Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems
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.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 March 2019
 DOI:
 10.1098/rspa.2018.0761
 arXiv:
 arXiv:1810.09928
 Bibcode:
 2019RSPSA.47580761P
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Mathematical Physics;
 Mathematics  Algebraic Geometry
 EPrint:
 14 pages, 3 figures