A note on spin rescalings in post-Newtonian theory

Usually, the reduced mass $\mu$ is viewed as a dropped factor in $\mu l$ and $\mu h$, where $l$ and $h$ are dimensionless Lagrangian and Hamiltonian functions. However, it must be retained in post-Newtonian systems of spinning compact binaries under a set of scaling spin transformations $\mathbf{S}_i=\mathbf{s}_iGM^{2}$ because $l$ and $h$ do not keep the consistency of the orbital equations and the spin precession equations but $(\mu/M) l$ and $(\mu/M) h$ do. When another set of scaling spin transformations $\mathbf{S}_i=\mathbf{s}_iG\mu M$ are adopted, the consistency of the orbital and spin equations is kept in $l$ or $h$, and the factor $\mu$ can be eliminated. In addition, there are some other interesting results as follows. The next-to-leading-order spin-orbit interaction is induced in the accelerations of the simple Lagrangian of spinning compact binaries with the Newtonian and leading-order spin-orbit contributions, and the next-to-leading-order spin-spin coupling is present in a post-Newtonian Hamiltonian that is exactly equivalent to the Lagrangian formalism. If any truncations occur in the Euler-Lagrangian equations or the Hamiltonian, then the Lagrangian and Hamiltonian formulations lose their equivalence. In fact, the Lagrangian including the accelerations with or without truncations can be chaotic for the two bodies spinning, whereas the Hamiltonian without the spin-spin term is integrable.