Second post-Newtonian Lagrangian dynamics of spinning compact binaries

TL;DR

This paper develops a second post-Newtonian Lagrangian framework for spinning compact binaries, incorporating spin-orbit and spin-spin interactions, and investigates their effects on chaotic dynamics.

Contribution

It introduces a 3PN spin-spin Hamiltonian derived from a Lagrangian formulation, highlighting the impact of spin interactions on chaos in binary systems.

Findings

3PN spin-spin Hamiltonian is small and sign-different from 2PN

Chaos probability is higher without spin-orbit coupling

Numerical evidence supports the influence of spin interactions on chaos

Abstract

The leading-order spin-orbit coupling is included in a post-Newtonian Lagrangian formulation of spinning compact binaries, which consists of the Newtonian term, first post-Newtonian (1PN) and 2PN non-spin terms and 2PN spin-spin coupling. This makes a 3PN spin-spin coupling occur in the derived Hamiltonian. The spin-spin couplings are mainly responsible for chaos in the Hamiltonians. However, the 3PN spin-spin Hamiltonian is small and has different signs, compared with the 2PN spin-spin Hamiltonian equivalent to the 2PN spin-spin Lagrangian. As a result, the probability of the occurrence of chaos in the Lagrangian formulation without the spin-orbit coupling is larger than that in the Lagrangian formulation with the spin-orbit coupling. Numerical evidences support the claim.