= Variational open-string boundary conditions
{title2=$J_m\delta X^m|_{\partial\Sigma}=0$}
The spatial integration-by-parts term in the <Nambu-Goto phase-space action> pairs endpoint displacement with momentum flux $J_m=eT^2X'_m+uP_m$. Free endpoint displacements require zero flux. Fixed normal coordinates instead set those variations to zero, while tangential variations retain a <Neumann boundary condition>. Boundary conditions must also be preserved by the equations of motion. In a boundary-preserving gauge with $u=0$, mixed <Neumann boundary conditions> and <Dirichlet boundary conditions> implement a planar <D-brane>.
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