A Maupertuis-type principle in relativistic mechanics and applications

We provide a Maupertuis-type principle for the following system of ODE, of interest in special relativity: $$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{m\dot{x}}{\sqrt{1-\mathrm{\mid }\dot{x}{\mathrm{\mid }}^2\mathrm{/}{c}^2}}\right)=\mathrm{\nabla }V(x),x\in \mathrm{\Omega }\subset {\mathbb{R}}^n,$$ where $m,c>0$ and $V:\mathrm{\Omega }\to \mathbb{R}$ is a function of class $C^1$. As an application, we prove the existence of multiple periodic solutions with prescribed energy for a relativistic $N$-centre type problem in the plane.