Abstract
A new canonical transformation is proposed to handle elliptic oscillators, that is, Hamiltonian systems made of two harmonic oscillators in a 1-1 resonance. Lissajous elements pertain to the ellipse drawn with a light pen whose coordinates oscillate at the same frequency, hence their name. They consist of two pairs of angle-action variables of which the actions and one angle refer to basic integrals admitted by an elliptic oscillator, namely, its energy, its angular momentum and its Runge-Lenz vector. The Lissajous transformation is defined in two ways: explicitly in terms of Cartesian variables, and implicitly by resolution of a partial differential equation separable in polar variables. Relations between the Lissajous variables, the common harmonic variables, and other sets of variables are discussed in detail.
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Deprit, A. The Lissajous transformation I. Basics. Celestial Mech Dyn Astr 51, 201–225 (1991). https://doi.org/10.1007/BF00051691
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DOI: https://doi.org/10.1007/BF00051691