  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.
