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Around a theorem of F. Dyson and A. Lenard: Energy Equilibria for Point Charge Distributions in Classical Electrostatics

Artem Abanov, Nathan Hayford, Dmitry Khavinson, Razvan Teodorescu

Abstract

We discuss several results in electrostatics: Onsager's inequality, an extension of Earnshaw's theorem, and a result stemming from the celebrated conjecture of Maxwell on the number of points of electrostatic equilibrium. Whenever possible, we try to provide a brief historical context and references.

Around a theorem of F. Dyson and A. Lenard: Energy Equilibria for Point Charge Distributions in Classical Electrostatics

Abstract

We discuss several results in electrostatics: Onsager's inequality, an extension of Earnshaw's theorem, and a result stemming from the celebrated conjecture of Maxwell on the number of points of electrostatic equilibrium. Whenever possible, we try to provide a brief historical context and references.

Paper Structure

This paper contains 10 sections, 5 theorems, 56 equations.

Table of Contents

  1. Introduction.
  2. Variations on Onsager's Inequality.
  3. F. Dyson and A. Lenard's Original Proof.
  4. Generalizations of the Onsager Inequality.
  5. Intersection Theorem.
  6. Intersection Theorem: A Necessary Condition for Equilibrium.
  7. Generalization to Compactly Supported Charge Distributions.
  8. Degeneracy in Maxwell's Problem with Planar Charge Distributions.
  9. Further observations
  10. Faraday's Problem.

Key Result

Proposition 2.1

Let $\{x_j\}_{j=1}^n$, $\{q_j\}_{j=1}^n$, and $\{\delta_j\}_{j=1}^n$ be defined as above. Then

Theorems & Definitions (18)

  • Definition 1
  • Remark 1
  • Proposition 2.1
  • proof
  • Remark 2
  • Proposition 2.2
  • Remark 3
  • Proposition 3.1
  • Remark 4
  • Remark 5
  • ...and 8 more