Open Systems in Classical Mechanics

TL;DR

This paper develops a categorical framework for modeling open systems in classical mechanics using new categories that formalize Lagrangian and Hamiltonian descriptions, enabling systematic composition and transformation of systems.

Contribution

It introduces the categories $ extsf{LagSy}$ and $ extsf{HamSy}$ for open classical systems and establishes a functor connecting Lagrangian and Hamiltonian formalisms.

Findings

Categories $ extsf{LagSy}$ and $ extsf{HamSy}$ formalize open systems.

Morphisms represent open systems; composition models system construction.

Legendre transformation induces a functor between the categories.

Abstract

Generalized span categories provide a framework for formalizing mathematical models of open systems in classical mechanics. We introduce categories $\mathsf{LagSy}$ and $\mathsf{HamSy}$ that respectively provide a categorical framework for the Lagrangian and Hamiltonian descriptions of open classical mechanical systems. The morphisms of $\mathsf{LagSy}$ and $\mathsf{HamSy}$ correspond to such open systems, and composition of morphisms models the construction of systems from subsystems. The Legendre transformation gives rise to a functor from $\mathsf{LagSy}$ to $\mathsf{HamSy}$ that translates from the Lagrangian to the Hamiltonian perspective.