Organizing Physics with Open Energy-Driven Systems

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.