Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems

Matteo Petrera, Jennifer Smirin, Yuri B. Suris

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

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 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.

Original languageEnglish
Article number20180761
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume475
Issue number2223
DOIs
StatePublished - 1 Mar 2019
Externally publishedYes

Keywords

  • Elliptic curve
  • Hamiltonian vector field
  • Integrable discretization
  • Integrable maps
  • Manin transformation

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