A Maupertuis-type principle in relativistic mechanics and applications

TL;DR

This paper establishes a Maupertuis-type variational principle for a relativistic system of ODEs relevant in special relativity and demonstrates its application in proving the existence of multiple periodic solutions with prescribed energy in a relativistic N-centre problem.

Contribution

It introduces a novel Maupertuis-type principle tailored for relativistic mechanics and applies it to find multiple periodic solutions in a relativistic N-centre problem.

Findings

Established a variational principle for relativistic ODEs.

Proved existence of multiple periodic solutions with fixed energy.

Applied the principle to a relativistic N-centre problem.

Abstract

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.