The Hamilton-Jacobi analysis for higher-order modified gravity

The Hamilton-Jacobi [HJ] study for the Chern-Simons [CS] modification of general relativity [GR] is performed. The complete structure of the constraints and the generalized brackets are reported, from these results the HJ fundamental differential is constructed and the symmetries of the theory are found. By using the constraints we remove an apparent Ostrogradsky's instability and the new structure of the canonical Hamiltonian is reported. In addition, the counting of physical degrees of freedom is carried out and some remarks are discussed.