An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps

TL;DR

This paper presents an elementary geometric construction of invariants for the KHK discretisation of 2D cubic Hamiltonian systems, revealing new insights into their structure and potential extensions.

Contribution

It introduces a simple geometric method to construct invariants of KHK maps, connecting them to hexagon side ratios and singular fibre structures.

Findings

Invariant expressed as product of ratios of affine polynomials

Application to multiple examples demonstrating the construction

Conjecture on possible extensions beyond initial hypotheses

Abstract

We show how to construct in an elementary way the invariant of the KHK discretisation of a cubic Hamiltonian system in two dimensions. That is, we show that this invariant is expressible as the product of the ratios of affine polynomials defining the prolongation of the three parallel sides of a hexagon. On the vertices of such a hexagon lie the indeterminacy points of the KHK map. This result is obtained analysing the structure of the singular fibres of the known invariant. We apply this construction to several examples, and we prove that a similar result holds true for a case outside the hypotheses of the main theorem, leading us to conjecture that further extensions are possible.