Equilibrium configuration in Lagrangian mechanics

[ralqs]

Suppose we have a system with scleronomic constraints. Is the condition that ∂V/∂q_j=0 for generalized coordinates qj a necessary condition for equilibrium? A sufficient condition?

I managed to "prove" that the above condition is necessary and sufficient for any type of holonomic constaint, sclerenomic or rheonomic. This must be a mistake, because I can find an example of a rheonomic system where the equilibrium points don't satisfy ∂V/∂q_j=0.

System is in equilibrium iff $\vec{F}_i=0$, where $\vec{F}_i$ is the total force on the ith particle.

Now, $Q_j=\sum_i \vec{F}_i\cdot\frac{\partial \vec{r}_i}{\partial q_j}$ where Q_j is the generalized force associated with the jth generalized coordinate. So, if $\vec{F}_i=0$ then Qj = 0. But $Q_j=-\frac{\partial V}{\partial q_j}$, so ∂V/∂q_j=0 is a necessary condition for equilibrium.

Now we prove that it is a sufficient condition. To do this, we find the $\vec{F}_i$'s as a function of the Q_js by making virtual displacements δq_j to the generalized coordinates. The the virtual work is
$\delta W = \sum_j Q_j \delta q_j = \sum_i \vec{F}_i \cdot \delta \vec{r}_i$. Writing $\delta q_j = \sum_i \nabla_i q_j\cdot\delta \vec{r}_i$ (we've tacitly expressed the generalized coordinates as functions of the ri's; $\nabla_i q_j$ stands for $\hat{x}_i\frac{\partial q_j}{\partial x_i}+\hat{y}_i\frac{\partial q_j}{\partial y_i}+\hat{z}_i\frac{\partial q_j}{\partial z_i}$).

From this, it follows that $\sum_i \vec{F}_i\cdot\delta \vec{r}_i = \sum_i (\sum_j Q_j \nabla_i q_j)\cdot \delta \vec{r}_i$, implying that $\vec{F}_i=\sum_j Q_j \nabla_i q_j$ herefore, if Q_j = 0, system is in equilibrium. QED?

Now, as far as I can tell I haven't used the assumption that the constraints are scleronomic, but maybe the assumption sneaked in there somewhere. However, there *must* be a mistake somewhere. Can anyone spot it?

 

[ralqs]

No one has answered my question. I can only assume that I was unclear in formulating it. So please, if there's something in my post that is confusing, let me know so I can clarify what I'm trying to ask.