Functional Derivative on Manifold?

Question

In M-theories, there are often Action functionals (in the physics sense), defined on manifold involving p-forms and such. Letting $\mathcal{M}$ denote the manifold with dimension $d$, one might encounter something of the form: $$S = a \int_\mathcal{M} A \wedge \mathrm{d}A + b\int_\mathcal{M} A \wedge A \wedge A $$ Where these are some sort of non-abelian differential forms. In the usual case, one defines the integrand as the Lagrangian $\mathcal{L}$ and can determine the equations obeyed by the dynamical variables by solving the Euler-Lagrange equation, for example if $\mathcal{L}= \mathcal{L}(t, x, dx/dt)$, then we solve: $$ \frac{\partial \mathcal{L}}{\partial x } - \frac{\mathrm{d}}{\mathrm{d} t }\left(\frac{\partial \mathcal{L}}{\partial (\frac{\mathrm{d}x}{\mathrm{d}t})} \right) = 0 $$ If there a mathematically rigorous way to generalize this to the case of differential forms, for example, in the action above, if I abuse notation assume that: $$ \frac{\partial \mathcal{L}}{\partial A}+ \mathrm{d}\left(\frac{\partial \mathcal{L}}{\partial (\mathrm{d} A)} \right)= 0 $$ Then I get something of the form: $$ a\mathrm{d}A + 3 b A\wedge A + a \mathrm{d}A = 2a\mathrm{d}A + 3 b A\wedge A = 0 $$ (Where $\mathrm{d}^2 =0$). Where this is in fact the correct equation of motion describing the system and is what one would compute by considering the variation $A \to A+ \delta A$. Is this a fluke? or can this method be trusted in general?