Chain Rule Confusion (Euler-Lagrange Equation)

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The discussion centers on the confusion regarding the differentiation notation used in the Euler-Lagrange equation. Participants clarify that the correct application of the chain rule leads to the expression dF/dx = (∂F/∂y) y' + (∂F/∂y') y'', emphasizing that total derivatives should be decomposed accordingly. There is a consensus that using dF/dy instead of ∂F/∂y is a mistake, as it conflates total and partial derivatives. The importance of recognizing y as a function of x is also highlighted, reinforcing the need for clarity in notation. Overall, the conversation underscores the significance of proper differentiation in the context of the Euler-Lagrange equation.
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Homework Statement
No explicit x dependence for E-L eq.
Relevant Equations
Euler-Lagrange.
1744095877659.png

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!
 
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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.
 
Orodruin said:
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!
 
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.
 
Orodruin said:
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.
 
Yeah ok, that’s just sloppy.
 

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