A general prescription for deriving rotation-vibration Hamiltonians that satisfy the Casimir condition is presented. This condition, achieved using the Eckart constraints, is that there is no vibrational angular momentum in the molecular equilibrium configuration. The Eckart condition, while useful for studying rotation-vibration inter-actions, is difficult to apply for coordinates other than rectilinear normal coordinates. The present derivation allows one to derive Hamiltonians in curvilinear coordinates, yet still take advantage of the most relevant properly of the Eckart frame, this being that Coriolis coupling is minimized in the limit of small vibrations.