TY - JOUR

T1 - Geometry of the Kahan discretizations of planar quadratic hamiltonian systems. II. Systems with a linear poisson tensor

AU - Petrera, Matteo

AU - Suris, Yuri B.

N1 - Publisher Copyright:
© American Institute of Mathematical Sciences.

PY - 2019

Y1 - 2019

N2 - 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 l(x,y) let B1,B2 be any two distinct points on the line ℓ(x,y) = -c, and let B,B4 be any two distinct points on the line l(x,y) = c. Set B0 = 1/2 (B1+B3) and B5 = 1/2 (B2 + B4); these points lie on the line l(x,y) = 0. Finally, let B∞ be the point at infinity on this line. Let E be the pencil of conics with the base points B1,B2,B3,B4. Then the composition of the B∞-switch and of the B0-switch on the pencil E is the Kahan discretization of a Hamiltonian vector field f = l(x,y) (∂H/∂y -∂H/∂x) Swith a quadratic Hamilton function H(x,y).

AB - 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 l(x,y) let B1,B2 be any two distinct points on the line ℓ(x,y) = -c, and let B,B4 be any two distinct points on the line l(x,y) = c. Set B0 = 1/2 (B1+B3) and B5 = 1/2 (B2 + B4); these points lie on the line l(x,y) = 0. Finally, let B∞ be the point at infinity on this line. Let E be the pencil of conics with the base points B1,B2,B3,B4. Then the composition of the B∞-switch and of the B0-switch on the pencil E is the Kahan discretization of a Hamiltonian vector field f = l(x,y) (∂H/∂y -∂H/∂x) Swith a quadratic Hamilton function H(x,y).

KW - Birational maps

KW - Hamiltonian systems

KW - Integrable map

KW - Kahan's discretization

KW - Pencil of conics

UR - http://www.scopus.com/inward/record.url?scp=85077569617&partnerID=8YFLogxK

U2 - 10.3934/jcd.2019020

DO - 10.3934/jcd.2019020

M3 - Article

AN - SCOPUS:85077569617

SN - 2158-2505

VL - 6

SP - 401

EP - 408

JO - Journal of Computational Dynamics

JF - Journal of Computational Dynamics

IS - 2

ER -