The Hopf--Rinow Theorem and Ma\~n\'e's Critical Value for Magnetic Geodesics on Half Lie-Groups

TL;DR

This paper extends the Hopf--Rinow theorem and Mañé's critical value concept to magnetic geodesic flows on half-Lie groups, generalizing finite-dimensional results to infinite-dimensional diffeomorphism groups.

Contribution

It introduces a framework for magnetic systems on half-Lie groups, defines Mañé's critical value in this context, and proves the Hopf--Rinow theorem holds above this energy threshold.

Findings

Lift of magnetic flow matches Finsler geodesic flow above critical energy

Hopf--Rinow theorem applies for energies above Mañé's critical value

Generalizes finite-dimensional magnetic geodesic results to infinite-dimensional groups

Abstract

In this article, we investigate \emph{right-invariant magnetic systems} on half-Lie groups, which consist of a strong right-invariant Riemannian metric and a right-invariant closed two-form. The main examples are groups of $H^s$ or $C^k$ diffeomorphisms of compact manifolds. In this setting, we define \emph{Ma\~n\'e's critical value} on the universal cover for weakly exact right-invariant magnetic fields. First, we prove that the lift of the magnetic flow to the universal cover coincides with a Finsler geodesic flow for energies above this threshold. Finally, we show that for energies above Ma\~n\'e's critical value, the full Hopf--Rinow theorem holds for such magnetic systems, thereby generalizing the work of Contreras and Merry from closed finite-dimensional manifolds to this infinite-dimensional context. Our work extends the recent results of Bauer, Harms, and Michor from geodesic flows to magnetic geodesic flows.