Document Zbl 07817671

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

A diagrammatic view of differential equations in physics. (English) Zbl 07817671

Math. Eng. (Springfield) 5, No. 2, Paper No. 36, 59 p. (2023).

Summary: 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.

Cited in 2 Documents

MSC:

81P68	Quantum computation
58A15	Exterior differential systems (Cartan theory)
18A10	Graphs, diagram schemes, precategories
78A30	Electro- and magnetostatics
14A10	Varieties and morphisms
82C70	Transport processes in time-dependent statistical mechanics

Keywords:

Tonti diagrams; category-theoretic diagrams; partial differential equations; exterior calculus; computational physics; multiphysics

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