Theoretical equivalence in classical mechanics and its relationship to duality

As a prolegomenon to understanding the sense in which dualities are theoretical equivalences, we investigate the intuitive 'equivalence' of hyper-regular Lagrangian and Hamiltonian classical mechanics. We show that the symplectification of these theories (via Tulczyjew׳s Triple) provides a sense in which they are (1) isomorphic, and (2) mutually and canonically definable through an analog of 'common definitional extension'.