A construction of commuting systems of integrable symplectic birational maps

TL;DR

This paper constructs integrable symplectic birational maps from Hamiltonian systems with cubic functions, showing they preserve integrals and symplectic structure, and are discretizations of commuting vector fields.

Contribution

It introduces a method to generate integrable birational maps via Kahan-Hirota-Kimura discretization applied to specific Hamiltonian systems with skew-Hamiltonian matrices.

Findings

The constructed maps are symplectic and integrable in the Liouville-Arnold sense.

The maps preserve a perturbed symplectic structure and integrals of motion.

Discretizations of commuting vector fields also commute and share invariants.

Abstract

We give a construction of completely integrable $(2m)$-dimensional Hamiltonian systems with cubic Hamilton functions. The construction depends on a constant skew-Hamiltonian matrix $A$, that is, a matrix satisfying $A^{\rm T}J=JA$, where $J$ is a non-degenerate skew-symmetric matrix defining the standard symplectic structure on the phase space $\mathbb R^{2m}$. Applying to any such system the so called Kahan-Hirota-Kimura discretization scheme, we arrive at a birational $(2m)$-dimensional map. We show that this map is symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure on $\mathbb R^{2m}$, and possesses $m$ independent integrals of motion, which are perturbations of the original Hamilton functions and are in involution with respect to the invariant symplectic structure. Thus, this map is completely integrable in the Liouville-Arnold sense. Moreover, under a suitable normalization of the original $m$-tuples of commuting vector fields, their Kahan-Hirota-Kimura discretizations also commute and share the invariant symplectic structure and the $m$ integrals of motion.