What Defines an Ideal Constraint in Physics?

[A_B]

Hi,

What exactly is the condition for a constraint to be ideal? Let's call the net force of constraint on particle i $\bar{N_i}$. Is the condition
$$\sum_i\bar{N_i}\cdot\delta\bar{r_i}=0?$$
Or is it
$$\bar{N_i}\cdot\delta\bar{r_i}=0$$ for each i? (from which the first follows immediately)

From the second, it follows that all constraint forces must be perpendicular to the allowed virtual displacements, while this does not follow from the first condition.Also , I don't understand why it is said we can't conclude from
$$\sum_i\bar{K_i}\cdot\delta\bar{r_i}=0$$ (equilibrium for statics)
that the $\bar{K_i}$ are zero BECAUSE the $\bar{r_i}$ are not independent. Suppose they are independent, isn't all we can say that the forces $\bar{K_i}$ would be perpendicular to the virtual displacement?
Is it because
$$\sum_i\bar{K_i}\cdot\delta\bar{r_i}=0$$
must hold for every possible virtual displacement that the $\bar{K_i}$ must be zero?

(i.e. all $\bar{r_i}$ independent -> no constraints -> $\bar{r_i}$ can be in any direction -> $\bar{K_i}$ must be zero)
Thanks

A_B

 

[Valerie Park]

Yes, the second condition is the one that defines an ideal constraint. All of the constraint forces must be perpendicular to the allowed virtual displacements for an ideal constraint to hold.The reason that you can't conclude from $\sum_i\bar{K_i}\cdot\delta\bar{r_i}=0$ that the $\bar{K_i}$ are zero is because the $\bar{r_i}$ are not independent. This means that when you change one $\bar{r_i}$, other $\bar{r_i}$ will also be affected. Thus, the direction of the force $\bar{K_i}$ may not be perpendicular to the virtual displacement of the particular $\bar{r_i}$. If all $\bar{r_i}$ were independent, then it would be possible to say that the $\bar{K_i}$ would be perpendicular to the virtual displacement, but this is not the case in general. In this situation, you can conclude that the $\bar{K_i}$ must be zero for every possible virtual displacement if $\sum_i\bar{K_i}\cdot\delta\bar{r_i}=0$ must hold for every possible virtual displacement.