Differential geometry methods in mathematical physics

数学物理中的微分几何方法

• 批准号: RGPIN-2016-04133
• 负责人: Mclenaghan, Raymond
• 金额: $ 1.09万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675527
• 关键词: Differential; geometry; methods; mathematical; physics

The main theme of my research is the study of differential equations arising in mathematical physics by the methods of differential geometry and Lie group theory. The work is being conducted in three interrelated areas, the common threads of which are the geometrical and group invariant methods used.***The first area is the study of completely integrable Hamiltonian systems. Such systems include a large number of important physical models described in general by nonlinear systems of differential equations that can be integrated by quadratures. To solve the problem of integrability for such systems defined on spaces of constant curvature the theory of warped products and concircular tensors (a special type of conformal Killing tensor) is being used to give an invariant characterization of all inequivalent separable coordinate systems for the Hamilton-Jacobi equation for the geodesics.***The second area, which is closely related to the first, concerns separability theory for Dirac's equation on pseudo-Riemannian background spaces. This study is important for the elaboration of particle creation mechanisms in black hole physics. The goal is the development of a comprehensive theory of separation of variables for the Dirac equation.***The third area of study is the problem of the validity of Huygens' principle in the strict sense for second order linear partial differential equations of normal hyperbolic type. The physical significance of the Huygens' property is that the wave phenomena governed by the equation propagate sharply without the presence wave tails. Work is being undertaken, in the physically important case of four-dimensions, on Hadamard's problem which consists in the determination up to equivalence of all the Huygens' equations. **

我的主要研究课题是用微分几何和李群理论的方法研究数学物理中的微分方程。 这项工作是在三个相互关联的领域进行的，其共同点是使用的几何和群不变方法。第一个领域是研究完全可积的哈密顿系统。 这样的系统包括大量的重要的物理模型，一般由微分方程的非线性系统描述，可以通过求积积分。 为了解决定义在常曲率空间上的这类系统的可积性问题，翘曲积和共圆张量（一种特殊类型的共形Killing张量）的理论被用来给出测地线的Hamilton-Jacobi方程的所有不等价可分坐标系的不变特征。第二个领域，这是密切相关的第一个，涉及分离理论的狄拉克方程的伪黎曼背景空间。 这项研究对于阐述黑洞物理学中的粒子产生机制具有重要意义。 目标是发展狄拉克方程的分离变量的综合理论。第三个研究领域是惠更斯原理在严格意义上对二阶常双曲型线性偏微分方程的有效性问题。 惠更斯性质的物理意义在于，由该方程控制的波动现象在不存在波尾的情况下急剧传播。 工作正在进行中，在物理上重要的情况下，四维，阿达玛的问题，其中包括在确定等价的所有惠更斯方程。 **