  We study the constrained Ostrogradski-Hamilton framework for the equations of
motion provided by mechanical systems described by second-order derivative
actions with a linear dependence in the accelerations. We stress out the
peculiar features provided by the surface terms arising for this type of
theories and we discuss some important properties for this kind of actions in
order to pave the way for the construction of a well defned quantum counterpart
by means of canonical methods. In particular, we analyse in detail the
constraint structure for these theories and its relation to the inherent
conserved quantities where the associated energies together with a Noether
charge may be identifed. The constraint structure is fully analyzed without the
introduction of auxiliary variables, as proposed in recent works involving
higher order Lagrangians. Finally, we also provide some examples where our
approach is explicitly applied and emphasize the way in which our original
arrangement results in propitious for the Hamiltonian formulation of covariant
feld theories.
