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?
Best Answer
Note that the treatment of Lagrange equations for non-holonomic constraints in Ref. 1 is inconsistent with Newton's laws, and has been retracted on the errata homepage for Ref. 1.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.
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