Virtual displacement in semi-holonomic constraints

Question

I am currently studying Lagrangian Mechanics for systems whose constraints equations have the form $$\sum_{k=1}^na_{\ell k}(q,t)\dot{q}_k+a_{\ell t}(q,t)=0\tag{1}$$ or, equivalently $$\sum_{k=1}^na_{\ell k}d{q}_k+a_{\ell t}dt=0\tag{2.66}$$ for $\ell=1,\dots,p$. The text I'm reading ("Analytical Mechanics", by Nivaldo Lemos, equation 2.68) states that, in terms of the virtual variations $\delta q_k$, this equation leads to $$\begin{equation} \sum_{k=1}^n a_{\ell k}\delta q_k=0\tag{2.68} \end{equation}$$ since $dt=0$. I tried to explicitly derive eq. (2.68) myself from eq. (1) \begin{align*} 0=&\,\,\delta\left(\sum_{k=1}^na_{\ell k}\dot{q}_k+a_{\ell t}\right)=\sum_{k=1}^n\left(\delta a_{\ell k}\dot{q}_k+a_{\ell k}\delta\dot{q}_k\right)+\delta a_{\ell t}=\\[5pt] =&\,\,\sum_{k=1}^n\left(\sum_{i=1}^n\frac{\partial a_{\ell k}}{\partial q_i}\delta q_i\dot{q}_k+a_{\ell k}\frac{d}{dt}(\delta q_k)\right)+\sum_{i=1}^n\frac{\partial a_{\ell t}}{\partial q_i}\delta q_i=\\[5pt] =&\,\,\sum_{i=1}^n\left(\sum_{k=1}^n\frac{\partial a_{\ell k}}{\partial q_i}\dot{q}_k+\frac{\partial a_{\ell t}}{\partial q_i}\right)\delta q_i+\sum_{k=1}^na_{\ell k}\frac{d}{dt}(\delta q_k)=\\[5pt] =&\,\,\sum_{i=1}^n\frac{\partial}{\partial q_i}\left(\sum_{k=1}^na_{\ell k}\dot{q}_k+a_{\ell t}\right)\delta q_i+\sum_{k=1}^na_{\ell k}\frac{d}{dt}(\delta q_k)=\\[5pt] =&\,\,\sum_{k=1}^na_{\ell k}\frac{d}{dt}(\delta q_k) \end{align*} which is almost what I wanted, except for the derivative of $\delta q_k$. I'm sure I misunderstood something, but I have no idea what it is. Can someone point it out?

Answer 1

Notice that eq. (2.68) follows immediately from eq. (2.66). However OP wants to derive eq. (2.68) from eq. (1). Unfortunately, OP's project is doomed.

1. On one hand, a virtual displacement $\delta$ (in the context of d'Alembert's principle) is only defined for the generalized positions $q^j$ at a fixed time $t$. So it makes sense to apply $\delta$ on eq. (2.66).

2. On the other hand, the virtual displacement $\delta$ of the generalized velocities $\dot{q}^j$ is not well-defined (since in d'Alembert's principle we have not specified an actual solution trajectory). So it does not make sense to apply $\delta$ on eq. (1).

Eq. (2.66) and eq. (1) are only equivalent for an actual solution trajectory, not for a virtual displacement.

References:

1. N.A. Lemos, Analytical Mechanics, 2018; Section 2.4.