Research article

A diagrammatic view of differential equations in physics

  • 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.

    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

    Related Papers:

  • 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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    [1] R. Abraham, J. E. Marsden, T. Ratiu, Manifolds, tensor analysis, and applications, 2 Eds., New York, NY: Springer, 1988. http://doi.org/10.1007/978-1-4612-1029-0
    [2] P. Alotto, F. Freschi, M. Repetto, Multiphysics problems via the cell method: The role of Tonti diagrams, IEEE Trans. Magn., 46 (2010), 2959–2962. http://doi.org/10.1109/TMAG.2010.2044487 doi: 10.1109/TMAG.2010.2044487
    [3] M. Arkowitz, Introduction to homotopy theory, New York, NY: Springer, 2011. http://doi.org/10.1007/978-1-4419-7329-0
    [4] J. Baez, J. P. Muniain, Gauge fields, knots and gravity, World Scientific, 1994. https://doi.org/10.1142/2324
    [5] J. Baez, M. Stay, Physics, topology, logic and computation: a Rosetta Stone, In: New structures for physics, Berlin, Heidelberg: Springer, 2010, 95–172. http://doi.org/10.1007/978-3-642-12821-9_2
    [6] J. C. Baez, K. Courser, Structured cospans, Theor. Appl. Categ., 35 (2020), 1771–1822.
    [7] J. C. Baez, F. Genovese, J. Master, M. Shulman, Categories of nets, In: 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), IEEE, 2021, 1–13. http://doi.org/10.1109/LICS52264.2021.9470566
    [8] J. C. Baez, D. Weisbart, A. M. Yassine, Open systems in classical mechanics, J. Math. Phys., 62 (2021), 042902. http://doi.org/10.1063/5.0029885 doi: 10.1063/5.0029885
    [9] W. Barham, P. J. Morrison, E. Sonnendrücker, A Hamiltonian model for the macroscopic Maxwell equations using exterior calculus, 2021, arXiv: 2108.07382.
    [10] A. Bossavit, A rationale for 'edge-elements' in 3-D fields computations, IEEE Trans. Magn., 24 (1988), 74–79. http://doi.org/10.1109/20.43860 doi: 10.1109/20.43860
    [11] A. Bossavit, On the geometry of electromagnetism (4): Maxwell's house, J. Japan Soc. Appl. Electromagn. & Mech, 6 (1998), 318–326.
    [12] R. Bott, L. W. Tu, Differential forms in algebraic topology, New York, NY: Springer, 1982. http://doi.org/10.1007/978-1-4757-3951-0
    [13] P. Boyland, Fluid mechanics and mathematical structures, In: An introduction to the geometry and topology of fluid flows, Dordrecht: Springer, 2001, 105–134. http://doi.org/10.1007/978-94-010-0446-6_6
    [14] G. E. Bredon, Sheaf theory, 2 Eds., New York, NY: Springer, 1997. http://doi.org/10.1007/978-1-4612-0647-7
    [15] W. L. Burke, Applied differential geometry, Cambridge University Press, 1985. http://doi.org/10.1017/CBO9781139171786
    [16] Q. Chen, Z. Qin, R. Temam, Treatment of incompatible initial and boundary data for parabolic equations in higher dimension, Math. Comput., 80 (2011), 2071–2096. http://doi.org/10.1090/S0025-5718-2011-02469-5 doi: 10.1090/S0025-5718-2011-02469-5
    [17] B. Coecke, E. O. Paquette, Categories for the practising physicist, In: New structures for physics, Berlin, Heidelberg: Springer, 2010, 173–286. http://doi.org/10.1007/978-3-642-12821-9_3
    [18] I. Cortes Garcia, S. Schöps, H. De Gersem, S. Baumanns, Systems of differential algebraic equations in computational electromagnetics, In: Applications of differential-algebraic equations: examples and benchmarks, Cham: Springer, 2018, 123–169. http://doi.org/10.1007/11221_2018_8
    [19] J. Crank, The mathematics of diffusion, 2 Eds., Oxford University Press, 1975.
