安培环路定理的拓扑学描述  

The topological description of Ampère s circuital law

作　　者：唐建立 李逸文 朱俊安 梁奇锋[1] TANG Jian-li;LI Yi-wen;ZHU Jun-an;LIANG Qi-feng(Department of Physics,Shaoxing University,Shaoxing,Zhejiang 312000,China)

机构地区：[1]绍兴文理学院物理系

出　　处：《大学物理》2020年第2期74-77,共4页College Physics

基　　金：国家自然科学基金(NSFC11574215);教育部留学回国人员启动项目

摘　　要：近年来拓扑学在量子力学中得到了广泛的运用.本文将安培环路定理积分式重新表达为一矢量场在轮胎参数面上的第一类陈数积分.数值模拟展示了该积分值为一整数即第一陈数,其代表矢量场的整体性质:当经历连续变换时,矢量场的局部数值发生改变但整体积分值即陈数仍保持不变;若陈数发生改变,则表明矢量场变换的连续性条件发生破坏,矢量场出现奇点.进一步通过高斯映射将该矢量场从参数轮胎面映射到单位球面上,并给出了第一陈数的直观几何意义.理论和数值结果揭示了安培环路定理的拓扑学本质,表明拓扑概念在经典物理学中也会有广泛应用.Topology has been widely discussed in quantum mechanics in recent years. In this paper,the integral formula of Ampère s circuital law is re-expressed as the integral formula of the first Chern number of a vector field defined on the torus parametric surface. The numerical simulation shows that the integral value is an integer,i.e.,the first Chen number,which represents the global property of vector field: When the vector field undergoes continuous transformation,the local value of the vector field changes,however the integral value,i. e.,the Chern number,remains unchanged;if the Chern number changes,it implies the vector field has experienced a discontinuous change and there rises the singularity in the distribution of vector field. Furthermore,the vector field is mapped from torus parametric surface to the unit sphere by Gauss mapping,and the intuitive geometric meaning of the first Chern number is given. The theoretical and numerical results reveal the topological nature of Ampère s circuital law,and show that the concept of topology can also be widely applied in classical physics.

关 键 词：安培环路定理 拓扑 陈数 

分 类 号：O413.1[理学—理论物理]