Organizing physics with open energy-driven systems

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Capucci, Matteo and Lynch, Owen and Spivak, David I. (2025) Organizing physics with open energy-driven systems. Electronic Proceedings in Theoretical Computer Science, 429. pp. 287-301. ISSN 2075-2180 (https://doi.org/10.4204/eptcs.429.16)

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Abstract

Organizing physics has been a long-standing preoccupation of applied category theory, going back at least to Lawvere. We contribute to this research thread by noticing that Hamiltonian mechanics and gradient descent depend crucially on a consistent choice of transformation -- which we call a reaction structure -- from the cotangent bundle to the tangent bundle. We then construct a compositional theory of reaction structures. Reaction-based systems offer a different perspective on composition in physics than port-Hamiltonian systems or open classical mechanics, in that reaction-based composition does not create any new constraints that must be solved for algebraically. The technical contributions of this paper are the development of symmetric monoidal categories of open energy-driven systems and open differential equations, and a functor between them, functioning as a "functorial semantics" for reaction structures. This approach echoes what has previously been done for open games and open gradient-based learners, and in fact subsumes the latter. We then illustrate our theory by constructing an n-fold pendulum as a composite of n-many pendula.

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