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In Goldstein's classical mechanics the following passage is confusing me:

We therefore have as the condition for equilibrium of a system that the virtual work of the applied forces vanishes: $$\sum_i \textbf F_i^{(a)}\cdot \delta \textbf r_i=0 \tag{1.43}.$$ Equation (1.43) is often called the principle of virtual work. Note that the coefficients of $\delta \mathbf r_i$ can no longer be set equal to zero; i.e., in general $\mathbf F_i^{(a)}\neq 0,$ since the $\delta \mathbf r_i$ are not completely independent but are connected by constraints.

It was my understanding that $\delta \mathbf r_i$ is a vector quantity, a change in one of the coordinates represented by a vector, which would make its "coefficients" its components. Perhaps this is what I'm misunderstanding, but if not, why would we be interested in setting these to zero? Surely that means we're setting the virtual displacement to zero in which case we're not actually doing anything.

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The phrase "the coefficient of $\delta {\bf r}_i$" refers to the corresponding vector ${\bf F}_i$. It does not mean the components of $\delta {\bf r}_i$.

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  • $\begingroup$ Ah I see, thank you. $\endgroup$
    – Charlie
    Jul 21, 2020 at 12:23

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