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Second order (conformally) superintegrable systems: classification, transformations and applications

二阶（共形）超可积系统：分类、转换和应用

基本信息

批准号：
353063958
负责人：
Dr. Andreas Vollmer
金额：
--
依托单位：
School of Mathematics and Statistics
依托单位国家：
德国
项目类别：
Research Fellowships
财政年份：
2017
资助国家：
德国
起止时间：
2016-12-31 至 2020-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/353063958?language=en
关键词：
Second order conformally superintegrable systems

项目摘要

Understanding spaces with a maximal number of functionally independent integrals of motion is a classical problem. A (maximally) superintegrable system is a Hamiltonian system on a n-dimensional manifold that admits 2n-1 functionally independent integrals of motion. The project aims to deepen our knowledge of 2-dimensional superintegrable systems, particularly those with quadratic and cubic integrals. New methods and techniques are used that have recently been established in related fields. The four major goals of the project are:(1) Provide an algebraic-geometric classification of 2-dimensional second-order (maximally) superintegrable systems on the sphere.The list of second-order superintegrable systems in dimension 2 is known, however ignoring the algebraic-geometric structure of the space of superintegrable systems. The project will fill this gap. The classification will provide insight into the properties of the variety of 2D second-order superintegrable systems on the sphere, which is a key problem in 2D and an important step towards a classification in arbitrary dimension.(2) Provide an organizing scheme for special functions originating from 2D second-order (maximally) superintegrable systems.There are several applications of the algebraic-geometric classification, e.g. quadratic algebras, transformations of superintegrable systems, or spectral theory. Moreover, the algebraic-geometric classification can be applied to special functions. The project focuses on this latter application because of the link between superintegrable systems in 2D and the so-called Askey scheme that organizes hypergeometric orthogonal polynomials. The project will improve the understanding of the link between superintegrable systems and special functions and likely reveal new properties and interconnections of special functions.(3) Provide a classification for Drach superintegrable systems on the 2-plane with one cubic and one quadratic integral of motion.(4) Provide such a classification for superintegrable systems with two cubic integrals of motion.Higher-order superintegrable systems have at least one integral of higher than quadratic degree in the velocities. Current knowledge about higher-order superintegrability is limited, but higher-order superintegrable systems have properties not found in second-order systems. The project aims to classify 2D superintegrable systems with cubic integrals in addition to the metric. To date, such systems are typically only understood if separation of variables in specific coordinate systems is assumed.

理解具有最大数量的函数独立运动积分的空间是一个经典问题。（极大）超可积系统是一个在n维流形上的哈密顿系统，它允许2n-1个函数独立的运动积分。该项目旨在加深我们对2维超可积系统的了解，特别是那些具有二次和三次积分的系统。使用了最近在相关领域建立的新方法和技术。该项目的四个主要目标是：（1）给出球面上2维二阶（极大）超可积系统的代数几何分类。2维二阶超可积系统的列表是已知的，但忽略了超可积系统空间的代数几何结构。该项目将填补这一空白。分类将提供洞察各种二维二阶超可积系统的性质，这是一个关键的问题，在2D和一个重要的一步，在任意维的分类。(2)给出一个由二维二阶（极大）超可积系统产生的特殊函数的组织方案。代数几何分类有几个应用，例如二次代数、超可积系统的变换或谱理论。此外，代数几何分类可以应用于特殊的功能。该项目的重点是后者的应用，因为超可积系统在2D和所谓的阿斯基计划，组织超几何正交多项式之间的联系。该项目将提高对超可积系统和特殊函数之间联系的理解，并可能揭示特殊函数的新性质和相互联系。(3)给出2-平面上具有一个三次和一个二次运动积分的Drach超可积系统的分类。(4)对具有两个三次运动积分的超可积系统给出这样的分类。高阶超可积系统至少有一个速度积分大于二次。目前关于高阶超可积性的知识是有限的，但高阶超可积系统具有二阶系统所没有的性质。该项目的目的是分类二维超可积系统与立方积分除了度量。到目前为止，这样的系统通常只有在假设特定坐标系中的变量分离时才能被理解。