  We provide a Maupertuis-type principle for the following system of ODE, of
interest in special relativity: $$ \frac{\rm d}{{\rm
d}t}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^2/c^2}}\right)=\nabla V(x),\qquad
x\in\Omega \subset \mathbb{R}^n, $$ where $m, c > 0$ and $V: \Omega \to
\mathbb{R}$ 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.
