Non-symplectic symmetries and bi-Hamiltonian structures of the rational Harmonic Oscillator

TL;DR

This paper explores the bi-Hamiltonian structures of the rational Harmonic Oscillator, revealing their origin in non-symplectic dynamical symmetries and deriving related recursion operators.

Contribution

It demonstrates that bi-Hamiltonian structures arise from non-symplectic symmetries and provides a geometric framework for understanding these structures in the rational Harmonic Oscillator.

Findings

Bi-Hamiltonian structures linked to non-symplectic symmetries

Recursion operators derived for the system

Geometric analysis of dynamical symmetries

Abstract

The existence of bi-Hamiltonian structures for the rational Harmonic Oscillator (non-central harmonic oscillator with rational ratio of frequencies) is analyzed by making use of the geometric theory of symmetries. We prove that these additional structures are a consequence of the existence of dynamical symmetries of non-symplectic (non-canonical) type. The associated recursion operators are also obtained.