Dual pairs and regularization of Kummer shapes in resonances

Ohsawa Tomoki

doi:10.3934/jgm.2019012

Article Contents

2019, Volume 11, Issue 2: 225-238. Doi: 10.3934/jgm.2019012

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Dual pairs and regularization of Kummer shapes in resonances

Ohsawa Tomoki

Department of Mathematical Sciences, The University of Texas at Dallas, 800 W Campbell Rd, Richardson, TX 75080-3021, USA

In honor of Darryl Holm's 70th birthday

Received: January 31, 2018

Revised: September 30, 2018

Published: May 2019

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Abstract

We present an account of dual pairs and the Kummer shapes for $ n:m $ resonances that provides an alternative to Holm and Vizman's work. The advantages of our point of view are that the associated Poisson structure on $ \mathfrak{su}(2)^{*} $ is the standard $ (+) $-Lie-Poisson bracket independent of the values of $ (n,m) $ as well as that the Kummer shape is regularized to become a sphere without any pinches regardless of the values of $ (n,m) $. A similar result holds for $ n:-m $ resonance with a paraboloid and $ \mathfrak{su}(1,1)^{*} $. The result also has a straightforward generalization to multidimensional resonances as well.

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Mathematics Subject Classification: 37J15, 53D17, 53D20, 70K30.

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Figure 1. The Kummer shape is regularized to be the sphere (green), and the reduced dynamics (red) (11) is at the intersection of the sphere and the level set (blue) of the Hamiltonian $h$.

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