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Can I take this product:

$\frac{dL}{dt}\frac{d L}{d \dot{x}}$

And factor out one of the $L$'s to get:

$L\frac{d}{dt} \left( \frac{d L}{d \dot{x}}\right)$

Where the operator $\frac{d}{dt}$ now operates on $\frac{d L}{d \dot{x}}$?

Is this allowed?

Thanks

2 Answers 2

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This is only allowed if $L$ is not a function of $t$. If $L$ is a function of $t$, then this is not allowed.

This is not factoring though, but using the identity that $\frac{d}{d\,x}(cf(x))=c\frac{d}{d\,x}f(x).$

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    I think you're making a mistake in interpreting $\big(\!\frac{dL}{dt}\!\big)\big(\!\frac{dL}{d\dot x}\!\big)$ as $\frac d{dt}\!\big(L\frac{dL}{d\dot x}\!\big)$.2012-07-25
2

Are you asking whether they are equal? Have you tried an example? Almost any one where the first expression is nonzero will do.