Action-angle variables of a binary black-hole with arbitrary eccentricity, spins, and masses at 1.5 post-Newtonian order*

Accurate and efficient modeling of the dynamics of binary black holes (BBHs) is crucial to their detection through gravitational waves (GWs), with LIGO/Virgo/KAGRA, and LISA in the future. Solving the dynamics of a BBH system with arbitrary parameters without simplifications (like orbit- or precession-averaging) in closed-form is one of the most challenging problems for the GW community. One potential approach is using canonical perturbation theory which constructs perturbed action-angle variables from the unperturbed ones of an integrable Hamiltonian system. Having action-angle variables of the integrable 1.5 post-Newtonian (PN) BBH system is therefore imperative. In this talk, we report on our latest results on the derivation of all five action variables and frequencies of a BBH system with arbitrary eccentricity, masses, and spins, at 1.5PN order. I also discuss how to construct closed-form solutions of such systems using action-angle variables at 1.5PN. All this is done using a novel method of extending the phase space by introducing unmeasurable phase space coordinates. This lays the groundwork to analytically solve the conservative dynamics of the BBH system with arbitrary masses, spins, and eccentricity, at higher PN order, by using canonical perturbation theory.