Chain Rule Confusion (Euler-Lagrange Equation)

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Homework Help Overview

The discussion revolves around the application of the chain rule in the context of the Euler-Lagrange equation, specifically regarding the differentiation of a function F with respect to its variables. Participants are examining the distinction between total and partial derivatives in this context.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand why the differentiation of F is expressed in terms of total derivatives rather than partial derivatives, questioning the implications of y being an independent variable. Other participants clarify the correct application of the chain rule and express confusion over the notation used by the lecturer.

Discussion Status

Participants are actively engaging with the concepts, with some providing clarifications regarding the application of the chain rule. There is a recognition of differing interpretations of notation, but no explicit consensus has been reached on the underlying confusion.

Contextual Notes

There is mention of potential inconsistencies in notation, specifically the use of total versus partial derivatives, which may contribute to the confusion. The discussion reflects a need for clarity on these definitions in the context of the Euler-Lagrange equation.

laser1
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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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