Periodic orbits in magnetic fields and Ricci curvature of Lagrangian systems

TL;DR

This paper proves the existence of periodic orbits for charged particles in magnetic fields by introducing Ricci curvature for Lagrangian systems and applying variational methods, under positive curvature conditions.

Contribution

It introduces a Ricci curvature concept for magnetic Lagrangian systems and demonstrates periodic solutions exist when this curvature is positive.

Findings

Periodic motions exist under positive Ricci curvature.

Ricci curvature provides a new criterion for particle trajectories.

Variational methods are effective in this setting.

Abstract

We consider a periodic problem for the motion of a charged particle in a magnetic field. Introducing a notion of Ricci curvature for such Lagrangian systems and using the methods of the calculus of variations in the large, we prove the existence of periodic motions for such particles under a condition of positivity of the Ricci curvature.