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: $\frac{\rm d}{{\rm d}t}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^2/c^2}}\right)=\nabla V(x),\qquad x\inΩ\subset \mathbb{R}^n,$ where $m, c > 0$ and $V: Ω\to \mathbb{R}$ 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.

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