Integrals and chaos in generalized H\'{e}non-Heiles Hamiltonians

TL;DR

This paper investigates the approximate integrals of motion in generalized Hénon-Heiles Hamiltonians, comparing theoretical and numerical surfaces of section, and explores how chaos emerges through resonance overlap, especially near the original Hénon-Heiles system.

Contribution

It extends the analysis of integrals of motion to a generalized Hénon-Heiles Hamiltonian, examining the transition from regular to chaotic dynamics and the resonance overlap mechanism.

Findings

Invariant curves are close for small perturbations but diverge as perturbation increases.

Chaos becomes dominant near the escape energy for certain parameter values.

Resonance overlap leads to chaos in the original Hénon-Heiles system.

Abstract

We study the approximate (formal) integrals of motion in the Hamiltonian $ H = \frac{1}{2}\left( \dot{x}^2 + \dot{y}^2 + x^2 + y^2 \right) + \epsilon\,\left( xy^2 + \alpha x^3\right)$ which is an extension of the usual H\'{e}non-Heiles Hamiltonian that has $\alpha = -1/3$. We compare the theoretical surfaces of section (at $y=0$) with the exact surfaces of section calculated by integrating numerically many orbits. For small $\epsilon$, the invariant curves of the theoretical and the exact surfaces of section are close to each other, but for large $\epsilon$ there are differences. The most important is the appearance of chaos in the exact case, which becomes dominant as $\epsilon$ approaches the escape perturbation for $\alpha<0$. We study in particular the cases $\alpha = 1/3$, which represents an integrable system, and $\alpha = 0$. Finally we examine the generation of chaos through the resonance overlap mechanism in the case $\alpha=-1/3$ (the original H\'{e}non-Heiles system) by showing both the homoclinic and the heteroclinic intersection of the asymptotic curves of the unstable periodic orbits.