Hi, I have some questions which I encountered during my thesis-writing, I hope some-one can help me out on this :)(adsbygoogle = window.adsbygoogle || []).push({});

First, I have some problems interpreting coordinate-transformations ( "active and passive") and the derivation of the Equations of Motion. We have

[tex]

S = \int L(\phi, \partial_{\mu} \phi) d^{4}x

[/tex]

and

[tex]

\delta S = 0

[/tex]

, in which the variation of the field is arbitrary. My question is: how exactly is this variation defined? One has 2 options:

[tex]

\delta \phi = \phi^{'}(x^{'}) - \phi (x)

[/tex]

or

[tex]

\delta \phi = \phi^{'}(x) - \phi (x)

[/tex]

where the difference lies in the argument. In notes of Aldrovandi and Pereira ( Notes for a classical course on fields ) option 1 is choosen. And in Inverno they say that option 2 is a coordinate transformation. Why is that? I tend to choose for option two, because here you actually change the field; I would say that a scalar quantity is invariant under transformation 1, so here you just state general covariance in stead of obtaining the Equations of Motion. At the other hand, you can always choose x=x'. In the end I want to look at why one is able to commute variations and partial derivatives, so I need the exact definition of the variation of the field.

Another question concerns Stokes theorem. I understand the theorem totally for n-forms ( where you define the boundary of your region with chains etc ), but why is it also valid for example in the derivation of the Euler-Lagrange equations or tensor densities?

Many thanks in forward :)

-edit My TeX-code is not working properly for some reason, but I hope it is clear.

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# Variations, Euler-Lagrange, and Stokes

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