Levin Maier, a mathematician at Heidelberg University, presents a research talk from the IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar. He starts from Arnold's geometric formulation of hydrodynamics and extends it to a wider class of Hamiltonian systems tied to PDEs from mathematical physics, which leads into infinite-dimensional manifolds and half Lie groups, where right multiplication is smooth but left multiplication only continuous, with diffeomorphism groups of compact manifolds as the key examples. Maier proves several Hopf-Rinow type theorems for right-invariant magnetic systems and certain Lagrangian systems on these half Lie groups, building on and extending work by Bauer, Harms, and Michor on geodesic flows. He closes by showing that any non-aspherical half Lie group with a strong Riemannian metric must admit a contractible periodic geodesic, drawing on joint work with M. Bauer and F. Ruscelli. The talk assumes graduate-level familiarity with differential geometry and infinite-dimensional analysis.