  We consider a class of finite-dimensional dynamical systems whose equations
of motion are derived from a non-local-in-time action principle. The action
functional has a zeroth order piece derived from a local Hamiltonian and a
perturbation in the form of a non-local functional of the trajectory on phase
space. We prove that the dynamics of these systems admits a local Hamiltonian
description to all order in the perturbation and we provide explicit formulae
for the $\mathcal{N}^{\text{th}}$ order Hamiltonian and symplectic form in
terms of the $(\mathcal{N}-1)^{\text{th}}$ order Hamiltonian flow. In the
context of general relativity, these systems arise in the study of binary
systems such as pairs of black holes or neutron stars in the small mass-ratio
and post-Newtonian approximations. We provide applications of the formalism to
binary systems in these regimes.
