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

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.