    [20] M. Desbrun, A. N. Hirani, M. Leok, J. E. Marsden, Discrete exterior calculus, 2005, arXiv: math/0508341.
    [21] G. A. Deschamps, Electromagnetics and differential forms, Proc. IEEE, 69 (1981), 676–696. http://doi.org/10.1109/PROC.1981.12048 doi: 10.1109/PROC.1981.12048
    [22] M. J. Dupré, S. I. Rosencrans, Classical and relativistic vorticity in a semi-Riemannian manifold, J. Math. Phys., 19 (1978), 1532–1535. http://doi.org/10.1063/1.523861 doi: 10.1063/1.523861
    [23] S. Eilenberg, S. Mac Lane, General theory of natural equivalences, Trans. Amer. Math. Soc., 58 (1945), 231–294. http://doi.org/10.1090/S0002-9947-1945-0013131-6 doi: 10.1090/S0002-9947-1945-0013131-6
    [24] B. Farkas, S.-A. Wegner, Variations on barbălat's lemma, The American Mathematical Monthly, 123 (2016), 825–830. http://doi.org/10.4169/amer.math.monthly.123.8.825 doi: 10.4169/amer.math.monthly.123.8.825
    [25] P. C. Fife, Mathematical aspects of reacting and diffusing systems, Berlin, Heidelberg: Springer, 1979. http://doi.org/10.1007/978-3-642-93111-6
    [26] B. Fong, Decorated cospans, Theor. Appl. Categ., 30 (2015), 1096–1120.
    [27] B. Fong, D. I. Spivak, Hypergraph categories, J. Pure Appl. Algebra, 223 (2019), 4746–4777. http://doi.org/10.1016/j.jpaa.2019.02.014 doi: 10.1016/j.jpaa.2019.02.014
    [28] P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory, Berlin, Heidelberg: Springer, 1967. http://doi.org/10.1007/978-3-642-85844-4
    [29] A. Grigor'yan, Introduction to analysis on graphs, American Mathematical Society, 2018.
    [30] P. W. Gross, P. R. Kotiuga, Electromagnetic theory and computation: a topological approach, Cambridge University Press, 2004. http://doi.org/10.1017/CBO9780511756337
    [31] R. Guitart, Sur le foncteur diagrammes, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 14 (1973), 181–182.
    [32] R. Guitart, Remarques sur les machines et les structures, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 15 (1974), 113–144.
    [33] R. Guitart, L. Van den Bril, Décompositions et lax-complétions, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 18 (1977), 333–407.
    [34] S. Gürer, P. Iglesias-Zemmour, Differential forms on manifolds with boundary and corners, Indagat. Math., 30 (2019), 920–929. http://doi.org/10.1016/j.indag.2019.07.004 doi: 10.1016/j.indag.2019.07.004
    [35] H. Halvorson, D. Tsementzis, Categories of scientific theories, In: Categories for the working philosopher, Oxford University Press, 2017, 402–429. http://doi.org/10.1093/oso/9780198748991.003.0017
    [36] A. N. Hirani, Discrete exterior calculus, PhD thesis, California Institute of Technology, 2003.
    [37] N. Johnson, D. Yau, Bimonoidal categories, $E_n$-monoidal categories, and algebraic $K$-theory, 2021. Available from: https://nilesjohnson.net/drafts/Johnson_Yau_ring_categories.pdf.
    [38] P. T. Johnstone, Fibrations and partial products in a 2-category, Appl. Categor. Struct., 1 (1993), 141–179. http://doi.org/10.1007/BF00880041 doi: 10.1007/BF00880041
    [39] P. T. Johnstone, Sketches of an elephant: A topos theory compendium, Oxford University Press, 2002.
