From Geometric Hydrodynamics to Periodic Geodesics on Manifolds of Mappings

In this talk, we begin by recalling Arnold’s geometric formulation of hydrodynamics and then extend this framework to a broader class of Hamiltonian systems, incorporating various PDEs arising in mathematical physics. This motivates the study of infinite-dimensional manifolds and, in particular, half Lie groups: topological groups in which right multiplication is smooth while left multiplication is only continuous. Important examples include groups of Hs- or Ck-diffeomorphisms of compact manifolds. Within this setting, we establish several Hopf–Rinow type theorems for right-invariant magnetic systems and for certain Lagrangian systems on half Lie groups, thereby extending recent results of Bauer–Harms–Michor from the case of geodesic flows to this more general context. Finally, we show that any non-aspherical half Lie group equipped with a strong Riemannian metric necessarily admits a contractible periodic geodesic. This talk is based partially on joint work with M. Bauer and F. Ruscelli.