Am I allowed to move around an operator like this?

Question

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

Answer 1

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 $$\frac{d}{d\,x}(cf(x))=c\frac{d}{d\,x}f(x).$$

Answer 2

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