Do parentheses around partial differentials$(\frac{\partial F}{\partial x}) $after the differential operator d/dt have any special meaning?

Question

Question on a part of the Euler-Lagrange-Equation.

Do the parentheses after the differentiation operator

$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial f'}\right)$

have any syntactic meaning or is it just optional, style, etc.?

I have seen the E-L-Eq. both with those parens and without.

Thanks

Answer 1

When mixed derivatives are taken (independent variable $x$ instead of $t$), the order of differentiation does not matter in.. in wholly partial or wholly full mode.

But such interchange leads to possible incorrect results.

To avoid possible confusion and incorrect result the correct order i.e., inside to outside is specifically and necessarily indicated by usage of parentheses.

E.g., for $ L=y \;y'$

$$\dfrac{d}{dx}\left(\dfrac{\partial L}{\partial y'}\right)= \dfrac{dy}{dx}$$

but

$$ \dfrac{\partial}{\partial y'}\left(\dfrac{dL}{dx}\right)= \dfrac{\partial}{\partial y'} (yy^{''}+y^{'2})= 2\dfrac{dy}{dx}.$$