Calculus in the derivation of Euler-Lagrange equation

  • I
  • Thread starter BearY
  • Start date
  • #1
53
8
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:

Answers and Replies

  • #2
Charles Link
Homework Helper
Insights Author
Gold Member
4,657
2,007
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.
 
  • Like
Likes BearY
  • #3
53
8
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.
 
  • Like
Likes Charles Link

Related Threads on Calculus in the derivation of Euler-Lagrange equation

Replies
3
Views
4K
  • Last Post
Replies
4
Views
3K
Replies
21
Views
23K
Replies
12
Views
4K
  • Last Post
Replies
2
Views
678
Replies
3
Views
1K
Top