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Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. II. Systems with a linear Poisson tensor | | Matteo Petrera
; Yuri B. Suris
; | | Date: |
13 Nov 2018 | | 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
$f=ell(x,y)egin{pmatrix}partial H/partial y \ -partial H/partial x
end{pmatrix}$ with a quadratic Hamilton function $H(x,y)$. This birational map
$Phi_f:mathbb C P^2dashrightarrowmathbb C P^2$ has three singular points
$B_0,B_2,B_4$, while the inverse map $Phi_f^{-1}$ has three singular points
$B_1,B_3,B_5$. | | Source: | arXiv, 1811.5791 | | Services: | Forum | Review | PDF | Favorites | |
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