Canonoid and Poissonoid Transformations, Symmetries and BiHamiltonian Structures

TL;DR

This paper characterizes canonoid and Poissonoid transformations on symplectic and Poisson manifolds, exploring their role in generating biHamiltonian structures, analyzing classical systems, and linking symmetries with constants of motion.

Contribution

It introduces a modern, coordinate-independent framework for canonoid transformations, extends them to Poisson manifolds, and connects these transformations with biHamiltonian structures and symmetries.

Findings

Generated biHamiltonian structures for mechanical systems.

Analyzed harmonic oscillator under canonoid transformations.

Linked Poissonoid transformations with constants of motion and symmetries.

Abstract

We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Utilizing 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 $\mathcal{so}^\ast (3) $ and $ 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.