[en] Two approximations are made, one essential and the other not so essential but convenient to keep the analytical treatment manageable: (1) only one nonlinear resonance is considered at a time so that the treatment is best suited when the tune is close to one resonance only. To improve this approximation, one must go to the next order which involves a canonical transformation of dynamical variables. Analytical treatment of more than one resonance is not possible for general cases; (2) in the formalism using the action-angle variables, the Hamiltonian can have terms which are independent of the angle variables. These terms are called ''phase-independent terms'' or ''shear terms.'' The tune is then a function of the oscillation amplitudes