On the general structure of mathematical models for physical systems

TL;DR

This paper proposes a unified framework for mathematical models of physical systems based on first principles, highlighting common structural elements like state spaces and duality principles, supported by various examples.

Contribution

It introduces a general structural framework for physical models, emphasizing common elements and duality, applicable across different fundamental systems.

Findings

Unified structure for physical models based on first principles

Identification of common elements like state spaces and duality

Examples from various fundamental physical systems

Abstract

It is proposed that the mathematical models for any physical systems that are based in first principles, such as conservation laws or balance principles, have some common elements, namely, a space of kinematical states, a space of dynamical states, a constitutive law that associates dynamical states with kinematical states, as well as a duality principle. The equations of motion or statics then come about from, on the one hand, specifying the integrability of the kinematical state, and on the other hand, specifying a statement that is dual to it for the dynamical states. Examples are given from various fundamental physical systems.