Interactions in the Lorentz force equation

In this paper we consider for arbitrary \(\mu \in \mathbb {R}\) $\mu \in \mathbb{R}$ the Lorentz force equation \(\begin{aligned} \left( \frac{q'}{\sqrt{1-|q'|^2}}\right) ' + \mu q= -\nabla _q V-\frac{\partial W}{\partial t}+q'\times \text {curl}_q\, W, \end{aligned}\) $$\begin{array}{c}\hfill {\left(\frac{q^{\prime }}{\sqrt{1-|q^{\prime }{|}^2}}\right)}^{\prime }+\mu q=-{\mathrm{\nabla }}_{q}V-\frac{\partial W}{\partial t}+q^{\prime }\times {\text{curl}}_{q}W,\end{array}$$ with a Kepler type electric potential \(V+\frac{\mu }{2}|q|^2\) $V+\frac{\mu }{2}{|q|}^2$ and a smooth magnetic potential W which are T-periodic in time. We show that two fundamentally different cases occurs: the case \(\mu = 0\) $\mu =0$ and the case \(\mu \ne 0.\) $\mu \ne 0.$ In both cases we show that under different types of interactions at infinity between the electric and the magnetic potentials, we have that the Lorentz force equation has a sequence \((q_n)\) $(q_n)$ of T-periodic solutions such that \(\mathcal I(q_n)\rightarrow +\infty \) $\mathcal{I}(q_n)\to +\infty$ as \(n\rightarrow \infty ,\) $n\to \infty ,$ where \(\mathcal I\) $\mathcal{I}$ is the action functional associated to the Lorentz force equation. To prove our main result–using the Lusternik–Schnirelman category and Ekeland’s variational principle—we develop a Lusternik–Schnirelman strategy for the nonsmooth action functional associated to the Lorentz force equation.