Infinitesimal transformations and equivalence of the Lagrangian and Hamiltonian descriptions

Starting from the infinitesimal point of view, the need to enlarge the Hamiltonian principle for Hamiltonian dynamics is analysed. So-called Lagrangian transformations are introduced to demonstrate the complete equivalence, not only of the Lagrangian and Hamiltonian equations of motion, but also of the corresponding formalisms. It is shown that the superiority of the Hamiltonian formalism lies exclusively in the passage from infinitesimal transformation generators to one-parameter subgroups of canonical transformation, since the passage preserves most of the properties acquired at the infinitesimal level.