Oscillators in resonance p:q:r

A canonical transformation is proposed to handle Hamiltonian systems made of the addition or subtraction of three harmonic oscillators in p:q:r resonance. This transformation is an extension of the classical Lissajous transformation for the 1:1 resonance. Our extended Lissajous variables consist of three pairs of action-angle variables, which makes possible the application of perturbation theories without encountering small divisors. A set of functions, related with the Lissajous variables, are found, and are used to describe the phase flow of reduced space after normalization.