I Understanding the Equations of Motion for the Dirac Lagrangian

JohnH
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Looking for a proof of
∂(∂L/∂[overline]Ψ[\overline])=0
and
∂L/∂Ψ=-[overline]Ψ[\overline]m
and
∂L/∂(∂[SUB]μ[/SUB]Ψ)=i[overline]Ψ[\overline]γ[SUP]μ[/SUP]
I'm having trouble following a proof of what happens when the Dirac Lagrangian is put into the Euler-Lagrange equation. This is the youtube video: and you can skip to 2:56 and pause to see all the math laid out. I understand the bird's eye results of the Dirac Lagrangian having an equation of motion that is either the Dirac equation or a [overline]Ψ[\overline] version of the Dirac equation, but I'm unclear about why, when putting the Dirac Lagrangian into the Euler-Lagrange equation, the following pieces are:

∂(∂L/∂[overline]Ψ[\overline])=0
and
∂L/∂Ψ=-[overline]Ψ[\overline]m
and
∂L/∂(∂μΨ)=i[overline]Ψ[\overline]γμ

Thanks for all replies.
 
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You get it by direct computation. The equations of motion for a field ##\psi## described by some Lagrangian ##\mathcal{L}## are the Euler-Lagrange equations $$\frac{\partial\mathcal{L}}{\partial\psi} - \frac{\partial}{\partial x^\mu}\frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi)}=0 \rm{.}$$ In units of ##\hbar=c=1##, the Dirac Lagrangian can be written as $$\mathcal{L} = \bar{\psi}\left(\mathrm{i}\gamma^\mu\partial_\mu - m\right)\psi = \bar{\psi}\mathrm{i}\gamma^\mu\partial_\mu\psi - m\bar{\psi}\psi \rm{.}$$ Now, you can plug this Lagrangian into the equations of motion above in which you differentiate with respect to ##\psi## - in this way you will obtain the equations of motion for the field ##\bar{\psi}##. Alternatively, you can differentiate with respect to ##\bar{\psi}## in the Euler-Lagrange equations, thereby obtaining the standard Dirac equation (i.e., the equations of motion for the field ##\psi##).

Observe that in the Lagrangian there is no term of the form ##\partial_\mu\bar{\psi}##, hence you get $$\frac{\partial\mathcal{L}}{\partial(\partial_\mu\bar{\psi})}=0 \rm{,}$$ which shows you that ##\frac{\partial\mathcal{L}}{\partial\bar{\psi}}=0## - this is one of the things you asked about. You can work out the rest of the identities from your Question yourself in the same manner.
 
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