Symmetries of the dynamics and Noether theorem in classical mechanics

The aim of this note is to discuss the relation between one-parameter continuous symmetries of the dynamics, defined on physical grounds, and conservation laws. In the Hamiltonian formulation, such symmetries of the dynamics in general leave the Hamiltonian invariant only up to a total derivative $d G(q)/d t$ . In this more general case, the corresponding formulation of Noether theorem gives that the conservation law displays a sort of ano\-ma\-ly, the constant of motion being the sum of the canonical generator of the symmetry transformations plus the generator $G$ of the gauge transformation $q_i \rightarrow q_i$, $p_i \rightarrow p_i - \partial G/ \partial q_i$.