Variational technique

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  • #1
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Main Question or Discussion Point

Suppose I try to find variation of the term \phi^2 under variation of \phi i.e., (\delta \phi). Then I take derivative of \phi^2 with respect to \phi and multiply by \delta \phi. In case of more complicated objects containing derivative of \phi what is the procedure? for example:
[tex]
\delta(\sqrt{\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi})
[/tex]
what is the variation of this quantity under variation of \phi?
 

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  • #2
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Is the rule here
[tex]
\delta(\sqrt{\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi})=\frac{\partial(\sqrt{\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi})}{\partial \phi}\delta(\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi)
[/tex]
But then the derivative part creates problem. The part under square root depends on derivative of \phi and not on \phi itself, so the result is zero. I am confused.
 
Last edited:
  • #3
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Always go back to the definition of the derivative:

[tex]\delta F[\phi] = F[\phi + \delta \phi] - F[\phi][/tex].

In any case, in your case the chain rule applies:
[tex]

\delta(\sqrt{\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi})=
\frac{1}{ 2\sqrt{\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi} }
\delta(\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi)

[/tex]
 

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