Second-order Lagrangian and symplectic current for gravitationally perturbed Dirac-Goto-Nambu strings and branes

We extend a recent analysis of gravitational perturbations on Dirac-Goto-Nambu-type strings, membranes and higher-dimensional branes. In an arbitrary gauge, it is shown that the relevant first-order equations governing the displacement vector of the worldsheet and metric perturbation are obtainable from a variational principle whose Lagrangian is constructed as a second-order perturbation of the standard Dirac-Goto-Nambu action density. A symplectic current functional is obtained as a by-product that is potentially useful for the derivation of conservation laws in particular circumstances.