Archive for Rational Mechanics and Analysis

, Volume 232, Issue 3, pp 1685–1724 | Cite as

Critical Point Theory for the Lorentz Force Equation

  • David Arcoya
  • Cristian Bereanu
  • Pedro J. TorresEmail author


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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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.


  1. 1.
    Acharya, S., Saxena, A.C.: The exact solution of the relativistic equation of motion of a charged particle driven by an elliptically polarized electromagnetic wave. IEEE Trans. Plasma Sci. 21, 257–259 (1993)ADSCrossRefGoogle Scholar
  2. 2.
    Alves, C.O., de Morais Filho, D.C.: Existence and concentration of positive solutions for a Schödinger logarithmic equation. Z. Angew. Math. Phys. 69, 144 (2018)CrossRefzbMATHGoogle Scholar
  3. 3.
    Andreev, S.N., Makarov, V.P., Rukhadze, A.A.: On the motion of a charged particle in a plane monochromatic electromagnetic wave. Quantum Electron. 1(39), 68–72 (2009)ADSCrossRefGoogle Scholar
  4. 4.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Arcoya, D., Boccardo, L.: Critical Points for Multiple Integrals of the Calculus of Variations. Arch. Rational Mech. Anal. 134, 249–274 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bartnik, R., Simon, L.: Spacelike hypersurfaces with prescribed boundary values and mean curvature. Commun. Math. Phys. 87, 131–152, 1982–1983Google Scholar
  7. 7.
    Benci, V., Rabinowitz, P.H.: Critical point theorems for indefinite functionals. Invent. Math. 52, 241–273 (1979)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bereanu, C., Mawhin, J.: Boundary value problems for some nonlinear systems with singular \(\phi \)-Laplacian. J. Fixed Point Theor. Appl. 4, 57–75 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bereanu, C., Jebelean, P., Mawhin, J.: Variational methods for nonlinear perturbations of singular \(\phi \)-Laplacians. Rend. Lincei Mat. Appl. 22, 89–111 (2011)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Brezis, H., Mawhin, J.: Periodic solutions of the forced relativistic pendulum. Differ. Integral Equ. 23, 801–810 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Damour, T.: Poincaré, the dynamics of the electron, and relativity. C. R. Phys. 18, 551–562 (2017)ADSCrossRefGoogle Scholar
  12. 12.
    Degiovanni, M., Marzocchi, M.: A critical point theory for nonsmooth functionals. Ann. Mat. Pura Appl. 167(4), 73–100 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc. (NS) 1, 443–474 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Felmer, P.L.: Periodic solutions of “superquadratic” Hamiltonian systems. J. Differ. Equ. 102, 188–207 (1993)Google Scholar
  15. 15.
    Feynman, R., Leighton, R., Sands, M.: The Feynman Lectures on Physics. Electrodynamics, Vol. 2. Addison-Wesley, Massachusetts, 1964Google Scholar
  16. 16.
    Jackson, J.D.: Classical Electrodynamics, 3rd edn. Wiley, 1999Google Scholar
  17. 17.
    Katriel, G.: Mountain pass theorems and global homeomorphism theorems. Ann. Inst. H. Poincaré Anal. Non Linéaire 11, 189–209, 1994Google Scholar
  18. 18.
    Landau, L.D., Lifschitz, E.M.: The Classical Theory of Fields, 4th edn, Vol. 2. Butterworth-Heinemann, 1980Google Scholar
  19. 19.
    Mawhin, J.: Stability and Bifurcation Theory for Non-Autonomous Differential Equations (Eds. Johnson R. and Pera M.P.) Springer Lecture Notes in Mathematica - 2065, 2013Google Scholar
  20. 20.
    Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989)CrossRefzbMATHGoogle Scholar
  21. 21.
    Minguzzi, E., Sánchez, M.: Connecting solutions of the Lorentz force equation do exist. Commun. Math. Phys. 264, 349–370 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Moser, J., Zehnder, E.J.: Notes on Dynamical Systems, Courant Lecture Notes, Vol. 12. AMS, 2005Google Scholar
  23. 23.
    Motreanu, D.: On the Proof of a Minimax Principle. Le Matematiche 58, 95–99 (2003)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Planck, M.: Das Prinzip der Relativität und die Grundgleichungen der Mechanik. Verh. Deutsch. Phys. Ges. 4, 136–141 (1906)zbMATHGoogle Scholar
  25. 25.
    Poincaré, H.: Sur la dynamique de l'électron. C. R. Hebd. séanc. Acad. Sci. Paris 140, 1504–1508 1905 (séance du 5 juin)Google Scholar
  26. 26.
    Poincaré, H.: Sur la dynamique de l'électron. Rend. Circ. Mat. Palermo 21, 129–176 (1906)CrossRefzbMATHGoogle Scholar
  27. 27.
    Rabinowitz, P.H.: Periodic solutions of Hamiltonian systems. Commun. Pure Appl. Math. 31, 157–184 (1978)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. CBMS Reg. Conf. Ser. Math., Vol. 65. AMS, Providence, 1986Google Scholar
  29. 29.
    Shebalin, J.V.: An exact solution to the relativistic equation of motion of a charged particle driven by a linearly polarized electromagnetic wave. IEEE Trans. Plasma Sci. 16, 390–392 (1988)ADSCrossRefGoogle Scholar
  30. 30.
    Szulkin, A.: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 3, 77–109, 1986Google Scholar

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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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