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## Main Question or Discussion Point

Hi, I hope I put this in the right place!

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

[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!

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!