Constructing the Lagrangian from spacetime translation symmetry closing a logical gap in Landau's Mechanics and building a unified classical–quantum symmetry picture

We present a geometric reconstruction of the Lagrangian formalism. It starts from the purely kinematic arc-length element in the q-t plane and introduces a local projection factor K. This framework naturally unifies classical mechanics, geometric optics (Fermat's principle), and optimal control (Pontryagin's Maximum Principle). Furthermore, we demonstrate that the failure of a single-valued Lagrangian for dissipative systems is a geometric manifestation of time-reversal symmetry breaking, structurally isomorphic to the Second Law of Thermodynamics.