Chain Rule Confusion (Euler-Lagrange Equation)

[laser1]

Homework Statement

No explicit x dependence for E-L eq.

Relevant Equations

Euler-Lagrange.

The above image is from my lecturer's notes.

My concern is when it seems like my lecturer has split up the dF/dx term into dF/dy y' + dF/dy' y''. Why is it this as opposed to $\frac{\partial F}{\partial y}$ etc.? Or would this not matter, because y is an independent variable, and hence, the partial F wrt y = total F wrt y? (although I guess y has to be a function of x...)

Thank you!

 

[Orodruin]

Not $dF/dx = (\partial F/\partial y) y’ - (\partial F/\partial y’) y’’$, it is $(\partial F/\partial y) y’ +(\partial F/\partial y’) y’’$. This is just the chain rule.

 

[laser1]

Not $dF/dx = (\partial F/\partial y) y’ - (\partial F/\partial y’) y’’$, it is $(\partial F/\partial y) y’ +(\partial F/\partial y’) y’’$. This is just the chain rule.

sorry, edited!

 

[Orodruin]

I don’t see the issue then. The total d/dx has to be decomposed using the chain rule.

It makes no sense to talk about a total derivative wrt y.

 

[laser1]

I don’t see the issue then. The total d/dx has to be decomposed using the chain rule.

It makes no sense to talk about a total derivative wrt y.

my lecturer uses the notation $dF/dy$ in the second last term. I am confused why it is not $\partial F/\partial y$ instead.

 

[Orodruin]

Yeah ok, that’s just sloppy.