Petrera Matteo, Smirin Jennifer, Suris Yuri B
Institut für Mathematik, MA 7-1, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany.
Proc Math Phys Eng Sci. 2019 Mar;475(2223):20180761. doi: 10.1098/rspa.2018.0761. Epub 2019 Mar 13.
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.
卡汉离散化适用于任何二次向量场，并产生一个双有理映射，该映射近似沿相流的平移。对于平面二次典范哈密顿向量场，已知此映射是可积的，并保持一族三次曲线。一般来说，这族曲线的九个基点包括三个无穷远点（对应于三次曲线的渐近方向）和位于一条二次曲线上的六个有限点。我们表明，卡汉离散化映射可以以六种不同方式表示为两个马尼恩对合的复合，分别对应一个无穷基点和一个有限基点。因此，可以对有限基点进行排序，使得得到的六边形有三对平行边，它们经过三个无穷基点。此外，基点上的这种几何条件结果是具有特征性的：如果满足该条件，那么相应曲线族中的三次曲线在平面二次典范哈密顿向量场的卡汉离散化下是不变的。
