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Equilibrium configuration in Lagrangian mechanics

  1. May 30, 2012 #1
    Suppose we have a system with scleronomic constraints. Is the condition that ∂V/∂qj=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/∂qj=0.

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

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

    Now we prove that it is a sufficient condition. To do this, we find the [itex]\vec{F}_i[/itex]'s as a function of the Qjs by making virtual displacements δqj to the generalized coordinates. The the virtual work is
    [itex]\delta W = \sum_j Q_j \delta q_j = \sum_i \vec{F}_i \cdot \delta \vec{r}_i[/itex]. Writing [itex]\delta q_j = \sum_i \nabla_i q_j\cdot\delta \vec{r}_i[/itex] (we've tacitly expressed the generalized coordinates as functions of the ri's; [itex]\nabla_i q_j[/itex] stands for [itex]\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}[/itex]).

    From this, it follows that [itex]\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[/itex], implying that [itex]\vec{F}_i=\sum_j Q_j \nabla_i q_j[/itex] 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?
  2. jcsd
  3. Jun 2, 2012 #2
    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.
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