homework and exercises - $SU(2)$ Yang-Mills EOM

I'm having trouble with some indices on my yang-mills lagrangian. I have a gauge group $SU(2)$ and a field strength tensor $$ F_{ab}^{i}=\partial_{a}A^{i}_{b}-\partial_{b}A^{i}_{a}+\epsilon^{i}_{\,\,jk}A^{j}_{a}A^{k}_{b}$$ and a lagrangian $$\mathcal{L}=-\frac{1}{4}F_{ab}^{i}F_{i}^{ab}$$ I have at the end of the exercise that the correct EOM are $$\partial^{a}F_{ab}^{i}+\epsilon^{ij}_{\;\;\;k}A^{a}_{j}F^{k}_{ab}=0$$ I am getting the first term on the left, but the second term I am getting something slightly different, instead, when I hit the lagrangian with $\partial \mathcal{L}/\partial A_{a}^{i}$ I get a term $\epsilon^{i}_{\,\,jk} A_{b}^{k}F_{i}^{ab}$ which leaves a free $j$ index. Now I know summed indices don't matter and can change letters freely, but order and contraction does matter, now I'm not sure how to make it come out with matching indices. When I do $\partial_a \partial \mathcal{L}/\partial (\partial_{a}A_{b}^{i})$ I come out with the typical $\partial_a F^{ab}_{i}$ but my free index there is $i$, not $j$, and I can't get them to match for the life of me.

Answer

Without seeing the details of how you're taking the derivative, I can't be sure, but my first thought is to wonder whether you're making multiple use of indices. For example, suppose you take the derivative with respect to $A_a^i$; you'll get

$$\frac{\partial\mathcal{L}}{\partial A_a^i} = -\frac{1}{4}\frac{\partial}{\partial A_a^i}\bigl[F_{\color{red}{ab}}^{\color{red}i} F_{\color{red}i}^{\color{red}{ab}}\bigr] = -\frac{1}{4}\frac{\partial F_{\color{red}{ab}}^{\color{red}i}}{\partial A_a^i}F_{\color{red}i}^{\color{red}{ab}} -\frac{1}{4}\frac{\partial F_{\color{red}i}^{\color{red}{ab}}}{\partial A_a^i}F_{\color{red}{ab}}^{\color{red}i}$$

In the process of evaluating $\frac{\partial F_{\color{red}{ab}}^{\color{red}i}}{\partial A_a^i}$, the indices in the numerator don't contract with the ones in the denominator, but it's going to be easy to miss that if you don't have the colors to distinguish them.

I'd suggest that when you label the field you're differentiating with respect to, use indices that don't show up anywhere in the Lagrangian, like this:

$$\frac{\partial\mathcal{L}}{\partial A_c^n} = -\frac{1}{4}\frac{\partial F_{ab}^{i}}{\partial A_c^n}F_{i}^{ab} -\frac{1}{4}\frac{\partial F_{i}^{ab}}{\partial A_c^n}F_{ab}^{i}$$

Then there's no danger of accidentally contracting the wrong indices. When you get down to the level of differentiating individual fields, you'll wind up with some Kronecker deltas,

$$\frac{\partial A_a^i}{\partial A_c^n} = \delta_a^c \delta_n^i$$

If you're differentiating a contravariant field, you can use the metric to raise and lower indices as needed,

$$\frac{\partial A_i^a}{\partial A_c^n} = \frac{\partial}{\partial A_c^n}\delta_{im}g^{ad}A_d^m = g^{ac} \delta_{in}$$

$\delta_{im}$ is the metric for $SU(2)$ configuration space, if I remember correctly.