The Journal of Geometric Mechanics (JGM)

Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures

Pages: 483 - 515, Volume 7, Issue 4, December 2015      doi:10.3934/jgm.2015.7.483

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Giovanni Rastelli - Dipartimento di Matematica, Università di Torino, Torino, via Carlo Alberto 10, Italy (email)
Manuele Santoprete - Department of Mathematics, Wilfrid Laurier University, 75 University Avenue West, Waterloo, ON, Canada (email)

Abstract: We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Using this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler's equations of the rigid body (on $ \mathfrak{ so}^\ast (3) $ and $ \mathfrak{ so}^\ast (4)$) and for an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid. We study the relationship between biHamiltonian structures and Poissonoid transformations for these examples. We analyze the link between Poissonoid transformations, constants of motion, and symmetries.

Keywords:  Canonoid transformations, biHamiltonian systems, symplectic geometry, Poisson geometry, Poissonoid transformations.
Mathematics Subject Classification:  Primary: 53D05, 37K10; Secondary: 53D17.

Received: July 2014;      Revised: July 2015;      Available Online: October 2015.