Symplectic structure of post-Newtonian Hamiltonian for spinning compact binaries

TL;DR

This paper reformulates the post-Newtonian Hamiltonian for spinning compact binaries into a fully symplectic form, enabling better analysis of their integrability and chaos properties across different spin configurations.

Contribution

It constructs conjugate spin variables to make the Hamiltonian symplectic, clarifying the integrability and chaos potential of various binary systems at all post-Newtonian orders.

Findings

Binary systems with one spin are typically integrable.

Two-spin systems with leading order spin-orbit interaction are integrable.

Chaos may occur in more complex spinning configurations.

Abstract

The phase space of a Hamiltonian system is symplectic. However, the post-Newtonian Hamiltonian formulation of spinning compact binaries in existing publications does not have this property, when position, momentum and spin variables $[X, P, S_1, S_2]$ compose its phase space. This may give a convenient application of perturbation theory to the derivation of the post-Newtonian formulation, but also makes classic theories of a symplectic Hamiltonian system be a serious obstacle in application, especially in diagnosing integrability and nonintegrability from a dynamical system theory perspective. To completely understand the dynamical characteristic of the integrability or nonintegrability for the binary system, we construct a set of conjugate spin variables and reexpress the spin Hamiltonian part so as to make the complete Hamiltonian formulation symplectic. As a result, it is directly shown with the least number of independent isolating integrals that a conservative Hamiltonian compact binary system with both one spin and the pure orbital part to any post-Newtonian order is typically integrable and not chaotic. And conservative binary system consisting of two spins restricted to the leading order spin-orbit interaction and the pure orbital part at all post-Newtonian orders is also integrable, independently on the mass ratio. For all other various spinning cases, the onset of chaos is possible.