TY - JOUR

T1 - Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems

AU - Petrera, Matteo

AU - Smirin, Jennifer

AU - Suris, Yuri B.

N1 - Publisher Copyright:
© 2019 The Author(s) Published by the Royal Society. All rights reserved.

PY - 2019/3/1

Y1 - 2019/3/1

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 canonical Hamiltonian vector field, this map is known to be integrable and to preserve a pencil of cubic curves. Generically, the nine base points of this pencil include three points at infinity (corresponding to the asymptotic directions of cubic curves) and six finite points lying on a conic. We show that the Kahan discretization map can be represented in six different ways as a composition of two Manin involutions, corresponding to an infinite base point and to a finite base point. As a consequence, the finite base points can be ordered so that the resulting hexagon has three pairs of parallel sides which pass through the three base points at infinity. Moreover, this geometric condition on the base points turns out to be characteristic: if it is satisfied, then the cubic curves of the corresponding pencil are invariant under the Kahan discretization of a planar quadratic canonical Hamiltonian vector field.

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 canonical Hamiltonian vector field, this map is known to be integrable and to preserve a pencil of cubic curves. Generically, the nine base points of this pencil include three points at infinity (corresponding to the asymptotic directions of cubic curves) and six finite points lying on a conic. We show that the Kahan discretization map can be represented in six different ways as a composition of two Manin involutions, corresponding to an infinite base point and to a finite base point. As a consequence, the finite base points can be ordered so that the resulting hexagon has three pairs of parallel sides which pass through the three base points at infinity. Moreover, this geometric condition on the base points turns out to be characteristic: if it is satisfied, then the cubic curves of the corresponding pencil are invariant under the Kahan discretization of a planar quadratic canonical Hamiltonian vector field.

KW - Elliptic curve

KW - Hamiltonian vector field

KW - Integrable discretization

KW - Integrable maps

KW - Manin transformation

UR - http://www.scopus.com/inward/record.url?scp=85064261874&partnerID=8YFLogxK

U2 - 10.1098/rspa.2018.0761

DO - 10.1098/rspa.2018.0761

M3 - Article

AN - SCOPUS:85064261874

SN - 1364-5021

VL - 475

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

IS - 2223

M1 - 20180761

ER -