Physics Asked on August 5, 2021
So I am trying to find equations of motion for the Lagrangian associated with a non-Abelian Gauge theory for $SU(N)$, and while I was doing the math, I was a bit confused the indices.
So I have $mathcal{L} = -frac{1}{4}F^i_{munu}F^{imunu}$
Where $F^i_{munu} = partial_mu A^i_nu -partial_nu A^i_mu +gf^{ijk}A^j_mu A^k_nu$
Ok, to find equation of motion, I did:
$frac{partialmathcal{L}}{partial A^a_mu} – partial_nu(frac{partialmathcal{L}}{partial(partial_nu A_mu^a)}) = 0 $
For my final answer, I got:
$gf^{abc}A_nu^cF^{bmunu} – partial_nu(F^{amunu}) = 0$
Then, I realized that this wikipedia article on Yang-Mills Theory has also written the equation of motion for the lagrangian, but they have:
$gf^{abc} A^{mu b} F^c_{munu} + partial^mu F^a_{munu} = 0$
I am not good with manipulating the four vector indices. Could someone help me understand if my equation of motion is same as wikipedia’s?
Also it is important to say that I do understand that while calculating the equation of motions, I decided to calculate the partial derivative w.r.t to $nu$ instead of $mu$ as they did in wikipedia. But the position of the four vector indices in the two answers is what is confusing me.
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