  We describe natural Hamiltonian systems using projective geometry. The null
lift procedure endows the tangent bundle with a projective structure where the
null Hamiltonian is identified with a projective conic and induces a Weyl
geometry. Projective transformations generate a set of known and new dualities
between Hamiltonian systems, as for example the phenomenon of coupling-constant
metamorphosis. We conclude outlining how this construction can be extended to
the quantum case for Eisenhart-Duval lifts.
