  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 couldn't 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 to 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 amongst all the three.
