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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?


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up vote 1 down vote accepted

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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Thanks. $L$ is a typical Lagrangian $L=L(x,\dot{x},t)$, so it is a function of $x$ which is in turn parameterized by $t$. I guess this means that the identity does not apply. – ben Jul 25 '12 at 3:10
That is correct, yes. – Daryl Jul 25 '12 at 3:27
I think you're making a mistake in interpreting $\big(\!\frac{dL}{dt}\!\big)\big(\!\frac{dL}{d\dot x}\!\big)$ as $\frac d{dt}\!\big(L\frac{dL}{d\dot x}\!\big)$. – Rahul Jul 25 '12 at 4:43

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

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