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How to set the variation of an integral to 0?

  1. Aug 17, 2015 #1
    So I have an integral:

    ## \delta W=\int_{-\Delta}^\Delta\left[x^2\left(\frac{d\xi}{dx}\right)^2−D_S\xi^2\right]dx ##
    Here ##\xi## is a function of ##x## and ##D_S## is a constant. ##\Delta## is just some small ##x##. Now I need to set the variation of ##\delta W## to 0. Do do this I differentiated whatever is inside the bracket and set it to 0. I get:

    ## x^2\xi″+x\xi′−D_S\xi = 0 ##

    However, the answer is:

    ## \frac{d}{dx}\left(x^2\frac{d\xi}{dx}\right)+D_S\xi = x^2\xi″+2x\xi′+D_S\xi = 0 ##

    Where the primes are derivatives with respect to x. As you can see the difference is a factor of 2 in the middle term and that minus sign.

    If anyone could point out where I am going wrong, it would be really appreciated.

    Thanks!
     
  2. jcsd
  3. Aug 17, 2015 #2

    Orodruin

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    You cannot simply differentiate the integrand with respect to the integration variable. You need to check out a textbook or lecture notes on variational calculus and apply Euler-Lagrange's equations.
     
  4. Aug 18, 2015 #3
    Thanks a lot! Got the answer!
     
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