论文标题
系统构建Painlevé-Type的非自主哈密顿方程。 I. Frobenius的整合性
Systematic construction of non-autonomous Hamiltonian equations of Painlevé-type. I. Frobenius integrability
论文作者
论文摘要
本文是三篇文章中的第一篇，探讨了Stäckel-Type和Painlevé-类型的动态系统之间的联系。在本文中，我们介绍了自动stäckel型系统的变形，向非自治的Frobenius可集成系统。首先，我们考虑具有与时间相关系数的可分离电势的二次化的准stäckel系统，然后我们提出了将这些方程式变形为非自主frobenius可集成系统的程序。然后，我们提出了一种将磁性分离电势与非自治的Frobenius集成系统变形的程序。我们还提供了所有$ 2 $ - 和$ 3 \，$ - 尺寸Frobenius的整合系统的完整列表，既有普通的和具有磁性电位，源自我们的建筑。此外，我们证明了两类系统之间的等效性。最后，我们展示了Painlevé方程$ p_ {i} -p_ {iv} $如何从我们的方案中得出。
This article is the first one in a suite of three articles exploring connections between dynamical systems of Stäckel-type and of Painlevé- type. In this article we present a deformation of autonomous Stäckel-type systems to non-autonomous Frobenius integrable systems. First, we consider quasi-Stäckel systems with quadratic in momenta Hamiltonians containing separable potentials with time dependent coefficients and then we present a procedure of deforming these equations to non-autonomous Frobenius integrable systems. Then, we present a procedure of deforming quasi-Stäckel systems with so called magnetic separable potentials to non-autonomous Frobenius integrable systems. We also provide a complete list of all $2$- and $3\,$-dimensional Frobenius integrable systems, both with ordinary and with magnetic potentials, that originate in our construction. Further, we prove the equivalence between both classes of systems. Finally we show how Painlevé equations $P_{I}-P_{IV}$ can be derived from our scheme.
