A note on the integrability of Hamiltonian 1 : 2 : 2 resonance

TL;DR

This paper investigates the integrability of the Hamiltonian normal form for 1:2:2 resonance, showing non-integrability for generic parameters and identifying a special case where integrability holds.

Contribution

It provides a rigorous non-integrability proof for the normal form up to order four using Morales-Ramis theory and highlights a unique integrable case.

Findings

Generic parameters lead to non-integrability.

A specific case of integrability is identified.

Normal form truncated to order three is integrable.

Abstract

We study the integrability of the Hamiltonian normal form of 1 : 2 : 2 resonance. It is known that this normal form truncated to order three is integrable. The truncated to order four normal form contains too many parameters. For a generic choice of parameters in the normal form up to order four we prove a non-integrability result using Morales-Ramis theory. We also isolate a non-trivial case of integrability.