Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures

Giovanni  Rastelli; Manuele  Santoprete

doi:10.3934/jgm.2015.7.483

Article Contents

2015, Volume 7, Issue 4: 483-515. Doi: 10.3934/jgm.2015.7.483

This issue Previous Article A unifying mechanical equation with applications to non-holonomic constraints and dissipative phenomena Next Article Invariant metrics on Lie groups

Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures

Giovanni Rastelli^1, and
Manuele Santoprete^2,

1.
Dipartimento di Matematica, Università di Torino, Torino, via Carlo Alberto 10
2.
Department of Mathematics, Wilfrid Laurier University, 75 University Avenue West, Waterloo, ON

Received: June 30, 2014

Revised: June 30, 2015

Published: November 2015

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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.

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Mathematics Subject Classification: Primary: 53D05, 37K10; Secondary: 53D17.

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