Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems

TL;DR

This paper explores the geometric structure of Kahan discretizations for planar quadratic Hamiltonian systems, revealing how their base points relate to integrability and invariance of cubic curve pencils.

Contribution

It demonstrates that the Kahan map can be expressed as compositions of Manin involutions and characterizes the geometric conditions for invariance of cubic curves.

Findings

Kahan discretization preserves a pencil of cubic curves in Hamiltonian systems.

The map can be represented as a composition of two Manin involutions.

A geometric condition on base points characterizes invariance of cubic curves.

Abstract

Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field, this map is known to be integrable and to preserve a pencil of cubic curves. Generically, the nine base points of this pencil include three points at infinity (corresponding to the asymptotic directions of cubic curves) and six finite points lying on a conic. We show that the Kahan discretization map can be represented in six different ways as a composition of two Manin involutions, corresponding to an infinite base point and to a finite base point. As a consequence, the finite base points can be ordered so that the resulting hexagon has three pairs of parallel sides which pass through the three base points at infinity. Moreover, this geometric condition on the base points turns out to be characteristic: if it is satisfied, then the cubic curves of the corresponding pencil are invariant under the Kahan discretization of a planar quadratic Hamiltonian vector field.