In both symplectic and contact geometry, the suspension of a Hamiltonian flow defines a germ of a symplectic/contact structure near a hypersurface. Additionally, questions about whether this germ extends to a compact set are related to questions about which flows are generated by positive Hamiltonians. We'll then explain how in the contact case this can be used to prove a general extension result, and why the same strategy fails in the symplectic case.