= Lagrange multiplier for antisymmetry
{c}
{title2=$\partial\mathcal L/\partial M^{[\mu\nu]}=0$}
To vary an <antisymmetric tensor> without treating its entries as independent physical variables, temporarily allow an unrestricted array $M^{\mu\nu}$ and add $\lambda_{\mu\nu}(M^{\mu\nu}+M^{\nu\mu})/2$ to the <Lagrangian density>, with symmetric <Lagrange multiplier> $\lambda$. Variation of $\lambda$ enforces antisymmetry. The antisymmetric part of the unrestricted $M$ equation eliminates $\lambda$ and gives the physical <Euler-Lagrange field equation>. Equivalently, vary only the six independent entries in four spacetime dimensions. Quadratic contractions must be differentiated before applying antisymmetry, or their factors and signs can be lost.
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