I am currently studying Analytical Mechanics having among one of the many references the book “Classical Mecanics” by Goldstein. And it is with respect to this that I have a doubt. My goal at the moment is to write down the equations of motions of non-holonomic systems by introducing the Lagrangian $$L^*=L+\sum_{\alpha=1}^k\lambda_\alpha f_\alpha,$$ where in $L$ the generalized coordinates $q_j$, $j=1,\dots, 3N$, are unconstrained. So, given $\alpha=1,\dots,k$ equations of constraint of the form $f_\alpha(q,\dot{q},t)=0$, if we suppose that these are linear in the velocities $\dot{q}_j$, for $j=1,\dots, 3N$, we can write the Lagrange's equations of motion as (eq. 1):
$$ \frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}_j}\right)-\frac{\partial L}{\partial q_j}=\sum_{\alpha=1}^k\frac{\partial f_\alpha}{\partial\dot{q}_j}\lambda_\alpha,\tag{1} $$
where $\lambda_\alpha(t)$ are the Lagrange's multipliers. Here, the Lagrangian $L\equiv L(q,\dot{q},t)$ has no constraints imposed on the generalised coordinates. Then, since:
$$ \frac{d}{dt}\sum_{\alpha=1}^k\frac{\partial f_\alpha}{\partial\dot{q}_j}\lambda_\alpha=\sum_{\alpha=1}^k\frac{d}{dt}\left(\frac{\partial f_\alpha}{\partial\dot{q}_j}\right)\lambda_\alpha+\sum_{\alpha=1}^k\frac{\partial f_\alpha}{\partial\dot{q}_j}\dot{\lambda}_\alpha, $$
by adding and subtracting the quantities:
$$ \frac{d}{dt}\sum_{\alpha=1}^k\frac{\partial f_\alpha}{\partial\dot{q}_j}\lambda_\alpha\hspace{3mm}\text{and}\hspace{3mm}-\sum_{\alpha=1}^k\frac{\partial f_\alpha}{\partial q_j}\lambda_\alpha $$
in equation (1) it is found that:
$$ \frac{d}{dt}\left(\frac{\partial L^*}{\partial\dot{q}_j}\right)-\frac{\partial L^*}{\partial q_j}=Q_j, $$
where $$L^*=L+\sum_{\alpha=1}^k\lambda_\alpha f_\alpha$$ is the new Lagrangian with constraints and:
$$ Q_j=\sum_{\alpha=1}^k\left[\frac{\partial f_\alpha}{\partial\dot{q}_j}\lambda_\alpha+\frac{d}{dt}\left(\frac{\partial f_\alpha}{\partial\dot{q}_j}\right)\lambda_\alpha+\frac{\partial f_\alpha}{\partial\dot{q}_j}\dot{\lambda}_\alpha-\frac{\partial f_\alpha}{\partial q_j}\lambda_\alpha\right] $$
are the generalized forces associated to the constraints. What I don't get is actually the expression for $Q_j$, because according to Goldstein's Classical Mechanics (3rd edition, page 47, equation 2.25), $Q_j$ should have the form:
$$ Q_j=\sum_{\alpha=1}^k\left[\frac{\partial f_\alpha}{\partial q_j}\lambda_\alpha-\frac{d}{dt}\left(\frac{\partial f_\alpha}{\partial\dot{q}_j}\right)\lambda_\alpha-\frac{\partial f_\alpha}{\partial\dot{q}_j}\dot{\lambda}_\alpha\right] \tag{2.25} $$
where there are some inconsistent signs and where the term $\frac{\partial f_\alpha}{\partial\dot{q}_j}\lambda_\alpha$ is not present anymore. Does anybody have some insights?
