Hamiltonian perturbation theory in $f(R)$ gravity

Hamiltonian perturbation theory is used to analyze the stability of $f(R)$ models. The Hamiltonian equations for the metric and its momentum conjugate are written for the $f(R)$ Lagrangian in the presence of perfect fluid matter. The perturbations examined are perpendicular to $R$. As perturbations are added to the metric and momentum conjugate to the induced metric instabilities are found, depending on the form of $f(R)$. Thus the examination of these instabilities is a way to rule out certain $f(R)$ models.