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I Calculus in the derivation of Euler-Lagrange equation

  1. Jun 19, 2017 #1
    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:oops:

    Edit: NM the second one I was stupid It's just chain rule :oops:
     
    Last edited: Jun 19, 2017
  2. jcsd
  3. Jun 19, 2017 #2

    Charles Link

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    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.
     
  4. Jun 19, 2017 #3
    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.
     
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