Integrable Henon-Heiles Hamiltonians: a Poisson algebra approach

TL;DR

This paper revisits 2D Henon-Heiles integrable systems, introduces new integrable perturbations, and constructs N-dimensional generalizations using Poisson algebra, providing explicit integrals of motion and broad applicability.

Contribution

It introduces a novel algebraic method to generate N-dimensional integrable systems from 2D potentials using Poisson algebra, including new perturbations.

Findings

Found new integrable perturbations of Henon-Heiles systems.

Constructed explicit N-dimensional integrable generalizations.

Provided a systematic algebraic approach for integrable potentials.

Abstract

The three integrable two-dimensional Henon-Heiles systems and their integrable perturbations are revisited. A family of new integrable perturbations is found, and N-dimensional completely integrable generalizations of all these systems are constructed by making use of sl(2,R)+h(3) as their underlying Poisson symmetry algebra. In general, the procedure here introduced can be applied in order to obtain N-dimensional integrable generalizations of any 2D integrable potential of the form V(q_1^2, q_2), and the formalism gives the explicit form of all the integrals of the motion. Further applications of this algebraic approach in different contexts are suggested.