On Mañé's Critical Value for Tonelli Lagrangians on Half Lie-Groups
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arXiv
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| Format: | Preprint |
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2025
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| author | Maier, Levin Ruscelli, Francesco |
| author_facet | Maier, Levin Ruscelli, Francesco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13428 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Mañé's Critical Value for Tonelli Lagrangians on Half Lie-Groups Maier, Levin Ruscelli, Francesco Differential Geometry Analysis of PDEs Dynamical Systems Symplectic Geometry 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. |
| title | On Mañé's Critical Value for Tonelli Lagrangians on Half Lie-Groups |
| topic | Differential Geometry Analysis of PDEs Dynamical Systems Symplectic Geometry |
| url | https://arxiv.org/abs/2511.13428 |