Perturbation dynamics for membranes and strings governed by the Dirac-Goto-Nambu action in curved space

It is shown how to set up a concise and fully covariant formalism, in terms of an appropriate (geodesically defined) displacement vector ${\xi }^{\mu }$, in such a way that the corresponding second order perturbation of the Dirac-Goto-Nambu action itself provides the action for a convenient secondary variation principle governing the linearized dynamics of first order perturbations of the relevant membrane and string models in an arbitrarily curved background, as well as determining the corresponding conserved symplectic current associated with any pair of distinct solutions ${\xi }^{\mu }$ and ${\stackrel{`}{\xi }}^{\mu }$ on the world sheet.