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Dual pairs and regularization of Kummer shapes in resonances

In honor of Darryl Holm's 70th birthday

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  • 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.

    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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