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I'm having trouble with some of the calculus in moving from the Klein-Gordin Lagrangian density to the equations of motion. The density is:

[tex]

L = \frac{1}{2}\left[ (\partial_μ\phi)(\partial^\mu \phi) - m^2\phi ^2 \right]

[/tex]

Now, to apply the Euler-Lagrange equations one needs to find:

[tex]

\frac{\partial L}{\partial(\partial_\mu\phi)}

[/tex]

Which to me, looked like it could be:

[tex]

\frac{\partial L}{\partial(\partial_\mu\phi)} = \frac{1}{2}\partial^\mu\phi

[/tex]

But that gives the wrong equation of motion - the half shouldn't be there. I guess my mistake is that this can

*sort of*be thought of as being like:

[tex]

L = \frac{1}{2}\left[ (\partial_μ\phi)^2 - m^2\phi ^2 \right]

[/tex]

Which works out fine, but I'm just not quite sure what's happening here. What is the best way to think of this? In particular, how does the positioning of μ change how to think of it. This seems really basic but I'm pretty lost! Can anyone help me understand how it works?

Thanks!