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Integration by parts

  1. Jul 20, 2011 #1
    Somebody could explain me, how of the second line arrive to the third one? in my book says that is integration by parts, please helpppp :eek:
     

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  2. jcsd
  3. Jul 20, 2011 #2
    the same is for this pleaseee :uhh:
     

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  4. Jul 20, 2011 #3

    hunt_mat

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    The first one is quite easy, you have [itex]\delta \dot{\phi}=\partial_{t} \delta \phi[/itex], now pat of the assumptions of calculus of variations is that the variation vanish at the boundaries. I think that will help you clear up your first problem.

    How is quantum field theory treating you?
     
  5. Jul 20, 2011 #4
    yes the variation vanish at the booundaries but i try and get a differente solution mmmmm

    I study classical field theory from Field Quantization-Greiner and Reinhardt
     
  6. Jul 21, 2011 #5

    hunt_mat

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    Okay you have:
    [tex]
    \delta S = \int_{t_{1}}^{t_{2}}\frac{\delta L}{\delta \phi}\delta \phi +\frac{\delta L}{\delta \dot{\phi}}\delta \dot{\phi} d^{3} \mathbf{x}
    [/tex]
    Using the hint that I gave:
    [tex]
    \delta S = \int_{t_{1}}^{t_{2}}\frac{\delta L}{\delta \phi}\delta \phi +\frac{\delta L}{\delta \dot{\phi}}\partial_{t}\delta \phi d^{3} \mathbf{x}
    [/tex]
    The second term can be integrated by parts to obtain:
    [tex]
    \int_{t_{1}}^{t_{2}}\frac{\delta L}{\delta \dot{\phi}}\partial_{t}\delta \phi d^{3}\mathbf{x} =\left[ \frac{\delta L}{\delta \dot{\phi}}\delta \phi\right]_{t_{1}}^{t_{2}}-\int_{t_{1}}^{t_{2}}\frac{\partial}{\partial t}\frac{\delta L}{\delta \dot{\phi}}\delta \phi d^{3}\mathbf{x}
    [/tex]
    Putting this back in the same integral we have:
    [tex]
    \delta S =\int_{t_{1}}^{t_{2}}\frac{\delta L}{\delta \phi}\delta \phi -\frac{\partial}{\partial t}\frac{\delta L}{\delta \dot{\phi}}\delta \phi d^{3}\mathbf{x}
    [/tex]
    So you see now?
     
    Last edited: Jul 21, 2011
  7. Jul 21, 2011 #6
    Thank youuuuuu so much, you're the best
    I'm really happy:biggrin:
     
  8. Jul 21, 2011 #7
    Now I'll try the other one
     
  9. Jul 21, 2011 #8

    hunt_mat

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    For that one, take a simple case of where the field has one space variable, i.e. when [itex]\phi \phi (t,x)[/itex] It will be easier and give you a good feeling for the general case.

    Glad to be of help...
     
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