Transformation properties of the Lagrange function

In the context of the Lagrangian and Hamiltonian mechanics, a generalized theory of coordinate transformations is analyzed. On the basis of such theory, a misconcept concerning the superiority of the Hamiltonian formalism with respect to the Lagrangian one is criticized. The consequent discussion introduces the relationship between the classical Hamilton action and the covariance properties of equations of motion, at the level of undergraduate teaching courses in Theoretical Mechanics. Moreover, a necessary and sufficient condition for a map to be canonoid is discussed.