- #1

timewalker

- 15

- 1

How to show the Proca equation by using the given Proca Lagrangian?

Surely, I know the Euler-Lagrange equation, but I can't solve this differentiation!(TT)

The given Proca lagrangian is,

[itex]\mathcal{L}= -\frac{1}{16\pi}(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})+ \frac{1}{8 \pi} (\frac{mc}{\hbar})^2 A^{\nu} A_{\nu}[/itex]

and the Euler-Lagrangian equation is,

[itex]\partial_{\mu}(\frac{\partial \mathcal{L}}{\partial(\partial_{\mu} A^{\nu})}) = \frac{\partial \mathcal{L}}{\partial A^\nu}[/itex]

At first, I just tried to solve

[itex]\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}A^{\nu})}= \frac{\partial}{\partial(\partial_{\nu}A^{\mu})}(-\frac{1}{16 \pi}(\partial^{\mu}A^{\nu}-\partial{^\nu}A^{\mu})(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})+\cdots)[/itex]

but I think I am misunderstand and not very well to handle these indices. So I think I can understand if I can see correct solving procedure. Please help me :(

Surely, I know the Euler-Lagrange equation, but I can't solve this differentiation!(TT)

The given Proca lagrangian is,

[itex]\mathcal{L}= -\frac{1}{16\pi}(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})+ \frac{1}{8 \pi} (\frac{mc}{\hbar})^2 A^{\nu} A_{\nu}[/itex]

and the Euler-Lagrangian equation is,

[itex]\partial_{\mu}(\frac{\partial \mathcal{L}}{\partial(\partial_{\mu} A^{\nu})}) = \frac{\partial \mathcal{L}}{\partial A^\nu}[/itex]

At first, I just tried to solve

[itex]\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}A^{\nu})}= \frac{\partial}{\partial(\partial_{\nu}A^{\mu})}(-\frac{1}{16 \pi}(\partial^{\mu}A^{\nu}-\partial{^\nu}A^{\mu})(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})+\cdots)[/itex]

but I think I am misunderstand and not very well to handle these indices. So I think I can understand if I can see correct solving procedure. Please help me :(

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