论文标题
物理中微分方程的示意图
A diagrammatic view of differential equations in physics
论文作者
论文摘要
以图表形式呈现微分方程的系统已在物理的某些部位，尤其是电磁和计算物理学变得普遍。在这项工作中，我们旨在将这种图表使用在牢固的数学基础上，同时还将广泛适用的框架系统化，以正式推理方程系统及其解决方案。我们的主要数学工具是类别理论图，它们是众所周知的，并且图表之间的形态是不太理解的。作为图解框架的应用，我们展示了如何通过基本的物理原理模块化构建复杂的多物理系统。包括电磁，运输现象，流体力学和其他领域的大量例子。
Presenting systems of differential equations in the form of diagrams has become common in certain parts of physics, especially electromagnetism and computational physics. In this work, we aim to put such use of diagrams on a firm mathematical footing, while also systematizing a broadly applicable framework to reason formally about systems of equations and their solutions. Our main mathematical tools are category-theoretic diagrams, which are well known, and morphisms between diagrams, which have been less appreciated. As an application of the diagrammatic framework, we show how complex, multiphysical systems can be modularly constructed from basic physical principles. A wealth of examples, drawn from electromagnetism, transport phenomena, fluid mechanics, and other fields, is included.
