We analyze the Brans-Dicke theory with a Gibbons-Hawking-York(GHY) boundary term and perform ADM decomposition both in Jordan and Einstein frames. For ω≠-3/2, we show that, at the Hamiltonian level, the Weyl (conformal) transformations from the Jordan to Einstein frames are not canonical transformations (in Hamiltonian sense). A set of canonical transformations is found. These are Anti-Gravity or Anti-Newtonian transformations and are different respect to the transformations from the Jordan to the Einstein frames. The case for ω=-3/2 with GHY boundary term is studied as well. The presence of the conformal invariance too in the Jordan frame, the Dirac’s constraint algebra of secondary first-class constraints is different in the Jordan frame respect to the Einstein frame. This inequivalence of the Dirac’s algebra between the two frames addresses, more strongly respect to the case ω≠-3/2, the non (Hamiltonian)-canonicity of the transformations from the Jordan to the Einstein Frames.