  We define Hamiltonian simplex differential graded algebras (DGA) with
differentials that deform the high energy symplectic homology differential and
wrapped Floer homology differential in the cases of closed and open strings in
a Weinstein manifold, respectively. The order $m$ term in the differential is
induced by varying natural degree $m$ co-products over an $(m-1)$-simplex,
where the operations near the boundary of the simplex are trivial. We show that
the Hamiltonian simplex DGA is quasi-isomorphic to the (non-equivariant)
contact homology algebra and to the Legendrian homology algebra of the ideal
boundary in the closed and open string cases, respectively.
