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Geometry of Ermakov Systems

  • C. Athorne
Conference paper
Part of the Research Reports in Physics book series (RESREPORTS)

Abstract

Ennakov systems are pairs of coupled nonlinear oscillators of the form
$${{{{\rm{d}}^{\rm{2}}}{\rm{x}}} \over {{\rm{d}}{{\rm{t}}^{\rm{2}}}}}{\omega ^2}({\rm{t}}){\rm{x = }}{{\rm{x}}^{{\rm{ - 3}}}}{\rm{f}}\left( {{{\rm{x}} \over {\rm{y}}}} \right),{{{{\rm{d}}^{\rm{2}}}{\rm{y}}} \over {{\rm{d}}{{\rm{t}}^{\rm{2}}}}}{\omega ^2}({\rm{t}}){\rm{y = }}{{\rm{y}}^{{\rm{ - 3}}}}{\rm{g}}\left( {{{\rm{y}} \over {\rm{x}}}} \right)$$
where ω2, f and g are arbitrary functions of their arguments. Such systems were first introduced in [1] and named after V. P. Ermakov who studied the case g=0 [2]. This case includes the so-called Pinney equation [3], f=const, studied in quantum mechanics [4] and nonlinear elasticity [5]. More general types of Ermakov system find application in shallow water wave theory [6] and in nonlinear optics [7].

Keywords

General Solution Projective Line Nonlinear Evolution Equation Nonlinear Elasticity Algebraic Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • C. Athorne
    • 1
  1. 1.Department of MathematicsUniversity of GlasgowGlasgowUK

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