On Ma\~n\'e's Critical Value for Tonelli Lagrangians on Half Lie-Groups

Abstract

In this article, we introduce Tonelli Lagrangians on half-Lie groups equipped with a strong right-invariant Riemannian metric. These are right-invariant Lagrangians defined on the tangent bundle of a half-Lie group with quadratic growth on each fiber. The main examples of half-Lie groups are groups of $H^s$ or $C^k$ diffeomorphisms of compact manifolds. We show that the Euler--Lagrange flow exists globally. We then introduce three thresholds of the energy, called the Ma~ne critical values, and prove that under mild regularity and completeness assumptions on the half-Lie group, any two points can be connected by a global Tonelli minimizer above the lowest of these energy thresholds. Under an additional assumption on the Lagrangian, such a minimizer is a flow line of the Euler--Lagrange flow. This extends the work of Contreras from closed finite-dimensional manifolds to the infinite-dimensional context. Moreover, our results also extend the recent work of Bauer, Harms, and Michor from geodesic flows to Euler--Lagrange flows of Tonelli Lagrangians. As an application, we obtain global well-posedness of all Euler--Poincare'e equations associated with Tonelli Lagrangians on half-Lie groups equipped with strong right-invariant Riemannian metrics.