Nambu–Goto equations of motion (${\partial }_a(\sqrt{-g}{g}^{ab}{\partial }_b{X}^m)=0$)

For a nondegenerate induced worldsheet metric $g_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$, variation gives ${\partial }_a(\sqrt{-g}{g}^{ab}{\partial }_b{X}^m)=0$. In conformal gauge this is the two-dimensional wave equation, supplemented by the Virasoro constraints.