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On Maxwell’s Conjecture for Coulomb Potential Generated by Point Charges

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Transactions on Computational Science XXVII

Part of the book series: Lecture Notes in Computer Science ((TCOMPUTATSCIE,volume 9570))

Abstract

The problem discussed herein is the one of finding the set of stationary points for the Coulomb potential function \( F(P)=\sum _{j=1}^K m_j / |PP_j | \) for the cases of \( K=3 \) and \( K=4 \) positive charges \( \{m_j\}_{j=1}^K \) fixed at the positions \( \{P_j\}_{j=1}^K \subset \mathbb R^2 \). Our approach is based on reducing the problem to that of evaluation of the number of real solution of an appropriate algebraic system of equations. We also investigate the bifurcation picture in the parameter domains.

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Notes

  1. 1.

    On excluding an extraneous factor.

  2. 2.

    Thanks to the open-source mathematical software system Sage, http://www.sagemath.org.

  3. 3.

    In the article [11], the expressions for S and \( \varPhi \) are provided with typos.

  4. 4.

    Thus, one should not be misled by the visual illusion: the triangle \( Q_1Q_2Q_3 \) is not an equilateral one!

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Acknowledgments

The authors are grateful to the anonymous referees for constructive suggestions and to Ivan Baravy for his help in drawing the figures. This work was supported by the St. Petersburg State University research grant # 9.38.674.2013.

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Authors and Affiliations

  1. Faculty of Applied Mathematics, St. Petersburg State University, Universitetskij pr. 35, Petrodvorets, 198504, St. Petersburg, Russia

    Alexei Yu. Uteshev & Marina V. Yashina

Authors
  1. Alexei Yu. Uteshev
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  2. Marina V. Yashina
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Correspondence to Alexei Yu. Uteshev .

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Editors and Affiliations

  1. University of Calgary, Calgary, Alberta, Canada

    Marina L. Gavrilova

  2. Sardina Systems, Tallinn, Estonia

    C.J. Kenneth Tan

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Uteshev, A.Y., Yashina, M.V. (2016). On Maxwell’s Conjecture for Coulomb Potential Generated by Point Charges. In: Gavrilova, M., Tan, C. (eds) Transactions on Computational Science XXVII. Lecture Notes in Computer Science(), vol 9570. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-50412-3_5

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  • DOI: https://doi.org/10.1007/978-3-662-50412-3_5

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-50411-6

  • Online ISBN: 978-3-662-50412-3

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