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Goldstein in "Classical Mechanics" (1ed) obtains

$-\int \sum_j \left(\frac {\partial V}{\partial q_j} \delta q_j + \frac{\partial V}{\partial \dot q_j} \delta \dot q_j\right) dt=-\delta \int V dt$

from

$- \int \sum_j \delta q_j \left( \frac {\partial V}{\partial q_j}- \frac {d}{dt} \frac {\partial V}{\partial \dot q_j}\right) dt.$

Could you explain me a little about the steps?

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    @wj32 Goldstein says that he has "reversed" integration by parts of the left-hand of first expression and so he has obtained the right-hand of first expression. And he also says that he started from the second expression. I don't know anything else.. :(2012-11-02

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This is a classical trick used in calculus of variation i.e. integrate by parts using $\,\frac d{dt} \delta q=\delta \dot q$ :

$\int \delta q \frac d{dt} \left(\frac{\partial V}{\partial \dot q}\right)\, dt=\left[ \delta q \frac{\partial V}{\partial \dot q}\right]-\int \frac{\partial V}{\partial \dot q}\frac d{dt}\delta q \, dt=-\int \frac{\partial V}{\partial \dot q} \delta \dot q \, dt$

(using too the hypothesis made for large values).

I wrote only the term at the right (the left one $\ \delta q \frac{\partial V}{\partial q}\ $ is unchanged). Hoping this clarified,

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    @sunrise: You are welcome !2012-11-02