N-dimensional integrability from two-photon coalgebra symmetry

TL;DR

This paper constructs a broad class of N-dimensional Hamiltonian systems with multiple integrals of motion using two-photon coalgebra symmetry, identifying new integrable systems including natural, geodesic, and electromagnetic Hamiltonians.

Contribution

It introduces a universal set of integrals of motion for these systems and explicitly finds new integrable Hamiltonians through sub-coalgebra analysis.

Findings

Universal (N-2) integrals of motion for broad Hamiltonian class

Explicit construction of new integrable N-dimensional Hamiltonians

Identification of integrable systems in natural, geodesic, and electromagnetic contexts

Abstract

A wide class of Hamiltonian systems with N degrees of freedom and endowed with, at least, (N-2) functionally independent integrals of motion in involution is constructed by making use of the two-photon Lie-Poisson coalgebra. The set of (N-2) constants of the motion is shown to be a universal one for all these Hamiltonians, irrespectively of the dependence of the latter on several arbitrary functions and N free parameters. Within this large class of quasi-integrable N-dimensional Hamiltonians, new families of completely integrable systems are identified by finding explicitly a new independent integral through the analysis of the sub-coalgebra structure of the two-photon algebra. In particular, new completely integrable N-dimensional Hamiltonians describing natural systems, geodesic flows and static electromagnetic Hamiltonians are presented.