A Maupertuis-type principle in relativistic mechanics and applications

Abstract

AbstractWe provide a Maupertuis-type principle for the following system of ODE, of interest in special relativity: $$\begin{aligned} \frac{\textrm{d}}{{\textrm{d}}t}\left( \frac{m\dot{x}}{\sqrt{1-|\dot{x}|^2/c^2}}\right) =\nabla V(x),\qquad x\in \Omega \subset \mathbb {R}^n, \end{aligned}$$ d d t m x ˙ 1 - | x ˙ | 2 / c 2 = ∇ V ( x ) , x ∈ Ω ⊂ R n , where $$m, c > 0$$ m , c > 0 and $$V: \Omega \rightarrow \mathbb {R}$$ V : Ω → R is a function of class $$C^1$$ 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.