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Real scalar field , Action, variation, deriving EoM

  1. Oct 29, 2016 #1
    ## L(x) = L(\phi(x), \partial_{u} \phi (x) ) = -1/2 (m^{2} \phi ^{2}(x) + \partial_{u} \phi(x) \partial^{u} \phi (x))## , the Lagrange density.
    ## S= \int d^{4}(x) L (x) ##, the action.

    ## \phi -> \phi + \delta \phi ## (just shortened the notation and dropped the x dependence)

    I have ## \delta S = \int d^{4} x ( -1/2( \delta (m^{2} \phi ^{2}) + \delta (\partial_{u} \phi \partial^{u} \phi )) =0 ## last equality by principle of least action .

    I have ##\delta (m^{2} \phi ^{2}) = m^{2} \phi \delta \phi ## , which is fine.

    I'm having problems with the next term:

    ##\delta (\partial_{u} \phi \partial^{u} \phi )) = \delta (\partial_{u} \phi ) \partial^{u} \phi + \partial_{u} \phi \delta ( \partial^{u} \phi ) ##

    I need to show that is equal to ##\partial_{u} \phi \partial^{u} (\delta \phi) ## I am then fine with the rest of the derivation, which involves doing integration by parts on this term to 'change' ## \partial^{u} (\delta \phi) ## to ##\delta \phi## and then loosing this arbitrary variation in ##\phi## to get the equations of motion.

    I am clueless how to get ## \delta (\partial_{u} \phi ) \partial^{u} \phi + \partial_{u} \phi \delta ( \partial^{u} \phi ) = \partial_{u} \phi \partial^{u} (\delta \phi) ##, any tips getting starting greatly appreciated.

    Many thanks in advance.
     
  2. jcsd
  3. Oct 29, 2016 #2

    Orodruin

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    This is not fine, you are missing a factor of two. You are also missing the same factor of two in the other relation.

    Try to use this to figure out how to solve your problem.
     
  4. Oct 30, 2016 #3
    apologies that was a typo, well it cancels with the 1/2 anyway.
    I don't see how it sheds any light on the issue I stated - the variation is acting seperately on ##\partial_{u}## and ##\partial^{u}##.
    I think that ## \delta ( \partial_{u} \phi ) = \delta \phi' ##, were ##\phi' = \partial x_{u} \phi ##, but I don't know what is ##partial^{u} \phi ##, actually...what is ## \partial^{u} ## ? , ##\partial_{u} x = \partial / \partial x^{u} ## and ##\partial x^{u} ## can be attained by raising a index in this? but I don't really know what it is, so I don't know what ##partial^{u} \phi ## is.

    Many thanks
     
  5. Oct 31, 2016 #4
  6. Nov 2, 2016 #5
    it's okay got it. thanks for the help guys :thumbup:
     
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