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Clebsch canonization of Lie–Poisson systems

  • *Corresponding author: Tomoki Ohsawa

    *Corresponding author: Tomoki Ohsawa

Dedicated to Professor Anthony Bloch on the occasion of his 65th birthday

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  • We propose a systematic procedure called the Clebsch canonization for obtaining a canonical Hamiltonian system that is related to a given Lie–Poisson equation via a momentum map. We describe both coordinate and geometric versions of the procedure, the latter apparently for the first time. We also find another momentum map so that the pair of momentum maps constitute a dual pair under a certain condition. The dual pair gives a concrete realization of what is commonly referred to as collectivization of Lie–Poisson systems. It also implies that solving the canonized system by symplectic Runge–Kutta methods yields so-called collective Lie–Poisson integrators that preserve the coadjoint orbits and hence the Casimirs exactly. We give a couple of examples, including the Kida vortex and the heavy top on a movable base with controls, which are Lie–Poisson systems on $ \mathfrak{so}(2,1)^{*} $ and $ (\mathfrak{se}(3) \ltimes \mathbb{R}^{3})^{*} $, respectively.

    Mathematics Subject Classification: Primary: 58F15, 53D20; Secondary: 37M15, 65P10, 70G65, 70H33.

    Citation:

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  • Figure 1.  (a) Time evolution of $ \mu $ computed using the canonized system (25). The solutions are shown for the time interval $ 0 \le t \le 100 $ with time step $ \Delta t = 0.1 $. (b) The red curve is the Lie–Poisson dynamics of the Kida vortex in $ \mathfrak{g}^{*} = {\mathfrak{so}}(2,1)^{*} \cong {\mathbb{R}}^{3} $ computed using the canonized system (25) and mapped by $ {\bf{M}}^{+} $ in (24). The green and orange surfaces are the level sets of the Hamiltonian $ h $ and the Casimir $ f_{1} $ from (22) and (19), respectively

    Figure 2.  Time evolutions of relative errors in Hamiltonian $ h $ and Casimir $ f_{1} $ from the Kida system. The dashed blue curve is the $ 4^{\rm th} $ order explicit Runge–Kutta method directly applied to Lie–Poisson equation (23) whereas the solid red curve is the $ 4^{\rm th} $ order Gauss–Legendre method applied to the canonized system (25). The solutions are shown for the time interval $ 0 \le t \le 1000 $ with time step $ \Delta t = 0.1 $. Note that, in (b), the red line is made thicker to make it visible; the actual variation is so small that it is barely visible if plotted with the same thickness as the blue line or as in (a)

    Figure 3.  Time evolutions of absolute or relative errors in components of momentum map $ {{\bf{J}}} $ from (16) and (20) computed by the $ 4^{\rm th} $ order Gauss–Legendre method applied to the canonized Kida system (25). The solutions are shown for the time interval $ 0 \le t \le 1000 $ with time step $ \Delta t = 0.1 $

    Figure 4.  Heavy top on a movable base

    Figure 5.  Time evolutions of relative errors in Hamiltonian $ h $ and three Casimirs $ f_{1} $, $ f_{2} $, $ f_{3} $ from the heavy top on a movable base system. The dashed blue curve is the Runge–Kutta method directly applied to Lie–Poisson equation (30) whereas the solid red curve is the $ 4^{\rm th} $ order Gauss–Legendre method applied to the canonized system. The solutions are shown for the time interval $ 0 \le t \le 30 $ with time step $ \Delta t = 0.01 $. Note that, in (b)–(d), the red line is made thicker to make it visible; the actual variation is so small that it is barely visible if plotted with the same thickness as the blue line or as in (a)

    Figure 6.  Time evolutions of errors in components of momentum map $ {\bf{J}} $ from (32) computed by the $ 4^{\rm th} $ order Gauss–Legendre method applied to the canonized system for heavy top on a movable base. Note that we used the absolute error for $ J_{0} $ because $ J_{0}(0) = 0 $, whereas all the others use relative errors. The solutions are shown for the time interval $ 0 \le t \le 30 $ with time step $ \Delta t = 0.01 $

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