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?

  • $\begingroup$ You will have $d=3$; google "Chern-Simons functional". $\endgroup$ Dec 7, 2017 at 0:23
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    $\begingroup$ yes this is a common action in generalized E&M, but am more concerned with the means of getting the EOM's. $\endgroup$ Dec 7, 2017 at 0:32
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    $\begingroup$ Since everything here lives in vector bundles, you can follow the classical derivation of the E-L equation: expand $\frac{d}{ds}|_{s=0}S(A+s \psi) = 0$ using the chain rule and integrate by parts to move the derivative off $\psi$, and you're left with $\int(\frac{\partial L}{\partial A}+\delta\frac{\partial L}{\partial d A})\psi = 0$ where $\delta$ is some dual/adjoint of $d$. Since this holds for every $\psi$; the parenthesized term must vanish. $\endgroup$ Dec 7, 2017 at 1:35
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    $\begingroup$ Thanks Anthony, do you know of any references I can look up to find this, particularly on the dual/adjoint of $\mathrm{d}$. Thanks. $\endgroup$ Dec 7, 2017 at 1:41
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    $\begingroup$ The point is not the setup, I don't care what the action is (I just arbitrarily chose the Chern-Simons term), what I care about is the method of getting the equations of motion, and if there is a general form of the euler lagrange equations to extremize a functional involving differential forms. $\endgroup$ Dec 7, 2017 at 19:34


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