Complex Representation of Dynamics of Coupled Nonlinear Oscillators

The complex representation of the classical equations of motion of a system of linear oscillators was used for the first time in a quantum mechanics and in the analysis of so-called coupled oscillations and waves in a mechanics, electronics engineering and solid state physics1, 2, 3, 4, 5, 6. The complex conjugate linear combinations of displacements and velocities of oscillators being sought-for functions in this representation, can be visually presented as vectors of equal length rotating in opposite directions. Actually it is enough to find only one complex function for each oscillator completely defining both displacement and velocity. Such choice of variables leads in particular to very simple and natural procedure of quantization: complex conjugate functions become operators of an annihilation and birth, and their squared module —by the number of elementary excitations1,5,8.

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