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The Nambu-Goto action (including normalization) is $$ S_{NG} = \frac{1}{2\pi \alpha'}\int dr d\theta \frac{L^2 r}{z^2} \sqrt{1 + z'^2}=\frac{L^2}{\alpha'}\int dr \frac{r}{z^2} \sqrt{1 + z'^2} $$ The corresponding Hamiltonian is not conserved as the Lagrangian is explicitly $r$-dependent. You can however see that a solution to the equations of motion $$ 0=\...



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