= DBI Hamiltonian bound for an electric spike
{c}
{title2=$\mathcal H^2/T_3^2=(1+\mathbf d\cdot\nabla X)^2+|\mathbf d-\nabla X|^2$}
For a static <D3-brane> with one scalar and no magnetic field, taking the Legendre transform of the <DBI action> gives the displayed identity. It bounds energy by the electric scalar charge and is saturated at $\mathbf d=\nabla X$. The opposite orientation uses the analogous identity with reversed signs. On saturation, the constitutive relation gives $\mathbf E=\mathbf d$ and the <Gauss law constraint in gauge theory> reduces to the <Laplace equation> for $X$.
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