Dual Pairs and Regularization of Kummer Shapes in Resonances

TL;DR

This paper offers a new perspective on dual pairs and Kummer shapes in $n:m$ resonances, simplifying the Poisson structure and regularizing the shapes to spheres or paraboloids, applicable to multidimensional cases.

Contribution

It introduces an alternative approach that standardizes the Poisson structure and regularizes Kummer shapes across different resonance values, extending to multidimensional resonances.

Findings

Poisson structure on $rak{su}(2)^*$ is the standard Lie--Poisson bracket.

Kummer shapes are regularized to spheres or paraboloids.

Results generalize to multidimensional resonances.

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.