Scaling symmetries and canonoid transformations in Hamiltonian systems

We investigate various types of symmetries and their mutual relationships in Hamiltonian systems defined on manifolds with different geometric structures: symplectic, cosymplectic, contact and cocontact. In each case we pay special attention to non-standard (non-canonical) symmetries, in particular scaling symmetries and canonoid transformations, as they provide new interesting tools for the qualitative study of these systems. Our main results are the characterizations of these non-standard symmetries and the analysis of their relation with conserved (or dissipated) quantities.