Electromagnetic curvature via Jacobi-Maupertuis and beyond

TL;DR

This paper introduces a new electromagnetic Ricci curvature based on Jacobi-Maupertuis reparametrization, demonstrating positivity under certain conditions and extending the existence of closed orbits near the maximum potential energy.

Contribution

It defines a novel electromagnetic Ricci curvature and proves its positivity under specific conditions, extending the existence of closed orbits in electromagnetic systems.

Findings

Electromagnetic Ricci curvature is positive when magnetic force is non-vanishing and potential is small.

Positivity of curvature leads to the extension of closed orbit existence.

Results apply to energies near the maximum potential value.

Abstract

In the setting of electromagnetic systems, we propose a new definition of electromagnetic Ricci curvature, naturally derived via the classical Jacobi-Maupertuis reparametrization from the recent works of Assenza [IMRN, 2024] and Assenza, Marshall Reber, Terek [Communications in Mathematical Physics, 2025]. On closed manifolds, we show that if the magnetic force is nowhere vanishing and the potential is sufficiently small in the $C^2$ norm, then this Ricci curvature is positive for energies close to the maximum value of the potential $e_0$. As a main application, under these assumptions, we extend the existence of contractible closed orbits at energy levels near $e_0$ from almost every to everywhere.