The phase coordinates transformation in Moyal noncommutative framework 2-dimensional harmonic oscillator and Painlevé second equation

Abstract

This work aims to explore the physical application of Moyal noncom- mutative formalism with the the presentation of new transformations which connect the old canonical coordinates to purly noncommuting phase coordi- nates through the parameters describe the Moyal noncommutative deforma- tion. These transformations are consistent with the Moyal noncommunica- tive brackets and build sixteen components traceless anti-symmetric tensor. This formalism incorporates space-space and space-momentum noncommu- taivity rather then noncommutativity (Heisenberg quantum commutation ) only for the canonical coordinates. The transformations imply to construct noncommutative analogs of 2-dimensional harmonic oscillator with its superintegrability and Painlevé second equation. The associated deformed Hamiltonians additionally involve some extra symmetries and reveal a deep physical understanding of about the additional symmetries with noncommunicative structure. The Painlevé second noncommutative coordinates transformation are also efficiently applied to generate the first Yablonskii Vorobev polynomial with Painlevé second parameter transformation.