- #1

- 80

- 0

**1. Hey,**

I have to find Maxwells equations using the variational principle and the electromagnetic action:

[tex]S=-\intop d^{4}x\frac{1}{4}F_{\mu\nu}F^{\mu\nu}[/tex]

by using

[tex]\frac{\delta s}{\delta A_{\mu(x)}}=0

[/tex]

therefore [tex]\partial_{\mu}F^{\mu\nu}=0

[/tex]

I have to find Maxwells equations using the variational principle and the electromagnetic action:

[tex]S=-\intop d^{4}x\frac{1}{4}F_{\mu\nu}F^{\mu\nu}[/tex]

by using

[tex]\frac{\delta s}{\delta A_{\mu(x)}}=0

[/tex]

therefore [tex]\partial_{\mu}F^{\mu\nu}=0

[/tex]

**3. I have had a go at the solution:**

[tex]S[\varphi]=-\intop d^{4}y\frac{1}{4}F_{\mu\nu}F^{\mu\nu}

[/tex]

[tex]-\int d^{4}y\frac{1}{4}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})[/tex]

[tex]\frac{\delta s}{\delta A_{\mu(x)}}=\frac{\delta s}{\delta A_{\mu(x)}}\int d^{4}y\frac{1}{4}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})

[/tex]

[tex]=-\frac{1}{4}\frac{\delta s}{\delta A_{\mu(x)}}\int2(\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}-\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu}

[/tex]

[tex]=\frac{1}{2}\int d^{4}y\eta^{\mu\alpha}\eta^{\nu\beta}\frac{\delta s}{\delta A_{\mu(x)}}(\partial_{\mu}A_{\nu}\partial_{\beta}A_{\alpha}-\partial_{\mu}A_{\nu}\partial_{\alpha}A_{\beta}

[/tex]

[tex]=\frac{1}{2}\int d^{4}y\eta^{\mu\alpha}\eta^{\nu\beta}A_{\nu}\frac{\delta s}{\delta A_{\mu(x)}}(-\partial_{\mu}\partial A_{\alpha}+\partial_{\mu}\partial_{\alpha}A_{\beta})

[/tex]

[tex]=\frac{1}{2}\int d^{4}yA_{\nu}\frac{\delta s}{\delta A_{\mu(x)}}(-\eta^{\mu\alpha}\eta^{\nu\beta}\partial_{\mu}\partial_{\beta}A_{\alpha}+\eta^{\mu\alpha}\eta^{\nu\beta}\partial_{\mu}\partial_{\alpha}A_{\beta})

[/tex]

[tex]=\frac{1}{2}\int d^{4}y\frac{\delta s}{\delta A_{\mu(x)}}(-A_{\nu}\partial_{\mu}\partial^{\nu}A^{\alpha}+A_{\nu}\partial_{\mu}\partial^{\mu}A^{\nu})

[/tex]

[tex]=\frac{1}{2}\int d^{4}yA_{\nu}\frac{\delta s}{\delta A_{\mu(x)}}(\partial_{\mu}\partial^{\nu}A^{\alpha}-\partial_{\mu}\partial^{\mu}A^{\nu})

[/tex]

[tex]=\frac{1}{2}\int d^{4}x\frac{\delta A_{\nu(y)}}{\delta A_{\mu(x)}}(\partial_{\mu}\partial^{\nu}\frac{\delta A^{\alpha(y)}}{\delta A_{\mu(x)}}-\partial_{\mu}\partial^{\mu}\frac{\delta A^{\nu(y)}}{\delta A_{\mu(x)}})

[/tex]

I don't know if what I have done is right... or not.... I've continued with the problem but it leads to the wrong answer...so yes I'd like help in checking what I've done so far...

[tex]S[\varphi]=-\intop d^{4}y\frac{1}{4}F_{\mu\nu}F^{\mu\nu}

[/tex]

[tex]-\int d^{4}y\frac{1}{4}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})[/tex]

[tex]\frac{\delta s}{\delta A_{\mu(x)}}=\frac{\delta s}{\delta A_{\mu(x)}}\int d^{4}y\frac{1}{4}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})

[/tex]

[tex]=-\frac{1}{4}\frac{\delta s}{\delta A_{\mu(x)}}\int2(\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}-\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu}

[/tex]

[tex]=\frac{1}{2}\int d^{4}y\eta^{\mu\alpha}\eta^{\nu\beta}\frac{\delta s}{\delta A_{\mu(x)}}(\partial_{\mu}A_{\nu}\partial_{\beta}A_{\alpha}-\partial_{\mu}A_{\nu}\partial_{\alpha}A_{\beta}

[/tex]

[tex]=\frac{1}{2}\int d^{4}y\eta^{\mu\alpha}\eta^{\nu\beta}A_{\nu}\frac{\delta s}{\delta A_{\mu(x)}}(-\partial_{\mu}\partial A_{\alpha}+\partial_{\mu}\partial_{\alpha}A_{\beta})

[/tex]

[tex]=\frac{1}{2}\int d^{4}yA_{\nu}\frac{\delta s}{\delta A_{\mu(x)}}(-\eta^{\mu\alpha}\eta^{\nu\beta}\partial_{\mu}\partial_{\beta}A_{\alpha}+\eta^{\mu\alpha}\eta^{\nu\beta}\partial_{\mu}\partial_{\alpha}A_{\beta})

[/tex]

[tex]=\frac{1}{2}\int d^{4}y\frac{\delta s}{\delta A_{\mu(x)}}(-A_{\nu}\partial_{\mu}\partial^{\nu}A^{\alpha}+A_{\nu}\partial_{\mu}\partial^{\mu}A^{\nu})

[/tex]

[tex]=\frac{1}{2}\int d^{4}yA_{\nu}\frac{\delta s}{\delta A_{\mu(x)}}(\partial_{\mu}\partial^{\nu}A^{\alpha}-\partial_{\mu}\partial^{\mu}A^{\nu})

[/tex]

[tex]=\frac{1}{2}\int d^{4}x\frac{\delta A_{\nu(y)}}{\delta A_{\mu(x)}}(\partial_{\mu}\partial^{\nu}\frac{\delta A^{\alpha(y)}}{\delta A_{\mu(x)}}-\partial_{\mu}\partial^{\mu}\frac{\delta A^{\nu(y)}}{\delta A_{\mu(x)}})

[/tex]

I don't know if what I have done is right... or not.... I've continued with the problem but it leads to the wrong answer...so yes I'd like help in checking what I've done so far...