Open systems in classical mechanics

Baez, John C.; Weisbart, David; Yassine, A.

doi:10.1063/5.0029885

Cited by 9 publications

(10 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Dually, open systems consisting of quantitative state variables on a geometrical space are modeled as spans, where the legs of a span give projections of the system's state space onto the state spaces of its boundaries. Composition is by pullback or variants thereof, such as in Baez, Weisbart, and Yassine's study of open systems in Lagrangian and Hamiltonian mechanics [BWY21].…”

Section: Composition Of Diagrams and Multiphysicsmentioning

confidence: 99%

A diagrammatic view of differential equations in physics

Patterson,

Baas,

Hosgood

et al. 2022

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Composition Of Diagrams and Multiphysicsmentioning

confidence: 99%

A diagrammatic view of differential equations in physics

Patterson,

Baas,

Hosgood

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Classical Mechanics. In the upcoming paper [4], we work in the categories RiemSurj, whose objects are Riemannian manifolds and whose morphisms are surjective Riemannian submersions, and SympSurj, whose objects are symplectic manifolds and whose morphisms are surjective Poisson maps. Unlike SurjSub, these categories are not subcategories of Diff.…”

Section: Diagram 12 Comparator Spanmentioning

confidence: 99%

“…Composition in this generalized span category is defined using F -pullbacks and appears to depend on the functor F . In a concurrent paper [4], we apply the tools that we develop here to the study of classical mechanics. Section 6 briefly discusses some examples of generalized span categories.…”

mentioning

confidence: 99%

Constructing Span Categories From Categories Without Pullbacks

Weisbart¹,

Yassine²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

Span categories provide an abstract framework for formalizing mathematical models of certain systems. The mathematical descriptions of some systems, such as classical mechanical systems, require categories that do not have pullbacks and this limits the utility of span categories as a formal framework. Given categories C and C ′ , we introduce the notion of span tightness of a functor F from C to C ′ as well as the notion of an F -pullback of a cospan in C . If F is span tight, then we can form a generalized span category Span(C , F ) and circumvent the technical difficulty of C failing to have pullbacks. Composition in Span(C , F ) uses F -pullbacks rather than pullbacks and in this way differs from the category Span(C ) but reduces to it when both C has pullbacks and F is the identity functor.Contents 19 References 19

show abstract

“…Dynamical systems are an extremely broad class of models, as reflected by previous work falling on many different points of the semantic axis. These points include circuit diagrams, Petri nets, Markov processes, finite state automata, ODEs, hybrid systems, and Lagrangian and Hamiltonian systems [5,2,1,21,11,10,3]. In this paper we focus on two kinds of dynamics: continuous flows and discrete transitions.…”

Section: Introductionmentioning

confidence: 99%

Operadic Modeling of Dynamical Systems: Mathematics and Computation

Libkind¹,

Baas²,

Patterson³

et al. 2021

Preprint

View full text Add to dashboard Cite

Dynamical systems are ubiquitous in science and engineering as models of phenomena that evolve over time. Although complex dynamical systems tend to have important modular structure, conventional modeling approaches suppress this structure. Building on recent work in applied category theory, we show how deterministic dynamical systems, discrete and continuous, can be composed in a hierarchical style. In mathematical terms, we reformulate some existing operads of wiring diagrams and introduce new ones, using the general formalism of -sets (copresheaves). We then establish dynamical systems as algebras of these operads. In a computational vein, we show that Euler's method is functorial for undirected systems, extending a previous result for directed systems. All the ideas in this paper are implemented as practical software using Catlab and the AlgebraicJulia ecosystem, written in the Julia programming language for scientific computing.

show abstract

Open systems in classical mechanics

Cited by 9 publications

References 16 publications

A diagrammatic view of differential equations in physics

A diagrammatic view of differential equations in physics

Constructing Span Categories From Categories Without Pullbacks

Operadic Modeling of Dynamical Systems: Mathematics and Computation

Contact Info

Product

Resources

About