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Critical Point Theory for the Lorentz Force Equation

  • David Arcoya
  • Cristian Bereanu
  • Pedro J. Torres
Article
  • 25 Downloads

Abstract

In this paper we study the existence and multiplicity of solutions of the Lorentz force equation
$$\left(\frac{q'}{\sqrt{1-|q'|^2}}\right)'=E(t,q) + q'\times B(t,q)$$
with periodic or Dirichlet boundary conditions. In Special Relativity, this equation models the motion of a slowly accelerated electron under the influence of an electric field E and a magnetic field B. We provide a rigourous critical point theory by showing that the solutions are the critical points in the Szulkin’s sense of the corresponding Poincaré non-smooth Lagrangian action. By using a novel minimax principle, we prove a variety of existence and multiplicity results. Based on the associated Planck relativistic Hamiltonian, an alternative result is proved for the periodic case by means of a minimax theorem for strongly indefinite functionals due to Felmer.

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Acknowledgements

This work is partially supported by MINECO (Spain) Grant with FEDER funds MTM2015-68210-P and MTM2017-82348-C2-1-P and Junta de Andalucía FQM-116 and FQM-183.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Análisis MatemáticoUniversidad de GranadaGranadaSpain
  2. 2.University of Bucharest, Faculty of MathematicsBucharestRomania
  3. 3.Institute of Mathematics “Simion Stoilow”Romanian AcademyBucharestRomania
  4. 4.Departamento de Matemática AplicadaUniversidad de GranadaGranadaSpain

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