I'm reading a book - and I've been stuck for a while on the same page. This is only a calculus question. We have the action:
[tex]S=\int d^4x \;\mathcal{L}[/tex]
with the Lagrangian (density):
[tex]\mathcal{L}=\mathcal{L}(\phi,\dot{\phi},\nabla\phi)[/tex]
We then vary S:
[tex]\delta S = \int d^4x \left[\frac{\partial \mathcal{L}}{\partial \phi}\delta\phi + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \delta(\partial_\mu \phi)\right] [/tex]
which is all fine and dandy, but now the next line says:
[tex] = \int d^4 x \left[\frac{\partial \mathcal{L}}{\partial \phi}-\partial_\mu \left(\frac{\partial \mathcal{L}}{\partial (\partial_\mu\phi)} \right) \right]\delta \phi + \partial_\mu \left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu\phi)} \delta \phi\right)[/tex]
Something like integration by parts must have befallen the second term...but I don't see it. I'm very inadequate in variational calculus, the mere sight of [itex]\delta[/itex] throws me off.
So what happened there between the two lines? Thanks
[tex]S=\int d^4x \;\mathcal{L}[/tex]
with the Lagrangian (density):
[tex]\mathcal{L}=\mathcal{L}(\phi,\dot{\phi},\nabla\phi)[/tex]
We then vary S:
[tex]\delta S = \int d^4x \left[\frac{\partial \mathcal{L}}{\partial \phi}\delta\phi + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \delta(\partial_\mu \phi)\right] [/tex]
which is all fine and dandy, but now the next line says:
[tex] = \int d^4 x \left[\frac{\partial \mathcal{L}}{\partial \phi}-\partial_\mu \left(\frac{\partial \mathcal{L}}{\partial (\partial_\mu\phi)} \right) \right]\delta \phi + \partial_\mu \left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu\phi)} \delta \phi\right)[/tex]
Something like integration by parts must have befallen the second term...but I don't see it. I'm very inadequate in variational calculus, the mere sight of [itex]\delta[/itex] throws me off.
So what happened there between the two lines? Thanks
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