# Calculus in the derivation of Euler-Lagrange equation

• I
In the derivation of Euler-Lagrange equation, when differentiating S with respect to α, there is a step:
$$\frac{\partial f(Y,Y',x)}{\partial\alpha}=\frac{\partial f}{\partial y}\frac{\partial y}{\partial\alpha}+\frac{\partial f}{\partial y'}\frac{\partial y'}{\partial\alpha}$$
Where $$Y = y(x)+\alphaη(x)$$

My puny math knowledge can't tell me 2 things:
1.why is it ##\frac{\partial y}{\partial\alpha}## instead of ##\frac{\partial Y}{\partial\alpha}##? Isn't the second one equal to η? Why is the first one equal to η? Did I skip something?

2. Where does the plus sign come from? I learned partial derivative before but I cannot recall anything like this. I have a feeling this is the result of forgetting something completely Edit: NM the second one I was stupid It's just chain rule Last edited:

Related Classical Physics News on Phys.org
Homework Helper
Gold Member
Wikipedia does a good write-up. They use the letter ## g ## instead of ## Y ##, but comparing the two, you are correct that it should be a capital ## Y ## and ## Y' ## in those terms.

• BearY
Wikipedia does a good write-up. They use the letter ## g ## instead of ## Y ##, but comparing the two, you are correct that it should be a capital ## Y ## and ## Y' ## in those terms.
That Wikipedia page really helped, thanks. I see that ##\frac{\partial S}{\partial\alpha}## when ##\alpha = 0## is the integrand we are looking for. My book didn't bother explaining that directly.

• 