Local Hamiltonian dynamics from non-local action principles and applications to binary systems in general relativity

Abstract

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.