  We consider Galileon models on curved spacetime, as well as the counterterms
introduced to maintain the second-order nature of the field equations of these
models when both the metric and the scalar are made dynamical. Working in a
gauge invariant framework, we first show how all the third-order time
derivatives appearing in the field equations -- both metric and scalar -- of a
Galileon model or one defined by a given counterterm can be eliminated to leave
field equations which contain at most second-order time derivatives of the
metric and of the scalar. The same is shown to hold for arbitrary linear
combinations of such models, as well as their k-essence-like/Horndeski
generalizations. This supports the claim that the number of degrees of freedom
in these models is only 3, counting 2 for the graviton and 1 for the scalar. We
comment on the arguments given previously in support of this claim. We then
prove that this number of degrees of freedom is strictly less that 4 in one
particular such model by carrying out a full-fledged Hamiltonian analysis. In
contrast to previous results, our analyses do not assume any particular gauge
choice of restricted applicability.
