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

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 $B_0=\frac{1}{2}(B_1+B_3)$ and $B_5=\frac{1}{2}(B_2+B_4)$; 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 $f=\ell (x,y)\left(\begin{array}{c}\partial H/\partial y\\ -\partial H/\partial x\end{array}\right)$ with a quadratic Hamilton function $H(x,y)$. This birational map ${\Phi }_f:\mathbb{C}{P}^2⇢\mathbb{C}{P}^2$ has three singular points $B_0,B_2,B_4$, while the inverse map ${\Phi }_f^{-1}$ has three singular points $B_1,B_3,B_5$.

8 pp, 1 figure. arXiv admin note: text overlap with arXiv:1810.09928

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• Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems