Hamiltonian normal forms for the post-Newtonian binary problem

TL;DR

This paper develops a Hamiltonian normal form approach for the post-Newtonian two-body problem, enabling detailed perturbative solutions for inspiraling compact binaries with both circular and eccentric orbits.

Contribution

It introduces a Lie series-based framework for deriving complete perturbative solutions in the conservative sector of the PN two-body problem, applicable up to 3PN order and beyond.

Findings

Recovered classical periapsis advance and orbital period corrections at 2PN.

Provided full orbital evolution in time coordinates.

Framework supports both circular and eccentric orbits.

Abstract

We revisit the dynamics of the post-Newtonian (PN) two-body problem for two inspiraling compact bodies. Starting from a matter-only reduced Hamiltonian, we present an adapted framework based on the Lie series approach, enabling the derivation of complete perturbative solutions within the conservative sector. Our framework supports both circular and eccentric orbits, and is applicable to any perturbation respecting rotational invariance and time-independence. In the context of the Arnowitt-Deser-Misner (ADM) canonical formalism, this includes up to at least 3PN order and local terms beyond. We provide an example application at 2PN, recovering classical periapsis advance and orbital period corrections, alongside the full orbital evolution in time coordinates. We discuss eventual extension to spinning and time-dependent systems.