~ \{1,\ldots, m\}. \tag{NHC}$$
Eqs. (NHC) and (NH1C) are equivalent for affine non-holonomic constraints, but not in general.


*Eq. (2.27) in Ref. 1 is essentially Chetaev's equations (CE) [5]
$$ \begin{align}\frac{d}{dt}\frac{\partial T}{\partial \dot{q}^j}-\frac{\partial T}{\partial q^j}~=~&Q_j+\sum_{\ell=1}^m\lambda^{\ell}\frac{\partial f_{\ell}}{\partial \dot{q}^j}, \cr j~\in~& \{1,\ldots, n\}. \end{align}\tag{CE}$$
DAP plus affine non-holonomic constraints (where $\frac{\partial f_{\ell}}{\partial \dot{q}^j}$ has maximal rank) imply CE, but not for general non-holonomic constraints [6].
References:

*

*H. Goldstein, Classical Mechanics, 3rd intl' ed, 2013; Section 2.4. Eq. (2.26) is wrong/misleading at best.


*H. Goldstein, Classical Mechanics, 3rd ed, 2001; 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.)


*M.R. Flannery, The enigma of nonholonomic constraints,
Am. J. Phys. 73 (2005) 265.


*E.J. Saletan & A.H. Cromer, A Variational Principle for Nonholonomic Systems, Am. J. Phys. 38 (1970) 892. Ref. 1 cites Ref. 4.


*N.G. Chetaev, Izv. Fiz.-Mat. Obsc. Kaz. Univ. 6 (1933) 68. The Chetaev term $\sum_{\ell=1}^m\lambda^{\ell} \frac{\partial f_{\ell}}{\partial \dot{q}^j}$ is invariant under reparametrizations of the constraints $f^{\prime}_k= f_{\ell} M^{\ell}{}_k$ and $\lambda^{\ell}=M^{\ell}{}_k\lambda^{\prime k}$.


*M.R. Flannery, D'Alembert-Lagrange analytical dynamics for
nonholonomic systems, J. Math. Phys. 52 (2011) 032705; p. 22.
--
$^1$ In this answer we will assume the commutativity rule
$$\frac{d}{dt}\delta q^j=\delta \frac{d}{dt}q^j,\tag{CR}$$
cf. e.g. this related Phys.SE post.
$^2$ Goldstein confusingly refers to Hamilton's principle, which goes against the main paradigm of using Newton's laws as a first principle.
