A diagrammatic view of differential equations in physics

TL;DR

This paper develops a rigorous mathematical framework using category theory to represent and analyze systems of differential equations in physics through diagrams, enabling modular construction of complex multiphysical systems.

Contribution

It introduces a systematic, diagrammatic approach based on category theory to formalize and reason about differential equations in physics, enhancing understanding and modularity.

Findings

Provides a category-theoretic diagrammatic framework for differential equations.

Demonstrates modular construction of complex physical systems.

Includes diverse examples from electromagnetism, fluid mechanics, and more.

Abstract

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.