In a lecture on classical mechanics, the professor derived a formula, which is a part of the D'Alambert principle: [tex]\nabla \Phi_{\alpha} \cdot \delta \vec{r} = 0[/tex] where [tex]\Phi_{\alpha}[/tex] are the restraints. He derived it in a strange way from the Taylor's formula:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\Phi_{\alpha} (\vec{r} + \delta \vec{r}) = \Phi_{\alpha} (\vec{r}) + \delta \vec{r} \cdot \nabla \Phi_{\alpha} + ...[/tex]

He said that the element [tex]\delta \vec{r}[/tex] is infinitesimal and that [tex]\vec{r} , \vec{r} + \delta \vec{r}[/tex] satisfy the restraints.

I understand that the term on the left side of Taylor's formula and the first term on the right side must be therefore equal to zero i.e:

[tex] 0 = \delta \vec{r} \cdot \nabla \Phi_{\alpha} + ...[/tex].

And here comes the critical point. He said that since [tex] \delta \vec{r}[/tex] is very small [tex] \delta \vec{r} \cdot \nabla \Phi_{\alpha} = 0[/tex]

My question is why - there are also the higher terms in the formula. For "very small" elements it might be almost true, but only approximately. So is this principle only approximately true? Can someone derive the last step in a more mathematical way? Thanks a lot.

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# Homework Help: Is the d'Alambert principle universal?

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