# Constrain forces and virtual displacements

1. May 4, 2012

### ralqs

I've read conflicting descriptions of the relationship between constraint forces and virtual displacements that I'd like to clarify.

Suppose $\vec{C}_i$ is the constraint force on the ith particle, and $\delta \vec{r}_i$ is its virtual displacement. Is it the case that $\vec{C}_i\cdot\vec{r}_i=0$? Or can one only say that $\sum_i \vec{C}_i\cdot\vec{r}_i=0$?

The first would be true, as I see it, if the virtual displacements were always perpendicular to the constraint forces, even if the constraints were moving...or does the movement of the constraint even matter if time is kept constant?

And if the first isn't true, how would one prove the second formula?

This evidently doesn't matter for the derivation of the Euler-Lagrange equations, but I would still like to clear this confusion up.

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