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On Force Interactions for Electrodynamics-Like Theories

Vladimir Gol'dshtein, Reuven Segev

TL;DR

The paper builds a metric-free framework for $p$-form electrodynamics within geometric continuum mechanics, deriving Maxwell equations as a consequence of a stress-theoretic construction and detailing how energy, power, and force distributions arise under virtual motions of regions in spacetime. By treating the potential field as a $p$-form and the stress object as an $(n-p-1)$-form, it obtains general expressions for energy variation and force distributions that are independent of constitutive relations, and then specializes to electrostatics and magnetostatics in $\mathbb{R}^{3}$ as concrete examples. A comprehensive p-form generalization is developed (including a metric-free Maxwell framework and explicit $p=0,1,2$ cases), along with transformations under motions and expressions for force densities in p-form electrodynamics-like theories. The work culminates in a magnetostatic reduction that recovers Lorentz- and Kelvin-type force densities within the differential-form formalism, offering a unified, geometry-based lens for force interactions in electrodynamics-like theories with potential extensions to higher-form theories and metric-free contexts.

Abstract

A framework for premetric p-form electrodynamics is proposed. Independently of particular constitutive relations, the corresponding Maxwell equations are derived as a special case of stress theory in geometric continuum mechanics. Expressions for the potential energy of a charged region in spacetime, as well as expressions for the force and stress interactions on the region, are presented. The expression for the force distribution is obtained by computing the rate of change of the proposed potential energy under a virtual motion of the region. These expressions differ from those appearing in the standard references. The cases of electrostatics and magnetostatics in R^3 are presented as examples.

On Force Interactions for Electrodynamics-Like Theories

TL;DR

The paper builds a metric-free framework for -form electrodynamics within geometric continuum mechanics, deriving Maxwell equations as a consequence of a stress-theoretic construction and detailing how energy, power, and force distributions arise under virtual motions of regions in spacetime. By treating the potential field as a -form and the stress object as an -form, it obtains general expressions for energy variation and force distributions that are independent of constitutive relations, and then specializes to electrostatics and magnetostatics in as concrete examples. A comprehensive p-form generalization is developed (including a metric-free Maxwell framework and explicit cases), along with transformations under motions and expressions for force densities in p-form electrodynamics-like theories. The work culminates in a magnetostatic reduction that recovers Lorentz- and Kelvin-type force densities within the differential-form formalism, offering a unified, geometry-based lens for force interactions in electrodynamics-like theories with potential extensions to higher-form theories and metric-free contexts.

Abstract

A framework for premetric p-form electrodynamics is proposed. Independently of particular constitutive relations, the corresponding Maxwell equations are derived as a special case of stress theory in geometric continuum mechanics. Expressions for the potential energy of a charged region in spacetime, as well as expressions for the force and stress interactions on the region, are presented. The expression for the force distribution is obtained by computing the rate of change of the proposed potential energy under a virtual motion of the region. These expressions differ from those appearing in the standard references. The cases of electrostatics and magnetostatics in R^3 are presented as examples.
Paper Structure (24 sections, 140 equations)

This paper contains 24 sections, 140 equations.

Table of Contents

  1. Introduction
  2. Electrostatics in $\mathbb{R}^{3}$
  3. Fields
  4. Transformations
  5. Mechanical Forces
  6. Magnetostatics in $\mathbb{R}^{3}$
  7. Fields
  8. Transformations
  9. Mechanical force distributions
  10. Traction Stresses in Geometric Continuum Mechanics
  11. Smooth $p$-Form Electrodynamics
  12. Metric free $p$-form electrodynamics on an $n$-dimensional spacetime
  13. Example: The case $p=0$ in $\mathbb{R}^{3}$
  14. Example: The case $p=1$ in $\mathbb{R}^{3}$
  15. Example: The case $p=2$ in $\mathbb{R}^{3}$
  16. ...and 9 more sections

Theorems & Definitions (3)

  • Remark 2.4
  • Remark 2.5
  • Remark 5.1