- #1
Irrational
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Hi,
I'm trying to work through something and it should be quite simple but somehow I've gotten a bit confused.
I've worked through the Euler Lagrange equations for the lagrangian:
<br /> \begin{align*}<br /> \mathcal{L}_{0} &= -\frac{1}{4}(\partial_{\mu}A_{\nu})(\partial^{\mu}A^{\nu}) \\<br /> &= \frac{1}{4}F_{\mu\nu}F^{\mu\nu}<br /> \end{align*}<br />
getting:
\Box A_{\nu} - \partial^{\nu}\partial_{\mu}A^{\mu} = 0
I'm ok with this.
Then considering the modified lagrangian:
\mathcal{L}_{\xi} = \mathcal{L}_{0} + \frac{\lambda}{2}(\partial_{\sigma}A^{\sigma})^2
I'm trying to work out the EL equation components and as part of one of these calculations, I've to calculate:
<br /> \begin{align*}<br /> \frac{\partial}{\partial(\partial_{\mu}A_{\nu})} \left[ \frac{\lambda}{2} (\partial_{\sigma}A^{\sigma})^2 \right] <br /> <br /> &= \frac{\lambda}{2} \frac{\partial}{\partial(\partial_{\mu}A_{\nu})} \left[ ( \partial_{\sigma}A^{\sigma} ) ( \partial_{\rho}A^{\rho} ) \right] \\<br /> <br /> &= \frac{\lambda}{2} \frac{\partial}{\partial(\partial_{\mu}A_{\nu})} \left[ ( \partial_{\sigma}A_{\alpha} \eta^{\sigma \alpha} ) ( \partial_{\rho}A_{\beta} \eta^{\rho \beta} ) \right] \\<br /> <br /> & = \frac{\lambda}{2} \eta^{\sigma \alpha} \eta^{\rho \beta} \frac{\partial}{\partial(\partial_{\mu}A_{\nu})} \left[ ( \partial_{\sigma}A_{\alpha} ) ( \partial_{\rho}A_{\beta} ) \right] \\<br /> <br /> & = \frac{\lambda}{2} \eta^{\sigma \alpha} \eta^{\rho \beta} \left[ ( \partial_{\sigma}A_{\alpha} ) \left( \frac{\partial}{\partial(\partial_{\mu}A_{\nu})} ( \partial_{\rho}A_{\beta} ) \right) + \left( \frac{\partial}{\partial(\partial_{\mu}A_{\nu})} ( \partial_{\sigma}A_{\alpha} ) \right) ( \partial_{\rho}A_{\beta} ) \right] \\<br /> <br /> & = \frac{\lambda}{2} \eta^{\sigma \alpha} \eta^{\rho \beta} \left[ ( \partial_{\sigma}A_{\alpha} ) \delta^{\mu}_{\rho} \delta^{\nu}_{\beta} + \delta^{\mu}_{\sigma} \delta^{\nu}_{\alpha} ( \partial_{\rho}A_{\beta} ) \right] \\<br /> <br /> & = \frac{\lambda}{2} \eta^{\sigma \alpha} \eta^{\rho \beta} ( \partial_{\sigma}A_{\alpha} ) \delta^{\mu}_{\rho} \delta^{\nu}_{\beta}<br /> + <br /> \frac{\lambda}{2} \eta^{\sigma \alpha} \eta^{\rho \beta} \delta^{\mu}_{\sigma} \delta^{\nu}_{\alpha} ( \partial_{\rho}A_{\beta} ) \\<br /> <br /> & = \frac{\lambda}{2} \eta^{\sigma \alpha} \eta^{\mu \nu} ( \partial_{\sigma}A_{\alpha} )<br /> + <br /> \frac{\lambda}{2} \eta^{\mu \nu} \eta^{\rho \beta} ( \partial_{\rho}A_{\beta} ) \\<br /> <br /> & = \frac{\lambda}{2} \eta^{\mu \nu} \left[ ( \partial_{\sigma}A^{\sigma} )<br /> + <br /> ( \partial_{\rho}A^{\rho} ) \right] \\<br /> <br /> & = \lambda \eta^{\mu \nu} ( \partial_{\sigma}A^{\sigma} ) \\<br /> <br /> \end{align*}<br />
Now I was hoping to get:
<br /> \lambda \partial^{\nu} A^{\mu}<br />
as ultimately I need the EL equations to give me:
<br /> \begin{align*}<br /> \frac{\partial \mathcal{L}_{\xi}}{\partial A_{\nu}} - \partial_{\mu} \left( \frac{\partial \mathcal{L}_{\xi}}{\partial (\partial_{\mu} A_{\nu})} \right)<br /> &=\Box A^{\nu} - \partial^{\nu} ( \partial_{\mu} A^{\mu} ) - \lambda \partial^{\nu}(\partial_{\mu} A^{\mu}) \\<br /> &= \Box A^{\nu} - ( 1 + \lambda ) \partial^{\nu} ( \partial_{\mu} A^{\mu} ) \\<br /> &= 0<br /> \end{align*}<br />
Can anyone show me where I've gone wrong? I didn't stick this in the homework section as it's not homework. I'm just trying to work through the through missing steps from the text I'm reading.
Thanks in advance
I'm trying to work through something and it should be quite simple but somehow I've gotten a bit confused.
