The Hamiltonian of the f(R) gravity

We derived conjugate momenta variables tensors and the Hamiltonian equation of the source free f(R) gravity from first principles using the Legendre transformation of these conjugate momenta variable tensors, conjugate coordinates variables - fundamental metric tensor and its first ordinary partial derivatives with respect to spacetime coordinates and second ordinary partial derivatives with respect to spacetime coordinates - and the Lagrangian of the f(R) gravity. Interpreting the derived Hamiltonian as the energy of the f(R) gravity we have shown that it vanishes for linear Lagrangians in Ricci scalar curvature without source (e.g. Einstein-Hilbert Lagrangian without matter fields) which is the same result obtained using the stress-energy tensor equation derived from variation of the matter field Lagrangian density. The resulting Hamiltonian equation forbids any model of negative power law in the dependence of the f(R) gravity on Ricci scalar curvature: f(R) = α R^-r, where r and α are positive real numbers, it also forbids any polynomial equation which contains terms with negative powers of the Ricci scalar curvature including a constant term, in which cases the Hamiltonian function in the Ricci scalar and therefore the energy of the f(R) gravity would attains a negative value and would not be bounded from below. The restrictions imposed by the non-negative Hamiltonian have far reaching consequences as the result of applying the f(R) gravity to the study of Black Holes and the Friedmann-Lemaître-Robertson-Walker model in Cosmology.