On the equivalence between different canonical forms of F(R) theory of gravity

Classical equivalence between Jordan's and Einstein's frame counterparts of F(R) theory of gravity has recently been questioned, since the two produce different Noether symmetries, which could not be translated back and forth using transformation relations. Here we add the Hamiltonian constraint equation, which is essentially the time-time component of Einstein's equation, through a Lagrange multiplier to the existence condition for Noether symmetry and show that all the three different canonical structures of F(R) theory of gravity, including the one which follows from Lagrange multiplier technique, admit each and every available symmetry independently. This establishes classical equivalence.