Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. II. Systems with a linear Poisson tensor

TL;DR

This paper demonstrates that the Kahan discretization of planar quadratic Hamiltonian systems with a linear Poisson tensor can be characterized as a composition of two involutions on a pencil of conics, revealing a reversible geometric structure.

Contribution

It shows that the Kahan discretization can be expressed as a composition of two involutions, providing a geometric interpretation and reversing previous results about its structure.

Findings

The Kahan map is an integrable birational map preserving a pencil of conics.

The map can be represented as a composition of two involutions on the pencil.

The inverse map has a symmetric structure with three singular points.

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 with a linear Poisson tensor and with a quadratic Hamilton function, this map is known to be integrable and to preserve a pencil of conics. In the paper `Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation' by P. van der Kamp et al., it was shown that the Kahan discretization can be represented as a composition of two involutions on the pencil of conics. In the present note, which can be considered as a comment to that paper, we show that this result can be reversed. For a linear form $\ell(x,y)$, let $B_1,B_2$ be any two distinct points on the line $\ell(x,y)=-c$, and let $B_3,B_4$ be any two distinct points on the line $\ell(x,y)=c$. Set $B_0=\tfrac{1}{2}(B_1+B_3)$ and $B_5=\tfrac{1}{2}(B_2+B_4)$; these points lie on the line $\ell(x,y)=0$. Finally, let $B_\infty$ be the point at infinity on this line. Let $\mathfrak E$ be the pencil of conics with the base points $B_1,B_2,B_3,B_4$. Then the composition of the $B_\infty$-switch and of the $B_0$-switch on the pencil $\mathfrak E$ is the Kahan discretization of a Hamiltonian vector field $f=\ell(x,y)\begin{pmatrix}\partial H/\partial y \\ -\partial H/\partial x \end{pmatrix}$ with a quadratic Hamilton function $H(x,y)$. This birational map $\Phi_f:\mathbb C P^2\dashrightarrow\mathbb C P^2$ has three singular points $B_0,B_2,B_4$, while the inverse map $\Phi_f^{-1}$ has three singular points $B_1,B_3,B_5$.