Group theory and nonlinear phenomena in physics

物理学中的群论和非线性现象

• 批准号: 4942-2011
• 负责人: Winternitz, Pavel
• 金额: $ 3.72万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568381
• 关键词: Group; theory; nonlinear; phenomena; physics

1.Continuous Symmetries of Discrete Equations. The objective of this research program is to turn Lie group theory into an effective tool for solving difference equations, as it already is for differential ones. The program has two aspects. In the first one, the difference equations and lattices are given a priori, i.e. come from the description of a discrete physical system and the aim is to find their symmetry group and use it to solve the difference system analytically. The second aspect concerns continuous phenomena described by differential equations with a nontrivial symmetry group. Here the aim is to discretize the equation while preserving its symmetries and then to use the obtained difference system to obtain improved numerical solutions of the original equation. During the coming five years my students and I will concentrate on this program for partial differential equations and multidimensional difference ones. In particular we will study discrete relativistic and nonrelativistic quantum systems on lattices. For linear systems we will use umbral calculus to impose Lorentz or Galilei invariance, for nonlinear one we will construct symmetry adapted (and transforming) lattices. An interesting byproduct of this research is the use of umbral calculus to produce a class of orthogonal polynomials particularly well suited to describe flattened optical beams in laser physics. 2. Superintegrable Systems in Classical and Quantum Mechanics. These are systems with more integrals of motion than degrees of freedom (like the Kepler-Coulomb system). All bounded trajectories in such systems in classical mechanics are periodic. In quantum mechanics they have degenerate energy levels and are exactly solvable. We have recently discovered an infinite family of such systems in a plane and will investigate its properties. I mention that the Hohmann Transfer, a fundamental procedure for the positioning of satellites and orbital maneuvering of interplanetary spacecraft is based on the superintegrability of the Kepler system.

1.离散方程的连续对称性。 这个研究项目的目标是把李群理论变成一个有效的工具来解决 差分方程，因为它已经是微分方程。该方案有两个方面。在第一种方法中，差分方程和格是先验给出的，即来自离散物理系统的描述，目的是找到它们的对称群，并用它来解析地求解差分系统。第二个方面涉及的连续现象所描述的微分方程的非平凡对称群。这里的目的是离散方程，同时保持其对称性，然后使用所获得的差分系统，以获得改进的数值解的原始方程。在未来的五年里，我和我的学生将专注于偏微分方程和多维差分方程的这个程序。特别是我们将研究离散的相对论和非相对论量子系统的晶格。对于线性系统，我们将使用本底演算来施加洛伦兹或伽利略不变性，对于非线性系统，我们将构造对称适应（和变换）格。 这项研究的一个有趣的副产品是使用本影演算来产生一类正交多项式，特别适合于描述激光物理中的扁平光束。 2.经典与量子力学中的超可积系统。 这些系统具有比自由度更多的运动积分（如开普勒-库仑系统）。经典力学中这类系统的所有有界轨线都是周期性的。在量子力学中，它们具有简并能级，并且是精确可解的。我们最近在平面上发现了一个这样的系统的无限族，并将研究它的性质。我提到 霍曼转移是卫星定位和行星际航天器轨道机动的基本程序，它基于开普勒系统的超可积性。