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Newton Algorithm on Constraint Manifolds and the 5-Electron Thomson Problem

Author

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  • Petre Birtea

    (West University of Timişoara)

  • Dan Comănescu

    (West University of Timişoara)

Abstract

We give a description of numerical Newton algorithm on a constraint manifold using only the ambient coordinates (usually Euclidean coordinates) and the geometry of the constraint manifold. We apply the numerical Newton algorithm on a sphere in order to find the critical configurations of the 5-electron Thomson problem. As a result, we find a new critical configuration of a regular pentagonal type. We also make an analytical study of the critical configurations found previously and determine their nature using Morse–Bott theory. The last section contains an analytical study of critical configurations for Riesz s-energy of 5-electron on a sphere, and their bifurcation behavior is pointed out.

Suggested Citation

  • Petre Birtea & Dan Comănescu, 2017. "Newton Algorithm on Constraint Manifolds and the 5-Electron Thomson Problem," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 563-583, May.
    • Handle: RePEc:spr:joptap:v:173:y:2017:i:2:d:10.1007_s10957-016-1049-0
    DOI: 10.1007/s10957-016-1049-0
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    References listed on IDEAS

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    1. Jin-Hua Wang & Jen-Chih Yao & Chong Li, 2012. "Gauss–Newton method for convex composite optimizations on Riemannian manifolds," Journal of Global Optimization, Springer, vol. 53(1), pages 5-28, May.
    2. Bittencourt, Tiberio & Ferreira, Orizon Pereira, 2015. "Local convergence analysis of Inexact Newton method with relative residual error tolerance under majorant condition in Riemannian manifolds," Applied Mathematics and Computation, Elsevier, vol. 261(C), pages 28-38.
    3. J. H. Wang, 2011. "Convergence of Newton’s Method for Sections on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 148(1), pages 125-145, January.
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    Cited by:

    1. Amore, Paolo & Jacobo, Martin, 2019. "Thomson problem in one dimension: Minimal energy configurations of N charges on a curve," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 519(C), pages 256-266.

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