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Toward the Landau–Lifshitz version of the Thomson electrostatics theorem
Journal article   Open access   Peer reviewed

Toward the Landau–Lifshitz version of the Thomson electrostatics theorem

Michael Grinfeld and Pavel Grinfeld
Results in physics, v 5(C)
2015
DOI: https://doi.org/10.1016/j.rinp.2015.07.006
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url
https://doi.org/10.1016/j.rinp.2015.07.006View
Published, Version of Record (VoR)CC BY-NC-ND V4.0,  Open

Abstract

Electrostatics The Thomson theorem (principle) Variational principles
In the classical textbook (Landau and Lifshitz, 1963), Landau and Lifshtz suggested their version of the famous Thomson variational principle (a.k.a Thomson theorem.) So far, their version has not gained the interest it deserves, either among physicists or among applied mathematicians. Partially, the lack of interest can be explained because of the quality of the suggested proof of the principle. It is considerably lower than the standards accepted in classical electrostatics and mathematical physics. Even more importantly, Landau and Lifshitz did not demostrate the minimum property of the electrostatic energy at equilibrium. In this note, we, first, modify and specify the Landau–Lifshitz formulation of the principle presenting it as the isoperimetric variational problem. Then, for this isoperimetric problem we calculate the first and second variations, and we prove that the first variation vanishes, whereas the second variation appears to be positive.

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Collaboration types
Domestic collaboration
Web of Science research areas
Materials Science, Multidisciplinary
Physics, Multidisciplinary
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