2014/07/31 by Giovanni Rastelli, Manuele Santoprete · 5 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Characterization (materials science) #Classical mechanics #Euler's formula #Geometry #Harmonic oscillator #Homogeneous space #Homotopy and Cohomology in Algebraic Topology #Integrable system #Lie algebra #Mathematical analysis #Mathematics #Motion (physics) #Nonlinear Waves and Solitons #Physics #Poisson bracket #Poisson manifold #Pure mathematics #Quantum mechanics #Rigid body #Symplectic geometry #math-ph #math.MP #msc:37K10 #msc:53D05 #msc:53D17
paper · pdf · doi:10.3934/jgm.2015.7.483
published in The Journal of Geometric Mechanics 7(4), 483-515 (American Institute of Mathematical Sciences) · 34 pages, no figures
openalex publication_date 2015/01/01 · arxiv created 2015/02/25 · arxiv updated 2017/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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^∗ (3) and \mathfrak so^∗ (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.