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In classical mechanics, when talking about the principle of virtual work, what is difference between $\delta r$ and $dr$? e.g. $W=\int \overrightarrow{F} \cdot \delta \overrightarrow{r} $ and $W=\int \overrightarrow{F} \cdot d \overrightarrow{r} $ .

Why can one exchange the place of $d\delta$ and $d$ in derivative calculation? e.g. $d\delta r=\delta d r$?

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In classical mechanics, $\delta$ is equitemporal variation. $\delta$ and $\mathrm{d}$ are practically the same for constant constraint, but when the constraint is time-varying, they are different.

For example, if a bead is constrained to a moving string, $\delta r$ will be along the string, while $\mathrm{d}r$ won't be.

Conceptually, $\delta$ is variation of a functional, while $\mathrm{d}$ is differential of a function. But in calculations, just change $\mathrm{d}$ to $\delta$, and set $\delta t=0$ and you will get the correct result.

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This is not correct, I stand by my answer. "Equitemporal variation" doesn't mean anything, and both $\delta r$ and $dr$ are along the tangent to the string. –  Ron Maimon Sep 8 '12 at 7:24
    
@RonMaimon: Maybe you should revisit a book on classical mechanics. –  C.R. Sep 8 '12 at 10:00
    
-1: This is not correct, you made it up. I don't read books, knowledge doesn't come from reading. It comes from thinking. –  Ron Maimon Sep 8 '12 at 15:16
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They are not different, they are the same, but "dr" is ossified notation, meaning the integration differential, and physicists often think about infinitesimal increments, so they use different letters to indicate smallness. You sort it out from context.

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