Foundations of the mathematical theory of electromagnetic waves
Foundations of the mathematical theory of electromagnetic waves
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DOI：
10.1007/978-3-662-11773-6
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发表时间：
1969
期刊：
影响因子：
--
通讯作者：
Claus Müller
中科院分区：
文献类型：
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作者：
Claus Müller
电磁波的技术应用创造了一个类似于牛顿势的经典理论的研究领域，该理论旨在建立电磁波的数学理论。这一趋势是由H. Hertz和G.在1880年到1890年之间。他们介绍的麦克斯韦的理论制定了许多数学问题的极大的普遍性和减少的理论描述的电磁现象的解决方案定义明确的数学问题。电磁波的快速技术发展始于狄利克雷和诺依曼问题首次解决的时候。继弗雷德霍尔姆的文件，1904年的线性积分方程的许多开放的问题，数学物理解决了迅速连续的D。Hilbert和H.庞加莱这些结果影响了电磁波理论，其中边值问题和本征值问题最为著名。因此，第一次数学研究与经典势理论密切相关。最有趣的结果，这一次是洛伦兹公设的公式有关的渐近行为的腔的本征频率，H。魏尔在1910年至1915年之间。这里很明显，电磁波理论的问题不能被理解为势理论问题的简单扩展，而是它们具有典型的困难，这是由于麦克斯韦方程的特殊形式造成的。与势理论的技术类似，发展了一些方法，这些方法遵循分离变量的思想，发现了麦克斯韦方程的特殊解。因此G.米氏在1908年解决了球面衍射问题。讨论了A。索末菲在世纪之交发现的使用了相关的结构。
The technical applications of the electromagnetic waves created a field of research similar to the classical theory of the Newtonian potential which aims at a mathematical theory of the electromagnetic waves. This trend was initiated by the strongly mathematical character of the fundamental papers published by H. Hertz and G. Heaviside between 1880 and 1890. Their presentation of Maxwell's theory formulated many mathematical problems of great generality and reduced the theoretical description of the electromagnetic phenomena to the solution of well defined mathematical problems. The rapid technical development of the electromagnetic waves began at the time when the Dirichlet and Neumann problems of potential theory were first solved. Following Fredholm's paper of 1904 on linear integral equations many of the open questions of mathematical physics were settled in quick succession by D. Hilbert and H. Poincaré. It seems natural that these results among which the boundary value and the eigenvalue problems are best known influenced the theory of electromagnetic waves. The first mathematical investigations are therefore closely related to the classical potential theory. The most interesting results of this time are the formulations of the Lorentz postulate regarding the asymptotic behavior of the eigenfrequences of cavities which H. Weyl gave between 1910 and 1915. Here it became obvious that the problems of the theory of electromagnetic waves can not be understood as simple extensions of the problems of potential theory, but that they possess typical difficulties which result from the special form of Maxwell's equations.In analogy to the techniques of potential theory, methods were developed which, following the idea of the separation of variables, discovered special solutions of Maxwell's equations. Thus G. Mie solved the problem of the diffraction by a sphere in 1908. The diffraction by the wedge and half-plane which A. Sommerfeld found at the turn of the century uses related structures.