Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems
arXiv:1810.09928 · doi:10.1098/rspa.2018.0761
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, 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 Hamiltonian vector field.
14 pages, 3 figures
References in corpus (2)
Cited by in corpus (9)
- Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems
- Manin involutions for elliptic pencils and discrete integrable systems
- A three-dimensional generalization of QRT maps
- On the singularity structure of Kahan discretizations of a class of quadratic vector fields
- An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps
- A new approach to integrals of discretizations by polarization
- Explicit isomorphisms for the symmetry algebras of continuous and discrete isotropic oscillators
- How one can repair non-integrable Kahan discretizations. II. A planar system with invariant curves of degree 6
- Kahan-Hirota-Kimura maps preserving original cubic hamiltonians