Kahan-Hirota-Kimura maps preserving original cubic hamiltonians

TL;DR

This paper investigates cubic Hamiltonian vector fields in low-dimensional spaces whose associated Kahan-Hirota-Kimura maps preserve the original Hamiltonian, revealing their integrability, symmetries, and symplectic properties.

Contribution

It extends the analysis of KHK maps preserving cubic Hamiltonians to $ extbf{R}^2$, $ extbf{R}^4$, and $ extbf{R}^6$, exploring their integrals, symmetries, and symplecticity.

Findings

Existence of additional first integrals in these systems.

KHK maps act as Lie symmetries for the vector fields.

The maps exhibit symplectic properties in the studied cases.

Abstract

We study the class of cubic Hamiltonian vector fields whose associated Kahan-Hirota-Kimura (KHK) maps preserve the original Hamiltonian function. Our analysis focuses on these fields in $\mathbb{R}^2$ and $\mathbb{R}^4$, extending to a family of fields in $\mathbb{R}^6$. Additionally, we investigate various properties of these fields, including the existence of additional first integrals of a specific type, their role as Lie symmetries of the corresponding KHK map, and the symplecticity of these maps.