  The general relativistic perturbations of scalar-tensor theories (STT) of
gravity are studied in a manifestly gauge invariant Hamiltonian formalism.
After the derivation of the Hamiltonian equations of motion in this framework,
the gauge invariant formalism is used to compute the evolution equations of
linear perturbations around a general relativistic spacetime background in the
Jordan frame. These equations are then specialized to the case of a flat FRW
cosmological background. Furthermore, the equivalence between the Jordan frame
and the Einstein frame of STT in the manifestly gauge invariant Hamiltonian
formalism is analyzed, and it is shown that also in this framework they can be
related by a conformal transformation. Finally, the obtained evolution
equations for the linear perturbations in our formalism are compared with those
in the standard cosmological perturbation theory. It turns out that the
perturbation equations in the two different formalisms coincide with each other
in a suitable limit.
