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Lusternik–Schnirelman theory for the action integral of the Lorentz force equation

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Abstract

In this paper we introduce new Lusternik–Schnirelman type methods for nonsmooth functionals including the action integral associated to the relativistic Lagrangian of a test particle under the action of an electromagnetic field

$$\begin{aligned} {\mathcal {L}}(t,q,q')=1-\sqrt{1-|q'|^2}+q'\cdot W(t,q) - V(t,q), \end{aligned}$$

where \(V:[0,T]\times {\mathbb {R}}^3\rightarrow {\mathbb {R}}\) and \(W:[0,T]\times {\mathbb {R}}^3\rightarrow {\mathbb {R}}^3\) are two \(C^1\)-functions with V even and W odd in the second variable. By applying them, we obtain various multiplicity results concerning T-periodic solutions of the relativistic Lorentz force equation in Special Relativity,

$$\begin{aligned} \left( \frac{q'}{\sqrt{1-|q'|^2}}\right) '=E(t,q) + q'\times B(t,q), \end{aligned}$$

where \( E=-\nabla _q V-\frac{\partial W}{\partial t}, B=\hbox {curl}_q\, W. \) The zero Dirichlet boundary value conditions are considered as well.

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Acknowledgements

This work is partially supported by MTM2017-82348-C2-1-P and PGC2018-096422-B-I00 (MCIU/AEI/FEDER, UE) and FQM-116 and FQM-183 (Junta de Andalucía).

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Correspondence to Pedro J. Torres.

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Communicated by P. Rabinowitz.

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Arcoya, D., Bereanu, C. & Torres, P.J. Lusternik–Schnirelman theory for the action integral of the Lorentz force equation. Calc. Var. 59, 50 (2020). https://doi.org/10.1007/s00526-020-1711-0

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Mathematics Subject Classification

  • 34B15
  • 58E05
  • 78A35
  • 83A05
  • 34C25