Noncommutative Kepler Dynamics: symmetry groups and bi-Hamiltonian structures

TL;DR

This paper explores the noncommutative Kepler problem, revealing its symmetry groups, deriving Hamiltonian dynamics in action-angle coordinates, and establishing a hierarchy of bi-Hamiltonian structures with Nijenhuis recursion operators.

Contribution

It introduces a novel analysis of noncommutative Kepler dynamics, identifying symmetry groups and constructing bi-Hamiltonian structures and recursion operators.

Findings

Identification of $SO(3),$ $SO(4),$ and $SO(1,3)$ symmetry groups.

Derivation of Hamiltonian vector field in action-angle coordinates.

Establishment of a hierarchy of bi-Hamiltonian structures.

Abstract

Integrals of motion are constructed from noncommutative (NC) Kepler dynamics, generating $SO(3),$ $SO(4),$ and $SO(1,3)$ dynamical symmetry groups. The Hamiltonian vector field is derived in action-angle coordinates, and the existence of a hierarchy of bi-Hamiltonian structures is highlighted. Then, a family of Nijenhuis recursion operators is computed and discussed.