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Deriving the EOM for Proca Lagrangian

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Homework Statement



Consider the Proca Lagrangian

[tex]L=-\frac{1}{16\pi}F^2-\frac{1}{c}J_{\mu}A^{\mu}+\frac{M^2}{8\pi}A_{\mu}A^{\mu}[/tex]​

in the Lorentz guage [tex]\partial_{\mu}A^{\mu}=0[/tex]

Find the equation of motion.


Homework Equations



[tex]F^2=F_{\mu\nu}F^{\mu\nu}[/tex]


The Attempt at a Solution



Well, first of all I'm not quite sure how E-L should look in this case. Clearly [tex]F_{\mu\nu}[/tex] is the part that is the derivatives of the dependent variables (A).
What I have gotten to is
[tex]\frac{\partial L}{\partial F_{\rho \sigma}} = -\frac{F^{\rho\sigma}}{8\pi}[/tex]
[tex]\frac{\partial L}{\partial A_{\sigma}}=-\frac{1}{c} J^{\sigma} + \frac{M^2}{4 \pi}A^{\sigma}[/tex]

So is this the E-L for 4D special relativity?
[tex]\frac{\partial L}{\partial A_\sigma}-\partial_\sigma \left( \frac{\partial L}{\partial F_{\rho\sigma}} \right)=0[/tex]​

Thanks for any help.
 

Answers and Replies

  • #2
George Jones
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I would more comfortable not taking a short-cut, i.e., I would expand the F's in term of A's, but, after substituting for the L's in your final equation, you seem to get the correct equation of motion.
 
  • #3
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Thanks so much for the quick reply.

What I'm mostly unsure about is how to formulate E-L in terms of covariant or contravariant components. Do I take the derivative with respect to [tex]A^\sigma[/tex] or [tex]A_\sigma[/tex].

I guess my question could be reformulated as:

Should E-L look like
[tex]\frac{\partial L}{\partial A_\sigma}-\partial^\sigma \left( \frac{\partial L}{\partial F_{\rho\sigma}} \right)=0[/tex]​
or
[tex]\frac{\partial L}{\partial A^\sigma}-\partial_\sigma \left( \frac{\partial L}{\partial F^{\rho\sigma}} \right)=0[/tex]​

And based on what do I choose between the two?
 
Last edited:
  • #4
George Jones
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790
I guess my question could be reformulated as:

Should E-L look like
[tex]\frac{\partial L}{\partial A_\sigma}-\partial^\sigma \left( \frac{\partial L}{\partial F_{\rho\sigma}} \right)=0[/tex]​
or
[tex]\frac{\partial L}{\partial A^\sigma}-\partial_\sigma \left( \frac{\partial L}{\partial F^{\rho\sigma}} \right)=0[/tex]​

And based on what do I choose between the two?
Neither, :smile:. In each equation, [itex]\sigma[/itex] is a free index in one term and a dummy summed index in the other term, which is a no-no. In terms of index placement, an upstairs index in a denominator is like a downstairs index in a numerator, and a downstairs index in a denominator is like an upstairs index in a numerator.
 

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