可积辛双有理映射交换系统的构造
A construction of commuting systems of integrable symplectic birational maps
摘要：
给出了具有三次Hamilton函数的完全可积$(2m)$维Hamilton系统的构造.该构造依赖于常数斜哈密顿矩阵$A$，即满足$A^{\rm T}J=JA$的矩阵，其中$J$是定义相空间$\mathbb R^{2m}$上的标准辛结构的非退化斜对称矩阵。应用到任何这样的系统的所谓的Kahan-Hirota-Kimura离散计划，我们得到一个双有理$(2m)$维映射。我们证明了这个映射是辛的关于一个辛结构，这是一个扰动的标准辛结构$\mathbb R^{2m}$，并具有$m$独立的运动积分，这是扰动的原始Hamilton函数和对合的不变辛结构。因此，这个映射在Liouville-Arnold意义下是完全可积的。此外，在对交换向量场的原始$m$ -元组进行适当的规范化后，它们的Kahan-Hirota-Kimura离散化也交换并共享不变辛结构和$m$运动积分.
英文摘要：
We give a construction of completely integrable $(2m)$-dimensional Hamiltonian systems with cubic Hamilton functions. The construction depends on a constant skew-Hamiltonian matrix $A$, that is, a matrix satisfying $A^{\rm T}J=JA$, where $J$ is a non-degenerate skew-symmetric matrix defining the standard symplectic structure on the phase space $\mathbb R^{2m}$. Applying to any such system the so called Kahan-Hirota-Kimura discretization scheme, we arrive at a birational $(2m)$-dimensional map. We show that this map is symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure on $\mathbb R^{2m}$, and possesses $m$ independent integrals of motion, which are perturbations of the original Hamilton functions and are in involution with respect to the invariant symplectic structure. Thus, this map is completely integrable in the Liouville-Arnold sense. Moreover, under a suitable normalization of the original $m$-tuples of commuting vector fields, their Kahan-Hirota-Kimura discretizations also commute and share the invariant symplectic structure and the $m$ integrals of motion.
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