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

Matteo Petrera, Yuri B. Suris

Research output: Contribution to journalArticlepeer-review

6 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 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).

Original languageEnglish
Pages (from-to)401-408
Number of pages8
JournalJournal of Computational Dynamics
Volume6
Issue number2
DOIs
StatePublished - 2019
Externally publishedYes

Keywords

  • Birational maps
  • Hamiltonian systems
  • Integrable map
  • Kahan's discretization
  • Pencil of conics

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