 Variational open-string boundary conditions

The spatial integration-by-parts term in the Nambu-Goto phase-space action pairs endpoint displacement with momentum flux $J_m=e{T}^2{X}_m^{\prime }+u{P}_m$. 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.