The n:m resonance dual pair

TL;DR

This paper constructs dual pairs of Poisson maps related to n:m resonance in Hamiltonian systems, revealing new structures beyond traditional momentum maps, with symplectic leaves shaped as Kummer surfaces.

Contribution

It introduces dual pairs of Poisson maps for n:m resonance, expanding the understanding of geometric structures in Hamiltonian dynamics beyond classical momentum maps.

Findings

Constructed dual pairs of Poisson maps for n:m resonance

Identified Kummer surfaces as symplectic leaves in the Poisson structure

Extended the geometric framework of resonance beyond momentum maps

Abstract

In this paper we build dual pairs of Poisson maps \[ \RR\stackrel{R_\pm}{\longleftarrow}(D,\om_\pm) \stackrel{\Pi_\pm}{\longrightarrow} B \] associated to $n:m$ resonance, as well as to $n:-m$ resonance. Except for the above mentioned cases $1:\pm1$, these are not pairs of momentum maps. Here $D$ is an open subset of $\CC^2$ with the above mentioned symplectic forms $\om_\pm$, and $B$ an open subset of $\RR^3$. The Poisson structure on $B$, which depends on the natural numbers $n$ and $m$, is not Lie-Poisson. Instead, its symplectic leaves are the Kummer shapes: bounded surfaces for $n:m$ resonance, and unbounded surfaces for $n:-m$ resonance