A diagrammatic view of differential equations in physics

Evan Patterson; Andrew Baas; Timothy Hosgood; James Fairbanks; Evan Patterson; Andrew Baas; Timothy Hosgood; James Fairbanks

doi:10.3934/mine.2023036

Mathematics in Engineering

2023, Volume 5, Issue 2: 1-59. doi: 10.3934/mine.2023036

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Research article

A diagrammatic view of differential equations in physics

1.
Topos Institute, Berkeley, CA, USA
2.
Georgia Tech Research Institute, Atlanta, GA, USA
3.
University of Florida, Gainesville, FL, USA

Received: 07 April 2022 Revised: 23 May 2022 Accepted: 23 May 2022 Published: 02 June 2022

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.
- Tonti diagrams,
- category-theoretic diagrams,
- partial differential equations,
- exterior calculus,
- computational physics,
- multiphysics
Citation: Evan Patterson, Andrew Baas, Timothy Hosgood, James Fairbanks. A diagrammatic view of differential equations in physics[J]. Mathematics in Engineering, 2023, 5(2): 1-59. doi: 10.3934/mine.2023036

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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.

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