Variation of the induced metric

Question

Nambu-Goto action

$$ S=-\mu\int\gamma^{1/2}d^2\xi $$ where $\gamma_{AB}=g_{\mu\nu}X^{\mu},_A X^{\nu},_B$ . Euler-Lagrange equations:

$$ \frac{\partial}{\partial\xi^A} \left(\frac{\partial\gamma^{1/2}}{\partial X^{\mu},_A}\right)-\frac{\partial\gamma^{1/2}}{\partial X^{\mu}}=0 $$

The variation of $\gamma^{1/2}$: $$ \delta\gamma^{1/2}=\frac{1}{2}\gamma^{1/2}\gamma^{AB}\delta\gamma_{AB} $$ My question is how to variate the induced metric $\gamma_{AB}$? I guess it should be variated with respect to spacetime coordinates and and spacetime metric. $$ \delta\gamma_{AB}=\delta(g_{\mu\nu})X^{\mu},_A X^{\nu},_B+2g_{\mu\nu}\delta(X^{\mu},_A) X^{\nu},_B $$ And the second question is, if the variation is true, then, when the derivative of $\gamma_{AB}$ with respect to $X^{\mu},_A$ and $X^{\mu}$ is evaluated must be $\delta g_{\mu\nu}=0$ and $\delta(X^{\mu},_A)=0$, similarly to the derivative of multivariable function?

Answer 1

OP is essentially asking how the Nambu-Goto (NG) action principle works. We should stress that there is no variation wrt. to the target space metric $g_{\mu\nu}(X)$ per se: It is treated as a background metric. The variation is purely wrt. the target space coordinates fields $X^{\mu}(\xi)$. It is straightforward exercise to derive the Euler-Lagrange (EL) equations for $X^{\mu}(\xi)$. For the Dirac-Bergmann constraint analysis of the NG action, see e.g. this Phys.SE post.