I've worked through the Euler Lagrange equations for the lagrangian:
<br /> \begin{align*}<br /> \mathcal{L}_{0} &= -\frac{1}{4}(\partial_{\mu}A_{\nu})(\partial^{\mu}A^{\nu}) \\<br /> &= \frac{1}{4}F_{\mu\nu}F^{\mu\nu}<br /> \end{align*}<br />
getting:
\Box A_{\nu} - \partial^{\nu}\partial_{\mu}A^{\mu} = 0
I'm ok with this.
Then considering the modified lagrangian:
\mathcal{L}_{\xi} = \mathcal{L}_{0} + \frac{\lambda}{2}(\partial_{\sigma}A^{\sigma})^2
I'm trying to work out the EL equation components and as part of one of these calculations, I've to calculate:
<br /> \begin{align*}<br /> \frac{\partial}{\partial(\partial_{\mu}A_{\nu})} \left[ \frac{\lambda}{2} (\partial_{\sigma}A^{\sigma})^2 \right] <br /> <br /> &= \frac{\lambda}{2} \frac{\partial}{\partial(\partial_{\mu}A_{\nu})} \left[ ( \partial_{\sigma}A^{\sigma} ) ( \partial_{\rho}A^{\rho} ) \right] \\<br /> <br /> &= \frac{\lambda}{2} \frac{\partial}{\partial(\partial_{\mu}A_{\nu})} \left[ ( \partial_{\sigma}A_{\alpha} \eta^{\sigma \alpha} ) ( \partial_{\rho}A_{\beta} \eta^{\rho \beta} ) \right] \\<br /> <br /> & = \frac{\lambda}{2} \eta^{\sigma \alpha} \eta^{\rho \beta} \frac{\partial}{\partial(\partial_{\mu}A_{\nu})} \left[ ( \partial_{\sigma}A_{\alpha} ) ( \partial_{\rho}A_{\beta} ) \right] \\<br /> <br /> & = \frac{\lambda}{2} \eta^{\sigma \alpha} \eta^{\rho \beta} \left[ ( \partial_{\sigma}A_{\alpha} ) \left( \frac{\partial}{\partial(\partial_{\mu}A_{\nu})} ( \partial_{\rho}A_{\beta} ) \right) + \left( \frac{\partial}{\partial(\partial_{\mu}A_{\nu})} ( \partial_{\sigma}A_{\alpha} ) \right) ( \partial_{\rho}A_{\beta} ) \right] \\<br /> <br /> & = \frac{\lambda}{2} \eta^{\sigma \alpha} \eta^{\rho \beta} \left[ ( \partial_{\sigma}A_{\alpha} ) \delta^{\mu}_{\rho} \delta^{\nu}_{\beta} + \delta^{\mu}_{\sigma} \delta^{\nu}_{\alpha} ( \partial_{\rho}A_{\beta} ) \right] \\<br /> <br /> & = \frac{\lambda}{2} \eta^{\sigma \alpha} \eta^{\rho \beta} ( \partial_{\sigma}A_{\alpha} ) \delta^{\mu}_{\rho} \delta^{\nu}_{\beta}<br /> + <br /> \frac{\lambda}{2} \eta^{\sigma \alpha} \eta^{\rho \beta} \delta^{\mu}_{\sigma} \delta^{\nu}_{\alpha} ( \partial_{\rho}A_{\beta} ) \\<br /> <br /> & = \frac{\lambda}{2} \eta^{\sigma \alpha} \eta^{\mu \nu} ( \partial_{\sigma}A_{\alpha} )<br /> + <br /> \frac{\lambda}{2} \eta^{\mu \nu} \eta^{\rho \beta} ( \partial_{\rho}A_{\beta} ) \\<br /> <br /> & = \frac{\lambda}{2} \eta^{\mu \nu} \left[ ( \partial_{\sigma}A^{\sigma} )<br /> + <br /> ( \partial_{\rho}A^{\rho} ) \right] \\<br /> <br /> & = \lambda \eta^{\mu \nu} ( \partial_{\sigma}A^{\sigma} ) \\<br /> <br /> \end{align*}<br />
Now I was hoping to get:
<br /> \lambda \partial^{\nu} A^{\mu}<br />
as ultimately I need the EL equations to give me:
<br /> \begin{align*}<br /> \frac{\partial \mathcal{L}_{\xi}}{\partial A_{\nu}} - \partial_{\mu} \left( \frac{\partial \mathcal{L}_{\xi}}{\partial (\partial_{\mu} A_{\nu})} \right)<br /> &=\Box A^{\nu} - \partial^{\nu} ( \partial_{\mu} A^{\mu} ) - \lambda \partial^{\nu}(\partial_{\mu} A^{\mu}) \\<br /> &= \Box A^{\nu} - ( 1 + \lambda ) \partial^{\nu} ( \partial_{\mu} A^{\mu} ) \\<br /> &= 0<br /> \end{align*}<br />
Can anyone show me where I've gone wrong? I didn't stick this in the homework section as it's not homework. I'm just trying to work through the through missing steps from the text I'm reading.
Thanks in advance
Last edited:
Well, up to a minus sign anyway. You've got λ ημν(∂σAσ). So plug this into the Euler-Lagrange equation: