Non-integrability of a Hamiltonian system and Legendre functions

TL;DR

The paper analyzes the non‑integrability of a two‑degree‑of‑freedom Hamiltonian with a sixth‑order homogeneous potential by studying the solvability of the associated Legendre equation through Galois and Ziglin–Morales–Ramis–Simo theory. By reducing the problem to the linearized variation equations along an invariant manifold, the authors obtain a Legendre‑type equation with parameters $p=-\frac{1}{2}\pm\frac{1}{2}\sqrt{\frac{4C+2A}{9C}}$ and $q=\frac{1}{6}$, and define $\tau=\pm\sqrt{\frac{2A+4C}{9C}}$ with $C\neq0$. Using Kimura’s criteria, they establish non‑integrability for broad parameter ranges, including irrational $\tau$ and many rational families, some accompanied by the constraint $D\neq0$. These results provide rigorous obstructions to meromorphic Liouville integrability for this class of degree‑6 Hamiltonians.

Abstract

We investigate the solvability of the Galois group of the associated Legendre equation and we apply it it for study integrability to a Hamiltonian system with a homogeneous potential of degree 6. In this paper, we study the Hamiltonian system with Hamiltonian \\ $H=\frac{1}{2}(p_r^2+p_z^2)+r^6+Ar^2z^4+Dr^3z^3+Br^4z^2+Cz^6$, ($A,\, B,\, C,\, D \in \mathbb{R}$) for meromorphic integrability. The technique is an application of the Ziglin-Moralez-Ruiz-Ramis-Simo Theory.

Theorems & Definitions (5)

• Proposition 2.1
• Theorem 2.2
• Proposition 3.1
• Proposition 3.2
• Theorem 4.1