Geodesics for Sub‑Finsler Structures over Principal Bundles

We begin by recalling basic concepts in sub‑Finsler geometry and Hamiltonian mechanics. We then investigate sub‑Finsler manifolds admitting a proper free action by isometries of a Lie group. Within this framework, we introduce the momentum map and the theory of symplectic reduction, and use these tools to analyze both normal and abnormal geodesics arising from the Pontryagin Maximum Principle. We apply this approach to establish the regularity of geodesics in metabelian sub‑Riemannian manifolds, to prove the Sard property in metabelian sub‑Riemannian Lie groups, and to derive several properties of normal curves in both metabelian sub‑Riemannian manifolds and sub‑Finsler Lie groups.