Nonconservative Lie series: post-Newtonian binary dynamics at 2.5PN

TL;DR

This work addresses nonconservative dynamics in gravitational two-body systems by formulating an initial value problem on a doubled phase-space and extending the Lie-series perturbation method to handle radiation-reaction at 2.5PN. The authors construct a resonant Birkhoff normal form using a weak nonconservative homological equation, yielding a secular sector that reproduces the Peters-Mathews evolution and an oscillatory sector that completes the time-domain solution. The complete time-domain dynamics combine conservative (2PN) motion with dissipative 2.5PN corrections, providing explicit evolution for Delaunay variables and yielding a framework applicable to high-precision gravitational-wave templates for detectors like LISA. Beyond the specific 2.5PN case, the approach offers a systematic route to cast arbitrary nonconservative systems into extended Hamiltonian form amenable to Lie perturbation, enabling higher-order extensions and improved modeling of dissipative quasi-integrable systems.

Abstract

We present a fully analytical solution to the dynamics of the non-spinning 2.5 post-Newtonian binary problem, accounting for both the long-term (secular) and short-term (oscillatory) temporal behavior, with no restriction on eccentricity. The radiative degrees of freedom are handled within the nonconservative Hamiltonian framework introduced in a companion paper. In this work, we apply the Lie series method to construct a resonant Birkhoff normal-form and the corresponding generator of the radiation-reaction dynamics. The secular piece reconstructs exactly the Peters-Mathews relations for semi-major axis and eccentricity. The oscillatory piece completes the dynamics and is well suited for gravitational wave templates. The procedure we present in this paper can be systematically employed to cast arbitrary nonconservative systems into extended Hamiltonian form so that the Lie method can be applied.