# [Physics] Hamilton’s principle with semiholonomic constraints in Goldstein

classical-mechanicsconstrained-dynamicslagrangian-formalismtextbook-erratumvariational-principle

I am studying from Goldstein's Classical Mechanics, 3rd edition. In section 2.4, he discussed Hamiltion's principle with semiholonomic constraints. The constraints can be written in the form $$f_\alpha(q_1,…,q_n;\dot{q_1},…,\dot{q_n};t)=0$$ 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?