  We provide a covariant framework to study classically the stability of small
perturbations on the so-called gonihedric string model by making precise use of
variational techniques. The local action depends of the square root of the
quadratic mean extrinsic curvature of the worldsheet swept out by the string,
and is reparametrization invariant. A general expression for the worldsheet
perturbations, guided by Jacobi equations without any early gauge fixing, is
obtained. This is manifested through a set of highly coupled nonlinear
differential partial equations where the perturbations are described by scalar
fields, $\Phi^i$, living in the worldsheet. This model contains, as a special
limit, to the linear model in the mean extrinsic curvature. In such a case the
Jacobi equations specialize to a single wave-like equation for $\Phi$.
