Geometry of Ermakov Systems

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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].