Ruling out chaos in comparable mass compact binary systems with one body spinning

Levin (2006, Phys. Rev. D, 74, 124027) has given two contrary claims on the chaotic behaviour of a system in which only one body of comparable mass binaries spins and spin effects are restricted to the leading order spin-orbit couplings. Chaos in one set of second post-Newtonian (2PN) harmonic coordinate Lagrangian equations of motion was allowed via the fractal basin boundary method. However, in another set of 2PN Arnowitt-Deser-Misner (ADM) Hamiltonian equations of motion no chaos was confirmed with the aid of parametric solutions. Is there chaos for conservative PN Lagrangian and Hamiltonian approaches to the dynamics of comparable mass binaries when only one object spins? This is still an open question. A paper on canonical, conjugate spin variables (Wu and Xie, 2010, Phys. Rev. D, 81, 084045) has directly shown that these Hamiltonian approaches are integrable and non-chaotic regardless of PN orders and spin effects. In this sense, what we are required to answer is only the question of whether the Lagrangian approaches allow chaos. As recently confirmed by Wu et al. (2015, Phys. Rev. D, 91, 024042), in ADM coordinates, any one of these Lagrangian approaches at a certain order generally has an analytical mathematical equivalent Hamiltonian at an infinite order from an analytical point of view or at a certain high enough finite order from a numerical point of view. The Hamiltonian is completely canonical and has four integrals of the total energy and total angular momentum in an eight-dimensional phase space, and therefore it is typically integrable. We use this to show the absence of chaos in the Lagrangian. On the other hand, we use the method of fast Lyapunov exponents to revisit the 2PN harmonic coordinate Lagrangian dynamics with the leading-order spin-orbit coupling of one body spinning. It is found that the fractal method is not sufficient to support chaos in unstable merging binaries, even if the radiation reaction is turned off. In summary, neither the PN conservative Lagrangian formulations nor the PN conservative Hamiltonian formulations can be chaotic in ADM/harmonic coordinates for the case of one body spinning. With this result, it is possible to end the dispute in the related literature regarding the two different claims on the chaotic behaviour of comparable mass binaries with one body spinning.