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

  • * Corresponding author: Yuri B. Suris

    * Corresponding author: Yuri B. Suris
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  • 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. [5], 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 $.

    Mathematics Subject Classification: Primary: 65P10, 37M15; Secondary: 37J35, 14E05.

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  • Figure 1.  The black curve is the conic $ C = 0 $, the red lines represent the reducible conic $ D = 0 $. The finite base points are $ B_1, \ldots, B_4 $. The points $ B_0 $, $ B_5 $ are singular points of the Kanan map $ \Phi_f $, resp. of $ \Phi_f^{-1} $, for the following data: $ \ell(x,y) = 3x-y $, $ H(x,y) = -x^2+(2/5)xy+(1/2)y^2-4x+4y $

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    [3] E. Celledoni, D. I. McLaren, B. Owren and G. R. W. Quispel, Geometric and integrability properties of Kahan's method: The preservation of certain quadratic integrals, J. Phys. A, 52 (2019), 065201, 9 pp. doi: 10.1088/1751-8121/aafb1e.
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    [6] W. Kahan, Unconventional Numerical Methods for Trajectory Calculations, Unpublished lecture notes, 1993.
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    [8] M. PetreraA. Pfadler and Y. B. Suris, On integrability of Hirota-Kimura type discretizations, Regular Chaotic Dyn., 16 (2011), 245-289.  doi: 10.1134/S1560354711030051.
    [9] M. Petrera, J. Smirin and Y. B. Suris, Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems, Proc. Royal Soc. A, 475 (2019), 20180761, 13 pp.
    [10] M. Petrera and Y. B. Suris, On the Hamiltonian structure of Hirota-Kimura discretization of the Euler top, Math. Nachr., 283 (2010), 1654-1663.  doi: 10.1002/mana.200711162.
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