Organizing Physics with Open Energy-Driven Systems

TL;DR

This paper develops a compositional categorical framework for reaction structures in physics, providing a new perspective on energy-driven systems and illustrating it with a multi-pendulum example.

Contribution

It introduces a symmetric monoidal category of open energy-driven systems and a functorial semantics for reaction structures, expanding the theoretical tools for organizing physics.

Findings

Constructed a functorial semantics for reaction structures

Developed symmetric monoidal categories for open systems

Demonstrated the theory with an n-fold pendulum example

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.