Note about Hamiltonian formulation of modified $F(R)$ Hořava-Lifshitz gravities and their healthy extension

This note is devoted to the study of Hamiltonian formulation of modified $F(R)$ Hořava-Lifshitz theories of gravity that were proposed recently by Chaichian and collaborators [arXiv:1001.4102]. We also study Hamiltonian formulation of the healthy extended Hořava-Lifshitz gravities and show that these theories have the Hamiltonian structure that is different from Hamiltonian formulation general relativity.