    [40] M. Kashiwara, P. Schapira, Categories and sheaves, Berlin, Heidelberg: Springer, 2006. http://doi.org/10.1007/3-540-27950-4
    [41] P. Katis, N. Sabadini, R. F. C. Walters, Span(Graph): A categorical algebra of transition systems, In: Algebraic methodology and software technology. AMAST 1997, Berlin, Heidelberg: Springer, 1997, 307–321. http://doi.org/10.1007/BFb0000479
    [42] D. E. Keyes, L. C. McInnes, C. Woodward, W. Gropp, E. Myra, M. Pernice, et al., Multiphysics simulations: Challenges and opportunities, The International Journal of High Performance Computing Applications, 27 (2013), 4–83. http://doi.org/10.1177/1094342012468181 doi: 10.1177/1094342012468181
    [43] A. Kock, Limit monads in categories, PhD thesis, The University of Chicago, 1967.
    [44] J. Kock, Whole-grain Petri nets and processes, 2022, arXiv: 2005.05108.
    [45] I. Kolár, P. W. Michor, J. Slovák, Natural operations in differential geometry, Berlin, Heidelberg: Springer, 1993. http://doi.org/10.1007/978-3-662-02950-3
    [46] E. A. Lacomba, F. Ongay, On a structural schema of physical theories proposed by E. Tonti, In: The mathematical heritage of C. F. Gauss, World Scientific, 1991, 432–453. http://doi.org/10.1142/9789814503457_0032
    [47] V. Lahtinen, P. R. Kotiuga, A. Stenvall, An electrical engineering perspective on missed opportunities in computational physics, 2018, arXiv: 1809.01002.
    [48] G. F. Lawler, Random walk and the heat equation, American Mathematical Society, 2010. http://doi.org/10.1090/stml/055
    [49] F. W. Lawvere, Functorial semantics of algebraic theories, PhD thesis, Columbia University, 1963.
    [50] T. Leinster, Higher operads, higher categories, Cambridge University Press, 2004. http://doi.org/10.1017/CBO9780511525896
    [51] T. Leinster, Basic category theory, Cambridge University Press, 2014. http://doi.org/10.1017/CBO9781107360068
    [52] S. Libkind, A. Baas, M. Halter, E. Patterson, J. Fairbanks, An algebraic framework for structured epidemic modeling, 2022, arXiv: 2203.16345.
    [53] S. Libkind, A. Baas, E. Patterson, J. Fairbanks, Operadic modeling of dynamical systems: mathematics and computation, 2021, arXiv: 2105.12282.
    [54] S. Mac Lane, Categories for the working mathematician, 2 Eds., New York, NY: Springer, 1978. http://doi.org/10.1007/978-1-4757-4721-8
    [55] R. B. Melrose, Differential analysis on manifolds with corners, unpublished work.
    [56] J. Meseguer, U. Montanari, Petri nets are monoids, Inform. Comput., 88 (1990), 105–155. http://doi.org/10.1016/0890-5401(90)90013-8 doi: 10.1016/0890-5401(90)90013-8
    [57] P. W. Michor, Topics in differential geometry, American Mathematical Society, 2008.
    [58] P. W. Michor, Manifolds of mappings for continuum mechanics, In: Geometric continuum mechanics, Cham: Birkhäuser, 2020, 3–75. http://doi.org/10.1007/978-3-030-42683-5_1
    [59] J. Moeller, C. Vasilakopoulou, Monoidal grothendieck construction, Theor. Appl. Categ., 35 (2020), 1159–1207.
    [60] M. S. Mohamed, A. N. Hirani, R. Samtaney, Discrete exterior calculus discretization of incompressible navier–stokes equations over surface simplicial meshes, J. Comput. Phys., 312 (2016), 175–191. http://doi.org/10.1016/j.jcp.2016.02.028 doi: 10.1016/j.jcp.2016.02.028
    [61] P. Mullen, A. McKenzie, D. Pavlov, L. Durant, Y. Tong, E. Kanso, et al., Discrete lie advection of differential forms, Found. Comput. Math., 11 (2011), 131–149. http://doi.org/10.1007/s10208-010-9076-y doi: 10.1007/s10208-010-9076-y
    [62] J. Peetre, Réctification à l'article: Une caractérisation abstraite des opérateurs différentiels, Math. Scand., 8 (1960), 116–120. http://doi.org/10.7146/math.scand.a-10598 doi: 10.7146/math.scand.a-10598
    [63] P. Perrone, W. Tholen, Kan extensions are partial colimits, Appl. Categor. Struct., in press. http://doi.org/10.1007/s10485-021-09671-9
    [64] G. Peschke, W. Tholen, Diagrams, fibrations, and the decomposition of colimits, 2020, arXiv: 2006.10890.
