# Classical Yang-Mills equation of motion with both electric and magnetic sources?

We know the classical Maxwell equation of motion (eom) with both electric and magnetic source can be written as:

(1) Explicit form

or more schematically as:

(2) Differential form $$d * F = * J_e$$ $$dF =* J_m$$

My question is that do we have such classical Yang-Mills equation of motion with both electric and magnetic source in both

(1) Explicit form?

(2) Differential form? Naively, we may write $$D * F = * J_e$$ $$D F =* J_m$$ where $$F= dA + A \wedge A$$ and $$D=d + [A, ]$$ as the covariant derivative version of exterior derivative $$d$$.

But: To be aware that for example, the $$SU(2)$$ Yang-Mills and $$SO(3)$$ Yang-Mills theory may have distinct constraint on the magnetic monopole (or the t Hooft loop). It does not seem to me that $$J_e$$ or $$J_m$$ contain such information?

• Please clarify your question. Are you asking why the same equations can be written in such different forms? – G. Smith May 19 at 23:59
• No. But I am interested in knowing both (1) Explicit form and (2) Differential form for YM with e and m source – annie heart May 20 at 0:02
• OK, that makes it clearer. Thank you. Do you really want the Yang-Mills equations written in terms of $\mathbf{E}$ and $\mathbf{B}$? They ate usually written in terms of $F^{\mu\nu}$. – G. Smith May 20 at 0:03
• written as $E^a$ and $B^a$ are preferred. But you can convert to $F$ first – annie heart May 20 at 1:16