  We develop in a consistent manner the Ostrogradski-Hamilton framework for
gonihedric string theory. The local action describing this model, being
invariant under reparametrizations, depends on the modulus of the mean
extrinsic curvature of the worldsheet swept out by the string, and thus we are
confronted with a genuine second-order in derivatives field theory. In our
geometric approach, we consider the embedding functions as the field variables
and, even though the highly non-linear dependence of the action on these
variables, we are able to complete the classical analysis of the emerging
constraints for which, after implementing a Dirac bracket, we are able to
identify both the gauge transformations and the proper physical degrees of
freedom of the model. The Ostrogradski-Hamilton framework is thus considerable
robust as one may recover in a straightforward and consistent manner some
existing results reported in the literature. Further, in consequence of our
geometrical treatment, we are able to unambiguously recover as a by-product the
Hamiltonian approach for a particular relativistic point-particle limit
associated with the gonihedric string action, that is, a model linearly
depending on the first Frenet-Serret curvature.
