论文标题
复杂的功能和几何结构与具有广义的runge-lenz运动积分赋予的可整合开普勒相关的系统家族相关
Complex functions and geometric structures associated to the superintegrable Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion
论文作者
论文摘要
研究了研究二维抗凝结$（k_1，k_2，k_3）$的quasi-bi-hamiltonian结构的存在。我们利用一种与某些复杂功能相关的方法，这些功能满足了有趣的泊松托架关系，并且以前应用于标准的开孔器问题，以及某些特定的社交系统，例如Smorodinsky-Winternitz（SW）系统，Tremblay-Turbiner-Turbiner-Winternitz（TTW）和Winternitz（Pwinternitz（Pwinternitz）系统。我们证明这些复杂函数很重要，原因有两个：首先，它们决定了运动的积分，其次它们决定了某些几何结构的存在（在这种特殊情况下，Quasi-bi-hamiltonian结构）。 所有结果取决于三个参数（$ k_1，k_2，k_3 $）的方式，在特定情况下，$ k_1 \ ne 0 $，$ k_2 = k_3 = 0 $，我们恢复了原始开普勒问题的结果（以前在Sigma 12，010（2016年）中研究了原始Kepler问题的结果）。本文可以被认为是分为两个部分，每个部分都呈现出不同的方法（不同的复杂函数和不同的准黑米尔顿结构）。
The existence of quasi-bi-Hamiltonian structures for a two-dimensional superintegrable $(k_1,k_2,k_3)$-dependent Kepler-related problem is studied. We make use of an approach that is related with the existence of some complex functions which satisfy interesting Poisson bracket relations and that was previously applied to the standard Kepler problem as well as to some particular superintegrable systems as the Smorodinsky-Winternitz (SW) system, the Tremblay-Turbiner-Winternitz (TTW) and Post-Winternitz (PW) systems. We prove that these complex functions are important for two reasons: first, they determine the integrals of motion, and second they determine the existence of some geometric structures (in this particular case, quasi-bi-Hamiltonian structures). All the results depend of three parameters ($k_1, k_2, k_3$) in such a way that in the particular case $k_1\ne 0$, $k_2= k_3= 0$, we recover the results of the original Kepler problem (previously studied in SIGMA 12, 010 (2016)). This paper can be considered as divided in two parts and every part present a different approach (different complex functions and different quasi-bi-Hamiltonian structures).
