On Mañé's Critical Value for Tonelli Lagrangians on Half Lie-Groups
Levin Maier, Francesco Ruscelli
TL;DR
This work extends Mañé’s critical-value framework and Hopf–Rinow-type completeness to Tonelli Lagrangians on half-Lie groups endowed with strong right-invariant metrics, bridging finite-dimensional Riemannian geometry with infinite-dimensional geometric hydrodynamics. It introduces and analyzes Mañé’s critical values $c(L)$, $c_u(L)$, $c_0(L)$, and $e_0(L)$ in this setting, and proves that for any pair of points and any energy level above $c(L)$ there exists a global minimizer of the time-free action; when the Lagrangian is strong Tonelli, such minimizers are Euler–Lagrange flow lines, yielding global well-posedness for the associated Euler–Poincaré and magnetic Euler–Arnold equations. The results generalize Contreras and Bauer–Harms–Michor from finite-dimensional manifolds to infinite-dimensional half-Lie groups, and provide a robust variational and geometric framework for analyzing EL-flows on groups of diffeomorphisms and related structures. Consequently, the paper delivers a foundational Hopf–Rinow-type completeness theorem in this infinite-dimensional context and offers concrete implications for geometric hydrodynamics and the global behavior of EL/Hamiltonian flows on Sobolev diffeomorphism 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.