  In this work, the warped product of Hamilton spaces is introduced and it is
shown that these spaces obtain Hamiltonian structure as well. Then, the
geometric properties of warped product Hamilton spaces such as their nonlinear
connections and natural cotangent bundle structures are studied. Moreover, we
prove some theorems that show geometric relation between warped product
Hamiltonian space and its base Hamiltonian manifolds. For example, we prove
that for non-constant warped function $f$, the sasaki lifted metric $G$ of
Hamiltonian warped product space is bundle-like for its vertical foliation if
and only if based Hamiltonian spaces are Riemannian manifolds.
