On the connections between symmetries and conservation rules of dynamical systems

The strict connection between Lie point-symmetries of a dynamical system and its constants of motion is discussed and emphasized through old and new results. It is shown in particular how the knowledge of the symmetry of a dynamical system can allow us to obtain conserved quantities that are invariant under the symmetry. In the case of Hamiltonian dynamical systems, it is shown that if the system admits a symmetry of a ‘weaker’ type (specifically, a lambda or a Lambda-symmetry), then the generating function of the symmetry is not a conserved quantity, but the deviation from the exact conservation is ‘controlled’ in a well-defined way. Several examples illustrate the various aspects