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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