    [65] S. Ramanan, Global calculus, American Mathematical Society, 2005.
    [66] E. Riehl, Factorization systems, 2008. Available from: https://math.jhu.edu/~eriehl/factorization.pdf.
    [67] E. Riehl, Categorical homotopy theory, Cambridge University Press, 2014. http://doi.org/10.1017/CBO9781107261457
    [68] E. Riehl, Category theory in context, Dover Publications, 2016.
    [69] M. Robinson, Sheaf and duality methods for analyzing multi-model systems, In: Recent applications of harmonic analysis to function spaces, differential equations, and data science, Cham: Birkhäuser, 2017, 653–703. http://doi.org/10.1007/978-3-319-55556-0_8
    [70] J. Rubinstein, Sine-Gordon equation, J. Math. Phys., 11 (1970), 258–266. http://doi.org/10.1063/1.1665057 doi: 10.1063/1.1665057
    [71] P. Selinger, A survey of graphical languages for monoidal categories, In: New structures for physics, Berlin, Heidelberg: Springer, 2010, 289–355. http://doi.org/10.1007/978-3-642-12821-9_4
    [72] M. Shulman, Framed bicategories and monoidal fibrations, Theor. Appl. Categ., 20 (2008), 650–738.
    [73] S. A. Socolofsky, G. H. Jirka, CVEN 489-501: Special topics in mixing and transport processes in the environment, Texas A & M University, Coastal and Ocean Engineering Division, Lecture notes.
    [74] D. I. Spivak, The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits, 2013, arXiv: 1305.0297.
    [75] D. I. Spivak, Database queries and constraints via lifting problems, Math. Struct. Comput. Sci., 24 (2014), e240602. http://doi.org/10.1017/S0960129513000479 doi: 10.1017/S0960129513000479
    [76] D. I. Spivak, Generalized lens categories via functors $\mathcal{C}^ \mathrm{op} \to \mathsf{Cat}$, 2019, arXiv: 1908.02202.
    [77] N. E. Steenrod, Cohomology operations, and obstructions to extending continuous functions, Adv. Math., 8 (1972), 371–416. http://doi.org/10.1016/0001-8708(72)90004-7 doi: 10.1016/0001-8708(72)90004-7
    [78] R. Street, R. F. C. Walters, The comprehensive factorization of a functor, Bull. Amer. Math. Soc., 79 (1973), 936–941. http://doi.org/10.1090/S0002-9904-1973-13268-9 doi: 10.1090/S0002-9904-1973-13268-9
    [79] E. Tonti, On the mathematical structure of a large class of physical theories, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, 52 (1972), 48–56.
    [80] E. Tonti, The mathematical structure of classical and relativistic physics, New York, NY: Birkhäuser, 2013. http://doi.org/10.1007/978-1-4614-7422-7
    [81] A. Vasy, Propagation of singularities for the wave equation on manifolds with corners, Ann. Math., 168 (2008), 749–812. http://doi.org/10.4007/annals.2008.168.749 doi: 10.4007/annals.2008.168.749
    [82] T. Wedhorn, Manifolds, sheaves, and cohomology, Wiesbaden: Springer, 2016. http://doi.org/10.1007/978-3-658-10633-1
    [83] C. Wells, Sketches: Outline with references, 1993. Available from: https://ncatlab.org/nlab/files/Wells_Sketches.pdf.
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