Petrera Matteo, Suris Yuri B
Institut für Mathematik, MA 7-1 , Technische Universität Berlin , Str. des 17. Juni 136, Berlin 10623, Germany.
Proc Math Phys Eng Sci. 2017 Feb;473(2198):20160535. doi: 10.1098/rspa.2016.0535.
We give a construction of completely integrable four-dimensional Hamiltonian systems with cubic Hamilton functions. Applying to the corresponding pairs of commuting quadratic Hamiltonian vector fields the so called Kahan-Hirota-Kimura discretization scheme, we arrive at pairs of birational four-dimensional maps. We show that these maps are symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure on [Formula: see text], and possess two independent integrals of motion, which are perturbations of the original Hamilton functions and which are in involution with respect to the perturbed symplectic structure. Thus, these maps are completely integrable in the Liouville-Arnold sense. Moreover, under a suitable normalization of the original pairs of vector fields, the pairs of maps commute and share the invariant symplectic structure and the two integrals of motion.
我们给出了具有三次哈密顿函数的完全可积四维哈密顿系统的一种构造。将所谓的卡汉 - 广田 - 木村离散化方案应用于相应的对易二次哈密顿向量场对，我们得到了双有理四维映射对。我们表明，这些映射相对于一个辛结构是辛的，该辛结构是对(\mathbb{R}^4)上标准辛结构的扰动，并且具有两个独立的运动积分，它们是原始哈密顿函数的扰动，并且相对于扰动后的辛结构是对合的。因此，这些映射在刘维尔 - 阿诺德意义下是完全可积的。此外，在对原始向量场对进行适当归一化的情况下，映射对是对易的，并且共享不变辛结构和两个运动积分。
