Integrability and action-angle variables of binary black holes

Two of the problems I worked on during my Ph.D. with Leo Stein led to

1. discovery of two new constants of motion for the 2nd post-Newtonian (PN) binary black hole (BBH) systems. This is the paper.

2. action-angle (AA) based closed-form solutins to the orbital dynamics of the 1.5PN BBH system. The relevant papers are this, this, and this.

For a set of lectures delivered to Prof. Nicolas Yunes’ group at the University of Illinois Urbana-Champaign, I prepared these set of lecture notes. To cite these, please use the arXiv version. NOTE: these notes do not re-present the material of my above papers on AA-based solution. They instead are meant to bring a beginning graduate student up to speed so that they can read these papers. These lecture notes assume the familiarity with the standard graduate level physics courses on the reader’s part.

This GitHub repo contains a Mathematica package which implements the above AA-based solution (among other things).

As a by-product, here is a Mathematica notebook (authored by Leo Stein), which lets you compute the Poisson brackets (PBs) between any two functions of the phase-space variables (position, momenta, and spins). The fundamental PB relations that this notebook assumes are Eqs. (6) of this paper. You must have xAct installed on your computer for this notebook to work.

Additionally, this Mathematica package generates the orbital solution as well as gravitational waves for non-spinning binary black hole systems up to 4PN (3PN) accuracy in the conservative (radiative) sector.