General conservation law for a class of physics field theories

TL;DR

This paper introduces a unified conservation law for a broad class of physics field theories using differential forms on Minkowski space, providing a foundational framework for multi-physics analysis and scientific computing.

Contribution

It formulates a general conservation law that encompasses various physics fields through differential forms, unifying their mathematical structure.

Findings

Provides a formal definition of general fields as differential forms.

Derives a universal conservation law applicable to multiple physics theories.

Facilitates the development of flexible scientific computing software.

Abstract

In this paper we form a general conservation law that unifies a class of physics field theories. For this we first introduce the notion of a general field as a formal sum differential forms on a Minkowski manifold. Thereafter, we employ the action principle to define the conservation law for such general fields. By construction, particular field notions of physics, such as electric field strength, stress, strain etc. become instances of the general field. Hence, the differential equations that constitute physics field theories become also instances of the general conservation law. Accordingly, the general field and the general conservation law together correspond to a large class of physics field models. The approach creates solid foundations for multi-physics analysis and is critical in developing software systems for scientific computing; the unifying structure shared by the class of field models makes it possible to implement software systems which are not restricted to finite lists of admissible problems.