The perturbed ideal resonance problem View Full Text


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Article Info

DATE

1973-01

AUTHORS

Alan H. Jupp

ABSTRACT

The author's previous studies concerning the Ideal Resonance Problem are enlarged upon in this article. The one-degree-of-freedom Hamiltonian system investigated here has the form\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{*{20}c} { - F = B(x) + 2\mu ^2 A(x)\sin ^2 y + \mu ^2 f(x,y),} \\ {\dot x = - F_y ,\dot y = F_x .} \\ \end{array}$$ \end{document}The canonically conjugate variablesx andy are respectively the momentum and the coordinate, andμ2 is a small positive constant parameter. The perturbationf is o (A) and is represented by a Fourier series iny. The vanishing of ∂B/∂x≡B(1) atx=x0 characterizes the resonant nature of the problem.With a suitable choice of variables, it is shown how a formal solution to this perturbed form of the Ideal Resonance Problem can be constructed, using the method of ‘parallel’ perturbations. Explicit formulae forx andy are obtained, as functions of time, which include the complete first-order contributions from the perturbing functionf. The solution is restricted to the region of deep resonance, but those motions in the neighbourhood of the separatrix are excluded. More... »

PAGES

91-106

References to SciGraph publications

  • 1969-03. Canonical transformations depending on a small parameter in CELESTIAL MECHANICS AND DYNAMICAL ASTRONOMY
  • 1972-01. A second-order solution of the Ideal Resonance Problem by Lie series in CELESTIAL MECHANICS AND DYNAMICAL ASTRONOMY
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    http://scigraph.springernature.com/pub.10.1007/bf01243510

    DOI

    http://dx.doi.org/10.1007/bf01243510

    DIMENSIONS

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