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Maniac_XOX

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- TL;DR Summary
- Im trying to find the laws of motion for a section of a lagrangian that would represent an extra operator to the proca theory for investigating additional properties of photons inside superconductors. Any help?

The euler lagrange equation I am using is:

$$\frac {\partial^\beta \partial L}{\partial(\partial^\beta A^\alpha) }= \frac {\partial L} {\partial A^\alpha}$$ Now the proca lagrangian i am using is $$L= -\frac {1}{16\pi} F_{\alpha\beta} F^{\alpha\beta} + \frac {\mu^2} {8\pi} A_\alpha A^\alpha - \frac {1} {c} J_\alpha A^\alpha$$ No problem in finding the laws of motion for this lagrangian, being $$\partial^\beta F_{\beta\alpha} + \mu^2 A_\alpha = \frac {4\pi}{c} J_\alpha$$ the extra operator that I have added to the proca lagrangian is $$-\beta A_\mu A^\mu (\partial_\rho A^\rho)$$ taken from https://arxiv.org/pdf/1402.6450.pdf, equation (3).

Using the euler lagrange equation, I found that:

(1) $$\partial^\beta \frac {\partial L}{\partial (\partial^\beta A^\alpha)}=-\partial^\beta(g_{\rho\sigma} \beta A_\mu A^\mu)$$ If i lower the index can this turn into ##-\partial^\beta(\beta A_\mu A_\mu)## ??

And also:

(2) $$\frac {\partial L}{\partial A^\alpha}=-2\beta A_\mu (\partial_\rho A^\rho)$$

Therefore

(3) $$\partial^\beta(\beta A_\mu A_\mu)=2\beta A_\mu (\partial_\rho A^\rho)$$ Is this okay?

$$\frac {\partial^\beta \partial L}{\partial(\partial^\beta A^\alpha) }= \frac {\partial L} {\partial A^\alpha}$$ Now the proca lagrangian i am using is $$L= -\frac {1}{16\pi} F_{\alpha\beta} F^{\alpha\beta} + \frac {\mu^2} {8\pi} A_\alpha A^\alpha - \frac {1} {c} J_\alpha A^\alpha$$ No problem in finding the laws of motion for this lagrangian, being $$\partial^\beta F_{\beta\alpha} + \mu^2 A_\alpha = \frac {4\pi}{c} J_\alpha$$ the extra operator that I have added to the proca lagrangian is $$-\beta A_\mu A^\mu (\partial_\rho A^\rho)$$ taken from https://arxiv.org/pdf/1402.6450.pdf, equation (3).

Using the euler lagrange equation, I found that:

(1) $$\partial^\beta \frac {\partial L}{\partial (\partial^\beta A^\alpha)}=-\partial^\beta(g_{\rho\sigma} \beta A_\mu A^\mu)$$ If i lower the index can this turn into ##-\partial^\beta(\beta A_\mu A_\mu)## ??

And also:

(2) $$\frac {\partial L}{\partial A^\alpha}=-2\beta A_\mu (\partial_\rho A^\rho)$$

Therefore

(3) $$\partial^\beta(\beta A_\mu A_\mu)=2\beta A_\mu (\partial_\rho A^\rho)$$ Is this okay?

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