  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 $q_i \rightarrow q_i$, $p_i \rightarrow p_i - \partial G/
\partial q_i$.
