The Lissajous transformation I. Basics View Full Text


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Article Info

DATE

1991-09

AUTHORS

André Deprit

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. More... »

PAGES

201-225

References to SciGraph publications

  • 1979. A survey of the Hénon-Heiles Hamiltonian with applications to related examples in STOCHASTIC BEHAVIOR IN CLASSICAL AND QUANTUM HAMILTONIAN SYSTEMS
  • 1931-12. Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche in MATHEMATISCHE ANNALEN
  • 1976-02. On resonant non linearly coupled oscillators with two equal frequencies in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1989. Simplify or Perish in APPLICATIONS OF COMPUTER TECHNOLOGY TO DYNAMICAL ASTRONOMY
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    http://scigraph.springernature.com/pub.10.1007/bf00051691

    DOI

    http://dx.doi.org/10.1007/bf00051691

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